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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:12:52 +0200 |
| parents | 22d6a63640c6 |
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<pb> <head>MATHEMATICAL COLLECTIONS AND TRANSLATIONS IN TWO TOMES <I>by</I> THOMAS SALUSBURY LONDON, 1661 AND 1665</head> <head>IN FACSIMILE WITH AN ANALYTICAL AND BIO-BIBLIOGRAPHICAL INTRODUCTION <I>by</I> STILLMAN DRAKE</head> <head>1967 DAWSONS OF PALL MALL LONDON</head> <head>ZEITLIN & VER BRUGGE LOS ANGELES</head> <pb> <head>MATHEMATICAL Collections <I>and</I> Tran$lations: <I>In two</I> TOMES.</head> <pb> <head>MATHEMATICAL COLLECTIONS AND TRANSLATIONS: THE FIRST TOME. <I>IN TWO PARTS.</I></head> <head>THE FIRST PART;</head> <head>Containing,</head> <P><I>I.</I> GALILEUS GALILEUS <I>His SYSTEM of the WORLD.</I></P> <P><I>II.</I> GALILEUS <I>His EPISTLE to the GRAND DUTCHESSE MOTHER, concerning the Au- thority of Holy SCRIPTURE in Philo$ophical Controver$ies.</I></P> <P><I>III.</I> JOHANNES KEPLERUS <I>His Reconcilings of SCRI- PTURE Texts, &c.</I></P> <P><I>IV.</I> DIDACUS à STUNICA <I>His Reconcilings of SCRI- PTURE Texts, &c.</I></P> <P><I>V.</I> P. A. FOSCARINUS <I>His Epi$tle to Father FANTONUS, reconciling the Authority of SCRIPTURE, and Judg- ments of Divines alledged again$t this SYSTEM.</I></P> <head>By <I>THOMAS SALUSBURY, E$q.</I></head> <head>LONDON, Printed by WILLIAM LEYBOURN, MDCLXI.</head> <pb> <head>MATHEMATICAL COLLECTIONS AND TRANSLATIONS. THE FIRST TOME.</head> <head>THE FIRST PART;</head> <head>Containing,</head> <P><I>I.</I> GALILEUS GALILEUS, <I>His SYSTEME of the World.</I></P> <P><I>II.</I> GALILEUS, <I>his EPISTLE to the GRAND DUTCHESSE Mother concerning the Authority of Sacred SCRIPTURE in Phylo$ophical Controver$ies.</I></P> <P><I>III.</I> JOHANNES KEPLERUS, <I>his Reconcilings of SCRI- PTURE Texts, &c.</I></P> <P><I>IV.</I> DIDACUS a STUNICA, <I>his Reconcilings of SCRI- PTURE Texts, &c.</I></P> <P><I>V.</I> P. A. FOSCARINUS, <I>his Epi$tle to Father FANTONUS, reconciling the Authority of Sacred SCRIPTURE, and Judgments of Divines alledged again$t, &c.</I></P> <head>By <I>THOMAS SALUSBURY, E$q.</I></head> <head>LONDON, Printed by WILLIAM LEYBOURNE, MDCLXI.</head> <head>Cancelled title-page of the original issue, in which the words “IN TWO PARTS” were omitted.</head> <head><I>Reproduced by permission of Yale University Library.</I></head> <pb> <head>To the Noble and mo$t perfectly Accompli$hed S^{t.} JOHN DENHAM Knight of the Noble Order of the BATH, And Surveyor General of his Ma^{ties} Works, &c.</head> <P>SIR,</P> <P>I humbly begge your Pardon for bringing this Book under your Pro- tection. Were it a Work of my own, or I any thing but the Tran$la- tour, I should ma$ter my Thoughts to a meaner Dedication; But being a Collection of $ome of the greate$t Ma$ters in the World, and never made English till now, I conceived I might $ooner procure their Welcome to a per$on $o eminent for Noble Candor, as well as for all tho$e Intellectual Excellencies wherewith Your Rich Soulis known to be furnished. I re$olv'd to be as kind to this Book as I could, <foot>and</foot> <pb> and $eriou$ly con$idering which way to effect it, I at la$t concluded to prefix Your Name, whom His Maje$ty and all his Subjects, (who have a higher Sen$e and Judgement of Excel- lent Parts) know be$t able to defend my Im- perfections. And yet I confe$s there's one thing makes again$t me, which is your eminent Integrity and great Affection to Truth, where- by my Lap$esin a Work of this Nature might ju$tly de$pair of Shelter, but that the Excel- lency of Your Native Candor $trives for Pre- dominancy over all Your great Abilities. For 'tis all-mo$t impo$$ible to think what Your Matchle$s Wit is not able to Conquer, would Your known Mode$ty but give leave: there- fore <I>Galileus, Kepler,</I> and tho$e other worthies in Learning are now brought before You in English Habit, having chang'd their Latine, Italian and French, whereby they were almo$t Strangers to our Nation, unle$s to $uch as You, who $o perfectly ma$ter the Originals. I know you have $o much and great imployment for His Maje$ty, and his good Subjects that I shall not robb you of another Minutes lo$s; be$ides the liberty of $ub$cribing my Self;</P> <P>SIR,</P> <P><I>Your Honours</I></P> <P>Mo$t Humble and Mo$t obedient Servant</P> <P>THOMAS SALUSBURY.</P> <pb> <P>READER,</P> <P>Mathematical Learning <I>(to $peak nothing touching the nece$sity & delight thereof) hath bin $o $paring- ly imparted to our Countrymen in their native Engli$h, e$pecially the nobler and $ublimer part, that in Compliance with the</I> Solicitations <I>of $everal of my noble and learned Friends, and the</I> Incli- nations <I>of $uch as are Mathematically di$po$ed, more e$pecially tho$e, who either want Time or Patience to look into the vulgar and un$tudied Languages, I did adventure upon this Work of Collecting & Tran$- lating from among$t the excellent Pieces that are $o abounding in the Italian and French Tongues, $ome of tho$e that my own ob$ervation and the intimation of Friends were mo$t u$efull and de$ired, and with all mo$t wanting in their Own.</I></P> <P><I>I was, indeed, at fir$t $eriou$ly Con$cious, and am now, by experience, fully convinced how di$proportionate the weight of the Enterprize is to the weakne$s of the Vndertaker, but yet the Pa$sion I ever had to be $ub$ervient to my Friends and Compatriots in their Inqui$ition after the$e Sublime Studies, and a Patience which I owe to the Flegme that is predominant in my Con$titution, joyned with a nine-years conver$ence in the$e Languages, as al$o an unhappy and long Vacation that the per$ecutions of the late Tyrants gave me from more advantagious employ- ments $o prevailed with me, that I re$olved to improve even my very Confinement to $erve tho$e Friends, whom, as the Times then $tood, I could not $ee.</I></P> <P><I>The Book being for Subject and De$ign intended chiefly for Gentlemen, I have hin as carele$s of u$ing a $tudied Pedantry in my Style; as careful in contriving a plea$ant and beautiful Impre$$ion. And when I had con$idered the hazard, and computed the charge of the undertaking, I found it to exceed the ability of a private Pur$e, e$pe- cially of mine, that had bin $o lately emptied by the hand of violent enemies, and perfidious friends; not to make mention here of the Sums that a Loyal Reflexion upon my Princes Affairs had at the $ame time drawn from me; and judg'd that the most $afe, ea$y, and rea$onable way was to invite tho$e Per$ons who had appeared de$irous of the Book, to be contributary to their own Contentment, by $ub$cribing towards the charge of this Pu- blication.</I></P> <P><I>And for the better management of the Work, I joyned to my $elf a Printer, who$e Genius having rendered him Mathematical, and my overtures of profit having intere$$ed his diligence, I was induced to promi$e my $elf a more than common A$$i$tance from him: and at his door I with rea$on lay all mi$carriages that concerns his Profe$$ion in the Bu$ine$s.</I></P> <P><I>In this Work I found more than ordinary Encouragement from that publick $pirited Per$on the Reverend and Learned Dr.</I> Thomas Barlow, <I>Provo$t of Queens Colledge Oxford, and</I> Margaret <I>Profe$$or in that Vniver- $ity, as al$o from tho$e two able Mathematicians and my Reall Friends Major</I> Miles Symner, <I>and Mr.</I> Robert Wood <I>of Trinity Colledge</I> Dublin, <I>and $ome few others who$e Mode$ty hath expre$ly enjoin'd me a concealment of their Names.</I></P> <P><I>Well, at length I have got to the end of my fir$t Stage; and if I have not rid Po$t, let my excu$e be that my long $tay for my Warrant cau$ed me to $et out late; and being ill mounted, and in a road full of rubbs, I could not with any $afety go fa$ter; but hope to get it up in the next Stage, for in that I intend to $hift my Hor$es.</I></P> <P><I>The names of tho$e Authors and Treatices which I judged would mo$t grace our Language, and gratify Stu- dents, are particularly expre$t in the General Title of the two Tomes. Di$tinct Tomes they are as con$i$ting of $everat Pieces: Collections I call them, becau$e they have bin $o publi$hed, di$per$t, and worn out of Print, that they very rarely meet in one hand: and Tran$lations I own them to be, as not pretending to any thing more than the di$po$ure and conver$ion of them: tho$e Tracts only excepted which compo$e the $econd Part of the $econd Tome.</I></P> <P><I>The fir$t Book which offers it $elf to your view in this Tome is that $ingular and unimitable Piece of Rea$on and Demon$tration the Sy$teme of</I> Galilco. <I>The $ubject of it is a new and Noble port of A$tronomy, to wit the Doctrine and Hypothe$is of the Mobility of the carth and the Stability of the Sun; the Hi$tory whereof I $hall hereafter give you at large in the Life of that famous Man. Only this by the by; that the Reader may not wonder why the$e Dialogues found $o various entertainment in Italy (for he cannot but have heard that though they have been with all veneration valued, read & applauded by the Iudicious yet they were with much dete$tation per$ecuted, $uppre$$ed & exploded by the Super$titious) I am to tell him that our Author having a$$igned his intimate Friends</I> Salviati <I>and</I> Sagredo <I>the more $ucceßfull Parts of the Challenger, and Moderater, he made the famous Commen- tator</I> Simplicius <I>to per$onate the Peripatetick. The Book coming out, and Pope</I> Urban <I>the</I> VIII. <I>taking his Ho- nour to be concern'd as having in his private Capacity bin very po$itive in declaiming against the Samian Philo- $ophy, and now (as he $uppo$ed) being ill delt with by</I> Galilco <I>who had $ummed up all his Arguments, and pur them into the mouth of</I> Simplicius; <I>his Holine$s thereupon conceived an implacable Di$plea$ure against our Au- thor, and thinking no other revenge $ufficient, he employed his Apo$tolical Authority, and deals with the Con$i$tory to condemn him and pro$cribe his Book as Heretical; pro$tituting the Cen$ure of the Church to his private revenge. This was</I> Galilco's <I>fortune in</I> Italy: <I>but had I not rea$on to hope that the Engli$h will be more ho$pitable, on the account of that Principle which induceth them to be civil to (I $ay not to dote on) Strangers, I $hould fear to be charged with imprudence for appearing an Interpreter to that great Philo$opher. And in this confidence I $hall forbear to make any large Exordium concerning him or his Book: & the rather in regard that $uch kind of Gau- deries become not the Gravity of the Subject; as al$o knowing how much (coming from me) they must fall $hort of the Merits of it, or him: but principally becau$e I court only per$ons of Judgement & Candor, that can di$tingui$h between a Native Beauty, and $purious Verni$h. This only let me premi$e, though more to excu$e my weakne$s in the menaging, than to in$inuate my ability in accompli$hing this $o arduous a Task, that the$e profound Dialogues have bin found $o unea$y to Tran$late, that neither affectation of Novelty could induce the French, nor the Tran$lating humour per$wade the Germans to undertake them. This difficulty, as I conceived, was charged either upon the Intricacy of this manner of Writing, or upon the $ingular Elegance in the $tile of</I> Galilco, <I>or el$e upon the</I> <foot>^{*} 2 <I>mi$car-</I></foot> <pb> <I>mi$carriage of the unfortunate</I> Mathias Berneggeius <I>who fir$t attempted to turn them into Latine for the benefit of the Learned World.</I></P> <P><I>I $hall not pre$ume to Cen$ure the Cen$ure which the Church of Rome pa$t upon this Doctrine and its A$$ectors. But, on the contrary, my Author having bin indefinite in his di$cour$e, I $hall forbear to exa$perate, and attempt to reconcile $uch per$ons to this Hypothe$is as devout e$teem for Holy Scripture, and dutifull Re$pect to Canonical Injunctions hath made to $tand off from this Opinion: and therefore for their $akes I have at the end of the Dia- logues by way of $upplement added an Epi$tle of</I> Galilco <I>to Her Most Serene Highne$s</I> Chri$tina Lotharinga <I>the Grand Dutche$$e Mother of</I> Tu$cany; <I>as al$o certain Ab$tracts of</I> John Kepler, <I>Mathematician to two Empe- rours, and</I> Didacus à Stunica <I>a famous Divine of Salam<*>nca, with an Epi$tle of</I> Paulo Antonio Fo$carini <I>a learn- ed Carmelite of Naples, that $hew the Authority of Sacred Scripture in determining of Philo$ophical and Natu- ral Controver$ies: hoping that the ingenious & impartial Reader will meet with full $atisfaction in the $ame. And lea$t what I have $poken of the prohibiting of the$e Pieces by the Inqui$ition may deterre any $crupulous per$on from reading of them, I have purpo$ely in$erted the Imprimatur by which that Office licenced them. And for a larger account of the Book or Author, I refer you to the Relation of his Life, which $hall bring up the Reare in the Second Tome.</I></P> <P><I>What remains of this, is that Excellent Di$cour$e of D.</I> Benedetto Ca$telli Abbate di San Benedetto Aloy$io, <I>concerning the Men$uration of Running waters, with other Treati$es of that Learned Prelate, & of the Superin- tendent</I> Cor$ini. <I>Some may alledge, and I doe confe$s that I promi$ed to publi$h the Life of</I> Galilco <I>in this place: But the great mi$carriages of Letters from $ome Friends in Italy and el$e where, to whom I am a Debtor for $e- veral Remarques, & from whom I daily expect yet greater Helps concerning the Hi$tory of that famous Per$onage: the$e di$appointments, I $ay, joyned with the undeniable Reque$t of $ome Friends, who were impatient to $ee</I> Ca$telli <I>in Engli$h, together with a con$ideration of the di$proportionate Bulk that would otherwi$e have bin betwixt the two Volumes, per$waded me to this exchange. This deviation from my Promi$e I hope is Venial, and for the ex- plating of it I plead Supererrogation: having in each Tome made $o large Aditions (though to my great ex- pen$e) that they make <*>er a third part more than I $tood by promi$e bound to Publi$h. That this is $o will appearby comparing the Contents I here prefix with the Adverti$ment I formerly Printed. For not to mention tho$e Epitomes of</I> Kepler <I>and</I> à Stunica, <I>the whole $econd and following Books of</I> Ca$tclli, <I>were not come to my hands at the time of my penning that Paper; yet knowing how imperfect the Volume would be without them, they being partly a $up- plement to the Theoremes and Problemes which the Abbot had formerly Printed, and partly experiments that had procured him and his Doctrine a very great Reputation, knowing this I $ay, I apprehended a nece$$ity of pu- bli$hing them with the re$t: and hope that if you think not the $ervice I have done therein worth your acknowledge- ment, you will yet at lea$t account the encrea$e of my expence a $ufficient extenuation of the Tre$pa$s that tho$e Additions have forced me to commit upon your Patience in point of Time.</I></P> <P><I>As for the $econd Tome, I have only this to a$$ure the Generous Readers; 1 that I am very confident I $hall be much more punctual in publi$hing that, than (for the rea$ons above related.) I was able to be in $etting forth this: 2 that they $hall not be abu$ed in advancing of their moneys, (as hath bin u$ed in the like ca$e) by $elling the remaining Copyes at an under rate; and <*> that I have a very great care that no di$e$teem may by my means a- ri$e unto this way of publi$hing Books, for that it is of excellent u$e in u$hering Great and Co$tly Volumes into the World.</I></P> <P><I>To $ay nothing of the di$advantages of Tran$lations in general, this of mine doubtle$s is not without it's Er- rours, and over$ights: but tho$e of the Printer di$counted, I hope the re$t may be allowed me upon the $core of</I> Hu- man Imbecilitic. <I>The truth is, I have a$$umed the Liberty to note the Mi$takes in the Florid Ver$ion of</I> Bernegge- rus <I>in the Margent, not $o much to reproach him, as to convince tho$e who told me that they accounted my pains needle$s, having his Latine Tran$lation by them. The like they $aid of the whole two Tomes: but they thereby cau$ed me to question their Under$tanding or Veracity. For $ome of the Books were yet never extant: As for in$tance; the Mcchanicks of Mon$ieur</I> Des Cartes, <I>a Manu$cript which I found among$t the many other Rarities that en- rich the well-cho$en Library of my Learned and Worthy Friend Dr.</I> Charles Scarburgh; <I>the Experiments of Gra- vity, and the Life of</I> Galileo, <I>both my own: Others were included in Volumes of great price, or $o di$per$ed that they were not to be purcha$ed for any money; as tho$e of</I> Kepler, à Stunica, Archimedes, Tartaglia, <I>and the Mecha- nicks of</I> Galileo: <I>And the remainder, though ea$yer to procure, were harder to be under$tood; as</I> Tartaglia <I>his notes on</I> Archimedes, Torricellio <I>his Doctrine of Projects,</I> Galileo <I>his Epi$tle to the Dutche$$e of</I> Tu$cany, <I>and above all his Dialogues</I> de Motu; <I>(never till now done into any Language) which were $o intermixt of Latine and Italian, that the difficulty of the Stile, joyned with the intricatne$$e of the Subject rendered them Unplea$ant, if not wholly Vnintelligible, to $uch as were not ab$olute Ma$ters of both the Tongues.</I></P> <P><I>To conclude; according to the entertainment that you plea$e to afford the$e Collections, I $hall be encouraged to proceed with the Publication of a large Body of Hydrography; declaring the Hi$tory, Art, Lawes, and Apendages of that Princely Study of Navigation, wherein I have omitted nothing of note that can be found either in</I> Dud- ley, Fournier, Aurigarius, Nonius, Snellus, Mar$ennus, Bay$ius, Mori$etu<*> Blondus, Wagoner, <I>abroad, or learnt amongst our Mariners at home, touching the Office of an Admiral, Commander, Pilot, Modelli$t, Shipwright, Gunner, &c.</I></P> <P><I>But order requiring that I $hould di$charge my fir$t Obligation before I contract a $econd; I $hall detein you no longer in the Portall, but put you into po$$e$$ion of the Premi$es,</I></P> <P>Novemb. 20, 1661.</P> <P><I>T. S.</I></P> <foot>THE</foot> <pb> <head>The CONTENTS of the FIRST TOME.</head> <head>PART THE FIRST.</head> <marg><I>Treati$e</I></marg> <P>I. GALILEUS GALILEUS, his SYSIEME of the WORLD: in Four DIALOGUES.</P> <P>II. HIS EPISTLE to her SERENE HIGHNESSE CHRISTIANA LOTHERINGA GRAND DUTCHESSE of TUSCANY, touching the Ancient and Modern DOCTRINE of HOLY FATHERS, and JUDICIOUS DIVINES, concerning the AUTHORITY of SACRED SCRIPTURE in PHYLOSOPHICAL CONTROVERSIES.</P> <P>III. JOHANNES KEPLERUS, his RECONCILINGS of TEXTS of SACRED SCRIPTURE that $eem to oppo$e the DOCTRINE of the EARTHS MOBILI- TY: ab$tracted from his INTRODUCTION unto his LEARNED COMMEN- TARIES upon the PLANET MARS.</P> <P>IV. DIDACUS A STUNICA, a learned SPANISH DIVINE, his RECONCILINGS of the $aid DOCTRINE with the TEXTS of SACRED SCRIPTURE; ab$tracted from his COMMENTARIE upon JOB.</P> <P>V. PAULUS ANTONIUS FOSCARINUS, a CARMELITE, his EPISTLE to SEBASTIANUS FANTONUS, the GENERAL of his ORDER, concerning the PYTHAGOREAN and COPERNICAN OPINION of the MOBILITY OF THE EARTH, and STABILITY OF THE SUN; and of the NEW SYSTEME or CONSTITUTION of the WORLD: in which he reconcileth the TEXTS OF SACRED SCRIPTURE, and ASSERTIONS of DIVINES, commonly alledged against this OPINION.</P> <P><I>A</I> Table <I>of the most ob$ervable</I> Per$ons <I>and</I> Matters <I>mentioned in the</I> Fir$t Part.</P> <head>PART THE SECOND.</head> <P>I. D. BENEDICTUS CASTELLUS, ABBOT OF S. BENEDICTUS ALOYSIUS, his DISCOURSE of the MENSURATION OF RUNNING WATERS: The Fir$t BOOK.</P> <P>II. HIS LETTER to GALILEUS, repre$enting the $tate of the Lake of PERUGIA in TUSCANY.</P> <P>III. HIS GEOMETRICAL DEMONSTRATIONS of the MEASURE of RUNNING WATERS.</P> <P>IV. HIS DISCOURSE of the MENSURATION OF RUNNING WATERS: The Second BOOK.</P> <P>V.<*> HIS CONSIDERATIONS concerning the LAKE OF VENICE. In two DISCOURSES.</P> <P>VI. HIS RULE for computing the quantity of MUD and SAND that LAND-FLOODS bring down to, and leave in the LAKE of VENICE.</P> <P>VII. HIS LETTER to Father FRANCESCO DI S. GIVSEPPE, wherein, at the in$tance of PRINCE LEOPALDO, he delivereth his judgment concerning the turning FIUME MORTO (a River near PISA in TUSCANY) into the SEA, and into the River SERCHIO.</P> <P>VIII. HIS $econd LETTER in anfwer to certain OBJECTIONS propo$ed, and DIFFICUL- TIES ob$erved by SIGNORE BARTOLOTTI, in that affair of the DIVERSION of FIUME MORTO.</P> <P>IX. HIS CONSIDERATION upon the DRAINING of the PONTINE FENNS in CALA- BRIA.</P> <P>X. HIS CONSIDERATION upon the DRAINING of the TERRITORIES of BOLOG- NA, FERRARA, and ROMAGNA.</P> <P>XI. HIS LETTER to D. FERRANTE CESARINI, applying his DOCTRINE to the MENSURATION of the LENGTH, and DISTRIBUTION of the QUANTITY of the WATERS of RIVERS, SPRINGS, AQUEDUCTS, &c.</P> <P>XII. D. CORSINUS, SUPERINTENDENT of the GENERAL DRAINS and PRESIDENT of ROMAGNA, his RELATION of the $tate of the WATERS in the TERRITORIES of BOLOGNA and FERRARA.</P> <P><I>A</I> Table <I>of the mo$t ob$ervable</I> Per$ons <I>and</I> Matters <I>mentioned in the</I> Second Part.</P> <foot>^{*} 3 The</foot> <pb> <head>The CONTENTS of the SECOND TOME,</head> <head>PART THE FIRST.</head> <marg><I>Treati$e</I></marg> <P>I. GALILEUS GALILEUS, his MATHEMATICAL DISCOURSES and DEMON- STRATIOMS touching two NEVV SCIENCES, pertaining to the MECHA- NICKS, and LOCAL MOTION: with an APPENDIX of the CENTRE of GRAVITY of $ome SOLIDS in Four DIALOGUES.</P> <P>II. HIS MECHANICKS; a New PEICE.</P> <P>III. RHENATUS DES CARTES, his MECHANICKS; tran$lated from his FRENCM MANUSCRIPT; a New PEICE.</P> <P>IV. ARCHIMEDES, his Tract DE INSIDENTIBUS HUMIDO; with the NOTES and DEMONSTRASIONS of NICOLAUS TARTALEUS, in Two BOOKS.</P> <P>V. GALILEUS his DISCOURSE of the things that move in or upon the WATER.</P> <P>VI. NICOLAUS TARTALEUS his INVENTIONS for DIVING UNDER WATER, RAISING OF SHIPS SUNK, &c. in Two BOOKS.</P> <head>PART THE SECOND.</head> <P>I. EVANGELISTA TORRICELLIUS, his DOCTRINE OF PROJECTS, and TABLES of the RANGES of GREAT GUNNS of all $orts; wherein he detects $undry ERRORS in GUNNERY: An EPITOME.</P> <P>II T. S. his EXPERIMENTS of the COMPARATIVE GRAVITY OF BODI<*>S in the AIRE and WATER.</P> <P>III. GALILEUS GALILEUS, his LIFE: in Five BOOKS,</P> <P>BOOK I. Containing Five Chapters.</P> <P><I>Chap.</I> 1. His Country.</P> <P>2. His Parents and Extraction.</P> <P>3. His time of Birth.</P> <P>4. His fir$t Education.</P> <P>5. His Ma$ters.</P> <P>II. Containing Three Chapters.</P> <P><I>Chap.</I> 1. His judgment in $everal Learnings.</P> <P>2. His Opinions and Doctrine.</P> <P>3. His Auditors and Scholars.</P> <P>III. Containing Four Chapters.</P> <P><I>Chap.</I> 1. His behaviour in Civil Affairs.</P> <P>2. His manner of Living.</P> <P>3. His morall Virtues.</P> <P>4. His misfortunes and troubles.</P> <P>IV. Containing Four Chapters.</P> <P><I>Chap.</I> 1. His per$on de$cribed.</P> <P>2. His Will and Death.</P> <P>3. His Inventions.</P> <P>4. His Writings.</P> <P>5. His Dialogues of the Sy$teme in particular, containing <I>Nine Sections.</I></P> <P><I>Section</I> 1. Of A$tronomy in General; its Definition, Prai$e, Original.</P> <P>2. Of A$tronomers: a Chronological Catalogue of the mo$t famous of them.</P> <P>3. Of the Doctrine of the Earths Mobility, <I>&c.</I> its Antiquity, and Progre$$e from <I>Pythagoras</I> to the time of <I>Copernicus.</I></P> <P>4. Of the Followers of <I>Copernicus,</I> unto the time of <I>Galileus.</I></P> <P>5. Of the $everall Sy$temes among$t A$tronomers.</P> <P>6. Of the Allegations again$t the <I>Copern.</I> Sy$teme, in 77 Arguments taken out of <I>Ricciolo,</I> with An$wers to them.</P> <P>7. Of the Allegations for the <I>Copern.</I> Sy$teme in so Arguments.</P> <P>8. Of the Scriptures Authorities produced again$t and for the Earths mobility.</P> <P>9. The Conclu$ion of the whole Chapter.</P> <P>V. Containing Four Chapters.</P> <P><I>Chap.</I> 1. His Patrons, Friends, and Emulators.</P> <P>2. Authors judgments of him.</P> <P>3. Authors that have writ for, or again$t him.</P> <P>4. A Conclu$ion in certain Reflections upon his whole Life.</P> <P><I>A</I> Table <I>of the whole</I> Second TOME.</P> <foot>The</foot> <pb> <head>THE SYSTEME OF THE WORLD: IN FOUR DIALOGUES. Wherein the Two GRAND SYSTEMES</head> <head>Of <I>PTOLOMY</I> and <I>COPERNICUS</I> are largely di$cour$ed of:</head> <head>And the <I>REASONS,</I> both <I>Phylo$ophical</I> and <I>Phy$ical,</I> as well on the one $ide as the other, <I>impartially</I> and <I>indefinitely</I> propounded:</head> <head>By <I>GALILEUS GALILEUS LINCEUS,</I> A <I>Gentleman</I> of <I>FLORENCE:</I> Extraordinary <I>Profe$$or</I> of the <I>Mathematicks</I> in the UNIVERSITY of <I>PISA</I>; and Chief <I>Mathematician</I> to the GRAND DUKE of <I>TVSCANY.</I></head> <head><I>Ingli$bed from the</I> Original <I>Italián</I> Copy, <I>by</I> THOMAS SALUSBURY.</head> <head>ALCINOUS, <G>*dei_ d) e)leuge/rion e<*>ai th_| gnwmh_| r\n me/llonta filosofei_n.</G></head> <head>SENECA, <I>Inter nullos magis quam inter PHILOSOPHOS e$$e debet aqua LIBERTAS.</I></head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURNE. MDCLXI.</head> <foot>^{*} 2</foot> <pb> <head>THE SYSTEME OF THE WORLD: IN FOUR DIALOGUES. Wherein the Two GRAND SYSTEMES</head> <head>Of <I>PTOLOMY</I> and <I>COPERNICUS</I> are largely di$cour$ed of:</head> <head>And the <I>REASONS,</I> both <I>Phylo$ophical</I> and <I>Phy$ical,</I> as well on the one $ide as the other, <I>impartially</I> and <I>indefinitely</I> propounded:</head> <head>By <I>GALILEUS GALILEUS LINCEUS,</I> A <I>Gentleman</I> of <I>FLORENCE:</I> Extraordinary <I>Profe$$or</I> of the <I>Mathematicks</I> in the UNIVERSITY of <I>PISA</I>; and Chief <I>Mathematician</I> to the GRAND DUKE of <I>TVSCANY.</I></head> <head><I>Ingli$bed from the</I> Original <I>Italián</I> Copy, <I>by</I> THOMAS SALUSBURY.</head> <head>ALCINOUS, <G>*dei_ d) e)leuge/rion e<*>ai th_| gnwmh_| r\n me/llonta filosofei_n.</G></head> <head>SENECA, <I>Inter nullos magis quam inter PHILOSOPHOS e$$e debet aqua LIBERTAS.</I></head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURNE. MDCLXI.</head> <pb> <head>THE SYSTEME OF THE WORLD: IN FOUR DIALOGUES. Wherein the Two GRAND SYSTEMES</head> <head>Of <I>PTOLOMY</I> and <I>COPERNICUS</I> are largely di$cour$ed of:</head> <head>And the <I>REASONS,</I> both <I>Phylo$ophical</I> and <I>Phy$ical,</I> as well on the one $ide as the other, <I>impartially</I> and <I>indefinitely</I> propounded:</head> <head>By <I>GALILEUS GALILEUS LINCEUS,</I> A <I>Gentleman</I> of <I>FLORENCE:</I> Extraordinary <I>Profe$$or</I> of the <I>Mathematicks</I> in the UNIVERSITY of <I>PISA</I>; and Chief <I>Mathematician</I> to the GRAND DUKE of <I>TVSCANY.</I></head> <head><I>Ingli$bed from the</I> Original <I>Italián</I> Copy, <I>by</I> THOMAS SALUSBURY.</head> <head>ALCINOUS, <G>*dei_ d) e)leuge/rion e<*>ai th_| gnwmh_| r\n me/llonta filosofei_n.</G></head> <head>SENECA, <I>Inter nullos magis quam inter PHILOSOPHOS e$$e debet aqua LIBERTAS.</I></head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURNE. MDCLXI.</head> <pb> <head>To the mo$t Serene Grand DUKE OF TUSCANY.</head> <P>Though the difference between Men and other living Creatures be very great, yet happly he that $hould $ay that he could $hew little le$s between Man and Man would not $peak more than he might prove. What proportion doth one bear to athou$and? and yet it is a common Proverb, <I>One Man is worth athou$and, when as a thou$and are not worth one.</I> This difference hath dependence upon the different abilities of their Intelle- ctuals; which I reduce to the being, or not being a Philo$o- pher; in regard that Philo$ophy as being the proper food of $uch as live by it, di$tingui$heth a Man from the common E$- $ence of the Vulgar in a more or le$s honourable degree accord- ing to the variety of that diet. In this $ence he that hath the highe$t looks, is of highe$t quality; and the turning over of the great Volume of Nature, which is the proper Object of Philo$ophy is the way to make one look high: in which Book, although what$oever we read, as being the Work of Al- mighty God, is therefore mo$t proportionate; yet notwith- $tanding that is more ab$olute and noble wherein we more plainly de$erne his art and skill. The <I>Con$titution</I> of the <I>Vnivers,</I> among all Phy$ical points that fall within Humane Compre- hen$ion, may, in my opinion, be preferred to the Precedency: for if that in regard of univer$al extent it excell all others, it ought as the Rule and Standard of the re$t to goe before them in Nobility. Now if ever any per$ons might challenge to be $ignally di$tingui$hed for Intellectuals from other men; <foot><I>Pto o-</I></foot> <pb> <I>Ptolomey</I> and <I>Copernicus</I> were they that have had the honour to $ee farthe$t into, and di$cour$e mo$t profoundly of the <I>Worlds Sy$teme.</I> About the Works of which famous Men the$e Dia- lous being chiefly conver$ant, I conceived it my duty to De- dicate them only to <I>Your Highne$s.</I> For laying all the weight upon the$e two, whom I hold to be the Able$t Wits that have left us their Works upon the$e Subjects; to avoid a Sole- ci$mein Manners, I was obliged to addre$s them to Him, who with me, is the Greate$t of all Men, from whom they can re- ceive either Glory or Patrociny. And if the$e two per$ons have $o farre illuminated my Under$tanding as that this my Book may in a great part be confe$$ed to belong to them, well may it al$o be acknowledged to belong to <I>Your Highne$s,</I> unto who$e Bounteous Magnificence I owe the time and lea$ure I had to write it, as al$o unto Your Powerful A$$i$tance, (never weary of honouring me) the means that at length I have had to publi$h it. May <I>Your Highne$s</I> therefore be plea$ed to accept of it according to Your accu$tomed Goodne$s; and if any thing $hall be found therein, that may be $ub$ervient towards the information or $atisfaction of tho$e that are Lovers of Truth; let them acknowledge it to be due to <I>Your Self,</I> who are $o expert in doing good, that Your Happy Dominion cannot $hew the man that is concerned in any of tho$e general Cala- mities that di$turb the World; $o that Praying for Your Pro$pe- rity, and continuance in this Your Pious and Laudable Cu- $tome, I humbly ki$s Your Hands;</P> <P><I>Your Mo$t Serene Highne$$es</I></P> <P>Mo$t Humble and mo$t devoted</P> <P>Servant and Subject</P> <P>GALILEO GALILEI.</P> <pb> <head>THE AUTHOR'S INTRODUCTION.</head> <P>Judicious Reader,</P> <P><I>There was publi$hed $ome years $ince in</I> Rome <I>a $alutiferous Edict, that, for the obviating of the dangerous Scandals of the pre$ent Age, impo$ed a $ea- $onable Silence upon the Pythagorean Opinion of the Mobility of the Earth. There want not $uch as unadvi$edly affirm, that that Decree was not the produ- ction of a $ober Scrutiny, but of an ill informed Pa$sion; & one may hear $ome mut- ter that Con$ultors altogether ignorant of A$tronomical Ob$ervations ought not to clipp the Wings of Speculative Wits with ra$h Prohibitions. My zeale can- not keep $ilence when I hear the$e incon$iderate complaints. I thought fit, as being thoroughly ac- quainted with that prudent Determination, to appear openly upon the Theatre of the World as a Wit- ne$s of the naked Truth. I was at that time in</I> Rome; <I>and had not only the audiences, but applauds of the mo$t Eminent Prelates of that Court; nor was that Decree Publi$hed without Previous Notice given me thereof. Therefore it is my re$olution in the pre$ent ca$e to give Foraign Nations to $ee that this point is as well under stood in</I> Italy, <I>and particularly in</I> Rome, <I>as Tran$alpine Diligence can imagine it to be: and collecting together all the proper Speculations that concern the</I> Copernican Sy$teme, <I>to let them know, that the notice of all preceded the Cen$ure of the</I> Roman Court; <I>and that there proceed from this Climate not only Doctrines for the health of the Soul, but al$o ingenious Di$coveries for the recreating of the Mind.</I></P> <P><I>To this end I have per$onated the</I> Copernican <I>in this Di$cour$e; proceeding upon an Hypothe$is purely Mathematical; $triving by all artificial wayes to repre$ent it Superiour, not to that of the Im- mobility of the Earth ab$olutely, but according as it is mentioned by $ome, that retein no more, but the name of</I> Peripateticks, <I>and are content, without going farther, to adore Shadows, not philo$ophizing with requi$it caution, but with the $ole remembrance of four</I> Principles, <I>but badly under $tood.</I></P> <P><I>We $hall treat of three principall heads. Fir$t I will endeavour to $hew that all Experiments that can be made upon the Earth are in$ufficient means to conclude it's Mobility, but are indifferently applicable to the Earth moveable or immoveable: and I hope that on this occa$ion we $hall di$cover many ob$er- vable pa$$ages unknown to the Ancients. Secondly we will examine the Cœle$tiall</I> Phœnomena <I>that make for the</I> Copernican Hypothe$is, <I>as if it were to prove ab$olutely victorious; adding by the way certain new Ob$ervations, which yet $erve only for the A$tronomical Facility, not for Natural Neceßity. In the third place I will propo$e an ingenuous Fancy. I remember that I have $aid many years $ince, that the unknown Probleme of the Tide might receive $ome light, admitting the Earths Motion. This Po$ition of mine pa$sing from one to another had found charitable Fathers that adopted it for the I$$ue of their own wit. Now, becau$e no $tranger may ever appear that defending him- $elf with our armes $hall charge us with want of caution in $o principal an Accident, I have thought good to lay down tho$e probabilities that would render it credible, admitting that the Earth did move. I hope, that by the$e Con$ider ations the World will come to know, that if other Nations have Navigated more than we, we have not $tudied le$s than they; & that our returning to a$$ert the Earths Stability, and to take the contrary only for a Mathematical</I> Capriccio, <I>proceeds not from inadvertency of what others have thought thereof, but (had we no other inducements) from tho$e Rea$ons that Pic- ty, Religion, the Knowledge of the Divine Omnipotency, and a con$ciou$ne$s of the incapacity of mans Vnder$tanding dictate unto us.</I></P> <foot>^{*} 3 <I>With</I></foot> <pb> <P><I>With all I conceived it very proper to expre$s the$e conceits by way of Dialogue, which, as not being bound up to the riggid ob$ervance of Mathematical Laws, gives place al$o to Digre$sions that are $ometimes no le$s curious than the principal Argument.</I></P> <P><I>I chanced to be $everal years $ince, at $everal times, in the Stupendious Citty of</I> Venice, <I>where I conver$ed with</I> Signore Giovan France$co Sagredo <I>of a Noble Extraction, and piercing wit. There came thither from</I> Florence <I>at the $ame time</I> Signore Filippo Salviati, <I>who$e lea$t glory was the Emi- nence of his Blood, and Magnificence of his E$tate: a $ublime Wit that fed not more hungerly upon any plea$ure than on elevated Speculations. In the company of the$e two I often di$cour$ed of the$e matters before a certain Peripatetick Philo$opher who $eemed to have no geater ob$tacle in under$tand- ing of the Truth, than the Fame he had acquired by Ari$totelical Interpretations.</I></P> <P><I>Now, $eeing that inexorable Death hath deprived</I> Venice <I>and</I> Florence <I>of tho$e two great Lights in the very Meridian of their years, I did re$olve, as far as my poor ability would permit, to perpetuate their lives to their honour in the$e leaves, bringing them in as Interlocutors in the pre$ent Controver$y. Nor $hall the Honest Peripatetick want his place, to whom for his exce$sive affection to wards the Com- mentaries of</I> Simplicius, <I>I thought fit, without mentioning his own Name, to leave that of the Author he $o much re$pected. Let tho$e two great Souls, ever venerable to my heart, plea$e to accept this pu- blick Monument of my never dying Love; and let the remembr ance of their Eloquence a$si$t me in delivering to Po$terity the Con$ider ations that I have promi$ed.</I></P> <P><I>There ca$ually happened (as was u$uall) $everal di$cour$es at times between the$e Gentlemen, the which had rather inflamed than $atisfied in their wits the thir$t they had to be learning; whereupon they took a di$creet re$olution to meet together for certain dayes, in which all other bu$ine$s $et a$ide, they might betake them$elves more methodically to contemplate the Wonders of God in Heaven, and in the Earth: the place appointed for their meeting being in the Palace of the Noble</I> Sagredo, <I>after the due, but very $hort complements</I>; Signore Salviati <I>began in this manner.</I></P> <fig> <foot>GALI-</foot> <p n=>1</p> <head>GALILÆUS Galilæus Lyncæus, HIS SYSTEME OF THE WORLD.</head> <head>The Fir$t Dialogue.</head> <head><I>INTERLOCVTORS.</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <head>SALVIATUS.</head> <P>It was our ye$terdayes re$olution, and a- greement, that we $hould to day di$cour$e the mo$t di$tinctly, and particularly we could po$$ible, of the natural rea$ons, and their efficacy that have been hitherto al- ledged on the one or other part, by the maintainers of the Po$itions, <I>Aristotelian,</I> and <I>Ptolomaique</I>; and by the followers <marg>Copernicus <I>repu- teth the earth œ Globe like to a Pla- net.</I></marg> of the <I>Copernican Sy$teme</I>: And becau$e <I>Copernicus</I> placing the Earth among the moveable Bodies of Hea- ven, comes to con$titute a Globe for the $ame like to a Planet; it would be good that we began our di$putation with the examina- tion of what, and how great the energy of the <I>Peripateticks</I> ar- guments is, when they demon$trate, that this <I>Hypothe$is</I> is impo$- <foot>A fible</foot> <p n=>2</p> $ible: Since that it is nece$$ary to introduce in Nature, $ub$tances <marg><I>Cœle$tial $ub$tan- ces that are inalte- rable, and Elemen- tary that be alte- rable, are nece$$ary in the opinion of</I> Ari$totle.</marg> different betwixt them$elves, that is, the Cœle$tial, and Elementa- ry; that impa$$ible and immortal, this alterable and corruptible. Which argument <I>Ari$totle</I> handleth in his book <I>De Cœlo,</I> in$inu- ating it fir$t, by $ome di$cour$es dependent on certain general a$- $umptions, and afterwards confirming it with experiments and per- ticular demon$trations: following the $ame method, I will pro- pound, and freely $peak my judgement, $ubmitting my $elf to your cen$ure, and particularly to <I>Simplicius,</I> a Stout Champion and contender for the <I>Ari$totelian</I> Doctrine.</P> <marg>Ari$totle <I>maketh the World perfect, becau$e it hath the threefold demen$i- on.</I></marg> <P>And the fir$t Step of the <I>Peripatetick</I> arguments is that, where <I>A- ri$totle</I> proveth the integrity and perfection of the World, telling us, that it is not a $imple line, nor a bare $uperficies, but a body adorned with Longitude, Latitude, and Profundity; and becau$e there are no more dimen$ions but the$e three; The World having them, hath all, and having all, is to be concluded perfect. And again, that by $imple length, that magnitude is con$tituted, which is called a Line, to which adding breadth, there is framed the Su- perficies, and yet further adding the altitude or profoundity, there re$ults the Body, and after the$e three dimen$ions there is no pa$$ing farther, $o that in the$e three the integrity, and to $o $peak, totality is terminated, which I might but with ju$tice have requi- red <I>Ari$totle</I> to have proved to me by nece$$ary con$equences, the rather in regard he was able to do it very plainly, and $peedily.</P> <P>SIMPL. What $ay you to the excellent demon$trations in the <marg>Ari$totles <I>demon- $trations to prove the dimen$ions to be three and no more.</I></marg> 2. 3. and 4. Texts, after the definition of <I>Continual</I>? have you it not fir$t there proved, that there is no more but three dimen$ions, for that tho$e three are all things, and that they are every where? And is not this confirmed by the Doctrine and Authority of the <marg><I>The number three celebrated among $t the</I> Pythagorians</marg> <I>Pythagorians,</I> who $ay that all things are determined by three, be- ginning, middle, and end, which is the number of All? And where leave you that rea$on, namely, that as it were by the law of Na- ture, this number is u$ed in the $acrifices of the Gods? And why being $o dictated by nature, do we atribute to tho$e things that are three, and not to le$$e, the title of all? why of two is it $aid both, and not all, unle$s they be three? And all this Doctrine you have in the $econd Text. Afterwards in the third, <I>Ad pleniorem</I> <marg>Omne, Totum & Perfectum.</marg> <I>$cientiam,</I> we read that <I>All,</I> the <I>Whole,</I> and <I>Perfect,</I> are formally one and the $ame; and that therefore onely the <I>Body,</I> among$t magnitudes is perfect: becau$e it is determined by three, which is All, and being divi$ible three manner of waies, it is every way di- vi$ible; but of the others, $ome are dividible in one manner, and $ome in two, becau$e according to the number a$$ixed, they have their divi$ion and continuity, and thus one magnitude is continu- <marg>Or Solid.</marg> ate one way, another two, a third, namely the Body, every way. <foot>More-</foot> <p n=>3</p> Moreover in the fourth Text; doth he not after $ome other Do- ctrines, prove it by another demon$tration? <I>Scilicet,</I> That no tran- $ition is made but according to $ome defect (and $o there is a tran- $ition or pa$$ing from the line to the $uperficies, becau$e the line is defective in breadth) and that it is impo$$ible for the perfect to want any thing, it being every way $o; therefore there is no tran- $ition from the Solid or Body to any other magnitude. Now think you not that by all the$e places he hath $ufficiently proved, how that there's no going beyond the three dimen$ions, Length, Breadth, and Thickne$s, and that therefore the body or $olid, which hath them all, is perfect?</P> <P>SALV. To tell you true, I think not my $elf bound by all the$e rea$ons to grant any more but onely this, That that which hath beginning, middle, and end, may, and ought to be called perfect: But that then, becau$e beginning, middle, and end, are Three, the num- ber Three is a perfect number, and hath a faculty of conferring <I>Perfection</I> on tho$e things that have the $ame, I find no inducement to grant; neither do I under$tand, nor believe that, for example, of feet, the number three is more perfect then four or two, nor do I conceive the number four to be any imperfection to the Ele- ments: and that they would be more perfect if they were three. Better therefore it had been to have left the$e $ubtleties to the <I>Rhetoricians,</I> and to have proved his intent, by nece$$ary demon$tra- tion; for $o it behoves to do in demon$trative $ciences.</P> <P>SIMPL. You $eem to $corn the$e rea$ons, and yet it is all the Doctrine of the <I>Pythagorians,</I> who attribute $o much to numbers; and you that be a <I>Mathematician,</I> and believe many opinions in the <I>Pythagorick</I> Philo$ophy, $eem now to contemn their My- $teries.</P> <P>SALV. That the <I>Pythagorians</I> had the $cience of numbers in high e$teem, and that <I>Plato</I> him$elf admired humane under$tand- ing, and thought that it pertook of Divinity, for that it under- <marg>Plato <I>held that humane under- $tanding partook ofaivi<*>ity, becau$e it understood num- bers.</I></marg> $tood the nature of numbers, I know very well, nor $hould I be far from being of the $ame opinion: But that the My$teries for which <I>Pythagoras</I> and his $ect, had the Science of numbers in $uch veneration, are the follies that abound in the mouths and writings <marg><I>The My$tery of</I> Pythagorick <I>num- bers fabulous.</I></marg> of the vulgar, I no waies credit: but rather becau$e I know that they, to the end admirable things might not be expo$ed to the con- tempt, and $corne of the vulgar, cen$ured as $acrilegious, the pub- <marg>De Papyrio p æ- textato, <I>Gellius</I> <*> 2. 3.</marg> li$hing of the ab$truce properties of Numbers, and incommen- $urable and irrational quantities, by them inve$tigated; and di- vulged, that he who di$covered them, was tormented in the other World: I believe that $ome one of them to deter the common $ort, and free him$elf from their inqui$itivene$s, told them that the my$teries of numbers were tho$e trifles, which afterwards did $o <foot>A 2 $pread</foot> <p n=>4</p> $pread among$t the vulgar; and this with a di$cretion and $ubtlety re$embling that of the prudent young man, that to be freed from the importunity of his inqui$itive Mother or Wife, I know not whether, who pre$$ed him to impart the $ecrets of the Senate, contrived that $tory, which afterwards brought her and many o- ther women to be derided and laught at by the $ame Senate.</P> <P>SIMPL. I will not be of the number of tho$e who are over curi- ous about the <I>Pythagorick</I> my$teries; but adhering to the point in hand; I reply, that the rea$ons produced by <I>Ari$totle</I> to prove the dimen$ions to be no more than three, $eem to me conclu- dent, and I believe, That had there been any more evident demon- $trations thereof, <I>Ari$totle</I> would not have omitted them.</P> <P>SAGR. Put in at lea$t, if he had known, or remembred any more. But you <I>Salviatus</I> would do me a great plea$ure to alledge unto me $ome arguments that may be evident, and clear enough for me to comprehend.</P> <P>SALV. I will; and they $hall be $uch as are not onely to be ap- prehended by you, but even by <I>Simplicius</I> him$elf: nor onely to be comprehended, but are al$o already known, although hap- ly unob$erved; and for the more ea$ie under$tanding thereof, we will take this Pen and Ink, which I $ee already prepared for <marg><I>A Geometrical de- mon$tration of the triple dimen$ion.</I></marg> $uch occa$ions, and de$cribe a few figures. And fir$t we will note [Fig. 1. <I>at the end of this Dialog.</I>] the$e two points AB, and draw from the one to the other the curved lines, ACB, and ADB, and the right line A B, I demand of you which of them, in your mind, is that which determines the di$tance between the terms AB, & why?</P> <P><I>S</I>AGR. I $hould $ay the right line, and not the crooked, as well becau$e the right is $horter, as becau$e it is one, $ole, and deter- minate, whereas the others are infinit, unequal, and longer; and my determination is grounded upon that, That it is one, and certain.</P> <P>SALV. We have then the right line to determine the length be- tween the two terms; let us add another right line and parallel to AB, which let be CD, [<I>Fig.</I> 2.] $o that there is put between them a $uperficies, of which I de$ire you to a$$ign me the breadth, therefore departing from the point A, tell me how, and which way you will go, to end in the line C D, and $o to point me out the breadth com- prehended between tho$e lines; let me know whether you will terminate it according to the quantity of the curved line A E, or the right line A F, or any other.</P> <P>SIMPL. According to the right A F, and not according to the crooked, that being already excluded from $uch an u$e.</P> <P>SAGR. But I would take neither of them, $eeing the right line A F runs obliquely; But would draw a line, perpendicular to C D, for this $hould $eem to me the $horte$t, and the propere$t of infinite that are greater, and unequal to one another, which may be <foot>pro-</foot> <p n=>5</p> produced from the term A to any other part of the oppo$ite line C D.</P> <P>SALV. Your choice, and the rea$on you bring for it in my judg- ment is mo$t excellent; $o that by this time we have proved that the fir$t dimen$ion is determined by a right line, the $econd name- ly the breadth with another line right al$o, and not onely right, but withall, at right-angles to the other that determineth the length, and thus we have the two dimen$ions of length and breadth, definite and certain. But were you to bound or termi- nate a height, as for example, how high this Roof is from the pave- ment, that we tread on, being that from any point in the Roof, we may draw infinite lines, both curved, and right, and all of di- ver$e lengths to infinite points of the pavement, which of all the$e lines would you make u$e of?</P> <P>SAGR. I would fa$ten a line to the Seeling, and with a plummet that $hould hang at it, would let it freely di$tend it $elf till it $hould reach well near to the pavement, and the length of $uch a thread being the $treighte$t and $horte$t of all the lines, that could po$sibly be drawn from the $ame point to the pavement, I would $ay was the true height of this Room.</P> <P>SALV. Very well, And when from the point noted in the pave- ment by this pendent thread (taking the pavement to be levell and not declining) you $hould produce two other right lines, one for the length, and the other for the breadth of the $uperficies of the$aid pavement, what angles $hould they make with the $aid thread?</P> <P>SAGR. They would doubtle$s meet at right angles, the $aid lines falling perpendicular, and the pavement being very plain and levell.</P> <P>SALV. Therefore if you a$$ign any point, for the term from whence to begin your mea$ure; and from thence do draw a right line, as the terminator of the fir$t mea$ure, namely of the length, it will follow of nece$$ity, that that which is to de$ign out the largene$s or breadth, toucheth the fir$t at right-angles, and that that which is to denote the altitude, which is the third dimen$ion, going from the $ame point formeth al$o with the other two, not oblique but right angles, and thus by the three perpendiculars, as by three lines, one, certain, and as $hort as is po$$ible, you have the three dimen$ions A B length, A C breadth, and A D height; and becau$e, clear it is, that there cannot concurre any more lines in the $aid point, $o as to make therewith right-angles, and the dimen$ions ought to be determined by the $ole right lines, which make between them- $elves right-angles; therefore the dimen$ions are no more but three, and that which hath three hath all, and that which hath all, is divi$ible on all $ides, and that which is $o, is perfect, <I>&c.</I></P> <foot>SIMP.</foot> <p n=>6</p> <P>SIMPL. And who $aith that I cannot draw other lines? why may not I protract another line underneath, unto the point A, that may be perpendicular to the re$t?</P> <P>SALV. You can doubtle$s, at one and the $ame point, make no more than three right lines concurre, that con$titute right angles between them$elves.</P> <P>SAGR. I $ee what <I>Simplicius</I> means, namely, that $hould the $aid D A be prolonged downward, then by that means there might be drawn two others, but they would be the $ame with the fir$t three, differing onely in this, that whereas now they onely touch, then they would inter$ect, but not produce new dimen$ions.</P> <marg><I>In phyfical proofs geometrical exact- ne$s is not nece$$a- ry.</I></marg> <P>SIMPL. I will not $ay that this your argument may not be con- cludent; but yet this I $ay with <I>Ari$totle,</I> that in things natural it is not alwaies nece$$ary, to bring <I>Mathematical</I> demon$trations.</P> <P>SAGR. Grant that it were $o where $uch proofs cannot be had, yet if this ca$e admit of them, why do not you u$e them? But it would be good we $pent no more words on this particular, for I think that <I>Salviatus</I> will yield, both to <I>Ari$totle,</I> and you, with- out farther demon$tration, that the World is a body, and perfect, yea mo$t perfect, as being the greate$t work of God.</P> <P>SALV. So really it is, therefore leaving the general contempla- <marg><I>Parts of the world are two, according to</I> Ari$totle, <I>Cœle- $tial and Elemen- tary contrary to one another.</I></marg> tion of the whole, let us de$cend to the con$ideration of its parts, which <I>Ari$totle,</I> in his fir$t divi$ion, makes two, and they very diffe- rent and almo$t contrary to one another; namely the Cœle$tial, and Elementary: that ingenerable, incorruptible, unalterable, un- pa$$ible, &c. and this expo$ed to a continual alteration, mutati- on, &c. Which difference, as from its original principle, he de- rives from the diver$ity of local motions, and in this method he proceeds.</P> <P>Leaving the $en$ible, if I may $o $peak, and retiring into the Ideal world, he begins Architectonically to con$ider that nature being the principle of motion, it followeth that natural bodies be <marg><I>Local motion of three kinds, right, circular, & mixt.</I></marg> indued with local motion. Next he declares local motion to be of three kinds, namely, circular, right, and mixt of right and cir- cular: and the two fir$t he calleth $imple, for that of all lines the <marg><I>Circular, and $treight motions are $imple, as pro- ceeding by $imple lines.</I></marg> circular, and right are onely $imple; and here $omewhat re- $training him$elf, he defineth anew, of $imple motions, one to be circular, namely that which is made about the <I>medium,</I> and the other namely the right, upwards, and downwards; upwards, that which moveth from the <I>medium</I>; downwards, that which goeth to- wards the <I>medium.</I> And from hence he infers, as he may by and ne- <marg><I>Ad medium, à me- dio, & circa medi- um.</I></marg> ce$$ary con$equence, that all $imple motions are confined to the$e three kinds, namely, to the <I>medium,</I> from the <I>medium,</I> and about the <I>medium</I>; the which corre$ponds $aith he, with what hath been $aid before of a body, that it al$o is perfected by three things, and $o <foot>is</foot> <p n=>7</p> is its motion. Having confirmed the$e motions, he proceeds $aying, that of natural bodies $ome being $imple, and $ome compo$ed of them (and he calleth $imple bodies tho$e, that have a principle of motion from nature, as the Fire and Earth) it follows that $imple motions belong to $imple bodies, and mixt to the com- pound; yet in $uch $ort, that the compounded incline to the part predominant in the compo$ition.</P> <P>SAGR. Pray you hold a little <I>Salviatus,</I> for I find $o many doubts to $pring up on all $ides in this di$cour$e, that I $hall be con$trained, either to communicate them if I would attentively hearken to what you $hall add, or to take off my attention from the things $poken, if I would remember objections.</P> <P>SALV. I will very willingly $tay, for that I al$o run the $ame hazard, and am ready at every $tep to lo$e my $elf whil$t I $ail be- tween Rocks, and boi$terous Waves, that make me, as they $ay, to lo$e my <I>Compa$s</I>; therefore before I make them more, propound your difficulties.</P> <marg><I>The definition of Nature, either im- perfect, or un$ea$o- nable, produced by</I> Ari$totle.</marg> <P>SAGR. You and <I>Ari$totle</I> together would at fir$t take me a little out of the $en$ible World, to tell me of the <I>Architecture,</I> wherewith it ought to be fabricated; and very appo$itly begin to tell me, that a natural body is by nature moveable, nature being (as el$ewhere it is defined) the principle of motion. But here I am $omewhat doubtfull why <I>Ari$totle</I> $aid not that of natural bo- dies, $ome are moveable by nature, and others immoveable, for that in the definition, nature is $aid to be the principle of Motion, and Re$t; for if natural bodies have all a principle of motion, either he might have omitted the mention of Re$t, in the definiti- on of nature: or not have introduced $uch a definition in this place. Next, as to the declaration of what <I>Ari$totle</I> intends by $imple motions, and how by Spaces he determines them, calling tho$e $im- ple, that are made by $imple lines, which are onely the right, and <marg><I>The Helix about the Cylinder may be $aid to be a $im- ple line.</I></marg> circular, I entertain it willingly; nor do I de$ire to tenter the in$tance of the Helix, about the Cylinder; which in that it is in e- very part like to it $elf, might $eemingly be numbred among $im- ple lines. But herein I cannot concurre, that he $hould $o re- $train $imple motions (whil$t he $eems to go about to repeat the $ame definition in other words) as to call one of them the motion about the <I>medium,</I> the others <I>Sur$um & Deor$um,</I> namely up- wards and downward; which terms are not to be u$ed, out of the World fabricated, but imply it not onely made, but already in- habited by us; for if the right motion be $imple, by the $implicity of the right line, and if the $imple motion be natural, it is made on every $ide, to wit, upwards, downwards, backwards, forwards, to the right, to the left, and if any other way can be imagined, pro- vided it be $traight, it $hall agree to any $imple natural body; or <foot>if</foot> <p n=>8</p> if not $o, then the $uppo$ion of <I>Ari$totle</I> is defective. It appears <marg>Ari$totle <I>accom- modates the rules of</I> Architecture <I>to the frame of the World, and not the frame to the rules.</I></marg> moreover that <I>Ari$totle</I> hinteth but one circular motion alone to be in the World, and con$equently but one onely Center, to which alone the motions of upwards and downwards, refer. All which are apparent proofs, that <I>Ari$totles</I> aim is, to make white black, and to accommodate <I>Architectur<*></I> to the building, and not to modle the building according to the precepts of <I>Arthitecture:</I> for if I $hould $ay that Nature in Univer$al may have a thou- $and Circular Motions, and by con$equence a thou$and Cen- ters, there would be al$o a thou$and motions upwards, and downwards. Again he makes as hath been $aid, a $imple motion, and a mixt motion, calling $imple, the circular and right; and mixt, the compound of them two: of natural bodies he calls $ome $imple (namely tho$e that have a natural principle to $imple mo- tion) and others compound: and $imple motions he attributes to $imple bodies, and the compounded to the compound; but by compound motion he doth no longer under$tand the mixt of right and circular, which may be in the World; but introduceth a mixt motion as impo$$ible, as it is impo$$ible to mixe oppo$ite motions made in the $ame right line, $o as to produce from them a motion partly upwards, partly downwards; and, to moderate $uch an ab- $urdity, and impo$$ibility, he a$$erts that $uch mixt bodies move <marg><I>Right motion, $ome- times $imple, ard $ometimes mixt ac- cording to</I> Ari$t.</marg> according to the $imple part predominant: which nece$$itates others to $ay, that even the motion made by the $ame right line is $ometimes $imple, and $ometimes al$o compound: $o that the $im- plicity of the motion, is no longer dependent onely on the $im- plicity of the line.</P> <P>SIMPL. How? Is it not difference $ufficient, that the $imple and ab$olute are more $wift than that which proceeds from predomi- nion? and how much fa$ter doth a piece of pure Earth de$cend, than a piece of Wood?</P> <P>SAGR. Well, <I>Simplicius</I>; But put ca$e the $implicity for this cau$e was changed, be$ides that there would be a hundred thou- $and mixt motions, you would not be able to determine the $im- ple; nay farther, if the greater or le$$e velocity be able to alter the $implicity of the motion, no $imple body $hould move with a $imple motion; $ince that in all natural right motions, the veloci- ty is ever encrea$ing, and by con$equence $till changing the $impli- city, which as it is $implicity, ought of con$equence to be immu- table, and that which more importeth, you charge <I>Ari$totle</I> with another thing, that in the definition of motions compounded, he hath not made mention of tardity nor velocity, which you now in$ert for a nece$$ary and e$$ential point. Again you can draw no advantage from this rule, for that there will be among$t the mixt bodies $ome, (and that not a few) that will move $wiftly, <foot>and</foot> <p n=>9</p> and others more $lowly than the $imple; as for example, Lead, and Wood, in compari$on of earth; and therefore among$t the$e mo- tions, which call you the $imple, and which the mixt?</P> <P>SIMPL. I would call that $imple motion, which is made by a $imple body, and mixt, that of a compound body.</P> <P>SAGR. Very well, and yet <I>Simplicius</I> a little before you $aid, that the $imple, and compound motions, di$covered which were mixt, and which were $imple bodies; now you will have me by $imple and mixt bodies, come to know which is the $imple, and which is the compound motion: an excellent way to keep us igno- rant, both of motions and bodies. Moreover you have al$o a little above declared, how that a greater velocity did not $uffice, but you $eek a third condition for the definement of $imple motion, for which <I>Ari$totle</I> contented him$elf with one alone, namely, of the $implicity of the Space, or <I>Medium</I>: But now according to you, the $imple motion, $hall be that which is made upon a $imple line, with a certain determinate velocity, by a body $imply moveable. Now be it as you plea$e, and let us return to <I>Ari$totle,</I> who defi- neth the mixt motion to be that compounded of the right, and cir- cular, but produceth not any body, which naturally moveth with $uch a motion.</P> <P>SALV. I come again to <I>Ari$totle,</I> who having very well, and Methodically begun his di$cour$e, but having a greater aim to re$t at, and hit a marke, predefigned in his minde, then that to which his method lead him, digre$$ing from the purpo$e, he comes to a$$ert, as a thing known and manife$t, that as to the motions directly upwards or downwards, they naturally agree to Fire, and Earth; and that therefore it is nece$$ary, that be$ides the$e bodies, which are neer unto us, there mu$t be in nature another, to which the circular motion may agree: which $hall be $o much the more excellent by how much the circular motion is more perfect, then the $treight, but how much more perfect that is than this, he deter- mines from the greatne$s of the circular lines perfection above the <marg><I>The circular line perfect, according to</I> Ari$totle, <I>and but the right im- perfect, and why.</I></marg> right line; calling that perfect, and this imperfect; imperfect, be- cau$e if infinite it wanteth a termination, and end: and if it be fi- nite, there is yet $omething beyond which it may be prolonged. This is the ba$is, ground work, and ma$ter-$tone of all the Fabrick of the <I>Aristotelian</I> World, upon which they $uper$truct all their other properties, of neither heavy nor light, of ingenerable incor- ruptible, exemption from all motions, $ome onely the local, &c. And all the$e pa$$ions he affirmeth to be proper to a $imple body that is moved circularly; and the contrary qualities of gravity, levity, corruptibility, &c. he a$$igns to bodies naturally moveable in a $treight line, for that if we have already di$covered defects in the foundation, we may rationally que$tion what $oever may far- <foot>B ther</foot> <p n=>10</p> ther built thereon. I deny not, that this which <I>Ari$totle</I> hitherto hath introduced, with a general di$cour$e dependent upon univer- $al primary principles, hathbeen $ince in proce$s of time, re-inforced with particular rea$ons, and experiments; all which it would be nece$$ary di$tinctly to con$ider and weigh; but becau$e what hath been $aid hitherto pre$ents to $uch as con$ider the $ame many and no $mall difficulties, (and yet it would be nece$$ary, that the pri- mary principles and fundamentals, were certain, firm, and e$tabli$h- ed, that $o they might with more confidence be built upon) it would not be ami$s, before we farther multiply doubts, to $ee if haply (as I conjecture) betaking our $elves to other waies, we may not light upon a more direct and $ecure method; and with better con$idered principles of Architecture lay our primary fundamen- tals. Therefore $u$pending for the pre$ent the method of <I>Ari$to- tle,</I> (which we will re-a$$ume again in its proper place, and parti- cularly examine;) I $ay, that in the things hitherto affirmed by <marg><I>The world is $up- po$ed by the Au- thor to be perfectly ordinate.</I></marg> him, I agree with him, and admit that the World is a body enjoy- ing all dimen$ions, and therefore mo$t perfect; and I add, that as $uch, it is nece$$arily mo$t ordinate, that is, having parts between them$elves, with exqui$ite and mo$t perfect order di$po$ed; which a$$umption I think is not to be denied, neither by you or any other.</P> <P>SIMPL. Who can deny it? the fir$t particular (of the worlds dimen$ions) is taken from <I>Ari$totle</I> him$elf, and its denominati- on of ordinate $eems onely to be a$$umed from the order which it mo$t exactly keeps.</P> <marg><I>Streight motion impo$$ible in the world exactly or- dinate.</I></marg> <P>SALV. This principle then e$tabli$hed, one may immediately conclude, that if the entire parts of the World $hould be by their nature moveable, it is impo$$ible that their motions $hould be right, or other than circular; and the rea$on is $ufficiently ea$ie, and manife$t; for that what$oever moveth with a right motion, changeth place; and continuing to move, doth by degrees more and more remove from the term from whence it departed, and from all the places thorow which it $ucce$$ively pa$$ed; and if $uch motion naturally $uited with it, then it was not at the be- ginning in its proper place; and $o the parts of the World were not di$po$ed with perfect order. But we $uppo$e them to be per- fectly ordinate, therefore as $uch, it is impo$$ible that they $hould by nature change place, and con$equently move in a right moti- <marg><I>Right motion by nature infinite.</I></marg> on. Again, the right motion being by nature infinite, for that the right line is infinite and indeterminate, it is impo$$ible that <marg><I>Motion by a right line naturally im- po$$ible.</I></marg> any moveable can have a natural principle of moving in a right line; namely toward the place whither it is impo$$ible to arrive, <marg><I>Nature attempts not things impo$$i- ble to be effected.</I></marg> there being no præ-$inite term; and nature, as <I>Ari$totle</I> him$elf $aith well, never attempts to do that which can never be done, <foot>nor</foot> <p n=>11</p> nor e$$aies to move whither it is impo$$ible to arrive. And if any one $hould yet object, that albeit the right line, and con$equent- ly the motion by it is producible <I>in infinitum,</I> that is to $ay, is in- terminate; yet neverthele$s Nature, as one may $ay, arbitrarily hath a$$igned them $ome terms, and given natural in$tincts to its natural bodies to move unto the $ame; I will reply, that this <marg><I>Right motion might perhaps be in the fir$t Chaos.</I></marg> might perhaps be fabled to have come to pa$s in the fir$t Chaos, where indi$tinct matters confu$edly and inordinately wandered; to regulate which, Nature very appo$itely made u$e of right mo- <marg><I>Right motion is commodious to range in order, things ous of or- der.</I></marg> tions, by which, like as the well-con$tituted, moving, di$dorder them$elves, $o were they which were before depravedly di$po$ed by this motion ranged in order: but after their exqui$ite di$tribu- tion and collocation, it is impo$$ible that there $hould remain na- tural inclinations in them of longer moving in a right motion, from which now would en$ue their removal from their proper and natural place, that is to $ay, their di$ordination; we may there- fore $ay that the right motion $erves to conduct the matter to erect the work; but once erected, that it is to re$t immoveable, or if <marg><I>Mundane bodies moved in the be- ginning in a right line, and after- wards circularly? according to</I> Plato.</marg> moveable, to move it $elf onely circularly. Unle$s we will $ay with <I>Plato,</I> that the$e mundane bodies, after they had been made and fini$hed, were for a certain time moved by their Maker, in a right motion, but that after their attainment to certain and de- terminate places, they were revolved one by one in Spheres, pa$- $ing from the right to the circular motion, wherein they have been ever $ince kept and maintained. A $ublime conceipt, and <marg>* Thus doth he co- vertly and mode$t- ly $tile him$elfe throughout this work.</marg> worthy indeed of <I>Plato</I>: upon which, I remember to have heard our common friend the ^{*}<I>Lyncean Academick</I> di$cour$e in this man- ner, if I have not forgot it. Every body for any rea$on con$titu- ted in a $tate of re$t, but which is by nature moveable, being $et <marg><I>A moveable be- ing in a $tate of re$t, $hall not move unle$s it have an inclination to $ome particular place.</I></marg> at liberty doth move; provided withal, that it have an inclina- tion to $ome particular place; for $hould it $tand indifferently af- fected to all, it would remain in its re$t, not having greater in- ducement to move one way than another. From the having of this inclination nece$$arily proceeds, that it in its moving $hall con- <marg><I>The moveable ac- celerates its moti- on, going towards the place whither it hath an inclina- tion.</I></marg> tinually increa$e its acceleration, and beginning with a mo$t $low motion, it $hall not acquire any degree of velocity, before it $hall have pa$$ed thorow all the degrees of le$s velocity, or grea- ter tardity: for pa$$ing from the $tate of quiet (which is the in- <marg><I>The moveable pa$- $ing from re$t, go- eth thorow all the degrees of tardity.</I></marg> finite degree of tardity of motion) there is no rea$on by which it $hould enter into $uch a determinate degree of velocity, before it $hall have entred into a le$s, and into yet a le$s, before it entred into that: but rather it $tands with rea$on, to pa$s fir$t by tho$e degrees neare$t to that from which it departed, and from tho$e to the more remote; but the degree from whence the moveable <marg><I>Re$t the in$inioe degree of tardity.</I></marg> began to move, is that of extreme tardity, namely of re$t. <foot>B 2 Now</foot> <p n=>12</p> <marg><I>The moveable doth not accelerate, $ave only as it approach- eth nearer to its term.</I></marg> Now this acceleration of motion is never made, but when the moveable in moving acquireth it; nor is its acqui$t other than an approaching to the place de$ired, to wit, whither its natural in- clination attracts it, and thither it tendeth by the $horte$t way; namely, by a right line. We may upon good grounds therefore $ay, That Nature, to confer upon a moveable fir$t con$tituted in re$t a determinate velocity, u$eth to make it move according to <marg><I>Nature, to intro- duce in the move- able a certain de- gree of velocity, made it move in a right line.</I></marg> a certain time and $pace with a right motion. This pre$uppo$ed, let us imagine God to have created the Orb <I>v. g.</I> of <I>Jupiter,</I> on which he had determined to confer $uch a certain velocity, which it ought afterwards to retain perpetually uniform; we may with <I>Plato</I> $ay, that he gave it at the beginning a right and accelerate motion, and that it afterwards being arrived to that intended de- <marg><I>Vniform velocity convenient to the circular motion.</I></marg> gree of velocity, he converted its right, into a circular motion, the velocity of which came afterwards naturally to be uniform.</P> <P>SAGR. I hearken to this Di$cour$e with great delight; and I believe the content I take therein will be greater, when you have $atisfied me in a doubt: that is, (which I do not very well com- prehend) how it of nece$$ity en$ues, that a moveable departing <marg><I>Betwixt re$t, and any a$$igned degree of velocity, infinite degrees of le$s ve- locity interpo$e.</I></marg> from its re$t, and entring into a motion to which it had a natural inclination, it pa$$eth thorow all the precedent degrees o$ tardity, comprehended between any a$$igned degree of velocity, and the $tate of re$t, which degrees are infinite? $o that Nature was not able to confer them upon the body of <I>Jupiter,</I> his circular moti- on being in$tantly created with $uch and $uch velocity.</P> <marg><I>Nature doth not immediately con- fer a determinate degree of velocity, howbeit $he could.</I></marg> <P>SALV. I neither did, nor dare $ay, that it was impo$$ible for God or Nature to confer that velocity which you $peak of, imme- diately; but this I $ay, that <I>de facto</I> $he did not doit; $o that the doing it would be a work extra-natural, and by confequence mi- raculous.</P> <P>SAGR. Then you believe, that a $tone leaving its re$t, and en- tring into its natural motion towards the centre of the Earth, pa$- $eth thorow all the degrees of tardity inferiour to any degree of velocity?</P> <P>SALV. I do believe it, nay am certain of it; and $o certain, that I am able to make you al$o very well $atisfied with the truth thereof.</P> <P>SAGR. Though by all this daies di$cour$e I $hould gain no more but $uch a knowledge, I $hould think my time very well be$towed.</P> <P>SALV. By what I collect from our di$cour$e, a great part of your $cruple lieth in that it $hould in a time, and that very $hort, pa$s thorow tho$e infinite degrees of tardity precedent to any ve- locity, acquired by the moveable in that time: and therefore be- fore we go any farther, I will $eek to remove this difficulty, which <foot>$hall</foot> <p n=>13</p> $hall be an ea$ie task; for I reply, that the moveable pa$$eth by the afore$aid degrees, but the pa$$age is made without $taying in <marg><I>The moveable de- parting from re$v pa$$eth thorow all degrees of velocity without $taying in any.</I></marg> any of them; $o that the pa$$age requiring but one $ole in$tant of time, and every $mall time containing infinite in$tants, we $hall not want enough of them to a$$ign its own to each of the infinite degrees of tardity; although the time were never $o $hort.</P> <P>SAGR. Hitherto I apprehend you; neverthele$s it is very much that that Ball $hot from a Cannon (for $uch I conceive the ca- dent moveable) which yet we $ee to fall with $uch a precipice, that in le$s than ten pul$es it will pa$s two hundred yards of al- titude; $hould in its motion be found conjoyned with $o $mall a degree of velocity, that, $hould it have continued to have moved at that rate without farther acceleration, it would not have pa$t the $ame in a day.</P> <P>SALV. You may $ay, nor yet in a year, nor in ten, no nor in a thou$and; as I will endeavour to $hew you, and al$o happily with- out your contradiction, to $ome $ufficiently $imple que$tions that I will propound to you. Therefore tell me if you make any que- $tion of granting that, that that ball in de$cending goeth increa- $ing its <I>impetus</I> and velocity.</P> <P>SAGR. I am mo$t certain it doth.</P> <P>SALV. And if I $hould $ay that the <I>impetus</I> acquired in any place of its motion, is $o much, that it would $uffice to re-carry it to that place from which it came, would you grant it?</P> <P>SAGR. I $hould con$ent to it without contradiction, provided al- waies, that it might imploy without impediment its whole <I>impetus</I> in that $ole work of re-conducting it $elf, or another equal toit, to <marg><I>The ponderous mo- ver de$cending ac- quireth</I> impetus <I>$ufficient to re- carry it to the like height.</I></marg> that $elf-$ame height as it would do, in ca$e the Earth were bored thorow the centre, and the Bullet fell a thou$and yards from the $aid centre, for I verily believe it would pa$s beyond the centre, a$cending as much as it had de$cended; and this I $ee plainly in the experiment of a plummet hanging at a line, which removed from the perpendicular, which is its $tate of re$t, and afterwards let go, falleth towards the $aid perpendicular, and goes as far be- yond it; or onely $o much le$s, as the oppo$ition of the air, and line, or other accidents have hindred it. The like I $ee in the wa- ter, which de$cending thorow a pipe, re-mounts as much as it had de$cended.</P> <P>SALV. You argue very well. And for that I know you will not $cruple to grant that the acqui$t of the <I>impetus</I> is by means of the receding from the term whence the moveable departed, and its ap- proach to the centre, whither its motion tendeth; will you $tick to yeeld, that two equal moveables, though de$cending by divers lines, without any impediment, acquire equal <I>impetus,</I> provided that the approaches to the centre be equal?</P> <foot>SAGR</foot> <p n=>14</p> <P>SAGR. I do not very well under$tand the que$tion.</P> <P>SALV. I will expre$s it better by drawing a Figure: therefore I will $uppo$e the line A B [in <I>Fig.</I> 3.] parallel to the Horizon, and upon the point B, I will erect a perpendicular B C; and after that I adde this $launt line C A. Under$tanding now the line C A to be an inclining plain exqui$itely poli$hed, and hard, upon which de$cendeth a ball perfectly round and of very hard matter, and $uch another I $uppo$e freely to de$cend by the perpendicular C B: will you now confe$s that the <I>impetus</I> of that which de- $cends by the plain C A, being arrived to the point A, may be equal to the <I>impetus</I> acquired by the other in the point B, after the de$cent by the perpendicular C B?</P> <marg><I>The impetuo$ity of moveables equally approaching to the centre, are equal.</I></marg> <P>SAGR. I re$olutely believe $o: for in effect they have both the $ame proximity to the centre, and by that, which I have already granted, their impetuo$ities would be equally $ufficient to re-carry them to the $ame height.</P> <P>SALV. Tell me now what you believe the $ame ball would do put upon the Horizontal plane A B?</P> <marg><I>Vpon an horizon- tall plane the move- able lieth $till.</I></marg> <P>SAGR. It would lie $till, the $aid plane having no declination.</P> <P>SALV. But on the inclining plane C A it would de$cend, but with a gentler motion than by the perpendicular C B?</P> <P>SAGR. I may confidently an$wer in the affirmative, it $eem- ing to me nece$$ary that the motion by the perpendicular C B $hould be more $wift, than by the inclining plane C A; yet ne- verthele$s, i$ this be, how can the Cadent by the inclination ar- rived to the point A, have as much <I>impetus,</I> that is, the $ame de- gree of velocity, that the Cadent by the perpendicular $hall have in the point B? the$e two Propo$itions $eem contradictory.</P> <marg><I>The veloeity by the inclining plane e- qual to the veloci- ty by the perpendi- oular, and the mo- tion by the perpen- dicular $wifter than by the incli- nation.</I></marg> <P>SALV. Then you would think it much more fal$e, $hould I $ay, that the velocity of the Cadents by the perpendicular, and inclination, are ab$olutely equal: and yet this is a Propo$ition mo$t true, as is al$o this that the Cadent moveth more $wiftly by the perpendicular, than by the inclination.</P> <P>SAGR. The$e Propo$itions to my ears $ound very har$h: and I believe to yours <I>Simplicius</I>?</P> <P>SIMPL. I have the $ame $en$e of them.</P> <P>SALV. I conceit you je$t with me, pretending not to compre- hend what you know better than my $elf: therefore tell me <I>Sim- plicius,</I> when you imagine a moveable more $wift than ano- ther, what conceit do you fancy in your mind?</P> <P>SIMPL. I fancie one to pa$s in the $ame time a greater $pace than the other, or to move equal $paces, but in le$$er time.</P> <P>SALV. Very well: and for moveables equally $wift, what's your conceit of them?</P> <P>SIMPL. I fancie that they pa$s equal $paces in equal times.</P> <foot>SALV-</foot> <p n=>15</p> <P>SALV. And have you no other conceit thereof than this?</P> <P>SIMPL. This I think to be the proper definition of equal mo- tions.</P> <marg><I>Velocities are $aid to be equal, when the $paces pa$$ed are proportionate to their time.</I></marg> <P>SAGR. We will add moreover this other: and call that equal velocity, when the $paces pa$$ed have the $ame proportion, as the times wherein they are pa$t, and it is a more univer$al definition.</P> <P>SALV. It is $o: for it comprehendeth the equal $paces pa$t in equal times, and al$o the unequal pa$t in times unequal, but pro- portionate to tho$e $paces. Take now the $ame Figure, and apply- ing the conceipt that you had of the more ha$tie motion, tell me why you think the velocity of the Cadent by C B, is greater than the velocity of the De$cendent by C A?</P> <P>SIMPL. I think $o; becau$e in the $ame time that the Cadent $hall pa$s all C B, the De$cendent $hall pa$s in C A, a part le$s than C B.</P> <P>SALV. True; and thus it is proved, that the moveable moves more $wiftly by the perpendicular, than by the inclination. Now con$ider, if in this $ame Figure one may any way evince the o- ther conceipt, and finde that the moveables were equally $wift by both the lines C A and C B.</P> <P>SIMPL. I $ee no $uch thing; nay rather it $eems to contradict what was $aid before.</P> <P>SALV. And what $ay you, <I>Sagredus</I>? I would not teach you what you knew before, and that of which but ju$t now you pro- duced me the definition.</P> <P>SAGR. The definition I gave you, was, that moveables may be called equally $wift, when the $paces pa$$ed are proportional to the times in which they pa$$ed; therefore to apply the defini- tion to the pre$ent ca$e, it will be requi$ite, that the time of de- $cent by C A, to the time of falling by C B, $hould have the $ame proportion that the line C A hath to the line C B; but I under$tand not how that can be, for that the motion by C B is $wifter than by C A.</P> <P>SALV. And yet you mu$t of nece$$ity know it. Tell me a little, do not the$e motions go continually accelerating?</P> <P>SAGR. They do; but more in the perpendicular than in the inclination.</P> <P>SALV. But this acceleration in the perpendicular, is it yet not- with$tanding $uch in compari$on of that of the inclined, that two equal parts being taken in any place of the $aid perpendicu- lar and inclining lines, the motion in the parts of the perpendicu- lar is alwaies more $wift, than in the part of the inclination?</P> <P>SAGR. I $ay not $o: but I could take a $pace in the inclinati- on, in which the velocity $hall be far greater than in the like $pace taken in the perpendicular; and this $hall be, if the $pace in the <foot>perpen-</foot> <p n=>16</p> perpendicular $hould be taken near to the end C, and in the in- clination, far from it.</P> <P>SALV. You $ee then, that the Propo$ition which $aith, that the motion by the perpendicular is more $wift than by the incli- nation, holds not true univer$ally, but onely of the motions, which begin from the extremity, namely from the point of re$t: without which re$triction, the Propo$ition would be $o deficient, that its very direct contrary might be true; namely, that the mo- tion in the inclining plane is $wifter than in the perpendicular: for it is certain, that in the $aid inclination, we may take a $pace pa$t by the moveable in le$s time, than the like $pace pa$t in the perpendicular. Now becau$e the motion in the inclination is in $ome places more, in $ome le$s, than in the perpendicular; there- fore in $ome places of the inclination, the time of motion of the moveable, $hall have a greater proportion to the time of the motion of the moveable, by $ome places of the perpendicular, than the $pace pa$$ed, to the $pace pa$$ed: and in other places, the pro- portion of the time to the time, $hall be le$s than that of the $pace to the $pace. As for example: two moveables departing from their quie$cence, namely, from the point C, one by the per- pendicular C B, [in <I>Fig.</I> 4.] and the other by the inclination C A, in the time that, in the perpendicular, the moveable $hall have pa$t all C B, the other $hall have pa$t C T le$$er. And therefore the time by C T, to the time by C B (which is equal) $hall have a greater proportion than the line C T to C B, being that the <I>$ame</I> to the <I>le$s,</I> hath a greater proportion than to the <I>greater.</I> And on the contrary, if in C A, prolonged as much as is requi- $ite, one $hould take a part equal to C B, but pa$t in a $horter time; the time in the inclination $hall have a le$s proportion to the time in the perpendicular, than the $pace to the $pace. If therefore in the inclination and perpendicular, we may $uppo$e $uch $paces and velocities, that the proportion between the $aid $paces be greater and le$s than the proportion of the times; we may ea$ily grant, that there are al$o $paces, by which the times of the motions retain the $ame proportion as the $paces.</P> <P>SAGR. I am already freed from my greate$t doubt, and con- ceive that to be not onely po$$ible, but nece$$ary, which I but now thought a contradiction: but neverthele$s I under$tand not as yet, that this whereof we now are $peaking, is one of the$e po$$ible or nece$$ary ca$es; $o as that it $hould be true, that the time of de$cent by C A, to the time of the fall by C B, hath the $ame proportion that the line C A hath to C B; whence it may without contradiction be affirmed, that the velocity by the incli- nation C A, and by the perpendicular C B, are equal.</P> <P>SALV. Content your $elf for this time, that I have removed <foot>your</foot> <p n=>17</p> your incredulity; but for the knowledge of this, expect it at $ome other time, namely, when you $hall $ee the matters concer- ning local motion demon$trated by our <I>Academick</I>; at which time you $hall find it proved, that in the time that the one movea- ble falls all the $pace C B, the other de$cendeth by C A as far as the point T, in which falls the perpendicular drawn from the point B: and to find where the $ame Cadent by the perpendi- cular would be when the other arriveth at the point A, draw from A the perpendicular unto C A, continuing it, and C B unto the interfection, and that $hall be the point $ought. Whereby you $ee how it is true, that the motion by C B is $wifter than by the inclination C A ($uppo$ing the term C for the beginning of the motions compared) becau$e the line C B is greater than C T, and the other from C unto the inter$ection of the perpendicular drawn from A, unto the line C A, is greater than C A, and therefore the motion by it is $wifter than by C A But when we compare the motion made by all C A, not with all the motion made in the $ame time by the perpendicular continued, but with that made in part of the time, by the $ole part C B, it hinders not, that the motion by C A, continuing to de$cend beyond, may arrive to A in $uch a time as is in proportion to the other time, as the line C A is to the line C B. Now returning to our fir$t purpo$e; which was to $hew, that the grave moveable leaving its quie$cence, pa$$eth defcending by all the degrees of tardity, precedent to any what$oever degree of velocity that it aequireth, re-a$$uming the $ame Figure which we u$ed before, let us remem- ber that we did agree, that the De$cendent by the inclination C A, and the Cadent by the perpendicular C B, were found to have acquired equal degrees of velocity in the terms B and A: now to proceed, I $uppo$e you will not $cruple to grant, that upon ano- ther plane le$s $teep than A C; as for example, A D [in <I>Fig.</I> 5.] the motion of the de$cendent would be yet more $low than in the plane A C. So that it is not any whit dubitable, but that there may be planes $o little elevated above the Horizon A B, that the moveable, namely the $ame ball, in any the longe$t time may reach the point A, which being to move by the plane A B, an infi- nite time would not $uffice: and the motion is made always more $lowly, by how much the declination is le$s. It mu$t be therefore confe$t, that there may be a point taken upon the term B, $o near to the $aid B, that drawing from thence to the point A a plane, the ball would not pa$s it in a whole year. It is requi$ite next for you to know, that the <I>impetus,</I> namely the degree of velo- city the ball is found to have acquired when it arriveth at the point A, is $uch, that $hould it continue to move with this $elf-$ame degree uniformly, that is to $ay, without accelerating or retarding; <foot>C in</foot> <p n=>18</p> in as much more time as it was in coming by the inclining plane, it would pa$s double the $pace of the plane inclined: namely (for example) if the ball had pa$t the plane D A in an hour, con- tinuing to move uniformly with that degree of velocity which it is found to have in its arriving at the term A, it $hall pa$s in an hour a $pace double the length D A; and becau$e (as we have $aid) the degrees of velocity acquired in the points B and A, by the moveables that depart from any point taken in the perpendicu- lar C B, and that de$cend, the one by the inclined plane, the o- ther by the $aid perpendicular, are always equal: therefore the cadent by the perpendicular may depart from a term $o near to B, that the degree of velocity acquired in B, would not $uffice ($till maintaining the $ame) to conduct the moveable by a $pace dou- ble the length of the plane inclined in a year, nor in ten, no nor in a hundred. We may therefore conclude, that if it be true, that according to the ordinary cour$e of nature a moveable, all external and accidental impediments removed, moves upon an in- clining plane with greater and greater tardity, according as the inclination $hall be le$s; $o that in the end the tardity comes to be infinite, which is, when the inclination concludeth in, and joyneth to the horizontal plane; and if it be true likewi$e, that the de- gree of velocity acquired in $ome point of the inclined plane, is equal to that degree of velocity which is found to be in the move- able that de$cends by the perpendicular, in the point cut by a parallel to the Horizon, which pa$$eth by that point of the incli- ning plane; it mu$t of nece$$ity be granted, that the cadent de- parting from re$t, pa$$eth thorow all the infinite degrees of tar- dity, and that con$equently, to acquire a determinate degree of velocity, it is nece$$ary that it move fir$t by right lines, de$cend- ing by a $hort or long $pace, according as the velocity to be acqui- red, ought to be either le$s or greater, and according as the plane on which it de$cendeth is more or le$s inclined; $o that a plane may be given with $o $mall inclination, that to acquire in it the a$$igned degree of velocity, it mu$t fir$t move in a very great $pace, and take a very long time; whereupon in the horizontal plane, any how little $oever velocity, would never be naturally acquired, $ince that the moveable in this ca$e will never move: but the <marg><I>The circular mo- tion is never ac- quired naturally, without right mo- tion precede it. Circular motion perpetually uni- form.</I></marg> motion by the horizontal line, which is neither declined or incli- ned, is a circular motion about the centre: therefore the circu- lar motion is never acquired naturally, without the right motion precede it; but being once acquired, it will continue perpetually with uniform velocity. I could with other di$cour$es evince and demon$trate the $ame truth, but I will not by $o great a digre$- fion interrupt our principal argument: but rather will return to it upon $ome other occa$ion; e$pecially $ince we now a$$umed the <foot>$ame</foot> <p n=>19</p> $ame, not to $erve for a nece$$ary demon$tration, but to adorn a <I>Platonick</I> Conceit; to which I will add another particular ob$er- vation of our <I>Academick,</I> which hath in it $omething of admira- ble. Let us $uppo$e among$t the decrees of the divine <I>Architect,</I> a purpo$e of creating in the World the$e Globes, which we be- hold continually moving round, and of a$$igning the centre of their conver$ions; and that in it he had placed the Sun immoveable, and had afterwards made all the $aid Globes in the $ame place, and with the intended inclinations of moving towards the Centre, till they had acquired tho$e degrees of velocity, which at fir$t $ee- med good to the $ame Divine Minde; the which being acquired, we la$tly $uppo$e that they were turned round, each in his Sphere retaining the $aid acquired velocity: it is now demanded, in what altitude and di$tance from the Sun the place was where the $aid Orbs were primarily created; and whether it be po$$ible that they might all be created in the $ame place? To make this inve- $tigation, we mu$t take from the mo$t skilfull A$tronomers the magnitude of the Spheres in which the Planets revolve, and like- wi$e the time of their revolutions: from which two cognitions is gathered how much (for example) <I>Jupiter</I> is $wifter than <I>Sa- turne</I>; and being found (as indeed it is) that <I>Jupiter</I> moves more $wiftly, it is requi$ite, that departing from the $ame altitude, <I>Ju- piter</I> be de$cended more than <I>Saturne,</I> as we really know it is, its Orbe being inferiour to that of <I>Saturne.</I> But by proceeding for- wards, from the proportions of the two velocities of <I>Jupiter</I> and <I>Saturne,</I> and from the di$tance between their Orbs, and from the proportion of acceleration of natural motion, one may finde in what altitude and di$tance from the centre of their revolutions, <marg><I>The magnitude of the Orbs, and the velocity of the mo- tion of the Planets, an$wer proportion- ably, as if de$cend- ed from the $ame place.</I></marg> was the place from whence they fir$t departed. This found out, and agreed upon, it is to be $ought, whether <I>Mars</I> de$cending from thence to his Orb, the magnitude of the Orb, and the ve- locity of the motion, agree with that which is found by calcula- tion; and let the like be done of the <I>Eartb,</I> of <I>Venus,</I> and of <I>Mercury</I>; the greatne$s of which Spheres, and the velocity of their motions, agree $o nearly to what computation gives, that it is very admirable.</P> <P>SAGR. I have hearkened to this conceit with extreme delight; and, but that I believe the making of the$e calculations truly would be a long and painfull task, and perhaps too hard for me to comprehend, I would make a trial of them.</P> <P>SALV. The operation indeed is long and difficult; nor could I be certain to finde it $o readily; therefore we $hall refer it to an- other time, and for the pre$ent we will return to our fir$t propo- $al, going on there where we made digre$$ion; which, if I well remember, was about the proving the motion by a right line of no <foot>C 2 u$e,</foot> <p n=>20</p> u$e, in the ordinate parts of the World; and we did proceed to $ay, that it was not $o in circular motions, of which that which is made by the moveable in it $elf, $till retains it in the $ame place, <marg><I>Finite and termi- nate circular mo- tions di$order not the parts of the World.</I></marg> and that which carrieth the moveable by the circumference of a circle about its fixed centre, neither puts it $elf, nor tho$e about it in di$order; for that $uch a motion primarily is finite and terminate (though not yet fini$hed and determined) but there is no point <marg><I>In the circular mo- tion, every point in the circumference is the begining and end.</I></marg> in the circumference, that is not the fir$t and la$t term in the cir- culation; and continuing it in the circumference a$$igned it, it leaveth all the re$t, within and without that, free for the u$e of others, without ever impeding or di$ordering them. This being a motion that makes the moveable continually leave, and con- <marg><I>Circular motion onely is uniform.</I></marg> tinually arrive at the end; it alone therefore can primarily be u- niform; for that acceleration of motion is made in the moveable, when it goeth towards the term, to which it hath inclination; and the retardation happens by the repugnance that it hath to leave and part from the $ame term; and becau$e in circular mo- tion, the moveable continually leaves the natural term, and con- tinually moveth towards the $ame, therefore, in it, the repug- nance and inclination are always of equal force: from which e- quality re$ults a velocity, neither retarded nor accelerated, <I>i. e.</I> an uniformity in motion. From this conformity, and from the being <marg><I>Circular motion may be continued perpetually.</I></marg> terminate, may follow the perpetual continuation by $ucce$$ively reiterating the circulations; which in an undeterminated line, and in a motion continually retarded or accelerated, cannot na- <marg><I>Right motion can- not naturally be perpetual.</I></marg> turally be. I $ay, naturally; becau$e the right motion which is retarded, is the violent, which cannot be perpetual; and the ac- celerate arriveth nece$$arily at the term, if one there be; and if there be none, it cannot be moved to it, becau$e nature moves not whether it is impo$$ible to attain. I conclude therefore, that the circular motion can onely naturally con$i$t with natural bo- dies, parts of the univer$e, and con$tituted in an excellent di$po- $ure; and that the right, at the mo$t that can be $aid for it, is <marg><I>Right motion a$- $igned to natural bodies, to reduce them to perfect or- der, when removed from their places.</I></marg> a$$igned by nature to its bodies, and their parts, at $uch time as they $hall be out of their proper places, con$tituted in a depraved di$po$ition, and for that cau$e needing to be redured by the $hort- e$t way to their natural $tate. Hence, me thinks, it may ratio- nally be concluded, that for maintenance of perfect order among $t the parts of the World, it is nece$$ary to $ay, that moveables are moveable onely circularly; and if there be any that move not <marg><I>Re$t onely, and circular motion are apt to con$erve or- der.</I></marg> circularly, the$e of nece$$ity are immoveable: there being no- thing but re$t and circular motion apt to the con$ervation of or- der. And I do not a little wonder with my $elf, that <I>Ari$totle,</I> who held that the Terre$trial globe was placed in the centre of the World, and there remained immoveable, $hould not $ay, that <foot>of</foot> <p n=>21</p> of natural bodies $ome are moveable by nature, and others immo- veable; e$pecially having before defined Nature, to be the prin- ciple of Motion and Re$t.</P> <P>SIMPL. <I>Ari$totle,</I> though of a very per$picacious wit, would not $train it further than needed: holding in all his argumen- <marg><I>Sen$ible experi- ments are to be pre- ferred before hu- mane argument a- tions.</I></marg> tations, that $en$ible experiments were to be preferred before any rea$ons founded upon $trength of wit, and $aid tho$e which $hould deny the te$timony of $en$e de$erved to be puni$hed with <marg><I>He who denies $en$e, de$erves to be deprived of it. Sen$e $heweth that things grave move to the</I> medium, <I>and the light to the concave.</I></marg> the lo$s of that $en$e; now who is $o blind, that $ees not the parts of the Earth and Water to move, as being grave, natural- ly downwards, namely, towards the centre of the Univer$e, a$- $igned by nature her $elf for the end and term of right motion <I>deor$ùm</I>; and doth not likewi$e $ee the Fire and Air to move right upwards towards the Concave of the Lunar Orb, as to the natural end of motion <I>$ur$ùm</I>? And this being $o manife$tly $een, and we being certain, that <I>eadem est ratio totius & partium,</I> why may we not a$$ert it for a true and manife$t propo$ition, that the natural motion of the Earth is the right motion <I>ad medium,</I> and that of the Fire, the right <I>à medio</I>?</P> <P>SALV. The mo$t that you can pretend from this your Di$- cour$e, were it granted to be true, is that, like as the parts of the Earth removed from the whole, namely, from the place where they naturally re$t, that is in $hort reduced to a depraved and di$- ordered di$po$ure, return to their place $pontaneou$ly, and there- fore naturally in a right motion, (it being granted, that <I>eadem $it ratio totius & partium</I>) $o it may be inferred, that the Terre$trial Globe removed violently from the place a$$igned <marg><I>It is que$tionable whether de$cending weights move in a right line.</I></marg> it by nature, it would return by a right line. This, as I have $aid, is the mo$t that can be granted you, and that onely for want of examination; but he that $hall with exactne$s revi$e the$e things, will fir$t deny, that the parts of the Earth, in returning to its whole, move in a right line, and not by a circular or mixt; and really you would have enough to do to demon$trate the contra- ry, as you $hall plainly $ee in the an$wers to the particular rea$ons and experiments alledged by <I>Ptolomey</I> and <I>Ari$totle.</I> Secondly, If another $hould $ay that the <I>parts</I> of the Earth, go not in their motion towards the Centre of the World, but to unite with its <I>Whole,</I> and that for that rea$on they naturally incline towards the centre of the Terre$trial Globe, by which inclination they con- $pire to form and pre$erve it, what other <I>All,</I> or what other Centre would you find for the World, to which the whole Terrene <marg><I>The Earth speri- cal by the con$pi- ration of its parts to its Centre.</I></marg> Globe, being thence removed, would $eek to return, that $o the rea$on of the <I>Whole</I> might be like to that of its <I>parts</I>? It may be added, That neither <I>Ari$totle,</I> nor you can ever prove, that the Earth <I>de facto</I> is in the centre of the Univer$e; but if any Centre <foot>may</foot> <p n=>22</p> <marg><I>The Sun more pro- bably in the centre of the Vniver$e, than the Earth.</I></marg> may be a$ligned to the Univer$e, we $hall rather find the Sun placed in it, as by the $equel you $hall under$tand.</P> <P>Now, like as from the con$entaneous con$piration of all the parts of the Earth to form its whole, doth follow, that they with <marg><I>Natural inclina- tion of the parts of all the globes of the World to go to their centre.</I></marg> equal inclination concurr thither from all parts; and to unite them$elves as much as is po$$ible together, they there $phelically adapt them$elves; why may we not believe that the Sun, Moon, and other mundane Bodies, be al$o of a round figure, not by o- ther than a concordant in$tinct, and natural concour$e of all the parts compo$ing them? Of which, if any, at any time, by any violence were $eparated from the whole, is it not rea$onable to think, that they would $pontaneou$ly and by natural in$tinct re- turn? and in this manner to infer, that the right motion agreeth with all mundane bodies alike.</P> <P>SIMPL. Certainly, if you in this manner deny not onely the Principles of Sciences, but manife$t Experience, and the Sen$es them$elves, you can never be convinced or removed from any o- pinion which you once conceit, therefore I will choo$e rather to be $ilent (for, <I>contra negantes principia non e$t di$putandum</I>) than contend with you. And in$i$ting on the things alledged by you even now ($ince you que$tion $o much as whether grave move- ables have a right motion or no) how can you ever rationally de- <marg><I>The right motion of grave bodies manife$t to $en$e.</I></marg> ny, that the parts of the Earth; or, if you will, that ponderous matters de$cend towards the Centre, with a right motion; when- as, if from a very high Tower, who$e walls are vcry upright and perpendicular, you let them fall, they $hall de$cend gliding and $liding by the Tower to the Earth, exactly in that very place where a plummet would fall, being hanged by a line fa$tned above, ju$t there, whence the $aid weights were let fall? is not this a more than evident argument of the motions being right, and to- <marg><I>Arguments of</I> A- ri$totle, <I>to prove that grave bodies move with an in- clination to arrive at the centre of the Vniver$e.</I></marg> wards the Centre? In the $econd place you call in doubt, whe- ther the parts of the Earth are moved, as <I>Ari$totle</I> affirms, to- wards the Centre of the World; as if he had not rationally de- mon$trated it by contrary motions, whil$t he thus argueth; The motion of heavie bodies is contrary to that of the light: but the motion of the light is manife$t to be directly upwards, namely, towards the circumference of the World, therefore the motion of the heavie is directly towards the Centre of the World: and it <marg><I>Heavie bodies move towards the centre of the Earth</I> per accidens.</marg> happens <I>per accidens,</I> that it be towards the centre of the Earth, for that this $triveth to be united to that. The $eeking in the next place, what a part of the Globe of the Sun or Moon would do, were it $eparated from its whole, is vanity; becau$e that there- <marg><I>To $eek what would follow upon an impo$$ibility, is folly.</I></marg> by that is $ought, which would be the con$equence of an impo$$i- bility; in regard that, as <I>Ari$totle</I> al$o demon$trates, the cœle$tial bodies are impa$$ible, impenetrable, and infrangible; $o that $uch <foot>a ca$e</foot> <p n=>23</p> a ca$e can never happen: and though it $hould, and that the $e- <marg><I>Cœle$tial bodies neither heavie nor light, according to</I> Ari$totle.</marg> parated part $hould return to its whole, it would not return as grave or light, for that the $ame <I>Ari$totle</I> proveth, that the Cœ- le$tial Bodies are neither heavie nor light.</P> <P>SALV. With what rea$on I doubt, whether grave bodies move by a right and perpendicular line, you $hall hear, as I $aid be- fore, when I $hall examine this particular argument. Touching the $econd point, I wonder that you $hould need to di$cover the <I>Paralogi$m</I> of <I>Ari$totle,</I> being of it $elf $o manife$t; and that you perceive not, that <I>Ari$totle</I> $uppo$eth that which is in que$ti- on: therefore take notice.</P> <P>SIMPL. Pray <I>Salviatus</I> $peak with more re$pect of <I>Ari$totle</I>: for who can you ever per$wade, that he who was the fir$t, only, and admirable explainer of the <I>Syllogi$tick</I> forms of demon$tration, <marg>Ari$totle <I>cannot e- quivocate, being the inventer of</I> Lo- gick.</marg> of <I>Elenchs,</I> of the manner of di$covering <I>Sophi$ms, Paralogi$ms,</I> and in $hort, of all the parts of <I>Logick,</I> $hould afterwards $o notoriou$ly equivocate in impo$ing that for known, which is in que$tion? It would be better, my Ma$ters, fir$t perfectly to under$tand him, and then to try, if you have a minde, to oppo$e him.</P> <P>SALV. <I>Simplicius,</I> we are here familiarly di$cour$ing among our $elves, to inve$tigate $ome truth; I $hall not be di$plea$ed that you di$cover my errors; and if I do not follow the mind of <I>Ari$totle,</I> freely reprehend me, and I $hall take it in good part. Onely give me leave to expound my doubts, and to reply $ome- thing to your la$t words, telling you, that <I>Logick,</I> as it is well under$tood, is the Organe with which we philo$ophate; but as it may be po$$ible, that an Arti$t may be excellent in making Or- gans, but unlearned in playing on them, thus he might be a great Logician, but unexpert in making u$e of <I>Logick</I>; like as we have many that theorically under$tand the whole Art of Poetry, and yet are unfortunate in compo$ing but meer four Ver$es; others <marg>* A famous <I>Italian</I> Painter.</marg> enjoy all the precepts of <I>Vinci</I>^{*}, and yet know not how to paint a Stoole. The playing on the Organs is not taught by them who know how to make Organs, but by him that knows how to play on them: Poetry is learnt by continual reading of Poets: Limn- ing is learnt by continual painting and de$igning: Demon$tration from the reading of Books full of demon$trations, which are the Mathematical onely, and not the Logical. Now returning to our purpo$e, I $ay, that that which <I>Ari$totle</I> $eeth of the motion of light bodies, is the departing of the Fire from any part of the Superficies of the Terre$trial Globe, and directly retreating from it, mounting upwards; and this indeed is to move towards a circumference greater than that of the Earth; yea, the $ame <I>A- ri$totle</I> makes it to move to the concave of the Moon, but that this circumference is that of the World, or concentrick to it, $o <foot>that</foot> <p n=>24</p> that to move towards this, is a moving towards that of the World, that he cannot affirm, unle$s he $uppo$eth, That the Centre of the <marg><I>Paralogi$m of</I> A- ri$totle, <I>in proving the Earth to be in the Centre of the World.</I></marg> Earth, from which we $ee the$e light a$cendent bodies to depart, be the $ame with the Centre of the World; which is as much as to $ay, that the terre$trial Globe is con$tituted in the mid$t of the World: which is yet that of which we were in doubt, and which <I>Aristotle</I> intended to prove. And do you $ay that this is not a <marg><I>The Paralogi$me of</I> Ari$totle <I>another way di$covered.</I></marg> manife$t <I>Paralogi$m</I>?</P> <P>SAGR. This Argument of <I>Ari$totle</I> appeared to me deficient al$o, and <I>non</I>-concludent for another re$pect; though it were granted, that that Circumference, to which the Fire directly mo- veth, be that which includeth the World: for that in a circle, not onely the centre, but any other point being taken, every move- able which departing thence, $hall move in a right line, and to- wards any what$oever part, $hall without any doubt go towards the circumference, and continuing the motion, $hall al$o arrive thither; $o that we may truly $ay, that it moveth towards the circumference: but yet it doth not follow, that that which mo- veth by the $ame line with a contrary motion, would go towards the centre, unle$s when the point taken were the centre it $elf, or that the motion were made by that onely line, which produced from the point a$$igned, pa$$eth thorow the centre. So that to $ay, that Fire moving in a right line, goeth towards the circumfe- rence of the World, therefore the parts of the Earth which by the $ame lines move with a contrary motion, go towards the cen- tre of the World, concludeth not, unle$s then when it is pre- $uppo$ed, that the lines of the Fire prolonged pa$s by the centre of the World; and becau$e we know certainly of them, that they pa$s by the centre of the Terre$trial Globe (being perpendicu- lar to its $uperficies, and not inclined) therefore to conclude, it mu$t be $uppo$ed, that the centre of the Earth is the $ame with the centre of the World; or at lea$t, that the parts of the Fire and Earth de$cend not, $ave onely by one $ole line which pa$$eth by the centre of the World. Which neverthele$s is fal$e, and re- pugnant to experience, which $heweth us, that the parts of Fire, not by one line onely, but by infinite, produced from the centre of the Earth towards all the parts of the World, a$cend always by lines perpendicular to the Superficies of the Terre$tri- al Globe.</P> <P>SALV. You do very ingeniou$ly lead <I>Ari$totle</I> to the $ame in- convenience, <I>Sagredus,</I> $hewing his manife$t equivoke; but withal you add another incon$i$tency. We $ee the Earth to be $pherical, and therefore are certain that it hath its centre, to which we $ee all its parts are moved; for $o we mu$t $ay, whil$t their motions are all perpendicular to the Superficies of the Earth; we <foot>mean,</foot> <p n=>25</p> mean, that as they move to the centre of the Earth, they move to their <I>Whole,</I> and to their Univer$al Mother: and we are $till far- ther $o free, that we will $uffer our $elves to be per$waded, that <marg><I>Grave bodies may more rationally be affirmed to tend to the Centre of the Earth, than of the Vniver$e.</I></marg> their natural in$tinct is, not to go towards the centre of the Earth, but towards that of the Univer$e; which we know not where to find, or whether it be or no; and were it granted to be, it is but an imaginary point, and a nothing without any quality. As to what <I>Simplicius</I> $aid la$t, that the contending whether the parts of the Sun, Moon, or other cœle$tial Body, $eparated from their <I>Whole,</I> $hould naturally return to it, is a vanity, for that the ca$e is impo$$ible; it being clear by the Demon$trations of <I>Ari$totle,</I> that the cœle$tial Bodies are impa$$ible, impenetrable, unparta- <marg><I>The conditions and attributes which differ the cœle$tial bodies from Ele- mentary, depend on the motions a$$ign- ed them by</I> Ari$t.</marg> ble, <I>&c.</I> I an$wer, that none of the conditions, whereby <I>Aristo- tle</I> di$tingui$heth the Cœle$tial Bodies from Elementary, hath o- ther foundation than what he deduceth from the diver$ity of the natural motion of tho$e and the$e; in$omuch that it being deni- ed, that the circular motion is peculiar to Cœle$tial Bodies, and affirmed, that it is agreeable to all Bodies naturally moveable, it is behoofull upon nece$$ary con$equence to $ay, either that the attributes of generable, or ingenerable, alterable, or unalterable, partable, or unpartable, <I>&c.</I> equally and commonly agree with all worldly bodies, namely, as well to the Cœle$tial as to the E- lementary; or that <I>Ari$totle</I> hath badly and erroneou$ly dedu- ced tho$e from the circular motion, which he hath a$$igned to Cœ- le$tial Bodies.</P> <P>SIMPL. This manner of argumentation tends to the $ubver$i- on of all Natural Philo$ophy, and to the di$order and $ubver$ion of Heaven and Earth, and the whole Univer$e; but I believe the Fundamentals of the <I>Peripateticks</I> are $uch, that we need not fear that new Sciences can be erected upon their ruines.</P> <P>SALV. Take no thought in this place for Heaven or the Earth, neither fear their $ubver$ion, or the ruine of Philo$ophy. As to Heaven, your fears are vain for that which you your $elf hold unalterable and impa$$ible; as for the Earth, we $trive to enoble and perfect it, whil$t we make it like to the Cœle$tial Bodies, and as it were place it in Heaven, whence your Philo$ophers have exiled it. Philo$ophy it $elf cannot but receive benefit from our <marg><I>The di$putes and contradictions of Philo$ophers may conduce to the benefit of Philo$o- phy.</I></marg> Di$putes, for if our conceptions prove true, new Di$coveries will be made; if fal$e, the fir$t Doctrine will be more confirmed. Rather be$tow your care upon $ome Philo$ophers, and help and defend them; for as to the Science it $elf, it cannot but improve. And that we may return to our purpo$e, be plea$ed freely to pro- duce what pre$ents it $elf to you in confirmation of that great dif- ference which <I>Ari$totle</I> puts between the Cœle$tial Bodies, and the Elementary parts of the World, in making tho$e ingenerable, <foot>D incor-</foot> <p n=>26</p> incorruptible, unalterable, <I>&c.</I> and this corruptible, alterable, <I>&c.</I></P> <P>SIMPL. I $ee not yet any need that <I>Ari$totle</I> hath of help, $tanding as he doth $toutly and $trongly on his feet; yea not be- ing yet a$$aulted, much le$s foiled by you. And what ward will you choo$e in this combate for this fir$t blow? <I>Aristotle</I> writeth, <marg>Ari$totles <I>di$cour$e to prove the incor- ruptibility of Hea- ven.</I></marg> that whatever is generated, is made out of a contrary in $ome $ubject, and likewi$e is corrupted in $ome certain $ubject from a <marg><I>Generation & cor- ruption is onely a- mong$t contraries, according to</I> Ari$t.</marg> contrary into a contrary; $o that (ob$erve) corruption and ge- neration is never but onely in contraries; If therefore to a Cœ- le$tial Body no contrary can be a$$igned, for that to the circular <marg><I>To the circular motion no other motion is contrary.</I></marg> motion no other motion is contrary, then Nature hath done very well to make that exempt from contraries, which was to be in- generable and incorruptible, This fundamental fir$t confirmed, it immediately followeth of con$equence, that it is inaugmenta- ble, inalterable, impa$$ible, and finally eternal, and a propor- <marg><I>Heaven an habi- tation for the imm- ortal Gods.</I></marg> tionate habitation to the immortal Deities, conformable to the opinion even of all men that have any conceit of the Gods. He <marg><I>Immutability of Heaven evident to $ex$e.</I></marg> afterwards confirmeth the $ame by $en$e; in regard, that in all times pa$t, according to memory or tradition, we $ee nothing re- moved, according to the whole outward Heaven, nor any of its <marg><I>He proveth that the circular motion hath no contrary.</I></marg> proper parts. Next, as to the circular motion, that no other is contrary to it, <I>Aristotle</I> proveth many ways; but without reci- ting them all, it is $ufficiently demon$trated, $ince fimple motions are but three, to the <I>medium,</I> from the <I>medium,</I> and about the <I>medium,</I> of which the two right, <I>$ur$um</I> and <I>deor$um,</I> are mani- fe$tly contrary; and becau$e one onely hath onely one for con- trary, therefore there re$ts no other motion which may be contra- ry to the circular. You $ee the $ubtle and mo$t concluding di$- cour$e of <I>Ari$totle,</I> whereby he proveth the incorruptibility of Heaven.</P> <P>SALV. This is nothing more, $ave the pure progre$s of <I>Ari$to- tle,</I> by me hinted before; wherein, be$ides that I affirm, that the motion which you attribute to the Cœle$tial Bodies agreeth al$o to the Earth, its illation proves nothing. I tell you therefore, that that circular motion which you a$$ign to Cœle$tial Bodies, $uiteth al$o to the Earth, from which, $uppo$ing that the re$t of your di$cour$e were concludent, will follow one of the$e three things, as I told you a little before, and $hall repeat; namely, either that the Earth it $elf is al$o ingenerable, and incorruptible, as the Cœle$tial bodies; or that the Cœle$tial bodies are, like as the Elementary generable, alterable &c. or that this difference of motion hath nothing to do with Generation and Corruption. The di$cour$e of <I>Ari$totle,</I> and yours al$o contain many Propo$i- tions not to be lightly admitted, and the better to examine them, it will be convenient to reduce them to the mo$t ab$tracted and <foot>di$tinct</foot> <p n=>27</p> di$tinct that can be po$$ible; and excu$e me <I>Sagredus,</I> if haply with $ome tediou$ne$s you hear me oft repeat the $ame things, and fancie that you $ee me rea$$ume my argument in the pub- lick circle of Di$putations. You $ay Generation and Corrupti- on are onely made where there are contraries; contraries are onely among$t $imple natural bodies, moveable with contrary motions; contrary motions are onely tho$e which are made by a right line between contrary terms; and the$e are onely two, that is to $ay, from the <I>medium,</I> and towards the <I>medium</I>; and $uch motions belong to no other natural bodies, but to the <I>Earth,</I> the <I>Fire,</I> and the other two Elements: therefore Generation and Corruption is onely among$t the Elements. And becau$e the third $imple motion, namely, the circular about the <I>medium,</I> hath no contrary, (for that the other two are contraries, and one onely, hath but onely one contrary) therefore that natural body with which $uch motion agreeth, wants a contrary; and having no contrary is ingenerable and incorruptible, &c. Becau$e where there is no contrariety, there is no generation or corruption, <I>&c.</I> But $uch motion agreeth onely with the Cœle$tial bodies; there- <marg><I>Its ea$ier to prove the Earth to move, than that corrupti- on is made by con- traries.</I></marg> fore onely the$e are ingenerable, incorruptible, <I>&c.</I> And to begin, I think it a more ea$ie thing, and $ooner done to re$olve, whether the Earth (a mo$t va$t Body, and for its vicinity to us, mo$t tractable) moveth with a $peedy motion, $uch as its revo- lution about its own axis in twenty four hours would be, than it is to under$tand and re$olve, whether Generation and Corruption ari$eth from contrariety, or el$e whether there be $uch things as generation, corruption and contrariety in nature. And if you, <I>Simplicius,</I> can tell me what method Nature ob$erves in working, when $he in a very $hort time begets an infinite number of flies from a little vapour of the Mu$t of wine, and can $hew me which are there the contraries you $peak of, what it is that corrupteth, and how; I $hould think you would do more than I can; for I profe$s I cannot comprehend the$e things. Be$ides, I would ve- ry gladly under$tand how, and why the$e corruptive contraries are $o favourable to Daws, and $o cruel to Doves; $o indulgent to Stags, and $o ha$ty to Hor$es, that they do grant to them many more years of life, that is, of incorruptibility, than weeks to the$e. Peaches and Olives are planted in the $ame $oil, expo$ed to the $ame heat and cold, to the $ame wind and rains, and, in a word, to the $ame contrarieties; and yet tho$e decay in a $hort time, and the$e live many hundred years. Furthermore, I never was thorowly $atisfied about this $ub$tantial tran$mutation ($till keep- ing within pure natural bounds) whereby a matter becometh $o transform'd, that it $hould be nece$$arily $aid to be de$troy'd, $o that nothing remaineth of its fir$t being, and that another body <foot>D2 quite</foot> <p n=>28</p> <marg><I>Bare tran$po$ition of parts may repre- $ent bodies under diver$e asp cts.</I></marg> quite differing there-from $hould be thence produced; and if I fancy to my $elf a body under one a$pect, and by and by under another very different, I cannot think it impo$$ible but that it may happen by a $imple tran$po$ition of parts, without corrupting or ingendring any thing a-new; for we $ee $uch kinds of Metamor- pho$es dayly: $o that to return to my purpo$e, I an$wer you, that ina$much as you go about to per$wade me that the Earth can not move circularly by way of corruptibility and generability, you have undertook a much harder task than I, that with argu- ments more difficult indeed, but no le$s concluding, will prove the contrary.</P> <P>SAGR. Pardon me, <I>Salviatus,</I> if I interrupt your di$cour$e, which, as it delights me much, for that I al$o am gravel'd with the $ame doubts; $o I fear that you can never conclude the $ame, without altogether digre$$ing from your chief de$ign: therefore if it be permitted to proceed in our fir$t argument, I $hould think that it were convenient to remit this que$tion of generation and corruption to another di$tinct and $ingle conference; as al$o, if it $hall plea$e you and <I>Simplicius,</I> we may do by other particular que$tions which may fall in the way of our di$cour$e; which I will keep in my mind to propo$e, and exactly di$cu$s them $ome other time. Now as for the pre$ent, $ince you $ay, that if <I>Ari- $totle</I> deny circular motion to the Earth in common with other bodies Cœle$tial, it chence will follow, that the $ame which be- falleth the Earth, <*> to ito being generable, alterable, <I>&c.</I> will hold al$o of Heaven, let us enquire no further if there be $uch things in nature, as generation and corruption, or not; but let us return to enquire what the Globe of the Earth doth.</P> <P>SIMPL. I cannot $uffer my ears to hear it que$tion'd, whether generation and corruption be in <I>rerum naturà,</I> it being a thing which we have continually before our eyes, and whereof <I>Ari$totle</I> <marg><I>By denying Prin- ciples in the Scien- ces, any Paradox may be maintain- ed.</I></marg> hath written two whole Books. But if you go about to deny the Principles of Sciences, and que$tion things mo$t manife$t, who knows not, but that you may prove what you will, and maintain any <I>Paradox</I>? And if you do not dayly $ee herbs, plants, ani- mals to generate and corrupt, what is it that you do $ee? Al$o, do you not continually behold contrarieties contend together, and the Earth change into Water, the Water turn to Air, the Air into Fire, and again the Air to conden$e into Clouds, Rains, Hails and Storms?</P> <P>SAGR. Yes, we $ee the$e things indeed, and therefore will grant you the di$cour$e of <I>Ari$totle,</I> as to this part of generation and corruption made by contraries; but if I $hall conclude by virtue of the $ame propo$itions which are granted to <I>Ari$totle,</I> that the Cœle$tial bodies them$elves are al$o generable and cor- <foot>ruptible</foot> <p n=>29</p> ruptible, a$well as the Elementary, what will you $ay then?</P> <P>SIMPL. I will $ay you have done that which is impo$$ible to be done.</P> <P>SAGR. Go to; tell me, <I>Simplicius,</I> are not the$e affections contrary to one another?</P> <P>SIMPL. Which?</P> <P>SAGR. Why the$e; Alterable, unalterable; pa$$ible, ^{*} impa$- <marg>* <I>Or,</I> Impatible.</marg> $ible; generable, ingenerable; corruptible, incorruptible?</P> <P>SIMPL. They are mo$t contrary.</P> <P>SAGR. Well then, if this be true, and it be al$o granted, that Cœle$tial Bodies are ingenerable and incorruptible; I prove that of nece$$ity Cœle$tial Bodies mu$t be generable and corru- ptible.</P> <P>SIMPL. This mu$t needs be a <I>Sophi$m.</I></P> <P>SAGR. Hear my Argument, and then cen$ure and re$olve it. <marg><I>Cœlestial Bodies are generable and corruptible, be- cau$e they are in- generable and in- corruptible.</I></marg> Cœle$tial Bodies, for that they are ingenerable and incorruptible, have in Nature their contraries, which are tho$e Bodies that be generable and corruptible; but where there is contrariety, there is al$o generation and corruption; therefore Cœle$tial Bodies are generable and corruptible.</P> <P>SIMPL. Did I not $ay it could be no other than a <I>Sophi$m</I>? This is one of tho$e forked Arguments called <I>Soritæ</I>: like that <marg><I>The forked Syllo- gi$m cal'd</I> <G>*cwri/h<*>hs.</G></marg> of the <I>Cretan,</I> who $aid that all <I>Cretans</I> were lyars; but he as being a <I>Cretan,</I> had told a lye, in $aying that the <I>Cretans</I> were ly- ars; it followed therefore, that the <I>Cretans</I> were no lyars, and con$equently that he, as being a <I>Cretan,</I> had $poke truth: And yet in $aying the <I>Cretans</I> were lyars, he had $aid true, and com- prehending him$elf as a <I>Cretan,</I> he mu$t con$equently be a lyar. And thus in the$e kinds of <I>Sophi$ms</I> a man may dwell to eternity, and never come to any conclu$ion.</P> <P>SAGR. You have hitherto cen$ured it, it remaineth now that you an$wer it, $hewing the fallacie.</P> <P>SIMPL. As to the re$olving of it, and finding out its fallacie, do you not in the fir$t place $ee a manife$t contradiction in it? Cœle$tial Bodies are ingenerable and incorruptible; <I>Ergo,</I> Cœle- $tial Bodies are generable and corruptible. And again, the con- <marg><I>Among$t Cœle$tial Bodies there is no contrariety.</I></marg> trariety is not betwixt the Cœle$tial Bodies, but betwixt the E- lements, which have the contrariety of the Motions, <I>$ur$ùm</I> and <I>deor$ùm,</I> and of levity and gravity; But the Heavens which move circularly, to which motion no other motion is contrary, want contrariety, and therefore they are incorruptible.</P> <P>SAGR. Fair and $oftly, <I>Simplicius</I>; this contrariety whereby you $ay $ome $imple Bodies become corruptible, re$ides it in the $ame Body which is corrupted, or el$e hath it relation to $ome o- other? I $ay, for example, the humidity by which a piece of Earth <foot>is</foot> <p n=>30</p> is corrupted, re$ides it in the $ame Earth or in $ome other bodie, which mu$t either be the Air or Water? I believe you will grant, that like as the Motions upwards and downwards, and gravity and levity, which you make the fir$t contraries, cannot be in the $ame Subject, $o neither can moi$t and dry, hot and cold: you mu$t therefore con$equently acknowledg that when a bodie cor- <marg><I>Contraries which are the cau$es of corruption, re$ide not in the $ame bo- dy that corrupteth.</I></marg> rupteth, it is occa$ioned by $ome quality re$iding in another con- trary to its own: therefore to make the Cœle$tial Body become corruptible, it $ufficeth that there are in Nature, bodies that have a contrariety to that Cœle$tial body; and $uch are the Elements, if it be true that corruptibility be contrary to incorruptibility.</P> <P>SIMPL. This $ufficeth not, Sir; The Elements alter and cor- rupt, becau$e they are intermixed, and are joyn'd to one another, <marg><I>Cœle$tial Bodies touch, but are not touched by the E- lements.</I></marg> and $o may exerci$e their contrariety; but Cœle$tial bodies are $eparated from the Elements, by which they are not $o much as toucht, though indeed they have an influence upon the Elements. It is requi$ite, if you will prove generation and corruption in Cœ- le$tial bodies, that you $hew, that there re$ides contrarieties be- tween them.</P> <P>SAGR. See how I will find tho$e contrarieties between them. The fir$t fountain from whence you derive the contrariety of the Elements, is the contrariety of their motions upwards and down- wards: it therefore is nece$$ary that tho$e Principles be in like <marg><I>Gravity & levity, varity and den$ity, are contrary qua- lities.</I></marg> manner contraries to each other, upon which tho$e motions de- pend. and becau$e that is moveable upwards by lightne$s, and this downwards by gravitv, it is nece$$ary that lightne$s and gravity are contrary to each other: no le$s are we to believe tho$e other Principles to be contraries, which are the cau$es that this is heavy, and that light: but by your own confe$$ion, levity and gravity follow as con$equents of rarity and den$ity; therefore <marg><I>The $tars infinitely $urpa$s the $ub- $tance of the re$t of Heaven in den$ity.</I></marg> rarity and den$ity $hall be contraries: the which conditions or affections are $o amply found in Cœle$tial bodies, that you e- $teem the $tars to be onely more den$e parts of their Heaven: and if this be $o, it followeth that the den$ity of the $tars exceeds that of the re$t of Heaven, by almo$t infinite degrees: which is manife$t, in that Heaven is infinitely tran$parent, and the $tars extremely opacous; and for that there are there above no other qualities, but more and le$s den$ity and rarity, which may be cau$es of the greater or le$s tran$parency. There being then $uch contrariety between the Cœleftial bodies, it is nece$$ary that they al$o be generable and corruptible, in the $ame manner as the Elementary bodies are; or el$e that contrariety is not the <marg><I>Rarity & den$ity in Cœle$tial bodies, is different from the rarity & den- $ity of the elements.</I></marg> cau$e of corruptibility, <I>&c.</I></P> <P>SIMPL. There is no nece$$ity either of one or the other, for that den$ity and rarity in Cœle$tial bodies, are not contraries to <foot>each</foot> <p n=>31</p> each other, as in Elementary bodies; for that they depend not on the primary qualities, cold and heat, which are contraries; but on the more or le$s matter in proportion to quantity: now much and little, $peak onely a relative oppo$ition, that is, the lea$t of oppo$itions, and which hath nothing to do with generation and corruption.</P> <P>SAGR. Therefore affirming, that den$ity and rarity, which a- mong$t the Elements $hould be the cau$e of gravity and levity, which may be the cau$es of contrary motions <I>$ur$ùm</I> and <I>deor- $ùm,</I> on which, again, dependeth the contrarieties for generation and corruption; it $ufficeth not that they be tho$e den$ne$$es and rarene$$es which under the $ame quantity, or (if you will) ma$s contain much or little matter, but it is nece$$ary that they be den$- ne$$es and rarene$$es cau$ed by the primary qualities, hot and cold, otherwi$e they would operate nothing at all: but if this be $o, <I>Ari$totle</I> hath deceived us, for that he $hould have told it us at <marg>Ari$totle <I>defective in a$$igning the cau$es why the ele- ments are genera- ble & corruptible.</I></marg> fir$t, and $o have left written that tho$e $imple bodies are gene- rable and corruptible, that are moveable with $imple motions upwards and downwards, dependent on levity and gravity, cau- $ed by rarity and den$ity, made by much or little matter, by rea$on of heat and cold; and not to have $taid at the $imple mo- tion <I>$ur$ùm</I> and <I>deor$ùm</I>: for I a$$ure you that to the making of bodies heavy or light, whereby they come to be moved with contrary motions, any kind of den$ity and rarity $ufficeth, whe- ther it proceed from heat and cold, or what el$e you plea$e; for heat and cold have nothing to do in this affair: and you $hall upon experiment find, that a red hot iron, which you mu$t grant to have heat, weigheth as much, and moves in the $ame manner as when it is cold. But to overpa$s this al$o, how know you but that Cœle$tial rarity and den$ity depend on heat and cold?</P> <P>SIMPL. I know it, becau$e tho$e qualities are not among$t Cœle$tial bodies, which are neither hot nor cold.</P> <P>SALV. I $ee we are again going about to engulph our $elves in a bottomle$s ocean, where there is no getting to $hore; for this is a Navigation without Compa$s, Stars, Oars or Rudder: $o that it will follow either that we be forced to pa$s from Shelf to Shelf, or run on ground, or to $ail continually in danger of being lo$t. Therefore, if according to your advice we $hall proceed in our main de$ign, we mu$t of nece$$ity for the pre$ent overpa$s this general con$ideration, whether direct motion be nece$$ary in Na- ture, and agree with $ome bodies; and come to the particular demon$trations, ob$ervations and experiments; propounding in the fir$t place all tho$e that have been hitherto alledged by <I>Ari- $totle, Ptolomey,</I> and others, to prove the $tability of the Earth, en- deavouring in the next place to an$wer them: and producing in <foot>the</foot> <p n=>32</p> the la$t place, tho$e, by which others may be per$waded, that the Earth is no le$s than the Moon, or any other Planet to be num- bered among$t natural bodies that move circularly.</P> <P>SAGR. I $hall the more willingly incline to this, in that I am better $atisfied with your Architectonical and general di$cour$e, than with that of <I>Ari$totle,</I> for yours convinceth me without the lea$t $cruple, and the other at every $tep cro$$eth my way with $ome block. And I $ee no rea$on why <I>Simplicius</I> $hould not be pre$ently $atisfied with the Argument you alledg, to prove that there can be no $uch thing in nature as a motion by a right line, if we do but pre$uppo$e that the parts of the Univer$e are di$po- $ed in an excellent con$titution and perfect order.</P> <P>SALV. Stay a little, good <I>Sagredus,</I> for ju$t now a way comes into my mind, how I may give <I>Simplicius</I> $atisfaction, provided that he will not be $o $trictly wedded to every expre$$ion of <I>A- ri$totle,</I> as to hold it here$ie to recede in any thing from him. Nor is there any que$tion to be made, but that if we grant the excel- lent di$po$ition and perfect order of the parts of the Univer$e, as to local $cituation, that then there is no other but the circular motion, and re$t; for as to the motion by a right line, I $ee not how it can be of u$e for any thing, but to reduce to their natural con$titution, $ome integral bodies, that by $ome accident were re- mov'd and $eparated from their whole, as we $aid above.</P> <P>Let us now con$ider the whole Terre$trial Globe, and enquire the be$t we can, whether it, and the other Mundane bodies are to con$erve them$elves in their perfect and natural di$po$ition. It is nece$$ary to $ay, either that it re$ts and keeps perpetually im- moveable in its place; or el$e that continuing always in its place, it revolves in its $elf; or that it turneth about a Centre, moving <marg>Ari$t. <I>&</I> Ptolomey <I>make the Terre- strial Globe immo- veable.</I></marg> by the circumference of a circle. Of which accidents, both <I>Ari- $totle</I> and <I>Ptolomey,</I> and all their followers $ay, that it hath ever ob$erved, and $hall continually keep the fir$t, that is, a perpetual <marg><I>It is better to $ay, that the Terre$tri- al Globe naturally resteth, than that it moveth directly downwards.</I></marg> re$t in the $ame place. Now, why, I pray you, ought they not to have $aid, that its natural affection is to re$t immoveable, ra- ther than to make natural unto it the motion ^{*} downwards, with which motion it never did or $hall move? And as to the motion <marg>*The word is, <I>all' ingiù,</I> which the Latine ver$ion ren- dreth <I>$ur$ùm,</I> which is quite con- trary to the Au- thors $en$e.</marg> by a right line, they mu$t grant us that Nature maketh u$e of it to reduce the $mall parts of the Earth, Water, Air, Fire, and every other integral Mundane body to their <I>Whole,</I> when any of them by chance are $eparated, and $o tran$ported out of their proper place; if al$o haply, $ome circular motion might not be found to be more convenient to make this re$titution. In my judg- ment, this primary po$ition an$wers much better, even according to <I>Ari$totles</I> own method, to all the other con$equences, than to attribute the $traight motion to be an intrin$ick and natural <foot>principle</foot> <p n=>33</p> principle of the Elements. Which is manife$t, for that if I aske the <I>Peripatetick,</I> if, being of opinion that Cœle$tial bodies are incorruptibe and eternal, he believeth that the Terre$tial Globe is not $o, but corruptible and mortal, $o that there $hall come a time, when the Sun and Moon and other Stars, continuing their beings and operations, the Earth $hall not be found in the World, but $hall with the re$t of the Elements be de$troyed and annihilated, I am certain that he would an$wer me, no: <marg><I>Right Motion with more rea$on attributed to the parts, than to the whole Elements.</I></marg> therefore generation and corruption is in the parts and not in the whole; and in the parts very $mall and $uperficial, which are, as it were, incen$ible in compari$on of the whole ma$$e. And becau$e <I>Ari$totle</I> deduceth generation and corruption from the contrariety of $treight motions, let us remit $uch motions to the parts, which onely change and decay, and to the whole Globe and Sphere of the Elements, let us a$cribe either the circular mo- tion, or a perpetual con$i$tance in its proper place: the only affections apt for perpetuation, and maintaining of perfect order. This which is $poken of the Earth, may be $aid with the $ame rea$on of Fire, and of the greate$t part of the Air; to which <marg><I>The Peripateticks improperly a$$ign tho$e motious to the Elements for Natural, with which they never were moved, and tho$e for Preter- natural with which they alwayes are moved.</I></marg> Elements, the <I>Peripateticks</I> are forced to a$cribe for intrin$ical and natural, a motion wherewith they were never yet moved, nor never $hall be; and to call that motion preternatural to them, wherewith, if they move at all, they do and ever $hall move. This I $ay, becau$e they a$$ign to the Air aud Fire the motion upwards, wherewith tho$e Elements were never moved, but only $ome parts of them, and tho$e were $o moved onely in or- der to the recovery of their perfect con$titution, when they were out of their natural places; and on the contrary they call the circular motion preternatural to them, though they are thereby ince$$antly moved: forgeting, as it $eemeth, what <I>Ari$totle</I> oft in- culcateth, that nothing violent can be permanent.</P> <P>SIMPL. To all the$e we have very pertinent an$wers, which <marg><I>Sen$ible experi- ments to be prefer- red to humane Arguments.</I></marg> I for this time omit, that we may come to the more particular rea$ons, and $en$ible experiments, which ought in conclu$ion to be oppo$ed, as <I>Ari$totle</I> $aitn well, to whatever humane rea$on can pre$ent us with.</P> <P>SAGR. What hath been $poken hitherto, $erves to clear up unto us which of the two general di$cour$es carrieth with it mo$t of probability, I mean that of <I>Ari$totle,</I> which would per$wade us, that the $ublunary bodies are by nature generable, and corru- ptible, <I>&c.</I> and therefore mo$t different from the e$$ence of Cœ- leftial bodies, which are impa$$ible, ingenerable, incorruptible, <I>&c.</I> drawn from the diver$ity of $imple motions; or el$e this of <I>Salviatus,</I> who $uppo$ing the integral parts of the World to be di$po$ed in a perfect con$titution, excludes by nece$$ary confe- <foot>E quence</foot> <p n=>34</p> quence the right or $traight motion of $imple natural bodies, as being of no u$e in nature, and e$teems the Earth it $elf al$o to be one of the Cœle$tial bodies adorn'd with all the prerogatives that agree with them; which la$t di$cour$e is hitherto much more likely, in my judgment, than that other. Therefore re- $olve, <I>Simplicius,</I> to produce all the particular rea$ons, experi- ments and ob$ervations, as well Natural as A$tronomical, that may $erve to per$wade us that the Earth differeth from the Cœ- le$tial bodies, is immoveable, and $ituated in the Centre of the World, and what ever el$e excludes its moving like to the Planets, as <I>Jupiter</I> or the <I>Moon, &c.</I> And <I>Salviatus</I> will be plea$ed to be $o civil as to an$wer to them one by one.</P> <P>SIMPL. See here for a beginning, two mo$t convincing Argu- ments to demon$trate the Earth to be mo$t different from the Cœle$tial bodies. Fir$t, the bodies that are generable, corru- ptible, alterable, <I>&c.</I> are quite different from tho$e that are in- generable, incorruptible, unalterable, <I>&c.</I> But the Earth is ge- nerable, corruptible, alterable, <I>&c.</I> and the Cœle$tial bodies in- generable, incorruptible, unalterable, <I>&c.</I> Therefore the Earth is quite different from the Cœle$tial bodies.</P> <P>SAGR. By your fir$t Argument you $pread the Table with the $ame Viands, which but ju$t now with much adoe were voided.</P> <P>SIMPL. Hold a little, Sir, and take the re$t along with you, and then tell me if this be not different from what you had be- fore. In the former, the <I>Minor</I> was proved <I>à priori,</I> & now you $ee it proved <I>à po$teriori:</I> Judg then if it be the $ame. I prove the <I>Minor,</I> therefore (the <I>Major</I> being mo$t manife$t) by $en$ible ex- perience, which $hews us that in the Earth there are made conti- nual generations, corruptions, alterations, <I>&c.</I> which neither our $en$es, nor the traditions or memories of our Ance$tors, ever $aw an in$tance of in Heaven; therefore Heaven is unalterable, <I>&c.</I> <marg><I>Heaven immuta- ble, becau$e there never was any mu- tation $een in it.</I></marg> and the Earth alterable, <I>&c.</I> and therefore different from Hea- ven. I take my $econd Argument from a principal and e$$ential accident, and it is this. That body which is by its nature ob- <marg><I>Bodies naturally lucid, are different from tho$e which are by nature ob- $cure.</I></marg> $cure and deprived of light, is divers from the luminous and $hi- ning bodies; but the Earth is ob$cure and void of light, and the Cœle$tial bodies $plendid, and full of light; <I>Ergo, &c.</I> An$wer to the$e Arguments fir$t, that we may not heap up too many, and then I will alledge others.</P> <P>SALV. As to the fir$t, the $tre$$e whereof you lay upon ex- perience, I de$ire that you would a little more di$tinctly produce me the alteration which you $ee made in the Earth, and not in Heaven; upon which you call the Earth alterable, and the Hea- vens not $o.</P> <P>SIMPL. I $ee in the Earth, plants and animals continually ge- <foot>nerating</foot> <p n=>35</p> nerating and decaying; winds, rains, tempe$ts, $torms ari$ing; and in a word, the a$pect of the Earth to be perpetually metamorpho- $ing; none of which mutations are to be di$cern'd in the Cœle$tial bodies; the con$titution and figuration of which is mo$t punctu- ally conformable to that they ever were time out of mind; without the generation of any thing that is new, or corruption of any thing that was old.</P> <P>SALV. But if you content your $elf with the$e vi$ible, or to $ay better, $een experiments, you mu$t con$equently account <I>China</I> and <I>America</I> Cœle$tial bodies, for doubtle$$e you never beheld in them the$e alterations which you $ee here in <I>Italy,</I> and that therefore according to your apprehen$ion they are inal- terable.</P> <P>SIMPL. Though I never did $ee the$e alterations $enfibly in tho$e places, the relations of them are not to be que$tioned; be$ides that, <I>cum eadem $it ratio totius, & partium,</I> tho$e Countreys being a part of the Earth, as well as ours, they mu$t of nece$$ity be alterable as the$e are.</P> <P>SALV. And why have you not, without being put to believe other mens relations, examined and ob$erved tho$e alterations with your own eyes?</P> <P>SIMPL. Becau$e tho$e places, be$ides that they are not ex- po$ed to our eyes, are $o remote, that our $ight cannot reach to comprehend therein $uch like mutations.</P> <P>SALV. See now, how you have unawares di$covered the falla- cy of your Argument; for, if you $ay that the alterations that are $een on the Earth neer at hand, cannot, by rea$on of the too great di$tance, be $een in <I>America,</I> much le$$e can you $ee them in the Moon, which is $o many hundred times more remote: And if you believe the alterations in <I>Mexico</I> upon the report of tho$e that come from thence, what intelligence have you from the Moon, to a$$ure you that there is no $uch alterations in it? Therefore, from your not $eeing any alterations in Heaven, whereas, if there were any $uch, you could not $ee them by rea- $on of their too great di$tance, and from your not having intel- ligence thereof, in regard that it cannot be had, you ought not to argue, that there are no $uch alterations; howbeit, from the $eeing and ob$erving of them on Earth, you well argue that therein $uch there are.</P> <P>SIMPL. I will $hew $o great mutations that have befaln on the Earth; that if any $uch had happened in the Moon, they might very well have been ob$erved here below. We find in <marg><I>The Mediterr ani- an Sea made by the $eparation of</I> Abi- la <I>and</I> Calpen.</marg> very antient records, that heretofore at the Streights of <I>Gibraltar,</I> the two great Mountains <I>Abila,</I> and <I>Calpen,</I> were continued to- gether by certain other le$$e Mountains which there gave check <foot>E 2 to</foot> <p n=>36</p> to the Ocean: but tho$e Hills, being by $ome cau$e or other $e- parated, and a way being opened to the Sea to break in, it made $uch an inundation, that it gave occa$ion to the calling of it $ince the Mid-land Sea: the greatne$s whereof con$idered, and the di- vers a$pect the $urface of the Water and Earth then made, had it been beheld afar off, there is no doubt but $o great a change might have been di$cerned by one that was then in the Moon; as al$o to us inhabitants of the Earth, the like alterations would be perceived in the Moon; but we find not in antiquity, that e- ver there was $uch a thing $een; therefore we have no cau$e to $ay, that any of the Cœle$tial bodies are alterable, <I>&c.</I></P> <P>SALV. That $o great alterations have hapned in the Moon, I dare not $ay, but for all that, I am not yet certain but that $uch changes might occur; and becau$e $uch a mutation could onely repre$ent unto us $ome kind of variation between the more clear, and more ob$cure parts of the Moon, I know not whether we have had on Earth ob$ervant Selenographers, who have for any con$iderable number of years, in$tructed us with $o exact Seleno- graphy, as that we $hould confidently conclude, that there hath no $uch change hapned in the face of the Moon; of the figura- tion of which I find no more particular de$cription, than the $ay- ing of $ome, that it repre$ents an humane face; of others, that it is like the muzzle of a lyon; and of others, that it is <I>Cain</I> with a bundle of thorns on his back: therefore, to $ay Heaven is un- alterable, becau$e that in the Moon, or other Cœle$tial bodies, no $uch alterations are $een, as di$cover them$elves on Earth, is a bad illation, and concludeth nothing.</P> <P>SAGR. And there is another odd kind of $cruple in this Argu- ment of <I>Simplicius,</I> running in my mind, which I would gladly have an$wered; therefore I demand of him, whether the Earth before the Mediterranian inundation was generable and corrupti- ble, or el$e began then $o to be?</P> <P>SIMPL. It was doubtle$s generable and corruptible al$o be- fore that time; but that was $o va$t a mutation, that it might have been ob$erved as far as the Moon.</P> <P>SAGR. Go to; if the Earth was generable and corruptible before that Inundation, why may not the Moon be $o like- wi$e without $uch a change? Or why $hould that be nece$$ary in the Moon, which importeth nothing on Earth?</P> <P>SALV. It is a $hrewd que$tion: But I am doubtfull that <I>Sim- plicius</I> a little altereth the Text of <I>Ari$totle,</I> and the other <I>Peri- patelicks,</I> who $ay, they hold the Heavens unalterable, for that they $ee therein no one $tar generate or corrupt, which is proba- bly a le$s part of Heaven, than a City is of the Earth, and yet innumerable of the$e have been de$troyed, $o as that no mark of them hath remain'd.</P> <foot>SAGR.</foot> <p n=>37</p> <P>SAGR. I verily believed otherwi$e, and conceited that <I>Sim- plicius</I> di$$embled this expo$ition of the Text, that he might not charge his Ma$ter and Con$ectators, with a notion more ab$urd than the former. And what a folly it is to $ay the Cœle$tial part is unalterable, becau$e no $tars do generate or corrupt there- in? What then? hath any $een a Terre$trial Globe corrupt, and another regenerate in its place? And yet is it not on all hands granted by Philo$ophers, that there are very few $tars in Heaven le$s than the Earth, but very many that are much bigger? So <marg><I>Its no le$s impo$$i- ble for a $tar to corrupt, than for the whole Terre- $trial Globe.</I></marg> that for a $tar in Heaven to corrupt, would be no le$s than if the whole Terre$trial Globe $hould be de$troy'd. Therefore, if for the true proof of generation and corruption in the Univer$e, it be nece$$ary that $o va$t bodies as a $tar, mu$t corrupt and regene- rate, you may $atisfie your $elf and cea$e your opinion; for I a$$ure you, that you $hall never $ee the Terre$trial Globe or any other integral body of the World, to corrupt or decay $o, that having been beheld by us for $o many years pa$t, they $hould $o di$$olve, as not to leave any foot$teps of them.</P> <P>SALV. But to give <I>Simplicius</I> yet fuller $atisfaction, and to reclaim him, if po$$ible, from his error; I affirm, that we have in <marg>Ari$totle <I>would change his opinion, did he $ee the no- velties of our age.</I></marg> our age new accidents and ob$ervations, and $uch, that I que$tion not in the lea$t, but if <I>Ari$totle</I> were now alive, they would make him change his opinion; which may be ea$ily collected from the very manner of his di$cour$ing: For when he writeth that he e- $teemeth the Heavens inalterable, &c. becau$e no new thing was $een to be begot therein, or any old to be di$$olved, he $eems im- plicitely to hint unto us, that when he $hould $ee any $uch acci- dent, he would hold the contrary; and confront, as indeed it is meet, $en$ible experiments to natural rea$on: for had he not made any reckoning of the $en$es, he would not then from the not $eeing of any $en$ible mutation, have argued immutability.</P> <P>SIMPL. <I>Ari$totle</I> deduceth his principal Argument <I>à priori,</I> $hewing the nece$$ity of the inalterability of Heaven by natural, manife$t and clear principles; and then $tabli$heth the $ame <I>à po- $teriori,</I> by $en$e, and the traditions of the antients.</P> <P>SALV. This you $peak of is the Method he hath ob$erved in delivering his Doctrine, but I do not bethink it yet to be that wherewith he invented it; for I do believe for certain, that he fir$t procured by help of the $en$es, $uch experiments and ob$er- vations as he could, to a$$ure him as much as it was po$$ible, of the <marg><I>The certaixty of the conclu$ion he<*>- peth by are$olutive method to $ind the demonstration.</I></marg> conclu$ion, and that he afterwards $ought out the means how to demon$trate it: For this, the u$ual cour$e in demon$trative Scien- ces, and the rea$on thereof is, becau$e when the conclu$ion is true, by help of re$olutive Method, one may hit upon $ome pro- po$ition before demon$trated, or come to $ome principle known <foot><I>per</I></foot> <p n=>38</p> <I>per $e</I>; but if the conclu$ion be fal$e, a man may proceed <I>in in- finitum,</I> and never meet with any truth already known; but ve- ry oft he $hall meet with $ome impo$$ibility or manife$t ab$urdi- <marg>Pythagoras <I>offered an Hecatomb for a Geometrical de- mon$tration which he found.</I></marg> ty. Nor need you que$tion but that <I>Pythagoras</I> along time be- fore he found the demon$tration for which he offered the Heca- tomb, had been certain, that the $quare of the $ide $ubtending the right angle in a rectangle triangle, was equal to the $quare of the other two $ides: and the certainty of the conclu$ion condu- ced not a little to the inve$tigating of the demon$tration, un- der$tanding me alwayes to mean in demon$trative Sciences. But what ever was the method of <I>Ari$totle,</I> and whether his arguing <I>à priori</I> preceded $en$e <I>à po$teriori,</I> or the contrary; it $ufficeth that the $ame <I>Ari$totle</I> preferreth (as hath been oft $aid) $en$ible ex- periments before all di$cour$es; be$ides, as to the Arugments <I>à priori</I> their force hath been already examined. Now returning to my purpo$ed matter, I $ay, that the things in our times di$- covered in the Heavens, are, and have been $uch, that they may give ab$olute $atisfaction to all Philo$ophers; fora$much as in the particular bodies, and in the univer$al expan$ion of Heaven, there have been, and are continually, $een ju$t $uch accidents as we call generations and corruptions, being that excellent A- $tronomers have ob$erved many Comets generated and di$$olved in parts higher than the Lunar Orb, be$ides the two new Stars, <marg><I>New $tars di$co- vered in Heaven.</I></marg> <I>Anuo</I> 1572, and <I>Anno</I> 1604, without contradiction much higher than all the Planets; and in the face of the Sun it $elf, by help <marg><I>Spots generate and di$$olve in the face of the Sun.</I></marg> of the <I>Tele$cope,</I> certain den$e and ob$cure $ub$tances, in $em- blance very like to the foggs about the Earth, are $een to be produced and di$$olved; and many of the$e are $o va$t, that they far exceed not only the Mediterranian Streight, but all <marg><I>Solar spots are bigger than all</I> A- $ia <I>and</I> Affrick.</marg> <I>Affrica</I> and <I>A$ia</I> al$o. Now if <I>Ari$totle</I> had $een the$e things, what think you he would have $aid, and done <I>Simplicius?</I></P> <P>SIMPL. I know not what <I>Ari$totle</I> would have done or $aid, that was the great Ma$ter of all the Sciences, but yet I know in part, what his Sectators do and $ay, and ought to do and $ay, unle$$e they would deprive them$elves of their guide, leader, and Prince in Philo$ophy. As to the Comets, are not tho$e Modern A$tronomers, who would make them Cœle$tial, convinced by <marg>* <I>A$tronomers con- futed by</I> Anti-Ty- cho.</marg> the ^{*}<I>Anti-Tycho,</I> yea, and overcome with their own weapons, I mean by way of Paralaxes and Calculations, every way tryed, concluding at the la$t in favour of <I>Aristotle,</I> that they are all Elementary? And this being overthrown, which was as it were their foundation, have the$e Novelli$ts any thing more where- with to maintain their a$$ertion?</P> <P>SALV. Hold a little, good <I>Simplicius,</I> this modern Author, what $aith he to the new Stars, <I>Anno</I> 1572, and 1604, and to <foot>the</foot> <p n=>39</p> the Solar $pots? for as to the Comets, I for my own particular little care to make them generated under or above the Moon; nor did I ever put much $tre$$e on the loquacity of <I>Tycho</I>; nor am I hard to believe that their matter is Elementary, and that they may elevate ($ublimate) them$elves at their plea$ure, with- out meeting with any ob$tacle from the impenetrability of the <I>Peripatetick</I> Heaven, which I hold to be far more thin, yielding, and $ubtil than our Air; and as to the calculations of the Pa- rallaxes, fir$t, the uncertainty whether Comets are $ubject to $uch accidents, and next, the incon$tancy of the ob$ervations, upon which the computations are made, make me equally $u$- pect both tho$e opinions: and the rather, for that I $ee him <marg>Anti-Tycho <I>wre- $teth A$tronomical ob$ervations to his own parpo$e.</I></marg> you call <I>Anti-Tycho,</I> $ometimes $tretch to his purpo$e, or el$e reject tho$e ob$ervations which interfere with his de$ign.</P> <P>SIMPL. As to the new Stars, <I>Anti-Tycho</I> extricates him$elf finely in three or four words; $aying, That tho$e mo- dern new Stars are no certain parts of the Cœle$tial bodies, and that the adver$aries, if they will prove alteration and genera- tion in tho$e $uperior bodies, mu$t $hew $ome mutations that have been made in the Stars de$cribed $o many ages pa$t, of which there is no doubt but that they be Cœle$tial bodies, which they can never be able to do: Next, as to tho$e mat- ters which $ome affirm, to generate and di$$ipate in the face of the Sun, he makes no mention thereof; wherefore I conclude, that he believed them fictious, or the illu$ions of the Tube, or at mo$t, $ome petty effecs cau$ed by the Air, and in brief, any thing rather than matters Cœle$tial.</P> <P>SALV. But you, <I>Simplicius,</I> what an$wer could you give to the oppo$ition of the$e importunate $pots which are $tarted up to di$turb the Heavens, and more than that, the <I>Peripatetick</I> Philo$ophy? It cannot be but that you, who are $o re$olute a Champion of it, have found $ome reply or $olution for the $ame, of which you ought not to deprive us.</P> <P>SIMPL. I have heard $undry opinions about this particular. One $aith: “They are Stars which in their proper Orbs, like as <marg><I>Sundry opinions touching the Solar $pots.</I></marg> <I>Venus</I> and <I>Mervury,</I> revolve about the Sun, and in pa$$ing un- der it, repre$ent them$elves to us ob$cure; and for that they are many, they oft happen to aggregate their parts together, and afterwards $eperate again. Others believe them to be aerial impre$$ions; others, the illu$ions of the chry$tals; and o- thers, other things: But I incline to think, yea am verily per- $waded, That they are an aggregate of many $everal opacous bodies, as it were ca$ually concurrent among them$elves. And therefore we often $ee, that in one of tho$e $pots one may number ten or more $uch $mall bodies, which are of irregu- <foot>lar</foot> <p n=>40</p> lar figures, and $eem to us like flakes of $now, or flocks of wooll, or moaths flying: they vary $ite among$t them$elves, and one while $ever, another while meet, and mo$t of all be- neath the Sun, about which, as about their Centre, they con- tinually move. But yet, mu$t we not therefore grant, that they are generated or di$$olved, but that at $ometimes they are hid behind the body of the Sun, and at other times, though remote from it, yet are they not $een for the vicinity of the immea$urable light of the Sun; in regard that in the eccentrick Orb of the Sun, there is con$tituted, as it were, an Onion, com- po$ed of many folds one within another, each of which, being <marg>* The Original $aith [<I>tempe$tata $i muove</I>] which the Latine Tran$lati- on, (Mi$taking <I>Tempectata,</I> aword in Heraldry, for <I>Tempe$tato,</I>) ren- dereth [<I>incitata movetur</I>] which $ignifieth a violent tran$portmeut, as in a $torm, that of a Ship.</marg> ^{*}$tudded with certain $mall $pots, doth move; and albeit their motion at fir$t $eemeth incon$tant and irregular, yet neverthe- le$$e, it is $aid at la$t, to be ob$erved that the very $ame $pots, as before,” do within a determinate time return again. This $eemeth to me the fitte$t an$wer that hath been found to a$$igne a rea$on of that $ame appearance, and withal to maintain the incorruptability and ingenerability of the Heavens; and if this doth not $uffice; there wants not more elevated wits, which will give you other, more convincing.</P> <P>SALV. If this of which we di$pute, were $ome point of Law, <marg><I>In natural Sci- ences, the art of Oratory is of no force.</I></marg> or other part of the Studies called <I>Humanity,</I> wherein there is neither truth nor fal$hood, if we will give $ufficient credit to the acutene$$e of the wit, readine$$e of an$wers, and the gene- ral practice of Writers, then he who mo$t aboundeth in the$e, makes his rea$on more probable and plau$ible; but in Natural Sciences, the conclu$ions of which are true and nece$$ary, and wherewith the judgment of men hath nothing to do, one is to be more cautious how he goeth about to maintain any thing that is fal$e; for a man but of an ordinary wit, if it be his good for- tune to be of the right $ide, may lay a thou$and <I>Demo$thenes</I> and a thou$and <I>Ari$totles</I> at his feet. Therefore reject tho$e hopes and conceits, wherewith you flatter your $elf, that there can be any men $o much more learned, read, and ver$ed in Authors, than we, that in de$pite of nature, they $hould be able to make that become true, which is fal$e. And $eeing that of all the opinions that have been hitherto alledged touching the e$- $ence of the$e Solar $pots, this in$tanced in by you, is in your judgment the true$t, it followeth (if this be $o) that all the re$t are fal$e; and to deliver you from this al$o, which doubtle$$e is a mo$t fal$e <I>Chimœra,</I> over-pa$$ing infinite other improbabilities that are therein, I $hall propo$e again$t it onely two experiments; <marg><I>An Argument that nece$$arily proveth the Solar $pots to generate and di$$olwe.</I></marg> one is, that many of tho$e $pots are $een to ari$e in the mid$t of the Solar ring, and many likewi$e to di$$olve and vani$h at a great di$tance from the circumference of the Sun; a nece$$ary Argu- <foot>ment</foot> <p n=>41</p> ment that they generate and di$$olve; for if without generating or corrrupting, they $hould appear there by onely local motion, they would all be $een to enter, and pa$s out by the extreme cir- <marg><I>A conclu$ive de- mon$tration, that the $pots are conti- guous to the body of the Sun.</I></marg> cumference. The other ob$ervation to $uch as are not $ituate in the lowe$t degree of ignorance in Per$pective, by the mutation of the appearing figures, and by the apparent mutations of the velocity of motion is nece$$arily concluding, that the $pots are contiguous to the body of the Sun, and that touching its $uperfi- cies, they move either with it or upon it, and that they in no wi$e move in circles remote from the $ame. The motion proves <marg><I>The motion of the spots towards the circumference of the Sun appears $low.</I></marg> it, which towards the circumference of the Solar Circle, appeareth very $low, and towards the mid$t, more $wift; the fi- gures of the $pots confirmeth it, which towards the circumference <marg><I>The figure of the spots appears nar- row towards the circumference of the Suns</I> di$cus, <I>& why.</I></marg> appear exceeding narrow in compari$on of that which they $eem to be in the parts nearer the middle; and this becau$e in the mid$t they are $een in their full lu$ter, and as they truly be; and towards the circumference by rea$on of the convexity of the glo- bous $uperficies, they $eem more compre$$'d: And both the$e diminutions of figure and motion, to $uch as know how to ob$erve and calculate them exactly, preci$ely an$wer to that which $hould appear, the $pots being contiguous to the Sun, and differ irrecon- cileably from a motion in circles remote, though but for $mal intervalls from the body of the Sun; as hath been diffu$ely de- <marg>* Under this word <I>Friend,</I> as al$o that of <I>Academick, & Common Friend, Galilœus</I> mode$tly conceals him$elf throughout the$e Dialogues.</marg> mon$trated by our ^{*} Friend, in his Letters about the Solar $pots, to <I>Marcus Vel$erus.</I> It may be gathered from the $ame muta- tion of figure, that none of them are $tars, or other bodies of $pherical figure; for that among$t all figures the $phere never appeareth compre$$ed, nor can ever be repre$ented but onely per- fectly round; and thus in ca$e any particular $pot were a round body, as all the $tars are held to be, the $aid roundne$s would as well appear in the mid$t of the Solar ring, as when the $pot is near the extreme: whereas, its $o great compre$$ion, and $hewing its $elf $o $mall towards the extreme, and contrariwi$e, $patious and large towards the middle, a$$ureth us, that the$e $pots are flat <marg><I>The Solar spots are not $pherical, but flat like thin plates.</I></marg> plates of $mall thickne$s or depth, in compari$on of their length and breadth. La$tly, whereas you $ay that the $pots after their determinate periods are ob$erved to return to their former a$pect, believe it not, <I>Simplicius,</I> for he that told you $o, will deceive you; and that I $peak the truth, you may ob$erve them to be hid in the face of the Sun far from the circumference; nor hath your Ob$ervator told you a word of that compre$$ion, which nece$$a- rily argueth them to be contiguous to the Sun. That which he tells you of the return of the $aid $pots, is nothing el$e but what is read in the forementioned Letters, namely, that $ome of them may $ometimes $o happen that are of $o long a duration, that <foot>F they</foot> <p n=>42</p> they cannot be di$$ipated by one $ole conver$ion about the Sun, which is accompli$hed in le$s than a moneth.</P> <P>SIMPL. I, for my part, have not made either $o long, or $o exact ob$ervations, as to enable me to boa$t my $elf Ma$ter of the <I>Quod ect</I> of this matter: but I will more accurately con$ider the $ame, and make tryal my $elf for my own $atisfaction, whether I can reconcile that which experience $hews us, with that which <I>Ari$totle</I> teacheth us; for it's a certain Maxim, that two Truths cannot be contrary to one another.</P> <P>SALV. If you would reconcile that which $en$e $heweth you, <marg><I>One cannot</I> (<I>$aith</I> Ari$totle) <I>$peak confidently of Hea- ven, by rea$on of its great di$tance.</I></marg> with the $olider Doctrines of <I>Ari$totle,</I> you will find no great dif- ficulty in the undertaking; and that $o it is, doth not <I>Ari$totle</I> $ay, that one cannot treat confidently of the things of Heaven, by rea$on of their great remotene$s?</P> <P>SIMPL. He expre$ly $aith $o.</P> <marg>Ari$totle <I>prefers $en$e before ratio- cination.</I></marg> <P>SALV. And doth he not likewi$e affirm, that we ought to pre- fer that which $en$e demon$trates, before all Arguments, though in appearance never $o well grounded? and $aith he not this without the lea$t doubt or hæ$itation?</P> <P>SIMPL. He doth $o.</P> <P>SALV. Why then, the $econd of the$e propo$itions, which are both the doctrine of <I>Ari$totle,</I> that $aith, that $en$e is to take <marg><I>Its a doctrine more agreeing with</I> A- ri$totle, <I>to $ay the Heavens are alter- able, than that which affirms them inalterable.</I></marg> place of Logick, is a doctrine much more $olid and undoubted, than that other which holdeth the Heavens to be unalterable; and therefore you $hall argue more <I>Ari$totelically,</I> $aying, the Hea- vens are alterable, for that $o my $en$e telleth me, than if you $hould $ay, the Heavens are u alterable, for that Logick $o per$wa- ded <I>Aristotle.</I> Furthermore, we may di$cour$e of Cœle$tial mat- <marg><I>We may by help of the</I> Tele$cope <I>di$- cour$e better of cœ- le$tial matters, than</I> Ari$tot. <I>him- $elf.</I></marg> ters much better than <I>Ari$totle</I>; becau$e, he confe$$ing the know- ledg thereof to be difficult to him, by rea$on of their remotene$s from the $en$es, he thereby acknowledgeth, that one to whom the $en$es can better repre$ent the $ame, may philo$ophate upon them with more certainty. Now we by help of the Tele$cope, are brought thirty or forty times nearer to the Heavens, than ever <I>Ari$totle</I> came; $o that we may di$cover in them an hundred things, which he could not $ee, and among$t the re$t, the$e $pots in the Sun, which were to him ab$olutely invi$ible; therefore we may di$cour$e of the Heavens and Sun, with more certainty than <I>Ari$tolte.</I></P> <P>SAGR. I $ee into the heart of <I>Simplicius,</I> and know that he is much moved at the $trength of the$e $o convincing Arguments; but on the other $ide, when he con$idereth the great authority which <I>Ari$totle</I> hath won with all men, and remembreth the great number of famous Interpreters, which have made it their bu$ine$s to explain his $en$e; and $eeth other Sciences, $o nece$$ary and <foot>profitable</foot> <p n=>43</p> profitable to the publick, to build a great part of their e$teem and reputation on the credit of <I>Ari$totle</I> he is much puzzled and perplexed: and methinks I hear him $ay, To whom then $hould <marg><I>The Declamation of</I> Simplicius.</marg> we repair for the deci$ion of our controver$ies, if <I>Ari$totle</I> were removed from the chair? What other Author $hould we follow in the Schools, Academies and Studies? What Philo$opher hath writ all the parts of Natural Philo$ophy, and that $o methodically without omitting $o much as one $ingle conclu$ion? Shall we then overthrow that Fabrick under which $o many pa$$engers find $helter? Shall we de$troy that <I>A$ylum,</I> that <I>Prytaneum,</I> where- in $o many Students meet with commodious harbour, where without expo$ing them$elves to the injuries of the air, with the onely turning over of a few leaves, one may learn all the $e- crets of Nature? Shall we di$mantle that fort in which we are $afe from all ho$tile a$$aults? But I pitie him no more than I do that Gentleman who with great expence of time and trea$ure, and the help of many hundred arti$ts, erects a very $umptu- ous Pallace, and afterwards beholds it ready to fall, by rea$on of the bad foundation; but being extremely unwilling to $ee the Walls $tript which are adorned with $o many beautifull Pictures; or to $uffer the columns to fall, that uphold the $tate- ly Galleries; or the gilded roofs, chimney-pieces, the freizes, the corni$hes of marble, with $o much co$t erected, to be rui- ned; goeth about with girders, props, $hoars, buttera$$es, to pre- vent their $ubver$ion.</P> <P>SALV. But ala$s, <I>Simplicius</I> as yet fears no $uch fall, and I would undertake to $ecure him from that mi$chief at a far le$s charge. There is no danger that $o great a multitude of <marg><I>Peripatetick Phi- lo$ophy unchange- able.</I></marg> $ubtle and wi$e Philo$ophers, $hould $uffer them$elves to be <I>Hector'd</I> by one or two, who make a little blu$tering; nay, they will rather, without ever turning the points of their pens again$t them, by their $ilence onely render them the object of univer$al $corn and contempt. It is a fond conceit for any one to think to introduce new Philo$ophy, by reproving this or that Author: it will be fir$t nece$$ary to new-mold the brains of men, and make them apt to di$tingui$h truth from fal$hood. A thing which onely God can do. But from one di$cour$e to another whither are we $tray'd? your memory mu$t help to guide me into the way again.</P> <P>SIMPL. I remember very well where we left. We were upon the an$wer of <I>Anti-Tycho,</I> to the objections again$t the immutability of the Heavens, among which you in$erted this of the Solar fpots, not $poke of by him; and I believe you intended to examine his an$wer to the in$tance of the New Stars.</P> <foot>F 2 SALV.</foot> <p n=>44</p> <P>SALV. Now I remember the re$t, and to proceed, Methinks there are $ome things in the an$wer of <I>Anti-Tycho,</I> worthy of reprehen$ion. And fir$t, if the two New Stars, which he can do no le$s than place in the uppermo$t parts of the Heavens, and which were of a long duration, but finally vani$hed, give him no ob$truction in maintaining the inalterability of Heaven, in that they were not certain parts thereof, nor mutations made in the antient Stars, why doth he $et him$elf $o vigorou$ly and earne$tly again$t the Comets, to bani$h them by all ways from the Cœle- $tial Regions? Was it not enough that he could $ay of them the $ame which he $poke of the New $tars? to wit, that in re- gard they were no certain parts of Heaven, nor mutations made in any of the Stars, they could no wi$e prejudice either Heaven, or the Doctrine of <I>Ari$totle</I>? Secondly, I am not very well $atis- fied of his meaning; when he $aith that the alterations that $hould be granted to be made in the Stars, would be de$tructive to the prerogative of Heaven; namely, its incorruptibility, <I>&c.</I> and this, becau$e the Stars are Cœle$tial $ub$tances, as is manife$t by the con$ent of every one; and yet is nothing troubled that <marg>* Ex tra Stellas.</marg> the $ame alterations $hould be made ^{*} without the Stars in the re$t of the Cœle$tial expan$ion. Doth he think that Heaven is no Cœle$tial $ub$tance? I, for my part, did believe that the Stars were called Cœle$tial bodies, by rea$on that they were in Hea- ven, or for that they were made of the $ub$tance of Heaven; and yet I thought that Heaven was more Cœle$tial than they; in like $ort, as nothing can be $aid to be more Terre$trial, or more fiery than the Earth or Fire them$elves. And again, in that he ne- ver made any mention of the Solar $pots, which have been evi- dently demon$trated to be produced, and di$$olved, and to be neer the Sun, and to turn either with, or about the $ame, I have rea$on to think that this Author probably did write more for others plea$ure, than for his own $atisfaction; and this I affirm, fora$- much as he having $hewn him$elf to be skilful in the Mathema- ticks, it is impo$$ible but that he $hould have been convinced by Demon$trations, that tho$e $ub$tances are of nece$$ity contigu- ous with the body of the Sun, and are $o great generations and corruptions, that none comparable to them, ever happen in the Earth: And if $uch, $o many, and $o frequent be made in the very Globe of the Sun, which may with rea$on be held one of the noble$t parts of Heaven, what $hould make us think that others may not happen in the other Orbs?</P> <marg><I>Generability and alteration is a greater perfection in the Worlds bo- dies than the con- trary qualities.</I></marg> <P>SAGR. I cannot without great admiration, nay more, deni- al of my under$tanding, hear it to be attributed to natural bodies, for a great honour and perfection that they are ^{*} impa$$ible, im- mutable, inalterable, <I>&c.</I> And on the contrary, to hear it to <marg>* Impatible.</marg> <foot>be</foot> <p n=>45</p> be e$teemed a great imperfection to be alterable, generable, mu- table, <I>&c.</I> It is my opinion that the Earth is very noble and ad- <marg><I>The Earth very noble, by rea$on of the many mutati- ons made therein.</I></marg> mirable, by rea$on of $o many and $o different alterations, mu- tations, generations, <I>&c.</I> which are ince$$antly made therein; and if without being $ubject to any alteration, it had been all one va$t heap of $and, or a ma$$e of <I>Ja$per,</I> or that in the time of the Deluge, the waters freezing which covered it, it had continued an immen$e Globe of Chri$tal, wherein nothing had <marg><I>The carth unpro- $itable and full of idlene$$e, its alte- rations taken away</I></marg> ever grown, altered, or changed, I $hould have e$teemed it a lump of no benefit to the World, full of idlene$$e, and in a word $uperfluous, and as if it had never been in nature; and $hould make the $ame difference in it, as between a living and dead creature: The like I $ay of the <I>Moon, Jupiter,</I> and all the other Globes of the World. But the more I dive into the con- $ideration of the vanity of popular di$cour$es, the more empty and $imple I find them. And what greater folly can there be imagined, than to call Jems, Silver and Gold pretious; and Earth and dirt vile? For do not the$e per$ons con$ider, that if there <marg><I>The Earth more noble than Gold and Jewels.</I></marg> $hould be as great a $carcity of Earth, as there is of Jewels and pretious metals, there would be no Prince, but would gladly give a heap of Diamonds and Rubies, and many Wedges of Gold, to purcha$e onely $o much Earth as $hould $uffice to plant a Ge$$e- mine in a little pot, or to $et therein a <I>China Orange,</I> that he might $ee it $prout, grow up, and bring forth $o goodly leaves, $o odi- riferous flowers, and $o delicate fruit? It is therefore $carcity and <marg><I>Scarcity and plen- ty enhan$e and de- ba$e the price of things.</I></marg> plenty that make things e$teemed and contemned by the vulgar; who will $ay that $ame is a mo$t beautiful Diamond, for that it re$embleth a cleer water, and yet will not part with it for ten Tun of water: The$e men that $o extol incorruptibility, inalte- <marg><I>Incorruptibility e- $teemed by the vul- gar out of their fear of death.</I></marg> rability, <I>&c.</I> $peak thus I believe out of the great de$ire they have to live long, and for fear of death; not confidering, that if men had been immortal, they $hould have had nothing to do in the World. The$e de$erve to meet with a <I>Medu$a</I>'s head, <marg><I>The di$paragers of corraptibility de- $erve to be turned into Statua's.</I></marg> that would transform them into Statues of <I>Dimond</I> and <I>Ja$per,</I> that $o they might become more perfect than they are.</P> <P>SALV. And it may be $uch a <I>Metamorpho$is</I> would not be al- together unprofitable to them; for I am of opinion that it is bet- ter not to di$cour$e at all, than to argue erroniou$ly.</P> <P>SIMPL. There is not the lea$t que$tion to be made, but that the Earth is much more perfect, being as it is alterable, mutable, <I>&c.</I> than if it had been a ma$$e of $tone; yea although it were one entire Diamond, mo$t hard and impa$$ile. But look how mueh <marg><I>The Cœle$tial bo- dies de$igned to $erve the Earth, need no more but motion and light.</I></marg> the$e qualifications enoble the Earth, they render the Heavenly bodies again on the other $ide $o much the more imperfect, in which, $uch conditions would be $uperfluous; in regard that the <foot>Cœle-</foot> <p n=>46</p> Cœle$tial bodies, namely, the Sun, Moon, and the other Stars, which are ordained for no other u$e but to $erve the Earth, need no other qualities for attaining of that end, $ave onely tho$e of light and motion.</P> <P>SAGR. How? Will you affirm that nature hath produced and de$igned $o many va$t perfect and noble Cœle$tial bodies, impa$- $ible, immortal, and divine, to no other u$e but to $erve the pa$- $ible, frail, and mortal Earth? to $erve that which you call the dro$$e of the World, and $ink of all uncleanne$$e? To what purpo$e were the Cœle$tial bodies made immortal, <I>&c.</I> to $erve a frail, <I>&c.</I> Take away this $ub$erviency to the Earth, and the in- numerable multitude of Cœle$tial bodies become wholly unu$e- <marg><I>Celestial bodies want an inter- changeable opera- tion upon each o- ther.</I></marg> ful, and $uperfluous, $ince they neither have nor can have any mutual operation betwixt them$elves; becau$e they are all unal- terable, immutable, impa$$ible: For if, for Example, the Moon be impa$$ible, what influence can the Sun or any other Star have upon her? it would doubtle$$e have far le$$e effect upon her, than that of one who would with his looks or imagination, lignifie a piece of Gold. Moreover, it $eemeth to me, that whil$t the Cœ- le$tial bodies concurre to the generation and alteration of the Earth, they them$elves are al$o of nece$$ity alterable; for other- wi$e I cannot under$tand how the application of the Sun or Moon to the Earth, to effect production, $hould be any other than to lay a marble Statue by a Womans $ide, and from that conjunction to expect children.</P> <marg><I>Alterability, &c. are not in the whole Terre$trial Globe, but in $ome of its parts.</I></marg> <P>SIMPL. Corruptibility, alteration, mutation, <I>&c.</I> are not in the whole Terre$trial Globe, which as to its whole, is no le$$e eter- nal than the Sun or Moon, but it is generable and corruptible as to its external parts; but yet it is al$o true that likewi$e in them ge- neration and corruption are perpetual, and as $uch require the heavenly eternal operations; and therefore it is nece$$ary that the Cœle$tial bodies be eternal.</P> <P>SAGR. All this is right; but if the corruptibility of the $uper- ficial parts of the Earth be nowi$e prejudicial to the eternity of its whole Globe, yea, if their being generable, corruptible, alter- able, <I>&c.</I> gain them great ornament and perfection; why can- <marg><I>Cœle$tial bodies alterable in their outward parts.</I></marg> not, and ought not you to admit alteration, generation, <I>&c.</I> like- wi$e in the external parts of the Cœle$tial Globes, adding to them ornament, without taking from them perfection, or berea- ving them of action; yea rather encrea$ing their effects, by grant- ing not onely that they all operate on the Earth, but that they mu- tually operate upon each other, and the Earth al$o upon them all?</P> <P>SIMPL. This cannot be, becau$e the generations, mutations, <I>&c.</I> which we $hould $uppo$e <I>v. g.</I> in the Moon; would be vain and u$ele$$e, <I>& natura nihil fru$tra facit.</I></P> <foot>SAGR.</foot> <p n=>47</p> <P>SAGR. And why $hould they be vain and u$ele$$e?</P> <P>SIMPL. Becau$e we cleerly $ee, and feel with our hands, that <marg><I>The generations & mutations happen- ing in the Earth, are all for the good of Man.</I></marg> all generations, corruptions, <I>&c.</I> made in the Earth, are all ei- ther mediately or immediately directed to the u$e, convenience, and benefit of man; for the u$e of man are hor$es brought forth, for the feeding of hor$es, the Earth produceth gra$$e, and the Clouds water it; for the u$e and nouri$hment of man, herbs, corn, fruits, bea$ts, birds, fi$hes, are brought forth; and in $um, if we $hould one by one dilligently examine and re$olve all the$e things, we $hould find the end to which they are all directed, to be the nece$$ity, u$e, convenience, and delight of man. Now of what u$e could the generations which we $uppo$e to be made in the Moon or other Planets, ever be to mankind? unle$$e you $hould $ay that there were al$o men in the Moon, that might enjoy the benefit thereof; a conceit either fabulous or impious.</P> <P>SAGR. That in the Moon or other Planets, there are genera- <marg><I>The Moon hath no generatings of things, like as we have, nor is it in- habited by men.</I></marg> ted either herbs, or plants, or animals, like to ours, or that there are rains, winds, or thunders there, as about the Earth, I nei- ther know, nor believe, and much le$$e, that it is inhabited by men: but yet I under$tand not, becau$e there are not genera- ted things like to ours, that therefore it nece$$arily followeth, that no alteration is wrought therein, or that there may not be other things that change, generate, and di$$olve, which are not <marg><I>In the Moon may be a generation of things different from ours.</I></marg> onely different from ours, but exceedingly beyond our imagina- tion, and in a word, not to be thought of by us. And if, as I am certain, that one born and brought up in a $patious Forre$t, among$t bea$ts and birds, and that hath no knowledg at all of the Element of Water, could never come to imagine another World <marg><I>He that had not heard of the Ele- ment of Water, could never fancy to him$elf Ships and Fi$hes.</I></marg> to be in Nature, different from the Eatth, full of living crea- tures, which without legs or wings $wiftly move, and not upon the $urface onely, as bea$ts do upon the Earth, but in the very bowels thereof; and not onely move, but al$o $tay them$elves and cea$e to move at their plea$ure, which birds cannot do in the air; and that moreover men live therein, and build Palaces and Cities, and have $o great convenience in travailing, that without the lea$t trouble, they can go with their Family, Hou$e, and whole Cities, to places far remote, like as I $ay, I am certain, $uch a per$on, though of never $o piercing an imagination, could never fancy to him$elf Fi$hes, the Ocean, Ships, Fleets, <I>Arma- do's</I> at Sea; thus, and much more ea$ily, may it happn, that in the Moon, remote from us by $o great a $pace, and of a $ub- $tance perchance very different from the Earth, there may be mat- ters, and operations, not only wide off, but altogether beyond all our imaginations, as being $uch as have no re$emblance to ours, and therefore wholly inexcogitable, in regard, that what we <foot>ima-</foot> <p n=>48</p> imagine to our $elves, mu$t nece$$arily be either a thing already $een, or a compo$ition of things, or parts of things $een at ano- ther time; for $uch are the <I>Sphinxes, Sirenes, Chimœra's, Cen- taurs,</I> &c.</P> <P>SALV. I have very often let my fancy ruminate upon the$e $pe- culations, and in the end, have thought that I had found $ome things that neither are nor can be in the Moon; but yet I have not found therein any of tho$e which I believe are, and may be there, $ave onely in a very general acceptation, namely, things that adorn it by operating, moving and living; and perhaps in a way <marg><I>There may be $ub- $tances in the Moon very diffe- rent from ours.</I></marg> very different from ours; beholding and admiring the greatne$s and beauty of the World, and of its Maker and Ruler, and with continual <I>Encomiums</I> $inging his pray$es; and in $umme (which is that which I intend) doing what $acred Writers $o frequently af- firm, to wit, all the creatures making it their perpetual imploy- ment to laud God.</P> <P>SAGR. The$e are the things, which $peaking in general terms, may be there; but I would gladly hear you in$tance in $uch as you believe neither are nor can be there; which perchance may be more particularly named.</P> <P>SALV. Take notice <I>Sagredus</I> that this will be the third time that we have unawares by running from one thing to another, lo$t our principal $ubject; and if we continue the$e digre$$ions, it will be longere we come to a conclu$ion of our di$cour$e; there- fore I $hould judg it better to remit this, as al$o $uch other points, to be decided on a particular occa$ion.</P> <P>SAGR. Since we are now got into the Moon, if you plea$e, let us di$patch $uch things as concern her, that $o we be not forced to $uch another tedious journey.</P> <P>SALV. It $hall be as you would have it. And to begin with things more general, I believe that the Lunar Globe is far diffe- rent from the Terre$trial, though in $ome things they agree. I will recount fir$t their re$emblances, and next their differences. The <marg><I>The</I> Fir$t <I>re$em- blance between the Moon and Earth; which is that of figure; is proved by the manner of be- ing illuminated by the Sun.</I></marg> Moon is manife$tly like to the Earth in figure, which undoubtedly is $pherical, as may be nece$$arily concluded from the a$pect of its $urface, which is perfectly Orbicular, and the manner of its re- ceiving the light of the Sun, from which, if its $urface were flat, it would come to be all in one and the $ame time illuminated, and likewi$e again in another in$tant of time ob$cured, and not tho$e parts fir$t, which are $ituate towards the Sun, and the re$t $ucce$- $ively, $o that in its oppo$ition, and not till then, its whole apparent circumference is enlightned; which would happen quite contrary, if the vi$ible $urface were concave; namely, the illu- <marg><I>The</I> Second <I>con- formity is the Moons being opa- cous as the Earth.</I></marg> mination would begin from the parts oppo$ite or aver$e to the Sun. Secondly $he is as the Earth, in her $elf ob$cure and opacous, by which opacity it is enabled to receive, and reflect the light of the <foot><*></foot> <p n=>49</p> Sun; which were it not $o, it could not do. Thirdly, I hold its <marg><I>Thirdly, The mat- ter of the Moon is den$e and mo ita- nous as the Earth.</I></marg> matter to be mo$t den$e and $olid as the Earth is, which I clearly argue from the unevenne$s of its $uperficies in mo$t places, by means of the many eminencies and cavities di$covered therein by help of the <I>$ele$cope</I>: of which eminencies there are many all over it, di- rectly re$embling our mo$t $harp and craggy mountains, of which you $hall there perceive $ome extend and run in ledges of an hun- dred miles long; others are contracted into rounder forms; and there are al$o many craggy, $olitary, $teep and cliffy rocks. But that of which there are frequente$t appearances, are certain Banks (I u$e this word, becau$e I cannot thing of another that better ex- pre$$eth them) pretty high rai$ed, which environ and inclo$e fields of $everal bigne$$es, and form $undry figures, but for the mo$t part circular; many of which have in the mid$t a mount rai$ed pretty high, and $ome few are repleni$hed with a matter $omewhat ob- $cure, to wit, like to the great $pots di$cerned by the bare eye, and the$e are of the greate$t magnitude; the number moreover of tho$e that are le$$er and le$$er is very great, and yet almo$t all circular. <marg><I>Fourthly, The Moon is di$tin- gui$hed into two different parts for clarity and ob$cu- rity, as the Terre- strial Globe into Sea and Land.</I></marg> Fourthly, like as the $urface of our Globe is di$tingui$hed into two principal parts, namely, into the Terre$trial and Aquatick: $o in the Lunar $urface we di$cern a great di$tinction of $ome great fields more re$plendant, and $ome le$s: who$e a$pect makes me believe, that that of the Earth would $eem very like it, beheld by any one from the Moon, or any other the like di$tance, to be illuminated <marg><I>The $urface of the Sea would $hew at a di$tance more ob- $oure than that of the Earth.</I></marg> by the Sun: and the $urface of the $ea would appear more ob- $cure, and that of the Earth more bright. Fifthly, like as we from the Earth behold the Moon, one while all illuminated, another <marg><I>Fiftly, Muta- tion of $igures in the Earth, like to tho$e of the Moon, and made with the $ame periods.</I></marg> while half; $ometimes more, $ometimes le$s; $ometimes horned, $ometimes wholly invi$ibly; namely, when its ju$t under the Sun beams; $o that the parts which look towards the Earth are dark: Thus in every re$pect, one $tanding in the Moon would $ee the illumination of the Earths $urface by the Sun, with the $ame periods to an hair, and under the $ame changes of figures. Sixtly, -----</P> <P>SAGR. Stay a little, <I>Salviatus</I>; That the illumination of the Earth, as to the $everal figures, would repre$ent it $elf to a per$on placed in the Moon, like in all things to that which we di$cover in the Moon, I under$tand very well, but yet I cannot conceive how it $hall appear to be done in the $ame period; $eeing that that which the Suns illumination doth in the Lunar $uperficies in a month, it doth in the Terre$trial in twenty four hours.</P> <P>SALV. Its true, the effect of the Sun about the illuminating the$e two bodies, and repleni$hing with its $plendor their whole $urfaces, is di$patch'd in the Earth in a Natural day, and in the Moon in a Month; but the variation of the figures in which the <foot>G illumi-</foot> <p n=>50</p> illuminated parts of the Terre$trial $uperficies appear beheld from the Moon, depends not on this alone, but on the divers a$pects which the Moon is $till changing with the Sun; $o that, if for in- $tance, the Moon punctually followed the motion of the Sun, and $tood, for example, always in a direct line between it and the Earth, in that a$pect which we call Conjunction, it looking always to the $ame Hemi$phere of the Earth which the Sun looks unto, $he would behold the $ame all light: as on the contrary, if it $hould always $tay in Oppo$ition to the Sun, it would never behold the Earth, of which the dark part would be continually turn'd towards the Moon, and therefore invi$ible. But when the Moon is in Quadrature of the Sun, that half of the Terre$trial Hemi$phere ex- po$ed to the $ight of the Moon which is towards the Sun, is lumi- nous; and the other towards the contrary is ob$cure: and there- fore the illuminated part of the Earth would repre$ent it $elf to the Moon in a $emi-circular figure.</P> <P>SAGR. I clearly perceive all this, and under$tand very well, that the Moon departing from its Oppo$ition to the Sun, where it $aw no part of the illumination of the Terre$trial $uperficies, and approaching day by day nearer the Sun, $he begins by little and little to di$cover $ome part of the face of the illuminated Earth; and that which appeareth of it $hall re$emble a thin $ickle, in regard the figure of the Earth is round: and the Moon thus acquiring by its motion day by day greater proximity to the Sun, $ucce$$ively di$covers more and more of the Terre$trial Hemi$phere enlightned, $o that at the Quadrature there is ju$t half of it vi$ible, in$omuch that we may $ee the other part of her: continuing next to proceed towards the Conjunction, it $ucce$$ively di$covers more and more of its $urface to be illuminated, and in fine, at the time of Conjun- ction $eeth the whole Hemi$phere enlightned. And in $hort, I very well conceive, that what befalls the Inhabitants of the Earth, in beholding the changes of the Moon, would happen to him that from the Moon $hould ob$erve the Earth; but in a contrary order, namely, that when the Moon is to us at her full, and in Oppo$ition to the Sun, then the Earth would be in Conjunction with the Sun, and wholly ob$cure and invi$ible; on the contrary, that po$ition which is to us a Conjunction of the Moon with the Sun, and for that cau$e a <I>M</I>oon $ilent and un$een, would be there an Oppo$ition of the Earth to the Sun, and, to $o $peak, <I>Full Earth,</I> to wit, all enlightned. And la$tly, look what part of the Lunar $urface ap- pears to us from time to time illuminated, $o much of the Earth in the $ame time $hall you behold from the Moon to be ob$cured: and look how much of the Moon is to us deprived of light, $o much of the Earth is to the Moon illuminated. In one thing yet the$e mutual operations in my judgment $eem to differ, and it is, that it <foot>being</foot> <p n=>51</p> being $uppo$ed, and not granted, that $ome one being placed in the Moon to ob$erve the Earth, he would every day $ee the whole Terre$trial $uperficies, by means of the Moons going about the Earth in twenty four or twenty five hours; but we never $ee but half of the Moon, $ince it revolves not in it $elf, as it mu$t do to be $een in every part of it.</P> <P>SALV. So that this, befals not contrarily, namely, that her re- volving in her $elf, is the cau$e that we $ee not the other half of her, for $o it would be nece$$ary it $hould be, if $he had the Epicy- cle. But what other difference have you behind, to exchange for this which you have named?</P> <P>SAGR. Let me $ee; Well for the pre$ent I cannot think of any other.</P> <P>SALV. And what if the Earth (as you have well noted) $eeth <marg><I>All the Earth $eeth half onely of the Moon, & the half onely of the Moon $eeth all the Earth.</I></marg> no more than half the Moon, whereas from the Moon one may $ee all the Earth; and on the contrary, all the Earth $eeth the Moon, and but onely half of it $eeth the Earth? For the inhabitants, to $o $peak, of the $uperior Hemi$phere of the Moon, which is to us invi$ible, are deprived of the $ight of the Earth: and the$e haply are the <I>Anticthones.</I> But here I remember a particular accident, newly ob$erved by our <I>Academian,</I> in the Moon, from whch are gathered <marg><I>From the Earth we $ee more than half the Lunar Globe.</I></marg> two nece$$ary con$equences; one is, that we $ee $omewhat more than half of the Moon; and the other is, that the motion of the Moon hath exact concentricity with the Earth: and thus he finds the <I>Phœnomenon</I> and ob$ervation. When the Moon hath a cor- re$pondence and natural $ympathy with the Earth, towards which it hath its a$pect in $uch a determinate part, it is nece$$ary that the right line which conjoyns their centers, do pa$$e ever by the $ame point of the Moons $uperficies; $o that, who $o $hall from the cen- ter of the Earth behold the $ame, $hall alwayes $ee the $ame <I>Di$cus</I> or Face of the Moon punctually determined by one and the $ame circumference; But if a man be placed upon the Terre- $trial $urface, the ray which from his eye pa$$eth to the centre of the Lunar Globe, will not pa$s by the $ame point of its $uperficies, by which the line pa$$eth that is drawn from the centre of the Earth to that of the Moon, $ave onely when it is vertical to him: but the Moon being placed in the Ea$t, or in the We$t, the point of incidence of the vi$ual ray, is higher than that of the line which conjoyns the centres; and therefore the ob$erver may di$cern $ome part of the Lunar Hemi$phere towards the upper circumfe- rence, and alike part of the other is invi$ible: they are di$cerna- ble and undi$cernable, in re$pect of the Hemi$phere beheld from the true centre of the Earth: and becau$e the part of the Moons circumference, which is $uperiour in its ri$ing, is nethermo$t in its $etting; therefore the difference of the $aid $uperiour and inferi- <foot>G 2 our</foot> <p n=>52</p> our parts mu$t needs be very ob$ervable; certain $pots and other notable things in tho$e parts, being one while di$cernable, and another while not. A like variation may al$o be ob$erved towards the North and South extremities of the $ame <I>Di$cus</I> (or Surface) according as the Moons po$ition is in one or the other Section of its Dragon; For, if it be North, $ome of its parts towards the North are hid, and $ome of tho$e parts towards the South are di$covered, and $o on the contrary. Now that the$e con$equen- <marg><I>Two $pots in the Moon, by which it is perceived that $he hath respect to the centre of the Earth in her mo- tion.</I></marg> ces are really true, is verified by the <I>Tele$cope,</I> for there be in the Moon two remarkable $pots, one of which, when the Moon is in the meridian, is $ituate to the Northwe$t, and the other is almo$t diametrically oppo$ite unto it; and the fir$t of the$e is vi- $ible even without the <I>Tele$cope</I>; but the other is not. That to- wards the Northwe$t is a rea$onable great $pot of oval figure, $e- parated from the other great ones; the oppo$ite one is le$$e, and al$o $evered from the bigge$t, and $ituate in a very cleer field; in both the$e we may manife$tly di$cern the fore$aid variations, and $ee them one after another; now neer the edge or limb of the Lunar <I>Di$cus,</I> and anon remote, with $o great difference that the di$tance betwixt the Northwe$t and the circumference of the <I>Di$cus</I> is more than twice as great at one time, as at the other; and as to the $econd $pot (becau$e it is neerer to the circumfe- rence) $uch mutation importeth more, than twice $o much in the former. Hence its manife$t, that the Moon, as if it were drawn by a magnetick vertue, con$tantly beholds the Terre$trial Globe with one and the $ame a$pect, never deviating from the $ame.</P> <P>SAGR. Oh! when will there be an end put to the new ob- $ervations aud di$coveries of this admirable In$trument?</P> <P>SALV. If this $ucceed according to the progre$$e of other great inventions, it is to be hoped, that in proce$$e of time, one may arrive to the $ight of things, to us at pre$ent not to be imagined. <marg><I>Sixthly, The Earth and Moon interchangeably do illuminate.</I></marg> But returning to our fir$t di$cour$e, I $ay for the $ixth re$emblance betwixt the Moon and Earth, that as the Moon for a great part of time, $upplies the want of the Suns light, and makes the nights, by the reflection of its own, rea$onable clear; $o the Earth, in recompence, affordeth it when it $tands in mo$t need, by reflecting the Solar rayes, a very cleer illumination, and $o much, in my opinion, greater than that which cometh from her to us, by how much the $uperficies of the Earth is greater than that of the Moon.</P> <P>SAGR. Hold there, <I>Salviatus</I> hold there, and permit me the plea$ure of relating to you, how at this fir$t hint I have penetrated the cau$e of an accident, which I have a thou$and times thought <marg><I>Light reflected from the Earth in- to the Moon.</I></marg> upon, but could never find out. You would $ay, that the imper- fect light which is $een in the Moon, e$pecially when it is horned, <foot>comes</foot> <p n=>53</p> comes from the reflection of the light of the Sun on the Superfi- cies of the Earth and Sea; and that light is more clear, by how much the horns are le$$e, for then the luminous part of the Earth, beheld by the Moon, is greater, according to that which was a little before proved; to wit, that the luminous part of the Earth, expo$ed to the Moon, is alway as great as the ob$cure part of the Moon, that is vi$ible to the Earth; whereupon, at $uch time as the Moon is $harp-forked, and con$equently its tenebrous part great, great al$o is the illuminated part of the Earth beheld from the Moon, and its reflection of light $o much the more potent.</P> <P>SALV. This is exactly the $ame with what I was about to $ay. In a word, it is a great plea$ure to $peak with per$ons judicious and apprehen$ive, and the rather to me, for that while$t others conver$e and di$cour$e touching Axiomatical truths, I have ma- ny times creeping into my brain $uch arduous Paradoxes, that though I have a thou$and times rehear$ed this which you at the ve- ry fir$t, have of your $elf apprehended, yet could I never beat it into mens brains.</P> <P>SIMPL. If you mean by your not being able to per$wade them to it, that you could not make them under$tand the $ame, I much wonder thereat, and am very confident that if they did not under$tand it by your demon$tration (your way of expre$$ion, being, in my judgment, very plain) they would very hardly have apprehended it upon the explication of any other man; but if you mean you have not per$waded them, $o as to make them be- lieve it, I wonder not, in the lea$t, at this; for I confe$$e my $elf to be one of tho$e who under$tand your di$cour$es, but am not $atisfied therewith; for there are in this, and $ome of the other $ix congruities, or re$emblances, many difficulties, which I $hall in$tance in, when you have gone through them all.</P> <P>SALV. The de$ire I have to find out any truth, in the acqui$t whereof the objections of intelligent per$ons ($uch as your $elf) may much a$$i$t me, will cau$e me to be very brief in di$patching that which remains. For a $eventh conformity, take their reci- <marg><I>Seventhly, The Earth and Moon do mutually eclip$e.</I></marg> procal re$pon$ion as well to injuries, as favours; whereby the Moon, which very often in the height of its illumination, by the interpo$ure of the Earth betwixt it and the Sun, is deprived of light, and eclip$ed, doth by way of revenge; in like manner, in- te<*>po$e it $elf between the Earth and the Sun, and with its $hadow ob$cureth the Earth; and although the revenge be not an$wer- able to the injury, for that the Moon often continueth, and that for a rea$onable long time, wholly immer$ed in the Earths $hadow, but never was the Earth wholly, nor for any long time, eclip$ed by the Moon; yet, neverthele$$e, having re$pect to the <foot>$mal-</foot> <p n=>54</p> $malne$$e of the body of this, in compari$on to the magnitude of the other, it cannot be denied but that the <I>will</I> and as it were <I>valour</I> of this, is very great. Thus much for their con- gruities or re$emblances. It $hould next follow that we di$cour$e touching their di$parity; but becau$e <I>Simplicius</I> will favour us with his objections again$t the former, its nece$$ary that we hear and examine them, before we proceed any farther.</P> <P>SAGR. And the rather, becau$e it is to be $uppo$ed that <I>Simplicius</I> will not any wayes oppo$e the di$parities, and incon- gruities betwixt the Earth and Moon, $ince that he accounts their $ub$tances extremely different.</P> <P>SIMPL. Among$t the re$emblances by you recited, in the pa- rallel you make betwixt the Earth and Moon, I find that I can admit none confidently $ave onely the fir$t, and two others; I grant the fir$t, namely, the $pherical figure; howbeit, even in this there is $ome kind of difference, for that I hold that of the Moon to be very $mooth and even, as a looking-gla$$e, where- as, we find and feel this of the Earth to be extraordinary montu- ous and rugged; but this belonging to the inequality of $uperfi- cies, it $hall be anon con$idered, in another of tho$e Re$emblan- ces by you alledged; I $hall therefore re$erve what I have to $ay thereof, till I come to the con$ideration of that. Of what you affirm next, that the Moon $eemeth, as you $ay in your $econd Re$emblance, opacous and ob$cure in its $elf, like the Earth; I admit not any more than the fir$t attribute of opacity, of which the Eclip$es of the Sun a$$ure me. For were the Moon tran$pa- rent, the air in the total ob$curation of the Sun, would not be- come $o duski$h, as at $uch a time it is, but by means of the tran$parency of the body of the Moon, a refracted light would pa$$e through it, as we $ee it doth through the thicke$t clouds. But as to the ob$curity, I believe not that the Moon is wholly depri- ved of light, as the Earth; nay, that clarity which is $een in the remainder of its <I>Di$cus,</I> over and above the $mall cre$cent en- lightened by the Sun, I repute to be its proper and natural light, <marg><I>The $econd clarity of the Moon e- $teemed to be its native light.</I></marg> and not a reflection of the Earth, which I e$teem unable, by rea$on of its a$perity (craggine$$e) and ob$curity, to reflect the raies of the Sun. In the third Parallel I a$$ent unto you in one <marg><I>The Earth unable to reflect the Suns raies.</I></marg> part, and di$$ent in another: I agree in judging the body of the Moon to be mo$t $olid and hard, like the Earth, yea much more; <marg><I>The $ub$tance of the Heavens impe- netrable, accord- ing to</I> Ari$totle.</marg> for if from <I>Ari$totle</I> we receive that the Heavens are impenetrable, and the Stars the mo$t den$e parts of Heaven, it mu$t nece$$arily follow, that they are mo$t $olid and mo$t impenetrable.</P> <P>SAGR. What excellent matter would the Heavens afford us for to make Pallaces of, if we could procure a $ub$tance $o hard and $o tran$parent?</P> <foot>SALV.</foot> <p n=>55</p> <P>SALV. Rather how improper, for being by its tran$parence, wholly invi$ible, a man would not be able without $tumbling at the thre$holds, and breaking his head again$t the Walls, to pa$s from room to room.</P> <P>SAGR. This danger would not befall him, if it be true, as $ome <marg><I>The $ubstance of Heaven intangi- ble.</I></marg> <I>Peripateticks</I> $ay, that it is intangible: and if one cannot touch it, much le$s can it hurt him.</P> <P>SALV. This would not $erve the turn, for though the matter of the Heavens cannot be toucht, as wanting tangible qualities: yet may it ea$ily touch the elementary bodies; and to offend us it is as $ufficient that it $trike us, nay wor$e, than if we $hould $trike it. But let us leave the$e <I>Pallaces,</I> or, to $ay better, the$e <I>Ca$tles</I> in the air, and not interrupt <I>Simplicius.</I></P> <P>SIMPL. The que$tion which you have $o ca$ually $tarted, is one of the mo$t difficulty that is di$puted in Philo$ophy; and I have on that $ubject mo$t excellent conceits of a very learned Doctor of <I>Padoua,</I> but it is not now time to enter upon them. Therefore returning to our purpo$e, I $ay that the Moon, in my opinion, is much more $olid than the Earth, but do not infer the $ame, as you do, from the craggine$s and montuo$ity of its $uperficies; but <marg><I>The $uperficies of the Moon more $leek than any Looking-glaß.</I></marg> rather from the contrary, namely, from its aptitude to receive (as we $ee it experimented in the harde$t $tones) a poli$h and lu$tre exceeding that of the $moothe$t gla$s, for $uch nece$$arily mu$t its $uperficies be, to render it apt to make $o lively reflection of the Suns rays. And for tho$e appearances which you mention, of Mountains, Cliffs, Hills, Valleys, <I>&c.</I> they are all illu$ions: and I have been pre$ent at certain publick di$putes, where I have heard it $trongly maintained again$t the$e introducers of novelties, <marg><I>The eminencies and cavities in the Moon are illu$ions of its opacous and perspicuous parts.</I></marg> that $uch appearances proceed from nothing el$e, but from the un- equal di$tribution of the opacous and per$picuous parts, of which the Moon is inwardly and outwardly compo$ed: as we $ee it often fall out in chry$tal, amber, and many other precious $tones of perfect lu$tre; in which by rea$on of the opacity of $ome parts, and the tran$parency of others, there doth appear $everal conca- vities and prominencies. In the fourth re$emblance, I grant, that the $uperficies of Terre$trial Globe beheld from afar, would make two different appearances, namely, one more clear, the other more dark; but I believe that $uch diver$ity would $ucceed quite con- trary to what you $ay; that is, I hold that the $urface of the wa- ter would appear lucid, becau$e that it is $mooth and tran$parent; and that of the Earth would appear ob$cure, by rea$on of its o- pacity and $cabro$ity, ill accommodated for reflecting the light of the Sun. Concernïng the fifth compari$on, I grant it wholly, and am able, in ca$e the Earth did $hine as the Moon, to $how the $ame to any one that $hould from thence above behold it, repre- <foot>$ented</foot> <p n=>56</p> $ented by figures an$werable to tho$e which we $ee in the Moon: I comprehend al$o, how the period of its illumination and varia- tion of figure, would be monthly, albeit the Sun revolves round about it in twenty four hours: and la$tly, I do not $cruple to admit, that the half onely of the Moon $eeth all the Earth, and that all the Earth $eeth but onely half of the Moon. For what remains, I repute it mo$t fal$e, that the Moon can receive light from the Earth, which is mo$t ob$cure, opacous, and utterly un- apt to reflect the Suns light, as the Moon doth reflect it to us: and as I have $aid, I hold that that light which we $ee in the remain- der of the Moons face (the $plendid cre$cents $ubducted) by the illumination, is the proper and natural light of the Moon, and no ea$ie matter would induce me to believe otherwi$e. The $eventh, touching the mutual Eclip$es, may be al$o admitted; howbeit that is wont to be called the eclip$e of the Sun, which you are plea$ed to phra$e the eclip$e of the Earth. And this is what <I>I</I> have at this time to $ay in oppo$ition to your $even congruities or re$emblances, to which objections, if you are minded to make any reply, <I>I</I> $hall willingly hear you.</P> <P>SALV. If I have well apprehended what you have an$wered, it $eems to me, that there $till remains in controver$ie between us, cer- tain conditions, which I made common betwixt the Moon & Earth, and they are the$e; You e$teem the Moon to be $mooth and poli$ht, as a Looking-gla$s, and as $uch, able to reflect the Suns light; and contrarily, the Earth, by rea$on of its montuo$ity, unable to make $uch reflection: You yield the Moon to be $olid and hard, and that you argue from its being $mooth and polite, and not from its being montuous; and for its appearing montuous, you a$$ign as the cau$e, that it con$i$ts of parts more and le$s opacous and per$pi- cuous. And la$tly, you e$teem that $econdary light, to be proper to the <I>M</I>oon, and not reflected from the Earth; howbeit you $eem not to deny the $ea, as being of a $mooth $urface, $ome kind of reflection. As to the convincing you of that error, that the reflection of the <I>M</I>oon is made, as it were, like that of a Looking-gla$s, <I>I</I> have $mall hope, whil$t <I>I</I> $ee, that what hath <marg>* <I>Il Saggiatore, & Lettere Solari,</I> two Treati$es of <I>Galilæus.</I></marg> been read in the ^{*} <I>Saggiator</I> and in the <I>Solar Letters</I> of our <I>Com- mon Friend,</I> hath profited nothing in your judgment, if haply you have attentively read what he hath there written on this $ub- ject.</P> <P>SIMPL. <I>I</I> have peru$ed the $ame $o $uperficially, according to the $mall time of lea$ure allowed me from more $olid $tudies; therefore, if you think you can, either by repeating $ome of tho$e rea$ons, or by alledging others, re$olve me the$e doubts, <I>I</I> will hearken to them attentively.</P> <P>SALV. <I>I</I> will tell you what comes into my mind upon the <foot>in$tant,</foot> <p n=>57</p> in$tant, and its po$$ible it may be a commixtion of my own con- ceipts; and tho$e which I have $ometime read in the fore-$aid Books, by which I well remember, that I was then perfectly $atisfied, although the conclu$ions, at fir$t $ight $eem'd unto me $trange Paradoxes. We enquire <I>Simplicius,</I> whether to the ma- king a reflection of light, like that which we receive from the Moon, it be nece$$ary that the $uperficies from whence the refle- ction commeth, be $o $mooth and polite, as the face of a Looking- Gla$$e, or whether a $uperficies not $mooth or poli$ht, but rough and uneven, be more apt for $uch a purpo$e. Now $uppo$ing two reflections $hould come unto us, one more bright, the other le$$e, from two $uperficies oppo$ite unto us, I demand of you, which of the two $uperficies you think would repre$ent it $elf to our $ight, to be the cleare$t, and which the ob$cure$t.</P> <P>SIMPL. I am very confident, that that $ame, which mo$t for- cibly reflected the light upon me, would $hew its $elf in its a$pect the clearer, and the other darker.</P> <P>SALV. Be plea$ed to take that Gla$$e which hangs on yonder <marg><I>It is proved at large that the Moons $urface is $harp.</I></marg> Wall, and let us go out into the Court-yard. Come <I>Sagredus.</I> Now hang the gla$$e yonder, again$t that $ame Wall, on which the Sun $hines, and now let us with-draw our $elves into the $hade. See yonder two $uperficies beaten by the Sun, namely, the Wall and the Gla$$e. Tell me now which appears cleare$t unto you, that of the Wall or that of the Gla$$e? Why do you not an$wer me?</P> <P>SAGR. I leave the reply to <I>Simplicius,</I> who made the que$ti- on; but I, for my own part, am per$waded upon this $mall be- ginning of the experiment, that the Moon mu$t be of a very un- poli$ht $urface.</P> <P>SALV. What $ay you <I>Simplicius,</I> if you were to depaint that Wall, and that Gla$$e fa$tened unto it, where would you u$e your darke$t colours, in de$igning the Wall, or el$e in painting the Looking-Gla$$e.</P> <P>SIMPL. Much the darker in depainting the Gla$$e.</P> <P>SALV. Now if from the $uperficies, which repre$ents it $elf more clear, there proceedeth a more powerful reflection of light, the Wall will more forcibly reflect the raies of the Sun, than the Gla$$e.</P> <P>SIMPL. Very well, Sir, have you ever a better experiment than this? you have placed us where the Gla$$e doth not rever- berate upon us; but come along with me a little this way; how, will you not $tir?</P> <P>SAGR. You perhaps $eek the place of the reflection, which the Gla$$e makth.</P> <P>SIMPL. I do $o.</P> <foot>H SAGR</foot> <p n=>58</p> <P>SAGR. Why look you, there it is upon the oppo$ite Wall, ju$t as big as the Gla$$e, and little le$$e bright than if the Sun had directly $hined upon it.</P> <P>SIMPL. Come hither therefore, and $ee from hence the $ur- face of the Gla$$e, and tell me whether you think it more ob- $cure than that of the Wall.</P> <P>SAGR. Look on it your $elf, for I have no mind at this time, to dazle my eyes; and I know very well, without $eeing it, that it there appears as $plendid and bright as the Sun it $elf, or little le$$e.</P> <P>SIMPL. What $ay you therefore, is the reflection of a Gla$$e le$$e powerful than that of a Wall? I $ee, that in this oppo$ite Wall, where the reflection of the other illuminated Wall comes, together with that of the Gla$$e, this of the Gla$$e is much clearer; and I $ee likewi$e, that, from this place where I $tand, the gla$$e it $elf appears with much more lu$tre than the Wall.</P> <P>SALV. You have prevented me with your $ubtlety; for I $tood in need of this very ob$ervation to demon$trate what remains. You $ee then the difference which happens betwixt the two refle- ctions made by the two $uperficies of the Wall and Gla$$e, per- cu'$t in the $elf-$ame manner, by the rayes of the Sun; and you $ee, how the reflection which comes from the Wall, diffu$eth it $elf towards all the parts oppo$ite to it, but that of the Gla$$e goeth towards one part onely, not at all bigger than the Gla$$e it $elf: you $ee likewi$e, how the $uperficies of the Wall, beheld from what part $oever, alwayes $hews it $elf of one and the $ame cleerne$$e, and every way, much clearer than that of the Gla$$e, excepting only in that little place, on which the Gla$$es reflection reverberates, for from thence indeed the Gla$$e appears much more lucid than the Wall. By the$e $o $en$ible, and palpable experi- ments, my thinks one may $oon come to know, whether the reflection which the Moon $ends upon us, proceed as from a Gla$$e, or el$e, as from a Wall, that is, from a $mooth $uperfi- cies, or a rugged.</P> <P>SAGR. If I were in the Moon it $elf, I think I could not with my hands more plainly feel the unevenne$$e of its $uperficies, than I do now perceive it, by apprehending your di$cour$e. The Moon beheld in any po$ture, in re$pect of the Sun and us, $heweth us its $uperficies, touch't by the Suns rayes, alwayes equally clear; an effect, which an$wers to an hair that of the Wall, which be- held from what place $oever, appeareth equally bright, and dif- fereth from the Gla$$e, which from one place onely appeareth lu- cid, and from all others ob$cure. Moreover, the light which cometh to me from the reflection of the Wall, is tollerable, and weak, in compari$on of that of the Gla$$e, which is little <foot>le$$e</foot> <p n=>59</p> le$$e forcible and offen$ive to the $ight, than that primary and direct light of the Sun. And thus without trouble do we behold the face of the Moon; which were it as a Gla$$e, it appearing to us by rea$on of its vicinity, as big as the Sun it $elf, its $plendor would be ab$olutely intollerable, and would $eem as if we beheld another Sun.</P> <P>SALV. A$cribe not, I be$eech you <I>Sagredus,</I> more to my de- mon$tration, than it produceth. I will oppo$e you with an in$tance, which I $ee not well how you can ea$ily re$olve. You in$i$t upon it as a grand difference between the Moon and Gla$$e, that it emits its reflection towards all parts equally, as doth the Wall; where- as the Gla$$e ca$ts it upon one onely determinate place; and from hence you conclude the Moon to be like to the Wall, and not to the Gla$$e: But I mu$t tell you, that that $ame Gla$$e ca$ts its <marg><I>Flat Looking- gla$$es ca$t forth the reflection to- wards but one place, but the $pherical every way.</I></marg> reflection on one place onely, becau$e its $urface is flat, and the reflex rayes being to depart at angles equal to tho$e of the rayes of incidence, it mu$t follow that from a plane or flat $uperficies, they do depart unitedly towards the $ame place; but in regard that the $uperficies of the Moon is not plain, but $pherical, and the incident rayes upon $uch a $uperficies, being to reflect them- $elves at angles equal to tho$e of the incidence towards all parts, by means of the infinity of the inclinations which compo$e the $pherical $uperficies, therefore the Moon may $end forth its reflecti- on every way; and there is no nece$$ity for its repercu$$ion upon one place onely, as that Gla$$e which is flat.</P> <P>SIMPL. This is one of the very $ame objections, which I in- tended to have made again$t him.</P> <P>SAGR. If this be one, you had need have more of them; yet I tell you, that as to this fir$t, it $eems to me to make more a- gain$t you, than for you.</P> <P>SIMPL. You have pronounced as a thing manife$t, that the refle- ction made by that Wall, is as cleer and lucid as that which the Moon $ends forth, and I e$teem it nothing in compari$on thereto. “For, in this bu$ine$$e of the illumination, its requi$ite to re$pect, and to di$tingui$h the <I>Sphere</I> of <I>Activity</I>; and who que$tions <marg><I>The $phere of Activity greater in the Cœle$tial bodies than in Ele- mentary.</I></marg> but the Cœle$tial bodies have greater Spheres of activity, than the$e our elementary, frail, and mortal ones? and that Wall, finally, what el$e is it but a little ob$cure Earth, unapt to $hine?”</P> <P><I>S</I>AGR. And here al$o I believe, that you very much deceive your felf. But I come to the fir$t objection moved by <I>Salviatus</I>; and I con$ider, that to make a body appear unto us luminous, it $uf- ficeth not that the rayes of the illuminating body fall upon it, but it is moreover requi$ite that the reflex rayes arrive to our eye; as is manife$tly $een in the example of that Gla$$e, upon <foot>H 2 which</foot> <p n=>60</p> which, without que$tion, the illuminating rayes of the Sun do come; yet neverthele$$e, it appears not to us bright and $hining, unle$$e we $et our eye in that particular place, where the refle- ction arriveth. Now let us con$ider what would $ucceed, were the gla$$e of a $pherical figure; for without doubt, we $hould find, that of the reflection made by the whole $urface illumina- ted, that to be but a very $mall part, which arriveth to the eye of a particular beholder; by rea$on that that is but an incon$ide- rable particle of the whole $pherical $uperficies, the inclination of which ca$ts the ray to the particular place of the eye; whence the part of the $pherical $uperficies, which $hews it $elf $hining to the eye, mu$t needs be very $mall; all the re$t being repre- $ented ob$cure. So that were the Moon $mooth, as a Looking- <marg><I>The Moon if it were $mooth, like a $pherical gla$$e, would be invi$ible.</I></marg> gla$$e, a very $mall part would be $een by any particular eye to be illu$trated by the Sun, although its whole Hemi$phere were ex- po$ed to the Suns rayes; and the re$t would appear to the eye of the beholder as not illuminated, and therefore invi$ible; and finally, the whole Moon would be likewi$e invi$ible, for $o much as that particle, whence the reflection $hould come, by rea$on of its $malne$$e and remotene$$e, would be lo$t. And as it would be invi$ible to the eye, $o would it not afford any light; for it is al- together impo$$ible, that a bright body $hould take away our darkne$$e by its $plendor, and we not to $ee it.</P> <P>SALV. Stay good <I>Sagredus,</I> for I $ee $ome emotions in the face and eyes of <I>Simplicius,</I> which are to me as indices that he is not either very apprehen$ive of, or $atisfied with this which you, with admirable proof, and ab$olute truth have $poken. And yet I now call to mind, that I can by another experiment remove all $cruple. I have $een above in a Chamber, a great $pherical Looking-gla$$e; let us $end for it hither, and while$t it is in bringing, let <I>Simplicius</I> return to con$ider, how great the clarity is which cometh to the Wall here, under the penthou$e, from the reflection of the flat gla$$e.</P> <P>SIMPL. I $ee it is little le$$e $hining, than if the Sun had di- rectly beat upon it.</P> <P>SALV. So indeed it is. Now tell me, if taking away that $mall flat gla$$e, we $hould put that great $pherical one in the $ame place, what effect (think you) would its reflection have upon the $ame Wall?</P> <P><I>S</I>IMPL. I believe that it would eject upon it a far greater and more diffu$ed light.</P> <P><I>S</I>ALV. But if the illumination $hould be nothing, or $o $mall, that you would $car$e di$cern it, what would you $ay then?</P> <P><I>S</I>IMPL. When I have $een the effect, I will bethink my $elf of an an$wer.</P> <foot>SALV.</foot> <p n=>61</p> <P>SALV. See here is the gla$$e, which I would have to be placed clo$e to the other. But fir$t let us go yonder towards the reflection of that flat one, and attentively ob$erve its clarity; $ee how bright it is here where it $hines, and how di$tinctly one may di$cern the$e $mall unevenne$$es in the Wall.</P> <P>SIMPL. I have $een and very well ob$erved the $ame, now place the other gla$$e by the $ide of the fir$t.</P> <P>SALV. See where it is. It was placed there a$$oon as you be- gan to look upon the Walls $mall unevenne$$es, and you percei- ved it not, $o great was the encrea$e of the light all over the re$t of the Wall. Now take away the flat gla$$e. Behold now all refle- ction removed, though the great convex gla$$e $till remaineth. Remove this al$o, and place it there again if you plea$e, and you $hall $ee no alteration of light in all the Wall. See here then de- mon$trated to $en$e, that the reflection of the Sun, made upon a $pherical convex gla$$e, doth not $en$ibly illuminate the places neer unto it. Now what $ay you to this experiment?</P> <P>SIMPL. I am afraid that there may be $ome <I>Leigerdemain,</I> u$ed in this affair; yet in beholding that gla$$e I $ee it dart forth a great $plendor, which dazleth my eyes; and that which im- ports mo$t of all, I $ee it from what place $oever I look upon it; and I $ee it go changing $ituation upon the $uperficies of the gla$$e, which way $oever I place my $elf to look upon it; a nece$$ary ar- gument, that the light is livelily reflected towards every $ide, and con$equently, as $trongly upon all that Wall, as upon my eye.</P> <P>SALV. Now you $ee how cautiou$ly and re$ervedly you ought to proceed in lending your a$$ent to that, which di$cour$e alone re- pre$enteth to you. There is no doubt but that this which you $ay, carrieth with it probability enough, yet you may $ee, how $en$i- ble experience proves the contrary.</P> <P>SIMPL. How then doth this come to pa$s?</P> <P>SALV. I will deliver you my thoughts thereof, but I cannot tell how you may be plea$'d therewith. And fir$t, that lively $plendor which you $ee upon the gla$s, and which you think occu- pieth a good part thereof, is nothing near $o great, nay is very ex- ceeding $mall; but its liveline$s occa$ioneth in your eye, (by means of the reflection made on the humidity of the extream parts of the eye-brows, which di$tendeth upon the pupil) an adventitious irradi- ation, like to that blaze which we think we $ee about the flame of a candle placed at $ome di$tance; or if you will, you may re$emble it to the adventitious $plendor of a $tar; for if you $hould <marg><I>The $mall body of the $tars fringed round about with rays, appeareth ve- ry much biggerthan plain and naked, and in its native clarity.</I></marg> compare the $mall body <I>v. g.</I> of the <I>Canicula,</I> $een in the day time with the <I>Tele$cope,</I> when it is $een without $uch irradiation, with the $ame $een by night by the eye it $elf, you will doubtle$s com- prehend that being irradiated, it appeareth above a thou$and <foot>times</foot> <p n=>62</p> times bigger than the naked and real body: and a like or greater augmentation doth the image of the Sun make, which you $ee in that gla$s. I $ay greater, for that it is more lively than the $tar, as is manife$t from our being able to behold the $tar with much le$s offence, than this reflection of the gla$s. The reverberation therefore which is to di$pere it $elf all over this wall, cometh from a $mall part of that gla$s, and that which even now came from the whole flat gla$s di$per$ed and re$train'd it $elf to a very $mall part of the $aid wall. What wonder is it then, that the fir$t re- flection very lively illuminates, and that this other is almo$t im- perceptible?</P> <P>SIMPL. I find my $elf more perplexed than ever, and there pre$ents it $elf unto me the other difficulty, how it can be that that wall, being of a matter $o ob$cure, and of a $uperficies $o un- poli$h'd, $hould be able to dart from it greater light, than a gla$s very $mooth and polite.</P> <P>SALV. Greater light it is not, but more univer$al; for as to the degree of brightne$s, you $ee that the reflection of that $mall flat gla$s, where it beamed forth yonder under the $hadow of the penthou$e, illuminateth very much; and the re$t of the wall which receiveth the reflection of the wall on which the gla$s is placed, is not in any great mea$ure illuminated, as was the $mall part on which the reflection of the gla$s fell. And if you would under- $tand the whole of this bu$ine$s, you mu$t con$ider that the $uper- <marg><I>The reflex light of uneven bodies, is more univer$al than that of the $mooth, & why.</I></marg> ficies of that wall's being rough, is the $ame as if it were compo- $ed of innumerable $mall $uperficies, di$po$ed according to in- numerable diver$ities of inclinations: among$t which it nece$$a- rily happens, that there are many di$po$ed to $end forth their reflex rays from them into $uch a place, many others into another: and in $um, there is not any place to which there comes not very many rays, reflected from very many $mall $uperficies, di$per$ed throughout the whole $uperficies of the rugged body, upon which the rays of the Sun fall. From which it nece$$arily follow- eth, That upon any, what$oever, part of any $uperficies, oppo$ed to that which receiveth the primary incident rays, there is produced reflex rays, and con$equently illumi- nation. There doth al$o follow thereupon, That the $ame body upon which the illuminating rays fall, beheld from what$oever place, appeareth all illuminated and $hining: and therefore the Moon, as being of a $uperficies rugged and <marg><I>The Moon, if it were $mooth and $leek, would be in- vi$ible.</I></marg> not $mooth, beameth forth the light of the Sun on every $ide, and to all beholders appeareth equally lucid. But if the $urface of it, being $pherical, were al$o $mooth as a gla$s, it would become wholly invi$ible; fora$much as that $mall part, from which the image of the Sun $hould be reflected unto the eye <foot>of</foot> <p n=>63</p> of a particular per$on, by rea$on of its great di$tance would be in- vi$ible, as I have $aid before.</P> <P>SIMPL. I am very apprehen$ive of your di$cour$e; yet me- thinks I am able to re$olve the $ame with very little trouble; and ea$ily to maintain, that the Moon is rotund and polite, and that it reflects the Suns light unto us in manner of a gla$s; nor there- fore ought the image of the Sun to be $een in the middle of it, “for- a$much as the $pecies of the Sun it $elf admits not its $mall figure to be $een at $o great a di$tance, but the light produced by the Sun may help us to conceive that it illuminateth the whole Lu- nar Body: a like effect we may $ee in a plate gilded and well polli$h'd, which touch't by a luminous body, appeareth to him that beholds it at $ome di$tance to be all $hining; and onely near at hand one may di$cover in the middle of it the $mall image of the luminous body.”</P> <P>SALV. Ingenuou$ly confe$$ing my dullne$s of apprehen$ion, I mu$t tell you, that I under$tand not any thing of this your di$- cour$e, $ave onely what concerns the gilt plate: and if you permit me to $peak freely, I have a great conceit that you al$o under$tand not the $ame, but have learnt by heart tho$e words written by $ome one out of a de$ire of contradiction, and to $hew him$elf more intel- ligent than his adver$ary; but it mu$t be to tho$e, which to appear al$o more wi$e, applaud that which they do not under$tand, and entertain a greater conceit of per$ons, the le$s they are by them under$tood: and the writer him$elf may be one of tho$e (of which there are many) who write what they do not under$tand, and <marg><I>Some write what they under$tand not, and therefore under$tand not what they write.</I></marg> con$equently under$tand not what they write. Therefore, o- mitting the re$t, I reply, as to the gilt plate, that if it be flat and not very big, it may appear at a di$tance very bright, whil$t a great light beameth upon it, but yet it mu$t be when the eye is in a de- terminate line, namely in that of the reflex rays: and it will ap- pear the more $hining, if it were <I>v. g.</I> of $ilver, by means of its being burni$hed, and apt through the great den$ity of the metal, to receive a perfect poli$h. And though its $uperficies, being very well brightned, were not exactly plain, but $hould have various in- clinations, yet then al$o would its $plendor be $een many ways; namely, from as many places as the various reflections, made by the $everal $uperficies, do reach: for therefore are Diamonds <marg><I>Diamonds ground to divers $ides, & why.</I></marg> ground to many $ides, that $o their plea$ing lu$tre might be beheld from many places. But if the Plate were very big, though it $hould be all plain, yet would it not at a di$tance appear all over $hining: and the better to expre$s my $elf, Let us $uppo$e a very large gilt plate expo$ed to the Sun, it will $hew to an eye far di$tant, the image of the Sun, to occupy no more but a certain part of the $aid plate; to wit, that from whence the reflection of the incident <foot>$olar</foot> <p n=>64</p> $olar rays come: but it is true that by the vivacity of the light, the $aid image will appear fringed about with many rays, and $o will $eem to occupie a far greater part of the plate, than really it doth. And to $hew that this is true, when you have noted the particular place of the plate from whence the reflection cometh, and concei- ved likewi$e how great the $hining place appeared to you, cover the greater part of that $ame $pace, leaving it only vi$ible about the mid$t; and all this $hall not any whit dimini$h the apparent $plen- dor to one that beholds it from afar; but you $hall $ee it largely di$pers'd upon the cloth or other matter, wherewith you covered it. If therefore any one, by $eeing from a good di$tance a $mall gilt plate to be all over $hining, $hould imagine that the $ame would al$o even in a plate as broad as the Moon, he is no le$s de- ceived, than if he $hould believe the Moon to be no bigger than the bottom of a tub. If again the plate were turn'd into a $phe- rical $uperficies, the reflection would be $een $trong in but one $ole particle of it; but yet by rea$on of its liveline$s, it will appear fringed about with many glittering rays: the re$t of the Ball would appear according as it was burni$hed; and this al$o onely then <marg><I>Silver burni$hed appears more ob- $cuee, than the not burni$hed, & why.</I></marg> when it was not very much poli$hed, for $hould it be perfectly brightned, it would appear ob$cure. An example of this we have dayly before our eyes in $ilver ve$$els, which whil$t they are only boyl'd in the <I>Argol</I> and <I>Salt,</I> they are all as white as $now, and do not reflect any image; but if they be in any part burni$h'd, they become in that place pre$ently ob$cure: and in them one may $ee the repre$entation of any thing as in Looking-gla$$es. And that chan- to ob$curity, proceeds from nothing el$e but the $moothing and plaining of a fine grain, which made the $uperficies of the $ilver rough, and yet $uch, as that it reflected the light into all parts, whereby it $eemed from all parts equally illuminated: which $mall unevenne$$es, when they come to be exqui$itely plained by the burni$h, $o that the reflection of the rays of incidence are all directed unto one determinate place; then, from that $ame place, the burni$h'd part $hall $hew much more bright and $hining than the re$t which is onely whitened by boyling; but from all other places it looks very ob$cure. And note, that the diver$ity of <marg><I>Burni$h'd Steel beheld from one place appears very bright, and from another, very ob- $cure.</I></marg> $ights of looking upon burni$h'd $uperficies, occa$ioneth $uch difference in appearances, that to imitate and repre$ent in picture, <I>v. g.</I> a poli$h'd Cuirace, one mu$t couple black plains with white, one $ideways to the other, in tho$e parts of the arms where the light falleth equally.</P> <P>SAGR. If therefore the$e great Philo$ophers would acquie$e in granting, that the Moon, <I>Venus</I> and the other Planets, were not of $o bright and $mooth a $urface as a Looking-gla$s, but wanted $ome $mall matter of it, namely, were as a $ilver plate, onely boyled <foot>white,</foot> <p n=>65</p> white, but not burni$hed; would this yet $uffice to the making of it vi$ible, and apt for darting forth the light of the Sun?</P> <P>SALV. It would $uffice in part; but would not give a light $o $trong, as it doth being mountainous, and in $um, full of eminencies and great cavities. But the$e Philo$ophers will never yield it to be le$$e polite than a gla$$e; but far more, if more it can be imagined; for they e$teeming that to perfect bodies perfect figures are mo$t $utable; it is nece$$ary, that the $phericity of tho$e Cœle$tial Globes be mo$t exact; be$ides, that if they $hould grant me $ome inequality, though never $o $mall, I would not $cruple to take any other greater; for that $uch perfection con$i$t- ing in indivi$ibles, an hair doth as much detract from its perfection as a mountain.</P> <P>SAGR. Here I meet with two difficulties, one is to know the rea$on why the greater inequality of $uperficies maketh the $tron- ger reflection of light; the other is, why the$e <I>Peripatetick</I> Gen- tlemen are for this exact figure.</P> <P>SALV. I will an$wer to the fir$t; and leave to <I>Simplieius</I> the <marg><I>The more rough $uperficies make greater reflection of light, than the le$s rough.</I></marg> care of making reply to the $econd. You mu$t know therefore, that the $ame $uperficies happen to be by the $ame light more or le$s illuminated, according as the rayes of illumination fall upon them <marg><I>Perpendicular rays illuminate more than the ob- lique, and why.</I></marg> more or le$$e obliquely; $o that the greate$t illumination is where the rayes are perpendicular. And $ee, how I will prove it to your $en$e. I bend this paper, $o, that one part of it makes an angle upon the other: and expo$ing both the$e parts to the reflection of the light of that oppo$ite Wall, you $ee how this $ide which re- ceiveth the rayes obliquely, is le$$e $hining than this other, where the reflection fals at right angles; and ob$erve, that as I by degrees receive the illumination more obliquely, it groweth weaker.</P> <P>SAGR. I $ee the effect, but comprehend not the cau$e.</P> <P>SALV. If you thought upon it but a minute of an hour, you would find it; but that I may not wa$te the time, $ee a kind of demon$tration thereof in <I>Fig.</I> 7.</P> <P>SAGR. The bare $ight of this Figure hath fully $atisfied me, therefore proceed.</P> <P>SIMPL. Pray you let me hear you out, for I am not of $o quick an apprehen$ion.</P> <P>SALV. Fancie to your $elf, that all the paralel lines, which you $ee to depart from the terms A. B. are the rays which fall upon the <marg><I>The more oblique Rayes illuminate leß, and why.</I></marg> line C. D. at right angles: then incline the $aid C. D. till it hang as D. O. now do not you $ee that a great part of tho$e rays which peirce C. D. pa$s by without touching D. O? If therefore D. O. be illuminated by fewer rays, it is very rea$onable, that the light received by it be more weak. Let us return now to the Moon, <foot>I which</foot> <p n=>66</p> which being of a $pherical figure, if its $uperficies were $mooth, as this paper, the parts of its hemi$phere illuminated by the Sun, which are towards its extremity, would receive much le$s light, than the middle parts; the rays falling upon them mo$t obliquely, and upon the$e at right angles; whereupon at the time of full Moon, when we $ee almo$t its whole Hemi$phere illuminated, the parts towards the mid$t, would $hew them$elves to us with more $plendor, than tho$e others towards the circumference: which is not $o in effect. Now the face of the Moon being repre$ented to me full of indifferent high mountains, do not you $ee how their tops and continuate ridges, being elevated above the convexity of the perfect $pherical $uperficies, come to be expo$ed to the view of the Sun, and accommodated to receive its rays much le$s ob- liquely, and con$equently to appear as luminous as the re$t?</P> <P>SAGR. All this I well perceive: and if there are $uch moun- tains, its true, the Sun will dart upon them much more directly than it would do upon the inclination of a polite $uperficies: but it is al$o true, that betwixt tho$e mountains all the valleys would become ob$cure, by rea$on of the va$t $hadows, which in that time would be ca$t from the mountains, whereas the parts towards the middle, though full of valleys and hills, by rea$on they have the Sun elevated, would appear without $hadow, and therefore more lucid by far than the extreme parts, which are no le$s diffu- $ed with $hadow than light, and yet we can perceive no $uch diffe- rence.</P> <P>SIMPL. I was ruminating upon the like difficulty.</P> <P>SALV. How much readier is <I>Simplicius</I> to apprehend the ob- jections which favour the opinions of <I>Ari$totle,</I> than their $oluti- ons? I have a kind of $u$pition, that he $trives al$o $ometimes to di$$emble them; and in the pre$ent ca$e, he being of him$elf able to hit upon the doubt, which yet is very ingenious, I cannot be- lieve but that he al$o was advi$'d of the an$wer; wherefore I will attempt to wre$t the $ame (as they $ay) out of his mouth. There- fore tell me, <I>Simplicius,</I> do you think there can be any $hadow, where the rays of the Sun do $hine?</P> <P>SIMPL. I believe, nay I am certain that there cannot; for that it being the grand luminary, which with its rays driveth away dark- ne$s, it is impo$$ible any tenebro$ity $hould remain where it com- eth; moreover, we have the definition, that <I>Tenebræ $unt priva- tio luminis.</I></P> <P>SALV. Therefore the Sun, beholding the Earth, Moon or o- ther opacous body, never $eeth any of its $hady parts, it not ha- ving any other eyes to $ee with, $ave its rays, the conveyers of light: and con$equently, one $tanding in the Sun would never $ee any thing of umbrage, fora$much as his vi$ive rays would ever <foot>go</foot> <p n=>67</p> go accompanied with tho$e illuminating beams of the Sun.</P> <P>SIMPL. This is true, without any contradiction.</P> <P>SALV. But when the Moon is oppo$ite to the Sun, what dif- ference is there between the tract of the rayes of your $ight, and that motion which the Suns rayes make?</P> <P>SIMPL. Now I under$tand you; for you would $ay, that the rayes of the $ight and tho$e of the Sun, moving by the $ame lines, we cannot perceive any of the ob$cure valleys of the Moon. Be plea$ed to change this your opinion, that I have either $imulation or di$$imulation in me; for I prote$t unto you, as I am a Gentle- man, that I did not gue$$e at this $olution, nor $hould I have thought upon it, without your help, or without long $tudy.</P> <P>SAGR. The re$olutions, which between you two have been alledged touching this la$t doubt, hath, to $peak the truth, $atisfi- ed me al$o. But at the $ame time this con$ideration of the vi- fible rayes accompanying the rayes of the Sun, hath begotten in me another $cruple, about the other part, but I know not whether I can expre$$e it right, or no: for it but ju$t now comming into my mind, I have not yet methodized it to my mind: but let us $ee if we can, all together, make it intelligible. There is no que$tion, but that the parts towards the circumference of that poli$h't, but not burni$h't Hemi$phere, which is illuminated by the Sun, receiving the rayes obliquely, receive much fewer thereof, than the middle- mo$t parts, which receive them directly. And its po$$ible, that a tract or $pace of <I>v. g.</I> twenty degrees in breadth, and which is to- wards the extremity of the Hemi$phere, may not receive more rays than another towards the middle parts, of but four degree broad: $o that that doubtle$s will be much more ob$cure than this; and $uch it will appear to whoever $hall behold them both in the face, or (as I may $ay) in their full magnitude. But if the eye of the beholder were con$tituted in $uch a place, that the breadth of the twenty degrees of the ob$cure $pace, appeared not to it longer than one of four degrees, placed in the mid$t of the Hemi$phere, I hold it not impo$$ible for it to appear to the $aid beholder e- qually clear and lucid with the other; becau$e, finally, between two equal angles, to wit, of four degrees apiece, there come to the eye the reflections of two equal numbers of rayes: namely, tho$e which are reflected from the middlemo$t $pace, four degrees in breadth, and tho$e reflected from the other of twenty degrees, but $een by compre$$ion, under the quantity of four degrees: and $uch a $ituation $hall the eye obtain, when it is placed between the $aid Hemi$phere, and the body which illuminates it; for then the $ight and rayes move in the $ame lines. It $eemeth not impo$$ible therefore, but that the Moon may be of a very equal $uperficies; and that neverthele$$e, it may appear when it is at the full, no le$s <foot>I 2 light</foot> <p n=>68</p> light in the extremities, than in the middle parts.</P> <P>SALV. The doubt is ingenious and worthy of con$ideration; and as it but ju$t now came into your mind unawares, $o I will like wi$e an$wer with what fir$t comes into my thoughts, and it may happily fall out, that by thinking more upon it, I may $tumble upon a better reply. But before, that I labyrinth my $elf any far- ther, it would be nece$$ary, that we a$$ure our $elves by $ome ex- periment, whether your objection prove in effect, what it $eemeth to conclude in appearance; and therefore taking once more the $ame paper, and making it to incline, by bending a little part thereof upon the remainder, let us try whether expo$ing it to the Sun, $o that the rayes of light fall upon the le$$er part directly, and upon the other obliquely; this which receiveth the rayes direct- ly appeareth more lucid; and $ee here by manife$t experience, that it is notably more clear. Now if your objection be conclu$ive, it will follow, that $tooping with our eye $o, that in beholding the other greater part, le$s illuminated, in compre$$ion or fore- $hortning, it appear unto us no bigger than the other, more $hining; and that con$equently, it be not beheld at a greater angle than that; it will nece$$arily en$ue, I $ay, that its light be encrea$ed, $o that it do $eem to us as bright as the other. See how I behold, and look upon it $o obliquely, that it appeareth to me narrower than the other; but yet, notwith$tanding its ob$curity, doth not to my perceiving, at all grow clearer. Try now if the $ame $ucceed to you.</P> <P>SAGR. I have look't upon it, and though I have $tooped with my eye, yet cannot I $ee the $aid $uperficies encrea$e in light or clarity; nay me thinks it rather grows more dusky.</P> <P>SALV. We are hitherto confident of the invalidity of the ob- jection; In the next place, as to the $olution, I believe, that, by rea$on the Superficies of this paper is little le$$e than $mooth, the rayes are very few, which be reflected towards the point of inci- dence, in compari$on of the multitude, which are reflected to- wards the oppo$ite parts; and that of tho$e few more and more are lo$t, the nearer the vi$ive rayes approach to tho$e lucid rayes of incidence; and becau$e it is not the incident rayes, but tho$e which are reflected to the eye, that make the object appear lu- minous; therefore, in $tooping the eye, there is more lo$t than got, as you your $elf confe$$e to have $een in looking upon the ob$cu- rer part of the paper.</P> <P>SAGR. I re$t $atisfied with this experiment and rea$on: It re- mains now, that <I>Simplicius</I> an$wer to my other que$tion, and tell me what moves the <I>Peripateticks</I> to require this $o exact rotundity in the Cœle$tial bodies.</P> <P>SIMPL. The Cœle$tial bodies being ingenerable, inalterable, im- <foot>pa$$ible,</foot> <p n=>69</p> pa$$ible, immortal, <I>&c.</I> they mu$t needs be ab$olutely perfect; and <marg><I>Perfect $phericity why a$cribed to Cœlestial bodies, by the</I> Peripate- ticks.</marg> their being ab$olute perfect, nece$$arily implies that there is in them all kinds of perfection; and con$equently, that their figure be al$o perfect, that is to $ay, $pherical; and ab$olutely and perfectly $pherical, and not rough and irregular.</P> <P>SALV. And this incorruptibility, from whence do you prove it?</P> <P>SIMPL. Immediately by its freedom from contraries, and me- diately, by its $imple circular motion.</P> <P>SALV. So that; by what I gather from your di$cour$e, in ma- <marg><I>The Figure is not the cau$e of incor- ruptibility, but of longer duration.</I></marg> king the e$$ence of the Cœle$tial bodies to be incorruptible, inal- terable, <I>&c,</I> there is no need of rotundity as a cau$e, or requi- $ite; for if this $hould cau$e inalterability, we might at our plea- $ure make wood, wax, and other Elementary matters, incorrup- tible, by reducing them to a $pherical figure.</P> <P>SIMPL. And is it not manife$t that a ball of Wood will better and longer be preferved, than an oblong, or other angular fi- gure, made of a like quantity of the $ame wood.</P> <P>SALV. This is mo$t certain, but yet it doth not of corruptible become incorruptible, but $till remains corruptible, though of a much longer duration. Therefore you mu$t note, that a thing cor- <marg><I>Corruptibility ad- mits of more or le$$e; $o doth noe incorruptibiliiy.</I></marg> ruptible, is capable of being more or le$$e $uch, and we may properly $ay this is le$$e corruptible than that; as for example, the <I>Ja$per,</I> than the <I>Pietra Sirena</I>; but incorruptibility admits not of more, or le$$e, $o as that it may be $aid this is more incorrupti- ble than that, if both be incorruptible and eternal. The diver- <marg><I>The perfection of figure, operateth in corruptible bo- dies, but not in the eternal.</I></marg> $ity of figure therefore cannot operate: $ave onely in matters ca- pable of more or le$$e duration; but in the eternal, which can- not be other than equally eternal, the operation of figure cea$eth. And therefore, $ince the Cœle$tial matter is not incorruptible by figure, but otherwayes no man needs to be $o $olicitous for this perfect $phericity; for if the matter be incorruptible, let it have what figure it will, it $hall be alwayes $uch.</P> <P>SAGR. But I am con$idering another thing, and $ay, that if <marg><I>If the $pherical fi- gure conferreth e- ternity, all bodies would be eternal.</I></marg> we $hould grant the $pherical figure a faculty of conferring incor- ruptibility, all bodies of what$oever figure, would be incorrupti- ble; fora$much as if the rotund body be incorruptible, corrupti- bility would then $ub$i$t in tho$e parts which alter the perfect ro- tundity; as for in$tance, there is in a <I>Die</I> a body perfectly round, and, as $uch, incorruptible; therefore it remaineth that tho$e an- gles be corruptible which cover and hide the rotundity; $o that the mo$t that could happen, would be, that tho$e angles, and (to $o $peak) excre$cencies, would corrupt. But if we proceed to a more inward con$ideration, that in tho$e parts al$o towards the angles, there are compri$ed other le$$er bals of the $ame matter; <foot>and</foot> <p n=>70</p> and therefore they al$o, as being round, mu$t be al$o incorrup- tible; and likewife in the remainders, which environ the$e eight le$$er Spheres, a man may under$tand that there are others: $o that in the end, re$olving the whole <I>Die</I> into innumerable balls, it mu$t nece$$arily be granted incorruptible. And the $ame di$- cour$e and re$olution may be made in all other figures.</P> <P>SALV. Your method in making the conclu$ion, for if <I>v. g.</I> a round Chry$tal were, by rea$on of its figure, incorruptible; namely, received from thence a faculy of re$i$ting all internal and external alterations, we $hould not find, that the joyning to it other Chry- $tal, and reducing it <I>v. g.</I> into a Cube, would any whit alter it within, or without; $o as that it would thereupon become le$$e apt to re$i$t the new ambient, made of the $ame matter, than it was to re$i$t the other, of a matter different; and e$pecially, if it be true, that corruption is generated by contraries, as <I>Ari- $totle</I> $aith; and with what can you enclo$e that ball of Cry$tal, that is le$$e contrary to it, than Cry$tal it $elf? But we are not a- ware how time flies away; and it will be too late before we come to an end of our di$pute, if we $hould make $o long di$cour$es, upon every particular; be$ides our memories are $o confounded in the multiplicity of notions, that I can very hardly recal to mind the Propot$iions, which I propo$ed in order to <I>Simplicius,</I> for our con$ideration.</P> <P>SIMPL. I very well remember them: And as to this particular que$tion of the montuo$ity of the Moon, there yet remains un- an$wered that which I have alledged, as the cau$e, (and which may very well $erve for a $olution) of that <I>Phænomenon,</I> $aying, that it is an illu$ion proceeding from the parts of the Moon, be- ing unequally opacous, and per$picuous.</P> <P>SAGR. Even now, when <I>Simplicius</I> a$cribed the apparent Pro- tnberancies or unevenne$$es of the Moon (according to the opinion of a certain <I>Peripatetick</I> his friend) to the diver$ly opacous, and <marg><I>Mother of Pearl accommodated to imitate the appa- rent unevenne$$es of the Moons $ur- face.</I></marg> per$picuous parts of the $aid Moon, conformable to which the like illu$ions are $een in Cry$tal, and Jems of divers kinds, I bethought my $elf of a matter much more commodious for the repre$enting $uch effects; which is $uch, that I verily believe, that that Philo$o- pher would give any price for it; and it is the mother of Pearl, which is wrought into divers figures, and though it be brought to an ex- treme evenne$$e, yet it $eemeth to the eye in $everal parts, $o vari- ou$ly hollow and knotty, that we can $carce credit our feeling of their evenne$$e.</P> <P>SALV. This invention is truly ingenious; and that which hath not been done already, may be done in time to come; and if there have been produced other Jems, and Cry$tals, which have nothing to do with the illu$ions of the mother of Pearl, the$e may <foot>be</foot> <p n=>71</p> be produced al$o; in the mean time, that I may not prevent any one, I will $uppre$$e the an$wer which might be given, and onely for this time betake my $elf to $atisfie the objections brought by <I>Simplicius.</I> I $ay therefore, that this rea$on of yours is too ge- neral, and as you apply it not to all the appearances one by one; which are $een in the Moon, and for which my $elf and others are induced to hold it mountainous, I believe you will not find any one that will be $atisfied with $uch a doctrine; nor can I think, that either you, or the Author him$elf, find in it any greater quietude, than in any other thing wide from the purpo$e. Of the <marg><I>The apparent un- evenne$$es of the Moon cannot be i- mitated by way of more and le$s opa- city & per$picuity.</I></marg> very many $everal appearances which are $een night by night in the cour$e of Moon, you cannot imitate $o much as one, by making a Ball at your choice, more or le$s opacous and per$picuous, and that is of a polite $uperficies; whereas on the contrary, one may <marg><I>The various a- $pects of the Moon, imitable with any opacous matter.</I></marg> make Balls of any $olid matter what$oever, that is not tran$parent, which onely with eminencies and cavities, and by receiving the il- lumination $everal ways, $hall repre$ent the $ame appearances and mutations to an hair, which from hour to hour are di$covered in <marg><I>Various appear an- ces from which the Moons montuo$ity is argued.</I></marg> the Moon. In them you $hall $ee the ledges of Hills expo$ed to the Suns light, to be very $hining, and after them the projections of their $hadows very ob$cure; you $hall $ee them greater and le$s, according as the $aid eminencies $hall be more or le$s di$tant from the confines which di$tingui$h the parts of the Moon illuminated, from the ob$cure: you $hall $ee the $ame term and confine, not equally diftended, as it would be if the Ball were poli$h'd, but craggie and rugged. You $hall $ee beyond the $ame term, in the dark parts of the Moon many bright prominencies, and di$tinct from the re$t of the illuminations: you $hall $ee the $hadows a- fore$aid, according as the illumination gradually ri$eth, to demi- ni$h by degrees, till they wholly di$appear; nor are there any of them to be $een when the whole Hemi$phere is enlightned. A- gain on the contrary, in the lights pa$$age towards the other He- mi$phere of the Moon, you $hall again ob$erve the $ame eminen- cies that were marked, and you $hall $ee the projections of their $hadows to be made a contrary way, and to decrea$e by degrees: of which things, once more I $ay, you cannot $hew me $o much as one in yours that are opacous and per$picuous.</P> <P>SAGR. One of them certainly he may imitate, namely, that of the Full-Moon, when by rea$on of its being all illuminated, there is not to be $een either $hadow, or other thing, which receiveth any alteration from its eminencies and cavities. But I be$eech you, <I>Salviatus,</I> let us $pend no more time on this Argument, for a per$on that hath had but the patience to make ob$ervation of but one or two Lunations, and is not $atisfied with this mo$t $en$ible truth, may well be adjudged void of all judgment; and upon <foot>$uch</foot> <p n=>72</p> $uch why $hould we throw away our time and breath in vain?</P> <P>SIMPI. I mu$t confe$s I have not made the ob$ervations, for that I never had $o much curio$ity, or the In$truments proper for the bu$ine$s; but I will not fail to do it. In the mean time, we may leave this que$tion in $u$pen$e, and pa$s to that point which follows, producing the motives inducing you to think that the Earth may reflect the light of the Sun no le$s forceably than the Moon, for it $eems to me $o ob$cure and opacous, that I judg $uch an effect altogether impo$$ible.</P> <P>SALV. The cau$e for which you repute the Earth unapt for illumination, may rather evince the contrary: And would it not be $trange, <I>Simplicius,</I> if I $hould apprehend your di$cour$es bet- ter than you your $elf?</P> <P>SIMPL. Whether I argue well or ill, it may be, that you may better under$tand the $ame than I; but be it ill or well that I di$cour$e, I $hall never believe that you can penetrate what I mean better than I my $elf.</P> <P>SALV. Well, I will make you believe the $ame pre$ently. Tell me a little, when the Moon is near the Full, $o that it may be $een by day, and al$o at midnight, at what do you think it more $plen- did, by day or by night?</P> <P>SIMPL. By night, without all compari$on. And methinks <marg><I>The Moon ap- pears brighter by night than by day.</I></marg> the Moon re$embleth that pillar of Clouds and pillar of Fire, which guided the <I>I$raelites</I>; which at the pre$ence of the Sun, appeared like a Cloud, but in the night was very glorious. Thus <marg><I>The Moon be- held in the day time, is like to a little cloud.</I></marg> I have by day ob$erved the Moon amid$t certain $mall Clouds, ju$t as if one of them had been coloured white, but by night it $hines with much $plendor.</P> <P>SALV. So that if you had never happened to $ee the Moon, $ave onely in the day time, you would not have thought it more $hining than one of tho$e Clouds.</P> <P>SIMPL. I verily believe I $hould not.</P> <P>SALV. Tell me now; do you believe that the Moon is really more $hining in the night than day, or that by $ome accident it $eemeth $o?</P> <P>SIMPL. I am of opinion, that it re$plends in it $elf as much in the day as night, but that its light appears greater by night, be- cau$e we behold it in the dark mantle of Heaven; and in the day time, the whole Atmo$phere being very clear, $o that $he little exceedeth it in lu$tre, $he $eems to us much le$s bright.</P> <P>SALV. Now tell me; have you ever at midnight $een the Ter- re$trial Globe illuminated by the Sun?</P> <P>SIMPL. This $eemeth to me a que$tion not to be ask'd, unle$s in je$t, or of $ome per$on known to be altogether void of $en$e.</P> <P>SALV. No, no; I e$teem you to be a very rational man, and <foot>do</foot> <p n=>73</p> do ask the que$tion $eriou$ly; and therefore an$wer me: and if afterwards you $hall think that I $peak impertinently, I will be content to be the $en$ele$s man: for he is much more a fool who interrogates $imply, than he to whom the que$tion is put.</P> <P>SIMPL. If then you do not think me altogether $imple, take it for granted that I have an$wered you already, and $aid, that it is impo$$ible, that one that is upon the Earth, as we are, $hould $ee by night that part of the Earth where it is day, namely, that is il- luminated by the Sun.</P> <P>SALV. Therefore you have never $een the Earth enlightned, $ave onely by day; but you $ee the Moon to $hine al$o in the dead of night. And this is the cau$e, <I>Simplicius,</I> which makes you believe that the Earth doth not $hine like the Moon; but if you could $ee the Earth illuminated, whil$t you were in $ome dark place, like our night, you would $ee it $hine brighter than the Moon. Now if you de$ire that the compari$on may proceed well, you mu$t compare the light of the Earth, with that of the Moon $een in the day time, and not with the $ame by night: for it is not in our power to $ee the Earth illuminated, $ave onely in the day. Is it not $o?</P> <P>SIMPL. So it ought to be.</P> <P>SALV. And fora$much as you your $elf have already confe$$ed to have $een the Moon by day among $ome little white Clouds, and very nearly, as to its a$pect, re$embling one of them; you did <marg><I>Clouds are no le$s apt than the Moon to be illuminated by the Sun.</I></marg> thereby grant, that tho$e Clouds, which yet are Elementary matters, are as apt to receive illumination, as the Moon, yea more, if you will but call to mind that you have $ometimes $een $ome Clouds of va$t greatne$s, and as perfect white as the Snow; and there is no que$tion, but that if $uch a Cloud could be con- tinued $o luminous in the deep of night, it would illuminate the places near about it, more than an hundred Moons. If therefore we were a$$ured that the Earth is illuminated by the Sun, like one of tho$e Clouds, it would be undubitable, but that it would be no le$s $hining than the Moon. But of this there is no que$tion to be made, in regard we $ee tho$e very Clouds in the ab$ence of the Sun, to remain by night, as ob$cure as the Earth: and that which is more, there is not any one of us, but hath $een many times $ome $uch Clouds low, and far off, and que$tioned whether they were Clouds or Mountains: an evident $ign that the Moun- tains are no le$s luminous than tho$e Clouds.</P> <marg><I>A wall illumina- ted by the Sun, compared to the Moon $hineth no le$s than it.</I></marg> <P>SAGR. But what needs more di$cour$e? See yonder the Moon is ri$en, and more than half of it illuminated; $ee there that wall, on which the Sun $hineth; retire a little this way, $o that you $ee the Moon $ideways with the wall: look now; which of them $hews more lucid? Do not you $ee, that if there is any advantage, <foot>K the</foot> <p n=>74</p> the wall hath it? The Sun $hineth on that wall; from thence it <marg><I>The third re$le- ction of a Wall illu- minates more than the fir$t of the Moon.</I></marg> is reverberated upon the wall of the Hall, from thence it's refle- cted upon that chamber, $o that it falls on it at the third reflection: and I am very certain, that there is in that place more light, than if the Moons light had directly faln upon it.</P> <P>SIMPL. But this I cannot believe; for the illumination of the Moon, e$pecially when it is at the full, is very great.</P> <P>SAGR. It $eemeth great by rea$on of the circumjacent dark <marg><I>The light of the Moon weaker than that of the twi- light.</I></marg> places; but ab$olutely it is not much, and is le$s than that of the twilight half an hour after the Sun is $et; which is manife$t, be- cau$e you $ee not the $hadows of the bodies illuminated by the Moon till then, to begin to be di$tingui$hed on the Earth. Whe- ther, again, that third reflection upon that chamber, illuminates more than the fir$t of the Moon, may be known by going thether, and reading a Book, and afterwards $tanding there in the night by the Moons light, which will $hew by which of them lights one may read more or le$s plainly, but I believe without further tryal, that one $hould $ee le$s di$tinctly by this later.</P> <P>SALV. Now, <I>Simplicius,</I> (if haply you be $atisfied) you may conceive, as you your $elf know very well, that the Earth doth $hine no le$s than the Moon; and the only remembring you of $ome things, which you knew of your $elf, and learn'd not of me, hath a$$ured you thereof: for I taught you not that the Moon $hews lighter by night than by day, but you under$tood it of your $elf; as al$o you could tell me that a little Cloud appeareth as lucid as the Moon: you knew al$o, that the illumination of the Earth can- not be $een by night; and in a word, you knew all this, without knowing that you knew it. So that you have no rea$on to be $cru- pulous of granting, that the dark part of the Earth may illuminate the dark part of the Moon, with no le$s a light than that where- with the Moon illuminates the ob$curities of the night, yea rather $o much the greater, ina$much as the Earth is forty times bigger than the Moon.</P> <P>SIMPL. I mu$t confe$s that I did believe, that that $econdary light had been the natural light of the Moon.</P> <P>SALV. And this al$o you know of your $elf, and perceive not that you know it. Tell me, do not you know without teaching, that the Moon $hews it $elf more bright by night than by day, in <marg><I>Luminous bodies appear the brighter in an ob$curer</I> am- bient.</marg> re$pect of the ob$curity of the $pace of the ambient? and con$e- quently, do you not know <I>in genere,</I> that every bright body $hews the clearer, by how much the ambient is ob$curer?</P> <P>SIMPL. This I know very well.</P> <P>SALV. When the Moon is horned, and that $econdary light $eemeth to you very bright, is it not ever nigh the Sun, and con- $equently, in the light of the <I>crepu$culum,</I> (twilight?)</P> <foot>SIMPL.</foot> <p n=>75</p> <P>SIMPL. It is $o; and I have oftentimes wi$h'd that the Air would grow thicker, that I might be able to $ee that $ame light more plainly; but it ever di$appeared before dark night.</P> <P>SALV. You know then very certainly, that in the depth of night, that light would be more con$picuous.</P> <P>SIMPL. I do $o; and al$o more than that, if one could but take away the great light of the cre$cent illuminated by the Sun, the pre$ence of which much ob$cureth the other le$$er.</P> <P>SALV. Why, doth it not $ometimes come to pa$s, that one may in a very dark night $ee the whole face of the Moon, without be- ing at all illuminated by the Sun?</P> <P>SIMPL. I know not whether this ever happeneth, $ave onely in the total Ecclip$es of the Moon.</P> <P>SALV. Why, at that time this its light would appear very clear, being in a mo$t ob$cure <I>medium,</I> and not darkned by the clarity of the luminous cre$cents: but in that po$ition, how light did it appear to you?</P> <P>SIMPL. I have $ometimes $een it of the colour of bra$s, and a little whiti$h; but at other times it hath been $o ob$cure, that I have wholly lo$t the $ight of it.</P> <P>SALV. How then can that light be $o natural, which you $ee $o cleer in the clo$e of the twilight, notwith$tanding the impediment of the great and contiguous $plendor of the cre$cents; and which again, in the more ob$cure time of night, all other light removed, appears not at all?</P> <P>SIMPL. I have heard of $ome that believed that $ame light to be participated to the$e cre$cents from the other Stars, and in par- ticular from <I>Venus,</I> the Moons neighbour.</P> <P>SALV. And this likewi$e is a vanity; becau$e in the time of its total ob$curation, it ought to appear more $hining than ever; for you cannot $ay, that the $hadow of the Earth intercepts the $ight of <I>Venus,</I> or the other Stars. But to $ay true, it is not at that in$tant wholly deprived thereof, for that the Terre$trial He- mi$phere, which in that time looketh towards the Moon, is that where it is night, that is, an intire privation of the light of the Sun. And if you but diligently ob$erve, you will very $en$ibly perceive, that like as the Moon, when it is $harp-horned, doth give very little light to the Earth; and according as in her the parts illumi- nated by the Suns light do encrea$e: $o likewi$e the $plendor to our $eeming encrea$eth, which from her is reflected towards us; thus the Moon, whil$t it is $harp-forked, and that by being between the Sun and the Earth, it di$covereth a very great part of the Ter- <marg>*<I>By the Moons two</I> Quadratures <I>you are to under$tand its fir$t and last quarters, as A- $trologers call them</I></marg> re$trial Hemi$phere illuminated, appeareth very clear: and depart- ing from the Sun, and pa$$ing towards the ^{*}Quadrature, you may $ee the $aid light by degrees to grow dim; and after the <foot>K2 Quadra-</foot> <p n=>76</p> Quadrature, the $ame appears very weak, becau$e it continually lo$eth more and more of the view of the luminous part of the Earth: and yet it $hould $ucceed quite contrary, if that light were its own, or communicated to it from the Stars; for then we $hould $ee it in the depth of night, and in $o very dark an ambient.</P> <P>SIMPL. Stay a little; for I ju$t now remember, that I have read in a little modern tract, full of many novelties; “That this $econdary light is not derived from the Stars, nor innate in the Moon, and lea$t of all communicated by the Earth, but that it is <marg><I>The $econdary light of the Moon cau$ed by the Sun, according to $ome.</I></marg> received from the $ame illumination of the Sun, which, the $ub- $tance of the Lunar Globe being $omewhat tran$parent, pene- trateth thorow all its body; but more livelily illuminateth the $uperficies of the Hemi$phere expo$ed to the rays of the Sun: and its pro$undity imbuing, and (as I may $ay) $wallowing that light, after the manner of a cloud or chry$tal, tran$mits it, and renders it vi$ibly lucid. And this (if I remember aright) he proveth by Authority, Experience and Rea$on; citing <I>Cleomedes, Vitellion, Macrobius,</I> and a certain other modern Author: and adding, That it is $een by experience to $hine mo$t in the days neare$t the Conjunction, that is, when it is horned, and is chiefly bright about its limb. And he farther writes, That in the Solar Ecclip$es, when it is under the <I>Di$cus</I> of the Sun, it may be $een tran$lucid, and more e$pecially towards its utmo$t Circle. And in the next place, for Arguments, as I think, he $aith, That it not being able to derive that light either from the Earth, or from the Stars, or from it $elf, it nece$$arily follows, that it cometh from the Sun. Be$ides that, if you do but grant this $uppo$ition, one may ea$ily give convenient rea$ons for all the particulars that occur. For the rea$on why that $ecundary light $hews more lively towards the outmo$t limb, is, the $hortne$s of the $pace that the Suns rays hath to penetrate, in regard that of the lines which pa$s through a circle, the greate$t is that which pa$$eth through the centre, and of the re$t, tho$e which are farthe$t from it, are always le$s than tho$e that are nearer. From the $ame principle, he $aith, may be $hewn why the $aid light doth not much dimini$h. And la$tly, by this way the cau$e is a$$igned whence it comes, that that $ame more $hining circle about the utmo$t edge of the Moon, is $een at the time of the Solar Ec- clip$e, in that part which lyeth ju$t under the <I>Di$cus</I> of the Sun, but not in that which is be$ide the <I>Di$cus</I>: which happeneth becau$e the rays of the Sun pa$s directly to our eye, through the parts of the Moon underneath: but as for the parts which are be$ides it, they fall be$ides the eye.”</P> <P>SALV. If this Philo$opher had been the fir$t Author of this o- pinion, I would not wonder that he $hould be $o affectionate to it, <foot>as</foot> <p n=>77</p> as to have received it for truth; but borrowing it from others, I cannot find any rea$on $ufficient to excu$e him for not perceiving its fallacies; and e$pecially after he had heard the true cau$e of that effect, and had it in his power to $atisfie him$elf by a thou$and experiments, and manife$t circum$tances, that the $ame proceeded from the reflection of the Earth, and from nothing el$e: and the more this $peculation makes $omething to be de$ired, in the judgment of this Author, and of all tho$e who give no credit to it: $o much the more doth their not having under$tood and remembred it, excu$e tho$e more rece$s Antients, who, I am very certain, did they now under$tand it, would without the lea$t repugnance admit thereof. And if I may freely tell you what I think, I cannot believe but that this <I>Modern</I> doth in his heart believe it; but I rather think, that the conceit he $hould not be the fir$t Author thereof, did a little move him to endeavour to $uppre$$e it, or to di$parage it at lea$t among$t the $imple, who$e number we know to be very great; and many there are, who much more affect the nume- rous applauds of the people, than the approbation of a few not vulgar judgments.</P> <P>SAGR. Hold good <I>Salviatus,</I> for me thinks, I $ee that you go not the way to hit the true mark in this your di$cour$e, for the$e that ^{*} confound all propriety, know al$o how to make them$elves <marg>* Tendono le pare- te al commune.</marg> Authors of others inventions, provided they be not $o $tale, and publick in the Schools and Market-places, as that they are more then notorious to every one.</P> <P>SALV. Ha! well aimed, you blame me for roving from the point in hand; but what have you to do with Schools and Mar- <marg><I>Its all one whe- ther opinions be new to men, or men new to opinions.</I></marg> kets? Is it not all one whether opinions and inventions be new to men, or the men new to them? If you ^{*} contend about the e- $teem of the Founders of Sciences, which in all times do $tart up, <marg>* <I>Conte$tare</I> fal$ly rendered in the Latine Tran$lation <I>content are.</I></marg> you may make your $elf their inventor, even to the Alphabet it $elf, and $o gain admiration among$t that illiterate rabble; and though in proce$$e of time your craft $hould be perceived, that would but little prejudice your de$igne; for that others would $ucceed them in maintaining the number of your fautors; but let us return to prove to <I>Simplicius</I> the invalidity of the rea$ons of his modern Author, in which there are $everal fal$ities, incon$equen- <marg><I>The $econdary light of the Moon appears in form of a Ring, that is to $ay, bright in the extreme circumfe- rence, and not in the mid$t, and why.</I></marg> cies, and incredible Paradoxes. And fir$t, it is fal$e that this $e- condary light is clearer about the utmo$t limb than in the middle parts, $o as to form, as it were, a ring or circle more bright than the re$t of its $pace or contence. True it is, indeed, that looking on the Moon at the time of twilight, at fir$t $ight there is the re- $emblance of $uch a circle, but by an illu$ion ari$ing from the di- ver$ity of confines that bound the Moons <I>Di$cus,</I> which are con- fu$ed by means of this $econdary light; fora$much as on the part <foot>towards</foot> <p n=>78</p> towards the Sun it is bounded by the lucid horns of the Moon, and on the other part, its confining term is the ob$cure tract of the twilight; who$e relation makes us think the candor of the Moons <I>Di$cus</I> to be $o much the clearer; the which happens to be ob- fu$cated in the oppo$ite part, by the greater clarity of the cre$- cents; but if this modern Author had e$$aied to make an inter- <marg><I>The may to ob- $erve the $econda- ry light of the Moon.</I></marg> po$ition between the eye and the primary $plendor, by the ridg of $ome hou$e, or $ome other $creen, $o as to have left vi$ible only the gro$e of the Moon, the horns excluded, he might have $een it all alike luminous.</P> <P>SIMPL, I think, now I remember, that he writes of his making u$e of $uch another Artifice, to hide from us the fal$e <I>Incidum.</I></P> <P>SALV. Oh! how is this (as I believed) inadvertency of his, changed into a lie, bordering on ra$hne$$e; for that every one may frequently make proof of the contrary. That in the next <marg><I>The Moons</I> Dif- cus <I>in a $olar E- clip$e can be $een onely by privation.</I></marg> place, at the Suns Eclip$e, the Moons <I>Di$cus</I> is $een otherwayes than by privation, I much doubt, and $pecially when the E- clip$e is not total, as tho$e mu$t nece$$arily have been, which were ob$erved by the Author; but if al$o he $hould have di$cove- red $omewhat of light, this contradicts not, rather favoureth our opinion; for that at $uch a time, the whole Terre$trial Hemi- $phere illuminated by the Sun, is oppo$ite to the Moon, $o that although the Moons $hadow doth ob$cure a part thereof, yet this is very $mall in compari$on of that which remains illuminated. That which he farther adds, that in this ca$e, the part of the limb, lying under the Sun, doth appear very lucid, but that which lyeth be$ides it, not $o; and that to proceed from the co- ming of the $olar rayes directly through that part to the eye, but not through this, is really one of tho$e fopperies, which di$co ver the other fictions, of him which relates them: For if it be requi$ite to the making a $econdary light vi$ible in the lunar <I>Di$- cus,</I> that the rayes of the Sun came directly through it to our eyes, doth not this pitiful Philo$opher perceive, that we $hould ne- ver $ee this $ame $econdary light, $ave onely at the Eclip$e of the Sun? And if a part onely of the Moon, far le$$e than half a de- gree, by being remote from the Suns <I>Di$cus,</I> can deflect or de- viate the rayes of the Sun, $o that they arrive not at our eye; what $hall it do when it is di$tant twenty or thirty degrees, as it is at its fir$t apparition? and what cour$e $hall the rayes of the Sun keep, which are to pa$$e thorow the body of the Moon, that <marg><I>The Author of the Book of conclu$i- ons, accommodates the things to his purpo$es, and not his purpo$es to the things.</I></marg> they may find out our eye? This man doth go $ucce$$ively con$i- dering what things ought to be, that they may $erve his purpo$e, but doth not gradually proceed, accommodating his conceits to the things, as really they are. As for in$tance, to make the light <foot>of</foot> <p n=>79</p> of the Sun capable to penetrate the $ub$tance of the Moon, he makes her in part diaphanous, as is <I>v. g.</I> the tran$parence of a cloud, or cry$tal: but I know not what he would think of $uch a tran- $parency, in ca$e the $olar rayes were to pa$$e a depth of clouds of above two thou$and miles; but let it be $uppo$ed that he $hould boldly an$wer, that might well be in the Cœle$tial, which are quite other things from the$e our Elementary, impure, and feculent bodies; and let us convict his error by $uch wayes, as admit him no reply, or (to $ay better) $ubter-fuge. If he will maintain, that the $ub$tance of the Moon is diaphanous, he mu$t $ay that it is $o, while$t that the rayes of the Sun are to pe- netrate its whole profundity, that is, more than two thou$and miles; but that if you oppo$e unto them onely one mile, or le$$e, they $hould no more penetrate that, than they penetrate one of our mountains.</P> <P>SAGR. You put me in mind of a man, who would have $old <marg><I>A je$t put upon one that would $ell a certain $ecret for holding corre$pon- dency with a per$on a thou$and miles off</I></marg> me a $ecret how to corre$pond, by means of a certain $ympathy of magnetick needles, with one, that $hould be two or three thou- $and miles di$tant; and I telling him, that I would willingly buy the $ame, but that I de$ired fir$t to $ee the experiment thereof, and that it did $uffice me to make it, I being in one Chamber, and he in the next, he an$wered me, that in $o $mall a di$tance one could not $o well perceive the operation; whereupon I turn'd him going, telling him, that I had no mind, at that time, to take a journey unto <I>Grand Cairo,</I> or to <I>Mu$covy,</I> to make the experi- ment; but that, if he would go him$elf, I would perform the other part, $taying in <I>Venice.</I> But let us hear whither the dedu- ction of our Author tendeth, and what nece$$ity there is, that he mu$t grant the matter of the Moon to be mo$t perforable by the rayes of the Sun, in a depth of two thou$and miles, but more opacous than one of our mountains, in a thickne$$e of one mile onely.</P> <P>SALV. The very mountains of the Moon them$elves are a proof thereof, which percu$$ed on one $ide of the Sun, do ca$t on the contrary $ide very dark $hadows, terminate, and more di- $tinct by much, than the $hadows of ours; but had the$e moun- tains been diaphanous, we could never have come to the know- ledg of any unevenne$$e in the $uperficies of the Moon, nor have $een tho$e luminous montuo$ities di$tingui$hed by the terms which $eparate the lucid parts from the dark: much le$$e, $hould we $ee this $ame term $o di$tinct, if it were true, that the Suns light did penetrate the whole thickne$$e of the Moon; yea rather, accord- ing to the Authors own words, we $hould of nece$$ity di$cern the pa$$age, and confine, between the part of the Sun $een, and the part not $een, to be very confu$ed, and mixt with light and <foot>dark-</foot> <p n=>80</p> darkne$$e; for that that matter which admits the pa$$age of the Suns rayes thorow a $pace of two thou$and miles, mu$t needs be $o tran$parent, that it would very weakly re$i$t them in a hun- dredth, or le$$er part of that thickne$$e; neverthele$$e, the term which $eparateth the part illuminated from the ob$cure, is inci- dent, and as di$tinct, as white is di$tinct from black; and e- $pecially where the Section pa$$eth through the part of the Moon, that is naturally more clear and montanous; but where the old $pots do part, which are certain plains, that by means of their $pherical inclination, receive the rayes of the Sun obliquely, there the term is not $o di$tinct, by rea$on of the more dimme il- lumination. That, la$tly, which he $aith, how that the $econdary light doth not dimini$h and langui$h, according as the Moon en- crea$eth, but con$erveth it $elf continually in the $ame efficacy; is mo$t fal$e; nay it is hardly $een in the quadrature, when, on the contrary, it $hould appear more $plendid, and be vi$ible after the <I>crepu$culum</I> in the dark of night. Let us conclude therefore, that the Earths reflection is very $trong upon the Moon; and that, which you ought more to e$teem, we may deduce from thence an- other admirable congruity between the Moon and Earth; name- <marg><I>The Earth may re- ciprocally operate upon Cœle$tial bo- dies, with its light.</I></marg> ly, that if it be true, the Planets operate upon the Earth by their motion and light, the Earth may probably be no le$$e potent in operating reciprocally upon them with the $ame light, and perad- venture, motion al$o. And though it $hould not move, yet may it retain the $ame operation; becau$e, as it hath been proved al- ready, the action of the light is the $elf $ame, I mean of the light of the Sun reflected; and motion doth nothing, $ave only vary the a$pects, which fall out in the $ame manner, whether we make the Earth move, and the Sun $tand $till, or the contrary.</P> <P>SIMPL. None of the Philo$ophers are found to have $aid, that the$e inferiour bodies operate on the Cœle$tial, nay, <I>Ari$totle</I> af- firmes the direct contrary.</P> <P>SALV. <I>Aristotle</I> and the re$t, who knew not that the Earth and Moon mutually illuminated each other, are to be excu$ed; but they would ju$tly de$erve our cen$ure, if while$t they de$ire that we $hould grant and believe with them, that the Moon operateth upon the Earth with light, they $hould deny to us, who have taught them that the Earth illuminates the Moon, the operation the Earth hath on the Moon.</P> <P>SIMPL. In $hort, I find in my $elf a great unwillingne$$e to admit this commerce, which you would per$wade me to be be- twixt the Earth and Moon, placing it, as we $ay, among$t the number of the Stars; for if there were nothing el$e, the great $eparation and di$tance between it and the Cœle$tial bodies, doth in my opinion nece$$arily conclude a va$t di$parity between them.</P> <foot>SALV.</foot> <p n=>81</p> <P>SALV. See <I>Simplicius</I> what an inveterate affection and radica- ted opinion can do, $ince it is $o powerful, that it makes you think that tho$e very things favour you, which you produce again$t your $elf. For if $eparation and di$tance are accidents $ufficient to per$wade with you a great diver$ity of natures, it mn$t follow that <marg><I>Affinity between he Earth & Moon in re$pect of their vicinity.</I></marg> proximity and contiguity import $imilitude. Now how much more neerer is the Moon to the Earth, than to any other of the Cœle$tial Orbs? You mu$t acknowledg therefore, according to your own con- ce$$ion (and you $hall have other Philo$ophers bear you company) that there is a very great affinity betwixt the Earth and Moon. Now let us proceed, and $ee whether any thing remains to be con- $idered, touching tho$e objections which you made again$t the re- $emblances that are between the$e two bodies.</P> <P>SIMPL. It re$ts, that we $ay $omething touching the $olidity of the Moon, which I argued from its being exqui$ite $mooth and polite, and you from its montuo$ity. There is another $cruple al- $o comes into my mind, from an opinion which I have, that the Seas reflection ought by the equality of its $urface, to be rendered $tronger than that of the Earth, who$e $uperficies is $o rough and opacous.</P> <P>SALV. As to the fir$t objection; I $ay, that like as among the parts of the Earth, which all by their gravity $trive to approach the <marg><I>Solidity of the Lunar Globe argu- ed from its being montainous.</I></marg> neare$t they can po$$ible to the center, $ome of them alwayes are more remote from it than the re$t, as the mountains more than the valleys, and that by rea$on of their $olidity and firmne$$e (for if they were of fluid, they would be even) $o the $eeing $ome parts of the Moon to be elevated above the $phericity of the low- er parts, argueth their hardne$$e; for it is probable that the mat- ter of the Moon is reduced into a $pherical form by the harmoni- ous con$piration of all its parts to the $ame $enten$e. Touching the $econd doubt, my thinks that the particulars already ob$erved to happen in the Looking-gla$$es, may very well a$$ure us, that the reflection of light comming from the Sea, is far weaker than that <marg><I>The Seas refle- ction of light much weaker than that of the Earth.</I></marg> which cometh from Land; under$tanding it alwayes of the univer$al reflection; for as to that particular, on which the wa- ter being calm, ca$teth upon a determinate place, there is no doubt, but that he who $hall $tand in that place, $hall $ee a very great reflection in the water, but every way el$e he $hall $ee the $urface of the Water more ob$cure than that of the Land; and to <marg><I>An experiment to prove the refle- ction of the Water le$$e clear than that of the Land.</I></marg> prove it to your $en$es, let us go into yonder Hall, and power forth a little water upon the Pavement. Tell me now, doth not this wet brick $hew more dull than the other dry ones? Doubt- le$$e it doth, and will $o appear, from what place $oever you be- hold it, except one onely, and this is that way which the light cometh, that entereth in at yonder window; go backwards therefore by a little and a little.</P> <foot>L SIMPL.</foot> <p n=>82</p> <P>SIMPL. Here I $ee the we$t part $hine more than all the re$t of the pavement, and I $ee that it $o hapneth, becau$e the refle- ction of the light which entereth in at the window, cometh to- wards me.</P> <P>SALV. That moi$ture hath done no more but filled tho$e little cavities which are in the brick with water, and reduced its $uper- ficies to an exact evene$$e; whereupon the reflex rayes i$$ue unitedly towards one and the $ame place; but the re$t of the pavement which is dry, hath its protuberances, that is, an innu- merable variety of inclinations in its $malle$t particles; whereup- on the reflections of the light $catter towards all parts, but more weakly than if they had gone all united together; and therefore, the $ame $heweth almo$t all alike, beheld $everal wayes, but far le$$e clear than the moi$tned brick. I conclude therefore, that the $urface of the Sea, beheld from the Moon, in like manner, as it would appear mo$t equal, (the I$lands and Rocks deducted) $o it would $hew le$$e clear than that of the Earth, which is montanous and uneven. And but that I would not $eem, as the $aying is, to harp too much on one $tring, I could tell you that I have ob- $erved in the Moon that $econdary light which I told you came to her from the reflection of the Terre$trial Globe, to be notably <marg><I>The $econdary light of the Moon clearer before the conjunction, than after.</I></marg> more clear two or three dayes before the conjunction, than after, that is, when we $ee it before break of day in the Ea$t, than when it is $een at night after Sun-$et in the We$t; of which dif- ference the cau$e is, that the Terre$trial Hemi$phere, which looks towards the Ea$tern Moon, hath little Sea, and much Land, to wit, all <I>A$ia,</I> whereas, when it is in the We$t, it beholds very great Seas, that is, the whole <I>Atlantick</I> Ocean as far as <I>America:</I> An Argument $ufficiently probable that the $urface of the water appears le$$e $plendid than that of the Earth.</P> <P>SIMPL. So that perhaps you believe, tho$e great $pots di$co- vered in the face of the Moon, to be Seas, and the other clearer parts to be Land, or $ome $uch thing?</P> <P>SALV. This which you ask me, is the beginning of tho$e in- congruities which I e$teem to be between the Moon and the Earth, out of which it is time to di$-ingage our $elves, for we have $tayed too long in the Moon. I $ay therefore, that if there were in nature but one way onely, to make two $uperficies illu$tra- ted by the Sun, to appear one more clear than the other, and that this were by the being of the one Earth, and the other Wa- ter; it would be nece$$ary to $ay that the $urface of the Moon were part earthy and part aquatick; but becau$e we know many wayes to produce the $ame effect (and others there may be which we know not of;) therefore I dare not affirm the Moon to con- $i$t of one thing more than another: It hath been $een already <foot>that</foot> <p n=>83</p> that a $ilver plate boiled, being toucht with the Burni$her, be- cometh of white ob$cure; that the moi$t part of the Earth $hews more ob$cure than the dry; that in the tops of Hills, the woody parts appear more gloomy than the naked and barren; which hapneth becau$e there falleth very much $hadow among the Trees, but the open places are illuminated all over by the Sun. And this mixtion of $hadow hath $uch operation, that in tu$ted velvet, the $ilk which is cut, is of a far darker colour than that which is not cut, by means of the $hadows diffu$ed betwixt thred and thred, and a plain velvet $hews much blacker than a Taffata, made of the $ame $ilk. So that if there were in the Moon things which $hould look like great Woods, their a$pect might repre$ent unto us the $pots which we di$cover; alike difference would be occa$ioned, if there were Seas in her: and la$tly, nothing hindreth, but that tho$e $pots may really be of an ob$curer colour than the re$t; for thus the $now makes the mountains $hew brighter. That which is plain- <marg><I>The ob$curer parts of the Moon are plains, and the more bright moun- tainous.</I></marg> ly ob$erved in the Moon is, that its mo$t ob$cure parts are all plains, with few ri$es and bancks in them; though $ome there be; the re$t which is of a brighter colour, is all full of rocks, moun- tains, hillocks of $pherical and other figures; and in particular, round about the $pots are very great ledges of mountains. That the <marg><I>Long ledges of mountaixs about the $pots of the Moon.</I></marg> $pots be plain $uperficies, we have a$$uredproof, in that we $ee, how that the term which di$tingui$heth the part illuminated from the ob$cure, in cro$$ing the $pots makes the inter$ection even, but in the clear parts it $hews all craggy and $hagged. But I know not as yet whether this evenne$$e of $uperficies may be $ufficient of it $elf alone, to make the ob$curity appear, and I rather think not. Be$ides, I account the Moon exceeding different from the Earth; for although I imagine to my $elf that tho$e are not idle and dead Regions, yet I affirm not, that there are in them motion and life, <marg><I>There are not generated in the Moon things like to ours, but if there be any pro- ductions, they are very different.</I></marg> much le$s that there are bred plants, animals or other things like to ours; but, if $uch there be, they $hould neverthele$s be very different, and remote from our imagination. And I am induced $o to think, becau$e in the fir$t place, I e$teem that the matter of the Lunar Globe con$i$ts not of Earth and Water; and this alone $ufficeth to take away the generations and alterations re$embling ours: but now $uppo$ing that there were in the Moon, Water and <marg><I>The Moon <*>os compo$ed of Water and Earth.</I></marg> Earth, yet would they not produce plants and animals like to ours; and this for two principal rea$ons: The fir$t is, that unto our <marg><I>Tho$e a$pects of the Sun nece$$ary for our generati- ons, are not $o in the Moon.</I></marg> productions there are required $o many variable a$pects of the Sun, that without them they would all mi$carry: now the habitudes of the Sun towards the Earth are far different from tho$e towards the Moon. We as to the diurnal illumination, have, in the greater part of the Earth, every twenty four hours part day, and part night, which effect in the Moon is monethly: and that annual decli- <foot>L2 nation</foot> <p n=>84</p> nation and elevation of the Sun in the Zodiack, by which it pro- <marg><I>Natural dayas in the Moon are of a Moneth long.</I></marg> duceth diver$ity of Sea$ons, and inequality of dayes and nights, are fini$hed in the Moon in a moneth; and whereas the Sun to us <marg><I>To the Moon the Sun a$eondeth and declineth with a difference of ten degrees, and to the Earth of forty $e- ven degrees.</I></marg> ri$eth and declineth $o much, that from the greate$t to the lea$t al- titude, there is a difference of almo$t 47 degrees, for $o much is the di$tance from one to the other Tropick; this is in the Moon but ten degrees only, or little more; namely, as much as the grea- te$t Latitudes of the Dragon on each $ide the Ecliptick. Now con$ider what effect the Sun would have in the torrid Zone, $hould it continually for fifteen dayes together beam forth its Rayes upon it; which without all que$tion would de$troy plants, herbs, and living creatures: and if it $hould chance that there were any production, it would be of herbs, plants, and creatures very diffe- <marg><I>There are no rains in the Moon.</I></marg> rent from tho$e which are now there. Secondly, I verily believe that in the Moon there are no rains, for if Clouds $hould gather in any part thereof, as they do about the Earth, they would there- upon hide from our $ight $ome of tho$e things, which we with the <I>Tele$cope</I> behold in the Moon, and in a word, would $ome way or other change its <I>Phœnomenon,</I> an effect which I could never by long and diligent ob$ervations di$cover; but alwayes beheld it in a even and pure $erenity.</P> <P>SAGR. To this may be an$wered, either that there might be great mi$ts, or that it might rain in the time of their night, that is, when the Sun doth not illuminate it.</P> <P>SALV. If other pa$$ages did but a$$ure us, that there were ge- nerations in it like to ours, and that there was onely wanting the concour$e of rains, we might find out this, or $ome other tempe- rament to $erve in$tead thereof, as it happens in <I>Egypt</I> by the in- undation of <I>Nile:</I> but not meeting with any accident, which cor- re$ponds with ours, of many that have been $ought out for the pro- duction of the like effects, we need not trouble our $elves to intro- duce one alone; and that al$o, not becau$e we have certain ob$er- vation of it, but for a bare non-repugnance that we find therein. Moreover, if I was demanded what my fir$t apprehen$ion, and pure natural rea$on dictated to me concerning the production of things like or unlike there above, I would alwayes reply, that they are mo$t different, and to us altogether unimaginable, for $o me thinks the riches of Nature, and the omnipotence of our Creator and Governour, do require.</P> <P>SAGR. I ever accounted extraordinary madne$$e that of tho$e, who would make humane comprehen$ion the mea$ure of what na- ture hath a power or knowledge to effect; whereas on the con- <marg><I>The having a perfect knowledg of nothing, maketh $ome believe they under$tand all things.</I></marg> trary there is not any the lea$t effect in Nature, which can be fully under$tood by the mo$t $peculative wits in the world. This their $o vain pre$umption of knowing all, can take beginning from no- <foot>thing</foot> <p n=>85</p> thing, unle$$e from their never having known any thing; for if one hath but once onely experienced the perfect knowledg of one onely thing, and but truly ta$ted what it is to know, he $hall per- ceive that of infinite other conclu$ions, he under$tands not $o much as one.</P> <P>SALV. Your di$cour$e is very concluding; in confirmation of which we have the example of tho$e who under$tand, or have known $ome thing, which the more knowing they are, the more they know, and freely confe$$e that they know little; nay, the wi$e$t man in all <I>Greece,</I> and for $uch pronounced by the Oracle, openly profe$$ed to know that he knew nothing.</P> <P>SIMPL. It mu$t be granted therefore, either that <I>Socrates</I> or that the <I>Oracle</I> it $elf was a lyar, <I>that declaring him to be mo$t wi$e, and he confe$$ing that he knew him$elf to be mo$t ig- norant.</I></P> <P>SALV. Neither one nor the other doth follow, for that both <marg><I>The an$wer of the Oracle true in judging</I> Socrates <I>the wi$eft of his time.</I></marg> the a$$ertions may be true. The <I>Oracle</I> adjudged <I>Socrates</I> the wi- $e$t of all men, who$e knowledg is limited; <I>Socrates</I> acknow- ledgeth that he knew nothing in relation to ab$olute wi$dome, which is infinite; and becau$e of infinite, much is the $ame part, as is little, and as is nothing (for to arrive <I>v. g.</I> to the infinite number, it is all one to accumulate thou$ands, tens, or ciphers,) therefore <I>Socrates</I> well perceived his wi$dom to be nothing, in compari$on of the infinite knowledg which he wanted. But yet, becau$e there is $ome knowledg found among$t men, and this not equally $hared to all, <I>Socrates</I> might have a greater $hare thereof than others, and therefore verified the an$wer of the <I>Oracle.</I></P> <P>SAGR. I think I very well under$tand this particular among$t men, <I>Simplicius</I> there is a power of operating, but not equally di$pen$ed to all; and it is without que$tion, that the power of an Emperor is far greater than that of a private per$on; but, both this and that are nothing in compari$on of the Divine Omnipo- tence. Among$t men, there are $ome that better under$tand Agriculture than many others; but the knowledg of planting a Vine in a trench, what hath it to do with the knowledg of ma- king it to $prout forth, to attract nouri$hment, to $elect this good part from that other, for to make thereof leaves, another to make $prouts, another to make grapes, another to make rai$ins, ano- ther to make the huskes of them, which are the works of mo$t wi$e Nature? This is one only particular act of the innumerable, which Nature doth, and in it alone is di$covered an infinite wi$- <marg><I>Divine Wi$dom infinitely infinise.</I></marg> dom, $o that Divine Wi$dom may be concluded to be infinitely infinite.</P> <P>SALV. Take hereof another example. Do we not $ay that the <foot>judi-</foot> <p n=>86</p> judicious di$covering of a mo$t lovely <I>Statua</I> in a piece of Marble, <marg>Buonarruotti, <I>a $tatuary of admi- rable ingenuity.</I></marg> hath $ublimated the wit of <I>Buonarruotti</I> far above the vulgar wits of other men? And yet this work is onely the imitation of a meer aptitude and di$po$ition of exteriour and $uperficial mem- bers of an immoveable man; but what is it in compari$on of a man made by nature, compo$ed of as many exteriour and inte- riour members, of $o many mu$cles, tendons, nerves, bones, which $erve to $o many and $undry motions? but what $hall we $ay of the $en$es, and of the powers of the $oul, and la$tly, of the under$tanding? May we not $ay, and that with rea$on, that the $tructure of a Statue fals far $hort of the formation of a living man, yea more of a contemptible worm?</P> <P>SAGR. And what difference think you, was there betwixt the Dove of <I>Architas,</I> and one made by Nature?</P> <P>SIMPL. Either I am none of the$e knowing men, or el$e there is a manife$t contradiction in this your di$cour$e. You ac- count under$tanding among$t the greate$t (if you make it not the chief of the) <I>Encomiums</I> a$cribed to man made by Nature, and a little before you $aid with <I>Socrates,</I> that he had no knowledg at all; therefore you mu$t $ay, that neither did Nature under$tand how to make an under$tanding that under$tandeth.</P> <P>SALV. You argue very cunningly, but to reply to your obje- ction I mu$t have recour$e to a Philo$ophical di$tinction, and $ay that the under$tanding is to be taken too ways, that is <I>inten$ivè,</I> or <marg><I>Man under$tand- eth very well</I> in- ten$ivè, <I>but little</I> exten$ivè.</marg> <I>exten$ivè</I>; and that <I>exten$ive,</I> that is, as to the multitude of intel- ligibles, which are infinite, the under$tanding of man is as no- thing, though he $hould under$tand a thou$and propo$itions; for that a thou$and, in re$pect of infinity is but as a cypher: but taking the under$tanding <I>inten$ive,</I> (in as much as that term imports) in- ten$ively, that is, perfectly $ome propo$itions, I $ay, that humane wi$- dom under$tandeth $ome propo$itions $o perfectly, and is as ab$o- lutely certain thereof, as Nature her $elf; and $uch are the pure Mathematical $ciences, to wit, Geometry and Arithmetick: in which Divine Wi$dom knows infinite more propo$itions, becau$e it knows them all; but I believe that the knowledge of tho$e few compre- hended by humane under$tanding, equalleth the divine, as to the certainty <I>objectivè,</I> for that it arriveth to comprehend the nece$- $ity thereof, than which there can be no greater certainty.</P> <P>SIMPL. This $eemeth to me a very bold and ra$h expre$$ion.</P> <P>SALV. The$e are common notions, and far from all umbrage of temerity, or boldne$s, and detract not in the lea$t from the Ma- je$ty of divine wi$dom; as it nothing dimini$heth the omnipotence thereof to $ay, that God cannot make what is once done, to be un- done: but I doubt, <I>Simplicius,</I> that your $cruple ari$eth from an o- pinion you have, that my words are $omewhat equivocal; there- <foot>fore</foot> <p n=>87</p> fore the better to expre$s my $elf I $ay, that as to the truth, of which Mathematical demon$trations give us the knowledge, it is the $ame, which the divine wi$dom knoweth; but this I mu$t grant you, that the manner whereby God knoweth the infinite propo- <marg><I>Gods manner of knowing different from that of men.</I></marg> $itions, of which we under$tand $ome few, is highly more excellent than ours, which proceedeth by ratiocination, and pa$$eth from con- <marg><I>Humane under- $tanding done by raciocination.</I></marg> clu$ion to conclu$ion, whereas his is done at one $ingle thought or intuition; and whereas we, for example, to attain the knowledg of $ome pa$$ion of the Circle, which hath infinite, beginning from one of the mo$t $imple, and taking that for its definition, do proceed with argumentation to another, and from that to a third, and then to a fourth, <I>&c.</I> the Divine Wi$dom, by the apprehen$ion of its e$$ence comprehends, without temporary raci- ocination, all the$e infinite pa$$ions; which notwith$tanding, are in effect virtually compri$ed in the definitions of all things; and, to <marg><I>Definitions con- tein virtually all the pa$$ions of the things defined.</I></marg> conclude, as being infinite, perhaps are but one alone in their nature, and in the Divine Mind; the which neither is wholly unknown to humane under$tanding, but onely be-clouded with thick and <marg><I>Infinite Pa$$ions are perhaps but one onely.</I></marg> gro$$e mi$ts; which come in part to be di$$ipated and clarified, when we are made Ma$ters of any conclu$ions, firmly demon- $trated, and $o perfectly made ours, as that we can $peedily run through them; for in $um, what other, is that propo$ition, that the $quare of the $ide $ubtending the right angle in any triangle, is equal to the $quares of the other two, which include it, but onely the Paralellograms being upon common ba$es, and between parallels equal among$t them$elves? and this, la$tly, is it not the $ame, as to $ay that tho$e two $uperficies are equal, of which equal parts applyed to equal parts, po$$e$$e equal place? Now <marg><I>The di$cour$es which humane rea$on makes in a certain time, the Divine Wi$dom re- $olveth in a mo- ment; that is, hath them alwayes pre- $ent.</I></marg> the$e inferences, which our intellect apprehendeth with time and a gradual motion, the Divine Wi$dom, like light, penetrateth in an in$tant, which is the $ame as to $ay, hath them alwayes pre- $ent: I conclude therefore, that our under$tanding, both as to the manner and the multitude of the things comprehended by us, is infinitely $urpa$t by the Divine Wi$dom; but yet I do not $o vilifie it, as to repute it ab$olutely nothing; yea rather, when I con$ider how many and how great mi$teries men have under$tood, di$covered, and contrived, I very plainly know and under$tand the mind of man to be one of the works, yea one of the mo$t ex- cellent works of God.</P> <P>SAGR. I have oft times con$idered with my $elf, in pur$uance <marg><I>The wit of man admirably ac<*>.</I></marg> of that which you $peak of, how great the wit of man is; and whil'$t I run thorow $uch and $o many admirable inventions found out by him, as well in the Arts, as Sciences; and again reflecting upon my own wit, $o far from promi$ing me the di$covery of any thing new, that I de$pair of comprehending what is already di$- <foot>covered,</foot> <p n=>88</p> covered, confounded with wonder, and $urpri$ed with de$pera- tion, I account my $elf little le$$e than mi$erable. If I behold a Statue of $ome excellent Ma$ter, I $ay with my $elf; When wilt thou know how to chizzle away the refu$e of a piece of Marble, and di$cover $o lovely a figure, as lyeth hid therein? When wilt thou mix and $pread $o many different colours upon a Cloth, or Wall, and repre$ent therewith all vi$ible objects, like a <I>Michael Angelo,</I> a <I>Raphaello,</I> or a <I>Tizvano</I>? If I behold what inventions men have in comparting Mu$ical intervals, in e$tabli$hing Pre- cepts and Rules for the management thereof with admirable de- light to the ear: When $hall I cea$e my a$toni$hment? What $hall I $ay of $uch and $o various In$truments of that Art? The reading of excellent Poets, with what admiration doth it $well any one that attentively con$idereth the invention of conceits, and their explanation? What $hall we $ay of Architecture? <marg><I>The invention of writing $tupendious above all others.</I></marg> What of Navigation? But, above all other $tupendious inventi- ons, what $ublimity of mind was that in him, that imagined to him$elf to find out a way to communicate his mo$t $ecret thoughts to any other per$on, though very far di$tant from him either in time, or place, $peaking with tho$e that are in the <I>India's</I>; $peak- ing to tho$e that are not yet born, nor $hall be this thou$and, or ten thou$and years? and with how much facility? but by the va- <marg>* For of $o many only the Italian Alphabet con$i$ts.</marg> rious collocation of ^{*} twenty little letters upon a paper? Let this be the Seal of all the admirable inventions of man, and the clo$e of our Di$cour$e for this day: For the warmer hours being pa$t, I $uppo$e that <I>Salviatus</I> hath a de$ire to go and take the air in his Gondelo; but too morrow we will both wait upon you, to con- tinue the Di$cour$es we have begun, <I>&c.</I></P> <foot>GAL-</foot> <pb> <fig> <cap><I>Place this Plate at the end of the first</I></cap> <cap>Dialogue ~</cap> <p n=>89</p> <head>GALILÆUS Galilæus Lyncæus, HIS SYSTEME OF THE WORLD.</head> <head>The Second Dialogue.</head> <head><I>INTERLOCVTORS.</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <P>SALV. The ye$ter-dayes diver$ions which led us out of the path of our principal di$cour$e, were $uch and $o many, that I know not how I can without your a$$i$tance reco- ver the track in which I am to proceed.</P> <P>SAGR. I wonder not, that you, who have your fancy charged and laden with both what hath been, and is to be $po- ken, do find your $elf in $ome confu$i- on; but I, who as being onely an Auditor, have nothing to bur- then my memory withal, but $uch things as I have heard, may happily by a $uccinct rehear$al of them, recover the fir$t thred of our Di$cour$e. As far therefore as my memory $erves me, the $um of ye$terdayes conferences were an examination of the Prin- <foot>M ciples</foot> <p n=>90</p> ciples of <I>Ptolomy</I> and <I>Copernicus,</I> and which of their opinions is the more probable and rational; that, which affirmeth the $ub- $tance of the Cœle$tial bodies to be ingenerable, incorruptible, un- alterable, impa$$ible, and in a word, exempt from all kind of change, $ave that of local, and therefore to be a <I>fifth e$$ence,</I> quite different from this of our Elementary bodies, which are generable, corrup- tible, alterable, <I>&c.</I> or el$e the other, which taking away $uch deformity from the parts of the World, holdeth the Earth to en- joy the $ame perfections as the other integral bodies of the uni- ver$e; and e$teemeth it a moveable and erratick Globe, no le$$e than the Moon, <I>Jupiter, Venus,</I> or any other Planet: And la$tly, maketh many particular parallels betwixt the Earth and Moon; and more with the Moon, than with any other Planet; hap- ly by rea$on we have greater and more certain notice of it, as being le$$e di$tant from us. And having, la$tly, concluded this $econd opinion to have more of probability with it than the fir$t, I $hould think it be$t in the $ub$equent di$cour$es to begin to exa- mine whether the Earth be e$teemed immoveable, as it hath been till now believed by mo$t men, or el$e moveable, as $ome ancient <I>Philo$ophers</I> held, and others of not very rece$$e times, were of opinion; and if it be moveable, to enquire of what kind its motion may be?</P> <P>SALV. I $ee already what way I am to take; but before we offer to proceed any farther, I am to $ay $omething to you touch- ing tho$e la$t words which you $pake, how that the opinion which holds the Earth to be endued with the $ame conditions that the Cœle$tial bodies enjoy, $eems to be more true than the contra- ry; for that I affirmed no $uch thing, nor would I have any of the Propo$itions in controver$ie, be made to $peak to any definitive $en$e: but I onely intended to produce on either part, tho$e rea- $ons and an$wers, arguments and $olutions, which have been hi- therto thought upon by others, together with certain others, which I have $tumbled upon in my long $earching thereinto, al- wayes remitting the deci$ion thereof to the judgment of others.</P> <P>SAGR. I was unawares tran$ported by my own $en$e of the thing; and believing that others ought to judg as I did, I made that conclu$ion univer$al, which $hould have been particular; and therefore confe$$e I have erred, and the rather, in that I know not what <I>Simplicius</I> his judgment is in this particular.</P> <P>SIMPL. I mu$t confe$$e, that I have been ruminating all this night of what pa$t ye$terday, and to $ay the truth, I meet there- in with many acute, new, aud plau$ible notions; yet neverthele$s, I find my $elf over-per$waded by the authority of $o many great <I>Writers,</I> and in particular ------- <I>&c.</I> I $ee you $hake your head <I>Sagredus,</I> and $mile to your $elf, as if I had uttered $ome great ab$urdity.</P> <foot>SAGR.</foot> <p n=>91</p> <P>SAGR. I not onely $mile, but to tell you true, am ready to bur$t with holding in my $elf from laughing outright, for you have put me in mind of a very pretty pa$$age, that I was a wit- ne$$e of, not many years $ince, together with $ome others of my worthy friends, which I could yet name unto you.</P> <P>SALV. It would be well that you told us what it was, that $o <I>Simplicius</I> may not $till think that he gave you the occa$ion of laughter.</P> <P>SAGR. I am content. I found one day, at home in his hou$e, at <I>Venice,</I> a famous Phi$ician, to whom $ome flockt for their $tudies, and others out of curio$ity, $ometimes came thither to $ee certain A- natomies di$$ected by the hand of a no le$$e learned, than careful and experienced Anatomi$t. It chanced upon that day, when I was <marg><I>The original of the Nerv s. ac- cording to</I> Ari$to- tle, <I>and according to Phi$icians.</I></marg> there, that he was in $earch of the original and ri$e of the Nerves, about which there is a famous controver$ie between the <I>Galeni$ts</I> and <I>Peripateticks</I>; and the Anatomi$t $hewing, how that the great number of Nerves departing from the Brain, as their root, and pa$$ing by the nape of the Neck, di$tend them$elves afterwards along by the Back-bone, and branch them$elves thorow all the Body; and that a very $mall filament, as fine as a thred went to the Heart; he turned to a Gentleman whom he knew to be a <I>Pe- ripatetick</I> Philo$opher, and for who$e $ake he had with extraor- dinary exactne$$e, di$covered and proved every thing, and demand- ed of him, if he was at length $atisfied and per$waded that the origi- nal of the Nerves proceeded from the Brain, and not from the Heart? To which the Philo$opher, after he had $tood mu$ing a <marg><I>The ridiculus an$wer of a Philo- $opher, determi- ning the original of the Nerves.</I></marg> while, an$wered; you have made me to $ee this bu$ine$$e $o plainly and $en$ibly, that did not the <I>Text</I> of <I>Ari$totle</I> a$$ert the contrary, which po$itively affirmeth the Nerves to proceed from the Heart, I $hould be con$trained to confe$$e your opinion to be true.</P> <P>SIMPL. I would have you know my Ma$ters, that this contro- ver$ie about the original of the Nerves is not yet $o proved and decided, as $ome may perhaps per$wade them$elves.</P> <P>SAGR. Nor que$tionle$$e ever $hall it be, if it find $uch like contradictors; but that which you $ay, doth not at all le$$en the extravagance of the an$wer of that <I>Peripatetick,</I> who again$t $uch $en$ible experience produced not other experiments, or rea- $ons of <I>Ari$totle,</I> but his bare authority and pure <I>ip$e dixit.</I></P> <P>SIMPL. <I>Ari$totle</I> had not gained $o great authority, but for the force of his Demon$trations, and the profoundne$$e of his arguments; but it is requi$ite that we under$tand him, and not onely under$tand him, but have $o great familiarity with his Books, that we form a perfect <I>Idea</I> thereof in our minds, $o as that every $aying of his may be alwayes as it were, pre$ent in our <foot>M 2 me-</foot> <p n=>92</p> memory for he did not write to the vulgar, nor is he obliged to $pin out his Sillogi$mes with the trivial method of di$putes; nay rather, u$ing a freedome, he hath $ometimes placed the proof of one Propo$ition among$t Texts, which $eem to treat of quite <marg><I>Requi$ites to fit a man to philo$o- phate well after the manner of</I> A- ri$totle.</marg> another point; and therefore it is requi$ite to be ma$ter of all that va$t <I>Idea,</I> and to learn how to connect this pa$$age with that, and to combine this Text with another far remote from it; for it is not to be que$tioned but that he who hath thus $tudied him, knows how to gather from his Books the demon$trations of every knowable deduction, for that they contein all things.</P> <P>SAGR. But good <I>Simplicius,</I> like as the things $cattered here and there in <I>Ari$totle,</I> give you no trouble in collecting them, but that you per$wade your $elf to be able by comparing and <marg><I>A cunning way to gather Philo$o- phy out of any book what$oever.</I></marg> connecting $everal $mall $entences to extract thence the juice of $ome de$ired conclu$ion, $o this, which you and other egregi- ous Philo$ophers do with the Text of <I>Ari$totle,</I> I could do by the <marg>* A word $ignify- ing works compo- $ed of many frag- ments of ver$es collected out of the Poets.</marg> ver$es of <I>Virgil,</I> or of <I>Ovid,</I> compo$ing thereof ^{*} <I>Centones,</I> and therewith explaining all the affairs of men, and $ecrets of Na- ture. But what talk I of <I>Virgil,</I> or any other Poet? I have a lit- tle Book much $horter than <I>Ari$totle</I> and <I>Ovid,</I> in which are con- teined all the Sciences, and with very little $tudy, one may gather out of it a mo$t perfect <I>Idea,</I> and this is the <I>Alphabet</I>; and there is no doubt but that he who knows how to couple and di$po$e aright this and that vowel, with tho$e, or tho$e other con$onants, may gather thence the infallible an$wers to all doubts, and de- duce from them the principles of all Sciences and Arts, ju$t in the $ame manner as the Painter from divers $imple colours, laid $eve- rally upon his <I>Pallate,</I> proceedeth by mixing a little of this and a little of that, with a little of a third, to repre$ent to the life men, plants, buildings, birds, fi$hes, and in a word, counterfeit- ing what ever object is vi$ible, though there be not on the <I>Pallate</I> all the while, either eyes, or feathers, or fins, or leaves, or $tones. Nay, farther, it is nece$$ary, that none of the things to be imita- ted, or any part of them, be actually among colours, if you would be able therewith to repre$ent all things; for $hould there be among$t them <I>v. gr.</I> feathers, the$e would $erve to repre$ent nothing $ave birds, and plumed creatures.</P> <P>SALV. And there are certain Gentlemen yet living, and in health, who were pre$ent, when a Doctor, that was Profe$$or in a fa- <marg><I>Invention of the</I> Tele$cope <I>taken from</I> Ari$totle.</marg> mous Academy, hearing the de$cription of the <I>Tele$cope,</I> by him not $een as then, $aid, that the invention was taken from <I>Ari- $totle,</I> and cau$ing his works to be fetch't, he turned to a place where the Philo$opher gives the rea$on, whence it commeth, that from the bottom of a very deep Well, one may $ee the $tars in Heaven, at noon day; and, addre$$ing him$elf to the company, <foot>$ee</foot> <p n=>93</p> $ee here, $aith he, the Well, which repre$enteth the Tube, $ee here the gro$s vapours, from whence is taken the invention of the Cry$tals, and $ee here la$tly the $ight fortified by the pa$$age of the rays through a diaphanous, but more den$e and ob$cure <I>medium.</I></P> <P>SAGR. This is a way to comprehend all things knowable, much like to that wherewith a piece of marble conteineth in it one, yea, a thou$and very beautiful Statua's, but the difficulty lieth in be- ing able to di$cover them; or we may $ay, that it is like to the prophe$ies of Abbot <I>Joachim,</I> or the an$wers of the Heathen <I>Oracles,</I> which are not to be under$tood, till after the things fore-told are come to pa$$e.</P> <P>SALV. And why do you not adde the predictions of the <I>Ge- nethliacks,</I> which are with like cleerne$$e $een after the event, in their Horo$copes, or, if you will, Configurations of the Heavens.</P> <P>SAGR. In this manner the Chymi$ts find, being led by their <marg><I>Chymi$ts inter- pret the Eables of the Poets to be $e- crets for making of Gold.</I></marg> melancholly humour, that all the $ublime$t wits of the World have writ of nothing el$e in reality, than of the way to make Gold; but, that they might tran$mit the $ecret to po$terity with- out di$covering it to the vulgar, they contrived $ome one way, and $ome another how to conceal the $ame under $everal maskes; and it would make one merry to hear their comments upon the ancient <I>Poets,</I> finding out the important mi$teries, which lie hid under their Fables; and the $ignification of the Loves of the <I>Moon,</I> and her de$cending to the Earth for <I>Endimion</I>; her di$plea$ure again$t <I>Acteon,</I> and what was meant by <I>Jupiters</I> turning him$elf into a $howre of <I>Gold</I>; and into flames of fire; and what great $ecrets of Art are conteined in that <I>Mercury</I> the <I>Interpreter</I>; in tho$e thefts of <I>Pluto</I>; and in tho$e <I>Branches</I> of <I>Gold.</I></P> <P>SIMPL. I believe, and in part know, that there want not in the World very extravagant heads, the vanities of whom ought not to redound to the prejudice of <I>Ari$totle,</I> of whom my thinks you $peak $ometimes with too little re$pect, and the onely antiquity and bare name that he hath acquired in the opinions of $o many famous men, $hould $uffice to render him honourable with all that profe$$e them$elves learned.</P> <P>SALV. You $tate not the matter rightly, <I>Simplicius</I>; There are $ome of his followers that fear before they are in danger, who give us occa$ion, or, to $ay better, would give us cau$e to e$teem him le$$e, $hould we con$ent to applaud their <I>Capricio's.</I> <marg><I>Some of</I> Ari$to- tles <I>Sectators im- pare the reputation of their Ma$ter, in going about to en- han$e it.</I></marg> And you, pray you tell me, are you for your part $o $imple, as not to know that had <I>Arictotle</I> been pre$ent, to have heard the Doctor that would have made him Author of the <I>Tele$cope,</I> he would have been much more di$plea$ed with him, than with tho$e, who laught at the Doctor and his Comments? Do you que$tion <foot>whe-</foot> <p n=>94</p> whether <I>Ari$totle,</I> had he but $een the novelties di$covered in Hea- ven, would not have changed his opinion, amended his Books, and embraced the more $en$ible Doctrine; rejecting tho$e $illy Gulls, which too $crupulou$ly, go about to defend what ever he hath $aid; not con$idering, that if <I>Ari$totle</I> were $uch a one as they fancy him to them$elves, he would be a man of an untracta- ble wit, an ob$tinate mind, a barbarous $oul, a $tubborn will, that accounting all men el$e but as $illy $heep, would have his Oracles preferred before the Sen$es, Experience, and Nature her $elf? They are the Sectators of <I>Aristotle</I> that have given him this Authority, and not he that hath u$urped or taken it upon him; and becau$e it is more ea$ie for a man to $culk under anothers $hield than to $hew him$elf openly, they tremble, and are affraid to $tir one $tep from him; and rather than they will admit $ome alterations in the Heaven of <I>Ari$totle,</I> they will impertinently de- ny tho$e they behold in the Heaven of <I>Nature.</I></P> <P>SAGR. The$e kind of Drolleries put me in mind of that Statu- <marg><I>A ridiculous pa$$age of a certain Statuary.</I></marg> ary which having reduced a great piece of Marble to the Image of an <I>Hercules,</I> or a thundring <I>Jupiter,</I> I know not whether, and given it with admirable Art $uch a vivacity and threatning fury, that it moved terror in as many as beheld it; he him$elf began al$o to be affraid thereof, though all its $prightfulne$$e, and life was his own workman$hip; and his affrightment was $uch, that he had no longer the courage to affront it with his Chizzels and Mallet.</P> <P>SALV. I have many times wondered how the$e nice maintain- ers of what ever fell from <I>Ari$totle,</I> are not aware how great a pre- judice they are to his reputation and credit; and how that the more they go about to encrea$e his Authority, the more they dimini$h it; for while$t I $ee them ob$tinate in their attempts to maintain tho$e Propo$itions which I palpably di$cover to be manife$tly fal$e; and in their de$ires to per$wade me that $o to do, is the part of a Philo$opher; and that <I>Ari$totle</I> him$elf would do the $ame, it much abates in me of the opinion that he hath rightly philo$ophated about other conclu$ions, to me more ab$tru$e: for if I could $ee them concede and change opinion in a manife$t truth, I would believe, that in tho$e in which they $hould per$i$t, they may have $ome $olid demon$trations to me un- known, and unheard of.</P> <P>SAGR. Or when they $hould be made to $ee that they have ha- zarded too much of their own and <I>Ari$totle</I>'s repuatation in con- fe$$ing, that they had not under$tood this or that conclu$ion found out by $ome other man; would it not be a le$s evil for them to $eek for it among$t his Texts, by laying many of them together, according to the art intimated to us by <I>Simplicius</I>? for if his <foot>works</foot> <p n=>95</p> works contain all things knowable, it mu$t follow al$o that they may be therein di$covered.</P> <P>SALV. Good <I>Sagredus,</I> make no je$t of this advice, which me thinks you rehear$e in too Ironical a way; for it is not long $ince that a very eminent Philo$opher having compo$ed a Book <I>de animà,</I> wherein, citing the opinion of <I>Ari$totle,</I> about its being or not be- ing immortal, he alledged many Texts, (not any of tho$e hereto- fore quoted by <I>Alexander ab Alexandro</I>: for in tho$e he $aid, that <I>Ari$totle</I> had not $o much as treated of that matter, much le$s de- termined any thing pertaining to the $ame, but others) by him$elf found out in other more ab$tru$e places, which tended to an er- roneous $en$e: and being advi$ed, that he would find it an hard matter to get a Licence from the Inqui$itors, he writ back unto <marg><I>A brave re$olu- tion of a certain</I> Peripatetick <I>Phi- lo$opher.</I></marg> his friend, that he would notwith$tanding, with all expedition procure the $ame, for that if no other ob$tacle $hould interpo$e, he would not much $cruple to change the Doctrine of <I>Ari$totle,</I> and with other expo$itions, and other Texts to maintain the con- trary opinion, which yet $hould be al$o agreeable to the $en$e of <I>Ari$totle.</I></P> <P>SAGR. Oh mo$t profound Doctor, this! that can command me that I $tir not a $tep from <I>Ari$totle,</I> but will him$elf lead him by the no$e, and make him $peak as he plea$eth. See how much it importeth to learn to take <I>Time</I> by the <I>Fore-top.</I> Nor is it $ea$onable to have to do with <I>Hercules,</I> whil'$t he is en- raged, and among$t the Furies, but when he is telling merry tales among$t the <I>Meonion Damo$els.</I> Ah, unheard of $ordidne$$e of <marg><I>The $ervile $pi- rit of $ome of</I> Ari- $totles <I>followers.</I></marg> $ervile $ouls! to make them$elves willing $laves to other mens opi- nions; to receive them for inviolable Decrees, to engage them- $elves to $eem $atisfied and convinced by arguments, of $uch effi- cacy, and $o manife$tly concludent, that they them$elves can- not certainly re$olve whether they were really writ to that pur- po$e, or $erve to prove that a$$umption in hand, or the contrary. But, which is a greater madne$$e, they are at variance among$t them$elves, whether the Author him$elf hath held the affirmative part, or the negative. What is this, but to make an Oracle of a Log, and to run to that for an$wers, to fear that, to reverence and adore that?</P> <P>SIMPL. But in ca$e we $hould recede from <I>Aristotle,</I> who have we to be our Guid in Philo$ophy? Name you $ome Author.</P> <P>SALV. We need a Guid in unknown and uncouth wayes, but in champion places, and open plains, the blind only $tand in need of a Leader; and for $uch, it is better that they $tay at home. But he that hath eyes in his head, and in his mind, him $hould a man choo$e for his Guid. Yet mi$take me not, thinking that I <marg><I>Too clo$e adhe- ring to</I> Ari$totle <I>is blameable.</I></marg> $peak this, for that I am again$t hearing of <I>Ari$totle</I>; for on the <foot>con-</foot> <p n=>96</p> contrary, I commend the reading, and diligently $tudying of him; and onely blame the $ervile giving ones $elf up a $lave unto him, $o, as blindly to $ub$cribe to what ever he delivers, and without $earch of any farther rea$on thereof, to receive the $ame for an in- violable decree. Which is an abu$e, that carrieth with it ano- ther great inconvenience, to wit, that others will no longer take pains to under$tand the validity of his Demon$trations. And what is more $hameful, than in the midde$t of publique di$putes, while$t one per$on is treating of demon$trable conclu$ions, to hear aother interpo$e with a pa$$age of <I>Ari$totle,</I> and not $el- dome writ to quite another purpo$e, and with that to $top the mouth of his opponent? But if you will continue to $tudy in this manner, I would have you lay a$ide the name of Philo$ophers; <marg><I>It is not ju$t, that tho$e who never philo$ophate, $hould u$urp the title of Philo$ophers.</I></marg> and call your $elves either Hi$torians or Doctors of Memory, for it is not $it, that tho$e who never philo$ophate, $hould u$urp the honourable title of Philo$ophers. But it is be$t for us to re- turn to $hore, and not lanch farther into a boundle$$e Gulph, out of which we $hall not be able to get before night. Therefore <I>Simplicius,</I> come either with arguments and demon$trations of your own, or of <I>Ari$totle,</I> and bring us no more Texts and na- <marg><I>The Sen$ible World.</I></marg> ked authorities, for our di$putes are about the Sen$ible World, and not one of Paper. And fora$much as in our di$cour$es ye$ter- day, we retrein'd the Earth from darkne$$e, and expo$ed it to the open skie, $hewing, that the attempt to enumerate it among$t tho$e which we call Cœle$tial bodies, was not a po$ition $o foil'd, and vanqui$h't, as that it had no life left in it; it followeth next, that we proceed to examine what probability there is for holding of it fixt, and wholly immoveable, <I>$cilicet</I> as to its entire Globe, what likelyhood there is for making it moveable with $ome motion, and of what kind that may be. And fora$much as in this $ame que$tion I am ambiguous, and <I>Simplicius</I> is re$olute, as likewi$e <I>Ari$totle</I> for the opinion of its immobility, he $hall one by one produce the arguments in favour of their opinion, and I will al- ledge the an$wers and rea$ons on the contrary part; and next <I>Sa- gredus</I> $hall tell us his thoughts, and to which $ide he finds him- $elf inclined.</P> <P>SAGR. Content; provided alwayes that I may re$erve the li- berty to my $elf of alledging what pure natural rea$on $hall $ome- times dictate to me.</P> <P>SALV. Nay more, it is that which I particularly beg of you; for, among$t the more ea$ie, and, to $o $peak, material con$idera- tions, I believe there are but few of them that have been omit- ted by Writers, $o that onely $ome of the more $ubtle, and re- mote can be de$ired, or wanting; and to inve$tigate the$e, what other ingenuity can be more $it than that of the mo$t acute and piercing wit of <I>Sagredus</I>?</P> <foot>SAGR.</foot> <p n=>97</p> <P>SAGR. I am what ever plea$eth <I>Salviatus,</I> but I pray you, let us not $ally out into another kind of digre$$ion complemental; for at this time I am a Philo$opher, and in the Schools, not in the Court.</P> <P>SALV. Let our contemplation begin therefore with this con$i- deration, that what$oever motion may be a$cribed to the Earth, it is nece$$ary that it be to us, (as inhabitants upon it, and con$e- quently partakers of the $ame) altogether imperceptible, and as if it were not at all, $o long as we have regard onely to terre$trial things; but yet it is on the contrary, as nece$$ary that the $ame <marg><I>The motions of the Earth are im- perceptible to its inhabitants.</I></marg> motion do $eem common to all other bodies, and vi$ible ob- jects, that being $eparated from the Earth, participate not of the $ame. So that the true method to find whether any kind of motion may be a$cribed to the Earth, and that found, to know what it is, is to con$ider and ob$erve if in bodies $eparated from the Earth, one may di$cover any appearance of motion, which e- <marg><I>The Earth can have no other mo- tions, than tho$e which to us appear commune to all the rest of the Vni- ver$e, the Earth excepted.</I></marg> qually $uiteth to all the re$t; for a motion that is onely $een, <I>v. gr.</I> in the <I>Moon,</I> and that hath nothing to do with <I>Venus</I> or <I>Jupiter,</I> or any other Stars, cannot any way belong to the Earth, or to any other $ave the Moon alone. Now there is a mo$t general and grand motion above all others, and it is that by which the Sun, <marg><I>The Diurnal Mo- tion, $eemeth com- mune to all the V- niver$e, $ave onely the Earth excepted.</I></marg> the Moon, the other Planets, and the Fixed Stars, and in a word, the whole Univer$e, the Earth onely excepted, appeareth in our thinking to move from the Ea$t towards the We$t, in the $pace of twenty four hours; and this, as to this fir$t appearance, hath no ob$tacle to hinder it, that it may not belong to the Earth alone, as well as to all the World be$ides, the Earth excepted; for the $ame a$pects will appear in the one po$ition, as in the other. Hence it is that <I>Ari$totle</I> and <I>Ptolomy,</I> as having hit upon this con- <marg>Ari$totle <I>and</I> Ptolomy <I>argue a- gain$t the Diur- nal Motion attri- buted to the Earth.</I></marg> $ideration, in going about to prove the Earth to be immoveable, argue not again$t any other than this <I>Diurnal</I> Motion; $ave onely that <I>Ari$totle</I> hinteth $omething in ob$cure terms again$t another Motion a$cribed to it by an <I>Ancient,</I> of which we $hall $peak in its place.</P> <P>SAGR. I very well perceive the nece$$ity of your illation: but I meet with a doubt which I know not how to free my $elf from, and this it is, That <I>Copernicus</I> a$$igning to the Earth another mo- tion be$ide the Diurnal, which, according to the rule even now laid down, ought to be to us, as to appearance, imperceptible in the Earth, but vi$ible in all the re$t of the World; me thinks I may nece$$arily infer, either that he hath manife$tly erred in a$$igning the Earth a motion, to which there appears not a general corre- $pondence in Heaven; or el$e that if there be $uch a congruity therein, <I>Ptolomy</I> on the other hand hath been deficient in not con- futing this, as he hath done the other.</P> <foot>N SALV.</foot> <p n=>98</p> <P>SALV. You have good cau$e for your doubt: and when we come to treat of the other Motion, you $hall $ee how far <I>Coper- nicus</I> excelled <I>Ptolomey</I> in clearne$s and $ublimity of wit, in that he $aw what the other did not, I mean the admirable harmony wherein that Motion agreed with all the other Cœle$tial Bodies. But for the pre$ent we will $u$pend this particular, and return to our fir$t con$ideration; touching which I will proceed to propo$e (begining with things more general) tho$e rea$ons which $eem to favour the mobility of the Earth, and then wait the an$wers which <marg><I>Why the diurnal motion more pro- bably $hould belong to the Earth, than to the re$t of the Vniver$e.</I></marg> <I>Simplicius</I> $hall make thereto. And fir$t, if we con$ider onely the immen$e magnitude of the Starry Sphere, compared to the $malne$s of the Terre$trial Globe, contained therein $o many mil- lions of times; and moreover weigh the velocity of the motion which mu$t in a day and night make an entire revolution thereof, I cannot per$wade my $elf, that there is any man who believes it more rea$onable and credible, that the Cœle$tial Sphere turneth round, and the Terre$trial Globe $tands $till.</P> <P>SAGR. If from the univer$ality of effects, which may in nature have dependence upon $uch like motions, there $hould indifferent- ly follow all the $ame con$equences to an hair, a$well in one <I>Hypo- the$is</I> as in the other; yet I for my part, as to my fir$t and general apprehen$ion, would e$teem, that he which $hould hold it more ra- tional to make the whole Univer$e move, and thereby to $alve the Earths mobility, is more unrea$onable than he that being got to the top of your Turret, $hould de$ire, to the end onely that he might behold the City, and the Fields about it, that the whole Country might turn round, that $o he might not be put to the trouble to $tir his head. And yet doubtle$s the advantages would be many and great which the <I>Copernican Hypothe$is</I> is attended with, above tho$e of the <I>Ptolomaique,</I> which in my opinion re- $embleth, nay $urpa$$eth that other folly; $o that all this makes me think that far more probable than this. But haply <I>Ari$totle, Ptolomey,</I> and <I>Simplicius</I> may find the advantages of their Sy- $teme, which they would do well to communicate to us al$o, if any $uch there be; or el$e declare to me, that there neither are or can be any $uch things.</P> <P>SALV. For my part, as I have not been able, as much as I have thought upon it, to find any diver$ity therein; $o I think I have found, that no $uch diver$ity can be in them: in $o much that I <marg><I>Motion, as to the things that equally move thereby, is as of it never were, & $o far operates as it hath relation to things deprived of motion.</I></marg> e$teem it to no purpo$e to $eek farther after it. Therefore ob- $erve: Motion is $o far Motion, and as Motion operateth, by how far it hath relation to things which want Motion: but in tho$e things which all equally partake thereof it hath nothing to do, and is as if it never were. And thus the Merchandi$es with which a $hip is laden, $o far move, by how far leaving <I>London,</I> they pa$s <foot>by</foot> <p n=>99</p> by <I>France, Spain, Italy,</I> and $ail to <I>Aleppo,</I> which <I>London, France, Spain &c.</I> $tand $till, not moving with the $hip: but as to the Che$ts, Bales and other Parcels, wherewith the $hip is $tow'd and and laden, and in re$pect of the $hip it $elf, the Motion from <I>Lon- don</I> to <I>Syria</I> is as much as nothing; and nothing-altereth the re- lation which is between them: and this, becau$e it is common to all, and is participated by all alike: and of the Cargo which is in the $hip, if a Bale were romag'd from a Che$t but one inch onely, this alone would be in that Cargo, a greater Motion in re$pect of the Che$t, than the whole Voyage of above three thou$and miles, made by them as they were $tived together.</P> <P>SIMPL. This Doctrine is good, $ound, and altogether <I>Peri- patetick.</I></P> <P>SALV. I hold it to be much more antient: and $u$pect that <I>A-</I> <marg><I>A propo$ition ta- ken by</I> Ari$totle <I>from the Antients, but $omewhat al- tered by him.</I></marg> <I>ri$totle</I> in receiving it from $ome good School, did not fully under- $tand it, and that therefore, having delivered it with $ome altera- tion, it hath been an occa$ion of confu$ion among$t tho$e, who would defend whatever he $aith. And when he writ, that what- $oever moveth, doth move upon $omething immoveable, I $uppo$e that he equivocated, and meant, that whatever moveth, moveth in re$pect to $omething immoveable; which propo$ition admitteth no doubt, and the other many.</P> <P>SAGR. Pray you make no digre$$ion, but proceed in the di$- $ertation you began.</P> <P>SALV. It being therefore manife$t, that the motion which is <marg><I>The fir$t di$cour$e to prove that the diurnal motion be- longs to the Earth.</I></marg> common to many moveables, is idle, and as it were, null as to the relation of tho$e moveables between them$elves, becau$e that a- mong them$elves they have made no change: and that it is ope- rative onely in the relation that tho$e moveables have to other things, which want that motion, among which the habitude is changed: and we having divided the Univer$e into two parts, one of which is nece$$arily moveable, and the other immoveable; for the obtaining of what$oever may depend upon, or be required from $uch a motion, it may as well be done by making the Earth alone, as by making all the re$t of the World to move: for that the operation of $uch a motion con$i$ts in nothing el$e, $ave in the relation or habitude which is between the Cœle$tial Bodies, and the Earth, the which relation is all that is changed. Now if for the obtaining of the $ame effect <I>ad unguem,</I> it be all one whe- ther the Earth alone moveth, the re$t of the Univer$e $tanding $till; or that, the Earth onely $tanding $till, the whole Univer$e <marg><I>Nature never doth that by many things, which may be done by a few.</I></marg> moveth with one and the $ame motion; who would believe, that Nature (which by common con$ent, doth not that by many things, which may be done by few) hath cho$en to make an innumerable number of mo$t va$t bodies move, and that with an unconceivable <foot>N 2 velocity,</foot> <p n=>100</p> velocity, to perform that, which might be done by the moderate motion of one alone about its own Centre?</P> <P>SIMPL. I do not well under$tand, how this grand motion $ig- ni$ieth nothing as to the Sun, as to the Moon, as to the other Pla- nets, and as to the innumerable multitude of fixed $tars: or why you $hould $ay that it is to no purpo$e for the Sun to pa$s from one Meridian to another; to ri$e above this Horizon, to $et beneath that other; to make it one while day, another while night: the like variations are made by the Moon, the other Planets, and the fixed $tars them$elves.</P> <P>SALV. All the$e alterations in$tanced by you, are nothing, $ave onely in relation to the Earth: and that this is true, do but i- <marg><I>The diurnal mo- tion cau$eth no mutation among$t the Cœle$tial Bo- dies, but all chan- ges have relation to the Earth.</I></marg> magine the Earth to move, and there will be no $uch thing in the World as the ri$ing or $etting of the Sun or Moon, nor Horizons, nor Meridians, nor days, nor nights; nor, in a word, will $uch a motion cau$e any mutation between the Moon and Sun, or any other $tar what$oever, whether fixed or erratick; but all the$e changes have relation to the Earth: which all do yet in $um import no other than as if the Sun $hould $hew it $elf now to <I>China,</I> anon to <I>Per$ia,</I> then to <I>Egypt, Greece, France, Spain, A- merica, &c.</I> and the like holdeth in the Moon, and the re$t of the Cœle$tial Bodies: which $elf $ame effect falls out exactly in the $ame manner, if, without troubling $o great a part of the Univer$e, <marg><I>A $ccond con- firmation that the diurnal motion be- longs to the Earth.</I></marg> the Terre$trial Globe be made to revolve in it $elf. But we will augment the difficulty by the addition of this other, which is a very great one, namely, that if you will a$cribe this <I>Great</I> Motion to Heaven, you mu$t of nece$$ity make it contrary to the particular motion of all the Orbs of the Planets, each of which without controver$ie hath its peculiar motion from the We$t towards the Ea$t, and this but very ea$ie and moderate: and then you make them to be hurried to the contrary part, <I>i. e.</I> from Ea$t to We$t, by this mo$t furious diurnal motion: whereas, on the contrary, making the Earth to move in it $elf, the contrariety of motions is taken away, and the onely motion from We$t to Ea$t is accom- modated to all appearances, and exactly $atisfieth every <I>Phœno- menon.</I></P> <P>SIMPL. As to the contrariety of Motions it would import lit- <marg><I>Circular moti- ons are not contra- ry, according to</I> Ari$totle.</marg> tle, for <I>Ari$totle</I> demon$trateth, that circular motions, are not con- trary to one another; and that theirs cannot be truly called con- trariety.</P> <P>SALV. Doth <I>Ari$totle</I> demon$trate this, or doth he not rather barely affirm it, as $erving to $ome certain de$ign of his? If con- traries be tho$e things, that de$troy one another, as he him$elf affirmeth, I do not $ee how two moveables that encounter each other in a circular line, $hould le$$e prejudice one another, than if they interfered in a right line.</P> <foot>SAGR.</foot> <p n=>101</p> <P>SAGR. Hold a little, I pray you. Tell me <I>Simplicius,</I> when two Knights encounter each other, tilting in open field, or when two whole Squadrons, or two Fleets at Sea, make up to grapple, and are broken and $unk, do you call the$e encounters contrary to one another?</P> <P>SIMPL. Yes, we $ay they are contrary.</P> <P>SAGR. How then, is there no contrariety in circular motions. The$e motions, being made upon the $uper$icies of the Earth or Water, which are, as you know, $pherical, come to be circular. Can you tell, <I>Simplicius,</I> which tho$e circular motions be, that are not contrary to each other? They are (if I mi$take not) tho$e of two circles, which touching one another without, one thereof being turn'd round, naturally maketh the other move the contra- ry ^{*} way; but if one of them $hall be within the other, it is im- <marg>As you $ee in a Mill, wherein the implicated cogs $et the wheels on mo- ving.</marg> po$$ible that their motion being made towards different points, they $hould not ju$tle one another.</P> <P>SALV. But be they contrary, or not contrary, the$e are but alterations of words; and I know, that upon the matter, it would be far more proper and agreeable with Nature, if we could $alve all with one motion onely, than to introduce two that are (if you will not call them contrary) oppo$ite; yet do I not cen$ure this introduction (of contrary motions) as impo$$ible; nor pretend I from the denial thereof, to inferre a nece$$ary Demon$tration, but onely a greater probability, of the other. A third rea$on <marg><I>A third confir- mation of the $ame Doctrine.</I></marg> which maketh the <I>Ptolomaique Hypothe$is</I> le$$e probable is, that it mo$t unrea$onably confoundeth the order, which we a$$uredly $ee to be among$t tho$e Cœle$tial Bodies, the circumgyration of which is not que$tionable, but mo$t certain. And that Order is, <marg><I>The greater Orbs make their conver- $ions in greater times.</I></marg> that according as an Orb is greater, it fini$heth its revolution in a longer time, and the le$$er, in $horter. And thus <I>Saturn</I> de$cri- bing a greater Circle than all the other Planets, compleateth the $ame in thirty yeares: <I>Jupiter</I> fini$heth his; that is le$$e, in twelve years: <I>Mars</I> in two: The Moon runneth thorow hers, $o much le$$e than the re$t, in a Moneth onely. Nor do we le$$e $en$ibly $ee that of the <I>Medicean Stars,</I> which is neare$t to <I>Ju-</I> <marg><I>The times of the</I> Medicean <I>Planets conver$ions.</I></marg> <I>piter,</I> to make its revolution in a very $hort time, that is, in four and forty hours, or thereabouts, the next to that in three dayes and an half, the third in $even dayes, and the mo$t remote in $ixteen. And this rate holdeth well enough, nor will it at all alter, while$t we a$$ign the motion of 24 hours to the Terre$trial Globe, for it to move round its own center in that time; but if you would have the Earth immoveable, it is nece$$ary, that when you have pa$t from the $hort period of the Moon, to the others $ucce$$ively bigger, until you come to that of <I>Mars</I> in two years, and from thence to that of the bigger Sphere of <I>Jupiter</I> in twelve years, and <foot>from</foot> <p n=>102</p> from this to the other yet bigger of <I>Saturn,</I> who$e period is of thirty years, it is nece$$ary, I $ay, that you pa$$e to another Sphere incomparably greater $till than that, and make this to ac- <marg><I>The motion of</I> 24 <I>hours a$cribed to the highe$t Sphere di$orders the period of the inferiour.</I></marg> compli$h an entire revolution in twenty four hours. And this yet is the lea$t di$order that can follow. For if any one $hould pa$$e from the Sphere of <I>Saturn</I> to the Starry Orb, and make it $o much bigger than that of <I>Saturn,</I> as proportion would require, in re$pect of its very $low motion, of many thou$ands of years, then it mu$t needs be a <I>Salt</I> much more ab$urd, to skip from this to another bigger, and to make it convertible in twenty four hours. But the motion of the Earth being granted, the order of the pe- riods will be exactly ob$erved, and from the very $low Sphere of <I>Saturn,</I> we come to the fixed Stars, which are wholly immovea- <marg><I>The fourth Con- firmation.</I></marg> ble, and $o avoid a fourth difficulty, which we mu$t of nece$$ity ad- mit, if the Starry Sphere be $uppo$ed moveable, and that is the <marg><I>Great di$parity among$t the moti- ons of the particu- lar fixed $tars, if their Sphere be moveable.</I></marg> immen$e di$parity between the motions of tho$e $tars them$elves; of which $ome would come to move mo$t $wiftly in mo$t va$t cir- cles, others mo$t $lowly in circles very $mall, according as tho$e or the$e $hould be found nearer, or more remote from the Poles; which $till is accompanied with an inconvenience, as well becau$e we $ee tho$e, of who$e motion there is no que$tion to be made, to move all in very immen$e circles; as al$o, becau$e it $eems to be an act done with no good con$ideration, to con$titute bodies, that are de$igned to move circularly, at immen$e di$tances from the centre, and afterwards to make them move in very $mall cir- cles. And not onely the magnitudes of the circles, and con$e- quently the velocity of the motions of the$e Stars, $hall be mo$t <marg><I>The fifth Con- firmation.</I></marg> different from the circles and motions of tho$e others, but (which $hall be the fifth inconvenience) the $elf-$ame Stars $hall $ucce$$ively vary its circles and velocities: For that <marg><I>The motions of the fixed $tars would accelerate and grow $low in divers times, if the $tarry Sphere were moueable.</I></marg> tho$e, which two thou$and years $ince were in the Equinoctial, and con$equently did with their motion de$cribe very va$t cir- cles, being in our dayes many degrees di$tant from thence, mu$t of nece$$ity become more $low of motion, and be reduced to move in le$$er circles, and it is not altogether impo$$ible but that a time may come, in which $ome of them which in aforetime had continually moved, $hall be reduced by uniting with the Pole, to a $tate of re$t, and then after $ome time of ce$$ation, $hall return to their motion again; whereas the other Stars, touching who$e motion none $tand in doubt, do all de$cribe, as hath been $aid, the great circle of their Orb, and in that maintain them$elves without any variation. The ab$urdity is farther enlarged (which <marg><I>The $ixth Con- firmatiox.</I></marg> let be the $ixth inconvenience) to him that more $eriou$ly exami- neth the thing, in that no thought can comprehend what ought to be the $olidity of that immen$e Sphere, who$e depth $o $tedfa$tly <foot>holdeth</foot> <p n=>103</p> holdeth fa$t $uch a multitude of Stars, which without ever chang- ing fite among them$elves, are with $o much concord carried a- bout, with $o great di$parity of motions. Or el$e, $uppo$ing the Heavens to be fluid, as we are with more rea$on to believe, $o as that every Star wandereth to and fro in it, by wayes of its own, what rules $hall regulate their motions, and to what pur- po$e, $o, as that being beheld from the Earth, they appear as if they were made by one onely Sphere? It is my opinion, that they might $o much more ea$ily do that, and in a more commodious manner, by being con$tituted immoveable, than by being made errant, by how much more facile it is to number the quarries in the Pavement of a <I>Piazza,</I> than the rout of boyes which run up and down upon them. And la$tly, which is the $eventh in$tance, if <marg><I>The Seventh Con- firmation.</I></marg> we atribute the Diurnal Motion to the highe$t Heaven, it mu$t be con$tituted of $uch a force and efficacy, as to carry along with it the innumerable multitude of fixed Stars, Bodies all of va$t magnitude, and far bigger than the Earth; and moreover all the Spheres of the Planets; notwith$tanding that both the$e and tho$e of their own nature move the contrary way. And be$ides all this, it mu$t be granted, that al$o the Element of Fire, and the great- er part of the Air, are likewi$e forcibly hurried along with the re$t, and that the $ole little Globe of the Earth pertinaciou$ly $tands $till, and unmoved again$t $uch an impul$e; a thing, which in my thinking, is very difficult; nor can I $ee how the Earth, a pendent body, and equilibrated upon its centre, expo$ed indif- <marg><I>The Earth a pendent Body, and equilibrated in a fluid</I> Medium <I>$eems unable to re$i$t the rapture of the Diurnal Motion.</I></marg> ferently to either motion or re$t, and environed with a liquid <I>am- bient,</I> $hould not yield al$o as the re$t, and be carried about. But we find none of the$e ob$tacles in making the Earth to move; a $mall body, and in$en$ible, compared to the Univer$e, and therefore unable to offer it any violence.</P> <P>SAGR. I find my fancy di$turbed with certain conjectures $o con- fu$edly $prung from your later di$cour$es; that, if I would be ena- bled to apply my $elf with atention to what followeth, I mu$t of ne- ce$$ity attempt whether I can better methodize them, and gather thence their true con$truction, if haply any can be made of them; and peradventure, the proceeding by interrogations may help me the more ea$ily to expre$$e my $elf. Therefore I demand fir$t of <I>Sim- plicius,</I> whether he believeth, that divers motions may natural- ly agree to one and the $ame moveable body, or el$e that it be requi$ite its natural and proper motion be onely one.</P> <P>SIMPL. To one $ingle moveable, there can naturally agree <marg><I>A $ingle move- able hath but onely one natural moti- on, and all the re$t are by partici- pation.</I></marg> but one $ole motion, and no more; the re$t all happen acciden- tally and by participation; like as to him that walketh upon the Deck of a Ship, his proper motion is that of his walk, his motion by participation that which carrieth him to his Port, whither he <foot>would</foot> <p n=>104</p> would never with his walking have arrived, if the Ship with its motion had not wafted him thither.</P> <P>SAGR. Tell me $econdly. That motion, which is communi- cated to any moveable by participation, while$t it moveth by it $elf, with another motion different from the participated, is it nece$$ary, that it do re$ide in $ome certain $ubject by it $elf, or el$e can it $ub$i$t in nature alone, without other $upport.</P> <P>SIMPL. <I>Ari$totle</I> giveth you an an$wer to all the$e que$tions, <marg><I>Motion cannot be made without its moveable $ub- ject.</I></marg> and tels you, that as of one $ole moveable the motion is but one; $o of one $ole motion the moveable is but one; and con$equent- ly, that without the inherence in its $ubject, no motion can ei- ther $ub$i$t, or be imagined.</P> <P>SAGR. I would have you tell me in the third place, whether you beblieve that the Moon and the other Planets and Cœle$tial bodies, have their proper motions, and what they are.</P> <P>SIMPL. They have $o, and they be tho$e according to which they run through the Zodiack, the Moon in a Moneth, the Sun in a Year, <I>Mars</I> in two, the Starry Sphere in tho$e $o many thou- $and. And the$e are their proper, or natural motions.</P> <P>SAGR. But that motion wherewith I $ee the fixed Stars, and with them all the Planets go unitedly from Ea$t to We$t, and re- turn round to the Ea$t again in twenty four hours, how doth it agree with them?</P> <P>SIMPL. It $uiteth with them by participation.</P> <P>SAGR. This then re$ides not in them, and not re$iding in them, nor being able to $ub$i$t without $ome $ubject in which it is re$ident, it mu$t of force be the proper and natural motion of $ome other Sphere.</P> <P>SIMPL. For this purpo$e A$tronomers, and Philo$ophers have found another high Sphere, above all the re$t, without Stars, to which Natural agreeth the Diurnal Motion; and this they call the <I>Primum mobile</I>; the which carrieth along with it all the in- feriour Spheres, contributing and imparting its motion to them.</P> <P>SAGR. But when, without introducing other Spheres unknown and hugely va$t, without other motions or communicated raptures, with leaving to each Sphere its $ole and $imple motion, without intermixing contrary motions, but making all turn one way, as it is nece$$ary that they do, depending all upon one $ole principle, all things proceed orderly, and corre$pond with mo$t perfect har- mony, why do we reject this <I>Phœnomenon,</I> and give our a$$ent to tho$e prodigious and laborious conditions?</P> <P>SIMPL. The difficulty lyeth in finding out this $o natural and expeditious way.</P> <foot>SAGR.</foot> <p n=>105</p> <P>SAGR. In my judgment this is found. Make the Earth the <I>Primum mobile,</I> that is, make it turn round its own <I>axis</I> in twenty four hours, and towards the $ame point with all the other Spheres; and without participating this $ame motion to any other Planet or Star, all $hall have their ri$ings, $ettings, and in a word, all their other appearances.</P> <P>SIMPL. The bu$ine$s is, to be able to make the Earth move without athou$and inconveniences.</P> <P>SALV. All the inconveniences $hall be removed as fa$t as you propound them: and the things $poken hitherto are onely the primary and more general inducements which give us to believe that the diurnal conver$ion may not altogether without probabi- lity be applyed to the Earth, rather than to all the re$t of the U- niver$e: the which inducements I impo$e not upon you as invio- lable Axioms, but as hints, which carry with them $omewhat of likelihood. And in regard I know very well, that one $ole ex- <marg><I>One $ingle ex- periment, or $ound demon$tration bat- tereth down all ar- guments meerly probable.</I></marg> periment, or concludent demon$tration, produced on the contrary part, $ufficeth to batter to the ground the$e and a thou$and other probable Arguments; therefore it is not fit to $tay here, but proceed forwards and hear what <I>Simplicius</I> an$wereth, and what greater probabilities, or $tronger arguments he alledgeth on the contrary.</P> <P>SIMPL. I will fir$t $ay $omething in general upon all the$e con- $iderations together, and then I will de$cend to $ome particulars. It $eems that you univer$ally bottom all you $ay upon the greater $implicity and facility of producing the $ame effects, whil$t you hold, that as to the cau$ing of them, the motion of the Earth a- lone, $erveth <I>as well</I> as that of all the re$t of the World, the Earth deducted: but as to the operations, you e$teem that much ea$ier than this. To which I reply, that I am al$o of the $ame opinion, $o long as I regard my own not onely finite, but feeble power; but having a re$pect to the $trength of the <I>Mover,</I> which is in- finite, its no le$$e ea$ie to move the Univer$e, than the Earth, yea than a $traw. And if his power be infinite, why $hould he not <marg><I>Of an infinite power one would think a greater part $hould rather be imploy'd than a le$$e.</I></marg> rather exerci$e a greater part thereof than a le$$e? Therefore, I hold that your di$cour$e in general is not convincing.</P> <P>SALV. If I had at any time $aid, that the Univer$e moved not for want of power in the <I>Mover,</I> I $hould have erred, and your reproof would have been $ea$onable; and I grant you, that to an infinite power, it is as ea$ie to move an hundred thou$and, as one. But that which I did $ay, concerns not the Mover, but one- ly hath re$pect to the Moveables; and in them, not onely to their re$i$tance, which doubtle$$e is le$$er in the Earth, than in the Univer$e; but to the many other particulars, but even now con$idered. As to what you $ay in the next place, that of an in- finite power it is better to exerci$e a great part than a $mall: I an- <foot>O $wer,</foot> <p n=>106</p> $wer, that of infinite one part is not greater than another, $ince <marg><I>Of infinity one part is no bigger than auother, al- though they are comparatively un- equal.</I></marg> both are infinite; nor can it be $aid, that of the infinite number, an hundred thou$and is a greater part than two, though that be fifty thou$and times greater than this; and if to the moving of the Univer$e there be required a finite power, though very great in compari$on of that which $ufficeth to move the Earth onely; yet is there not implied therein a greater part of the infinite power, nor is that part le$$e infinite which remaineth unimploy'd. So that to apply unto a particular effect, a little more, or a little le$$e power, importeth nothing; be$ides that the operation of $uch vertue, hath not for its bound or end the Diurnal Motion onely; but there are $everal other motions in the World, which we know of, and many others there may be, that are to us unknown. Therefore if we re$pect the Moveables, and granting it as out of que$tion, that it is a $horter and ea$ier way to move the Earth, than the Univer$e; and moreover, having an eye to the $o many other abreviations, and facilities that onely this way are to be ob- tained, an infallible Maxime of <I>Ari$totle,</I> which he teacheth us, that, <I>fru$tra fit per plura, quod pote$t fieri per pauciora,</I> ren- dereth it more probable that the Diurnal Motion belongs to the Earth alone, than to the Univer$e, the Earth $ubducted.</P> <P>SIMPL. In reciting that Axiom, you have omitted a $mall clau$e, which importeth as much as all the re$t, e$pecially in our ca$e, that is to $ay, the words <I>æquè bene.</I> It is requi$ite therefore to examine whether this <I>Hypothe$is</I> doth <I>equally well</I> $atisfie in all particulars, as the other.</P> <P>SALV. The knowledg whether both the$e po$itions do <I>æquè bene,</I> $atisfie, may be comprehended from the particular exami- nation of the appearances which they are to $atisfie; for hitherto we have di$cour$ed, and will continue to argue <I>ex hypothe$i,</I> namely, $uppo$ing, that as to the $atisfaction of the appearances, <marg><I>In the Axiome</I> Fru$tra fit per plu- ra, &c. <I>the addi- tion of</I> æque benè, <I>is $uperfluous.</I></marg> both the a$$umptions are equally accomodated. As to the clau$e which you $ay was omitted by me, I have more rea$on to $u$pect that it was $uperfluou$ly in$erted by you. For the expre$$ion <I>æquè bene,</I> is a relative that nece$$arily requireth two terms at lea$t, for a thing cannot have relation to its $elf, nor do we $ay, <I>v. gr.</I> re$t to be <I>equally good,</I> as re$t. And becau$e, when we $ay, <I>that is done in vain by many means, which may be done with fewer,</I> we mean, that that which is to be done, ought to be the $ame thing, not two different ones; and becau$e the $ame thing can- not be $aid to be done as well as its $elf; therefore, the addition of the Phra$e <I>æquè bene</I> is $uperfluous, and a relation, that hath but one term onely.</P> <P>SAGR. Unle$$e you will have the $ame befal us, as did ye$ter- day, let us return to our matter in hand; and let <I>Simplicius</I> be- <foot>gin</foot> <p n=>107</p> gin to produce tho$e difficulties that $eem in his opinion, to thwart this new di$po$ition of the World.</P> <P>SIMPL. That di$po$ition is not new, but very old, and that you may $ee it is $o, <I>Ari$totle</I> confuteth it; and his confutations are the$e: “Fir$t if the Earth moveth either in it felf about its <marg>Ari$totles <I>Ar- guments for the Earths quie$$ence.</I></marg> own Centre, or in an Excentrick Circle, it is nece$$ary that that $ame motion be violent; for it is not its natural motion, for if it were, each of its parts would partake thereof; but each of them moveth in a right line towards its Centre. It being therefore violent and pteternatural, it could never be perpetu- al: But the order of the World is perpetual. Therefore, <I>&c.</I> Secondly, all the other moveables that move circularly, $eem to ^{*} $tay behind, and to move with more than one motion, the <marg>* <I>Re$tino indietzo,</I> which is meant here of that moti- on which a bowl makes when its born by its by as to one $ide or other, and $o hindered in its direct motion.</marg> <I>Primum Mobile</I> excepted: Whence it would be nece$$ary that the Earth al$o do move with two motions; and if that $hould be $o, it would inevitably follow, that mutations $hould be made in the Fixed Stars, the which none do perceive; nay without any variation, the $ame Stars alwayes ri$e from towards the $ame places, and in the $ame places do $et. Thirdly, the mo- tion of the parts is the $ame with that of the whole, and natural- ly tendeth towards the Centre of the Univer$e; and for the $ame cau$e re$t, being arrived thither. He thereupon moves the que- $tion whether the motion of the parts hath a tendency to the centre of the Univer$e, or to the centre of the Earth; and conclu- deth that it goeth by proper in$tinct to the centre of the Univer$e, and <I>per accidence</I> to that of the Earth; of which point we largely di$cour$ed ye$terday. He la$tly confirmeth the $ame with a fourth argument taken from the experiment of grave bodies, which fal- ing from on high, de$cend perpendicularly unto the Earths$urface; and in the $ame manner <I>Projections</I> $hot perpendicularly upwards, do by the $ame lines return perpendicularly down again, though they were $hot to a very great height. All which arguments nece$- $arily prove their motion to be towards the Centre of the Earth, which without moving at all waits for, and receiveth them. He intimateth in the la$t place that the A$tronomers alledg other rea$ons in confirmation of the $ame conclu$ions, I mean of the Earths being in the Centre of the Univer$e, and immoveable; and in$tanceth onely in one of them, to wit, that all the <I>Phæ- nomena</I> or appearances that are $een in the motions of the Stars, perfectly agree with the po$ition of the Earth in the Centre; which would not be $o, were the Earth $eated otherwi$e. The re$t produced by <I>Ptolomy</I> and the other A$tronomers, I can give you now if you plea$e, or after you have $poken what you have to $ay in an$wer to the$e of <I>Ari$totle.”</I></P> <P>SALV. The arguments which are brought upon this occa$ion <foot>O 2 are</foot> <p n=>108</p> are of two kinds: $ome have re$pect to the accidents Terre$trial, <marg><I>Two kindes of Arguments tou- ching the Earths motion or rest.</I></marg> without any relation to the Stars, and others are taken from the <I>Phænomena</I> and ob$ervations of things Cœle$tial. The arguments of <I>Ari$totle</I> are for the mo$t part taken from things neer at hand, and he leaveth the others to <I>A$tronomers</I>; and therefore it is the be$t way, if you like of it, to examine the$e taken from experi- ments touching the Earth, and then proceed to tho$e of the other kind. And becau$e <I>Ptolomy, Tycho,</I> and the other <I>A$tronomers</I> <marg><I>Arguments of</I> Ptolomy <I>and</I> Ty- cho, <I>and other per- $ons, over and a- bove tho$e of</I> Ari- $totle.</marg> and <I>Philo$ophers,</I> be$ides the arguments of <I>Ari$totle</I> by them a$$u- med, confirmed, and made good, do produce certain others; we will put them all together, that $o we may not an$wer twice to the $ame, or the like objections. Therefore <I>Simplicius,</I> choo$e whether you will recite them your $elf, or cau$e me to ea$e you of this task, for I am ready to $erve you.</P> <P>SIMPL. It is better that you quote them, becau$e, as having taken more pains in the $tudy of them, you can produce them with more readine$$e, and in greater number.</P> <marg><I>The fir$t argu- ment taken from grave bodies fal- ling from on high to the ground.</I></marg> <P>SALV. All, for the $tronge$t rea$on, alledge that of grave bo- dies, which falling downwards from on high, move by a right line, that is perpendicular to the $urface of the Earth, an argument which is held undeniably to prove that the Earth is immoveable: for in ca$e it $hould have the diurnal motion, a Tower, from the top of which a $tone is let fall, being carried along by the conver- $ion of the Earth, in the time that the $tone $pends in falling, would be tran$ported many hundred yards Ea$tward, and $o far di$tant from the Towers foot would the $tone come to ground. The which effect they back with another experiment; to wit, by let- <marg><I>Which is confir- med by the experi- ment of a body let fall from the round top of a Ship.</I></marg> ting a bullet of lead fall from the round top of a Ship, that lieth at anchor, and ob$erving the mark it makes where it lights, which they find to be neer the ^{*} partners of the Ma$t; but if the $ame bullet <marg>* That is, at the foot of the Ma$t, upon the upper deck.</marg> be let fall from the $ame place when the $hip is under $ail, it $hall light as far from the former place, as the $hip hath run in the time of the leads de$cent; and this for no other rea$on, than becau$e the natural motion of the ball being at liberty is by a right line to- <marg><I>The $econd ar- gument taken from a Projection $hot very high.</I></marg> wards the centre of the Earth. They forti$ie this argument with the experiment of a projection $hot on high at a very great di- $tance; as for example, a ball $ent out of a Cannon, erected per- pendicular to the horizon, the which $pendeth $o much time in a$- cending and falling, that in our parallel the Cannon and we both $hould be carried by the Earth many miles towards the Ea$t, $o that the ball in its return could never come neer the Peece, but <marg><I>The third argu- ment taken from the $hots of a Can- non, towards the Ea$t, and towards the West.</I></marg> would fall as far We$t, as the Earth had run Ea$t. They againe adde a third, and very evident experiment, <I>$cilicet,</I> that $hooting a bullet point blank (or as Gunners $ay, neither above nor under me- tal) out of a Culverin towards the Ea$t, and afterwards another, <foot>with</foot> <p n=>109</p> with the $ame charge, and at the $ame elevation or di$port towards the We$t, the range towards the We$t $hould be very much grea- ter then the other towards the Ea$t: for that whil'$t the ball goeth We$tward, and the Peece is carried along by the Earth Ea$tward, the ball will fall from the Peece as far di$tant as is the aggregate of the two motions, one made by it $elf towards the We$t, and the other by the Peece carried about by the Earth towards the Ea$t; and on the contrary, from the range of the ball $hot Ea$tward you are to $ub$tract the $pace the Peece moved, being carried after it. Now $uppo$e, for example, that the range of the ball $hot We$t were five miles, and that the Earth in the $ame parallel and in the time of the Bals ranging $hould remove three miles, the Ball in this ca$e would fall eight miles di$tant from the Culverin, namely, its own five We$tward, and the Culverins three miles Ea$tward: but the range of the $hot towards the Ea$t would be but two miles long, for $o much is the remainder, after you have $ub$tracted from the five miles of the range, the three miles which the Peece had moved towards the $ame part. But experience $heweth the Ranges to be equal, therefore the Culverin, and con$equently the Earth are immoveable. And the $tability of the Earth is no le$fe <marg><I>This argument is confirmed by two $hots towards the South and towards the North.</I></marg> confirmed by two other $hots made North and South; for they would never hit the mark, but the Ranges would be alwayes wide, or towards the We$t, by meanes of the remove the mark would make, being carried along with the Earth towards the Ea$t, whil'$t the ball is flying. And not onely $hots made by the Meridians, <marg><I>And it is like- wi$e confirmed by two $hots towards the Ea$t, and to- wards the We$t.</I></marg> but al$o tho$e aimed Ea$t or We$t would prove uncertain; for tho$e aim'd Ea$t would be too high, and tho$e directed We$t too low, although they were $hot point blank, as I $aid. For the Range of the Ball in both the $hots being made by the Tangent, that is, by a line parallel to the Horizon, and being that in the di- urnal motion, if it be of the Earth, the Horizon goeth continually de$cending towards the Ea$t, and ri$ing from the We$t (therefore the Oriental Stars $eem to ri$e, and the Occidental to decline) $o that the Oriental mark would de$cend below the aime, and there- upon the $hot would fly too high, and the a$cending of the We$t- ern mark would make the $hot aimed that way range too low; $o that the Peece would never carry true towards any point; and for that experience telleth us the contrary, it is requi$ite to $ay, that the Earth is immoveable.</P> <P>SIMPL. The$e are $olid rea$ons, and $uch as I believe no man can an$wer.</P> <P>SALV. Perhaps they are new to you?</P> <P>SIMPL. Really they are; and now I $ee with how many ad- mirable experiments Nature is plea$ed to favour us, wherewith to a$$i$t us in the knowledge of the Truth. Oh! how exactly one <foot>truth</foot> <p n=>110</p> truth agreeth with another, and all con$pire to render each other inexpugnable!</P> <P>SAGR. What pity it is that Guns were not u$ed in <I>Ari$totles</I> age, he would with help of them have ea$ily battered down ig- norance, and $poke without hæ$itation of the$e mundane points.</P> <P>SALV. I am very glad that the$e rea$ons are new unto you, that $o you may not re$t in the opinion of the <I>major</I> part of <I>Peripate- ticks,</I> who believe, that if any one for$akes the Doctrine of <I>Ari- $totle,</I> it is becau$e they did not under$tand or rightly apprehend his demon$trations. But you may expect to hear of other Novel- <marg>Copernicus <I>his followers are not moved through ig- nor ance of the ar- guments on the o- ther part.</I></marg> ties, and you $hall $ee the followers of this new Sy$teme produce a- gain$t them$elves ob$ervations, experiences, and rea$ons of farre greater force than tho$e alledged by <I>Aristotle, Ptolomy,</I> and other oppo$ers of the $ame conclu$ions, and by this means you $hall come to a$certain your $elf that they were not induced through want of knowledge or experience to follow that opinion.</P> <P>SAGR. It is requi$ite that upon this occa$ion I relate unto you $ome accidents that befell me, $o $oon as I fir$t began to hear $peak of this new doctrine. Being very young, and having $carcely fi- ni$hed my cour$e of Philo$ophy, which I left off, as being $et upon other employments, there chanced to come into the$e parts a cer- tain Foreigner of <I>Ro$tock,</I> who$e name, as I remember, was <I>Chri-</I> <marg>Chri$tianus Vur- $titius <I>read certain Lectures touching the opinion of</I> Co- pernicus, <I>& what en$ued thereupon.</I></marg> <I>$tianus Vur$titius,</I> a follower of <I>Copernicus,</I> who in an <I>Academy</I> made two or three Lectures upon this point, to whom many flock't as Auditors; but I thinking they went more for the novelty of the $ubject than otherwi$e, did not go to hear him: for I had conclu- ded with my $elf that that opinion could be no other than a $olemn madne$$e. And que$tioning $ome of tho$e who had been there, I perceived they all made a je$t thereof, execpt one, who told me that the bu$ine$$e was not altogether to be laugh't at, and becau$e this man was reputed by me to be very intelligent and wary, I re- pented that I was not there, and began from that time forward as oft as I met with any one of the <I>Copernican</I> per$wa$ion, to demand of them, if they had been alwayes of the $ame judgment; and of as many as I examined, I found not $o much as one, who told me not that he had been a long time of the contrary opinion, but to have changed it for this, as convinced by the $trength of the rea$ons pro- ving the $ame: and afterwards que$tioning them, one by one; to $ee whether they were well po$$e$t of the rea$ons of the other $ide; <marg><I>The followers of</I> Copernicus <I>were all fir$t again$t that opinion, but the Sectators of</I> Ari$totle <I>&</I> Pto- lomy, <I>were never of the other $ide.</I></marg> I found them all to be very ready and perfect in them; $o that I could not truly $ay, that they had took up this opinion out of ig- norance, vanity, or to $hew the acutene$$e of their wits. On the contrary, of as many of the <I>Peripateticks</I> and <I>Ptolomeans</I> as I have asked (and out of curio$ity I have talked with many) what pains they had taken in the Book of <I>Copernicus,</I> I found very <foot>few</foot> <p n=>111</p> few that had $o much as $uperficially peru$ed it; but of tho$e whom, I thought, had under$tood the $ame, not one; and more- over, I have enquired among$t the followers of the <I>Peripatetick</I> Doctrine, if ever any of them had held the contrary opinion, and likewi$e found none that had. Whereupon con$idering that there was no man who followed the opinion of <I>Copernicus,</I> that had not been fir$t on the contrary $ide, and that was not very well ac- quainted with the rea$ons of <I>Ari$totle</I> and <I>Ptolomy</I>; and, on the contrary, that there is not one of the followers of <I>Ptolomy</I> that had ever been of the judgment of <I>Copernicus,</I> and had left that, to imbrace this of <I>Ari$totle,</I> con$idering, I $ay, the$e things, I began to think, that one, who leaveth an opinion imbued with his milk, and followed by very many, to take up another owned by very few, and denied by all the Schools, and that really $eems a very great Paradox, mu$t needs have been moved, not to $ay forced, by more powerful rea$ons. For this cau$e, I am become very curious to dive, as they $ay, into the bottom of this bu$ine$$e, and account it my great good fortune that I have met you two, from whom I may without any trouble, hear all that hath been, and, haply, can be $aid on this argument, a$$uring my $elf that the $trength of your rea$ons will re$olve all $cruples, and bring me to a certainty in this $ubject.</P> <P>SIMPL. But its po$$ible your opinion and hopes may be di$ap- pointed, and that you may find your $elves more at a lo$$e in the end than you was at fir$t.</P> <P>SAGR. I am very confident that this can in no wi$e befal me.</P> <P>SIMPL. And why not? I have a manife$t example in my $elf, that the farther I go, the more I am confounded.</P> <P>SAGR. This is a $ign that tho$e rea$ons that hitherto $eemed concluding unto you, and a$$ured you in the truth of your opi- nion, begin to change countenance in your mind, and to let you by degrees, if not imbrace, at lea$t look towards the contrary te- nent; but I, that have been hitherto indifferent, do greatly hope to acquire re$t and $atisfaction by our future di$cour$es, and you will not deny but I may, if you plea$e but to hear what per$wa- deth me to this expectation.</P> <P>SIMPL. I will gladly hearken to the $ame, and $hould be no le$$e glad that the like effect might be wrought in me.</P> <P>SAGR. Favour me therefore with an$wering to what I $hall ask you. And fir$t, tell me, <I>Simplicius,</I> is not the conclu$ion, which we $eek the truth of, Whether we ought to hold with <I>Ari$totle</I> and <I>Ptolomy,</I> that the Earth onely abiding without motion in the Centre of the Univer$e, the Cœle$tial bodies all move, or el$e, Whether the Starry Sphere and the Sun $tanding $till in the Centre, <foot>the</foot> <p n=>112</p> the Earth is without the $ame, and owner of all tho$e motions that in our $eeming belong to the Sun and fixed Stars?</P> <P>SIMPL. The$e are the conclu$ions which are in di$pute.</P> <P>SAGR. And the$e two conclu$ions, are they not of $uch a na- ture, that one of them mu$t nece$$arily be true, and the other fal$e?</P> <P>SIMPL. They are $o. We are in a <I>Dilemma,</I> one part of which mu$t of nece$$ity be true, and the other untrue; for between Mo- tion and Re$t, which are contradictories, there cannot be in$tanced a third, $o as that one cannot $ay the Earth moves not, nor $tands $till; the Sun and Stars do not move, and yet $tand not $till.</P> <P>SAGR. The Earth, the Sun, and Stars, what things are they in nature? are they petite things not worth our notice, or grand and worthy of con$ideration?</P> <P>SIMPL They are principal, noble, integral bodies of the Uni- ver$e, mo$t va$t and con$iderable.</P> <P>SAGR. And Motion, and Re$t, what accidents are they in Nature?</P> <marg><I>Motion and re$t principal accidents in nature.</I></marg> <P>SIMPL. So great and principal, that Nature her $elf is defined by them.</P> <P>SAGR. So that moving eternally, and the being wholly immo- veable are two conditions very con$iderable in Nature, and indi- cate very great diver$ity; and e$pecially when a$cribed to the principal bodies of the Univer$e, from which can en$ue none but very different events.</P> <P>SIMPL. Yea doubtle$$e.</P> <P>SAGR. Now an$wer me to another point. Do you believe that in <I>Logick, Rhethorick,</I> the <I>Phy$icks, Metaphy$icks, Mathematicks,</I> and finally, in the univer$ality of Di$putations there are arguments $ufficient to per$wade and demon$trate to a per$on the fallacious, no le$$e then the true conclu$ions?</P> <marg><I>Vntruths cannot be demonstrated, as Truths are.</I></marg> <P>SIMPL. No Sir; rather I am very confident and certain, that for the proving of a true and nece$$ary conclu$ion, there are in <marg><I>For proof of true conclu$ions, many $olid arguments may be produced, but to prove a fal- $ity, none.</I></marg> nature not onely one, but many very powerfull demon$trations: and that one may di$cu$$e and handle the $ame divers and $undry wayes, without ever falling into any ab$urdity; and that the more any Sophi$t would di$turb and muddy it, the more clear would its certainty appear: And that on the contrary to make a fal$e po$i- tion pa$$e for true, and to per$wade the belief thereof, there can- not be any thing produced but fallacies, Sophi$ms, Paralogi$mes, Equivocations, and Di$cour$es vain, incon$i$tant, and full of re- pugnances and contradictions.</P> <P>SAGR. Now if eternal motion, and eternal re$t be $o princi- pal accidents of Nature, and $o different, that there can depend on them only mo$t different con$equences, and e$pecially when <foot>applied</foot> <p n=>113</p> applyed to the Sun, and to the Earth, $o va$t and famous bodies of the Univer$e; and it being, moreover, impo$$ible, that one of two contradictory Propo$itions, $hould not be true, and the other fal$e; and that for proof of the fal$e one, any thing can be pro- duced but fallacies; but the true one being per$wadeable by all kind of concluding and demon$trative arguments, why $hould you think that he, of you two, who $hall be $o fortunate as to maintain the true Propo$ition ought not to per$wade me? You mu$t $uppo$e me to be of a $tupid wit, perver$e judgment, dull mind and intellect, and of a blind rea$on, that I $hould not be able to di$tingui$h light from darkne$$e, jewels from coals, or truth from fal$hood.</P> <P>SIMPL. I tell you now, and have told you upon other occa$ions, that the be$t Ma$ter to teach us how to di$cern So- phi$mes, Paralogi$mes, and other fallacies, was <I>Ari$totle,</I> who in this particular can never be deceived.</P> <P>SAGR. You in$i$t upon <I>Aristotle,</I> who cannot $peak. Yet I tell you, that if <I>Ari$totle</I> were here, he would either yield him- <marg>Ari$totle <I>would either refute his adver$aries argu- ments, or would alter his opinion.</I></marg> $elf to be per$waded by us, or refuting our arguments, convince us by better of his own. And you your $elf, when you heard the experiments of the Suns related, did you not acknowledg and admire them, and confe$$e them more concludent than tho$e of <I>Ari$totle?</I> Yet neverthele$$e I cannot perceive that <I>Salviatus,</I> who hath produced them, examined them, and with exqui$ite care $can'd them, doth confe$$e him$elf per$waded by them; no nor by others of greater force, which he intimated that he was about to give us an account of. And I know not on what grounds you $hould cen$ure Nature, as one that for many Ages hath been lazie, and forgetful to produce $peculative <I>wits</I>; and that knoweth not how to make more $uch, unle$$e they be $uch kind of men as $lavi$hly giving up their judgments to <I>Ari$totle,</I> do under$tand with his brain, and re$ent with his $en$es. But let us hear the re$idue of tho$e rea$ons which favour his opinion, that we may thereupon proceed to $peak to them; comparing and weighing them in the ballance of impartiality.</P> <P>SALV. Before I proceed any farther, I mu$t tell <I>Sagredus,</I> that in the$e our Di$putations, I per$onate the <I>Copernican,</I>, and imi- tate him, as if I were his <I>Zany</I>; but what hath been effected in my private thoughts by the$e arguments which I $eem to alledg in his favour, I would not have you to judg by what I $ay, whil'$t I am in the heat of acting my part in the Fable; but after I have laid by my di$gui$e, for you may chance to find me different from what you $ee me upon the Stage. Now let us go on.</P> <P><I>Ptolomy</I> and his followers produce another experiment like to <marg><I>An argument taken from the Clouds, and from Birds.</I></marg> that of the Projections, and it is of things that being $eparated <foot>P from</foot> <p n=>114</p> from the Earth, continue a good $pace of time in the Air, $uch as are the Clouds, Birds of flight; and as of them it cannot be $aid that they are rapt or tran$parted by the Earth, having no ad- he$ion thereto, it $eems not po$$ible, that they $hould be able to keep pace with the velocity thereof; nay it $hould rather $eem to us, that they all $wiftly move towards the We$t: And if being carried about by the Earth, pa$$e our parallel in twenty four hours, which yet is at lea$t $ixteen thou$and miles, how can Birds follow $uch a cour$e or revolution? Whereas on the con- trary, we $ee them fly as well towards the Ea$t, as towards the We$t, or any other part, without any $en$ible difference. More- <marg><I>An argument taken from the air which we feel to beat upon us when we run a Hor$e at full $peed.</I></marg> over, if when we run a Hor$e at his $peed, we feel the air beat vehemently again$t our face, what an impetuous bla$t ought we perpetually to feel from the Ea$t, being carried with $o rapid a cour$e again$t the wind? and yet no $uch effect is perceived. Take another very ingenious argument inferred from the following ex- <marg><I>An argument taken from the whirling of circu- lar motion, which hath a faculty to extrude and di$$i- pate.</I></marg> periment. The circular motion hath a faculty to extrude and di$- $ipate from its Centre the parts of the moving body, when$oever either the motion is not very $low, or tho$e parts are not very well fa$tened together; and therefore, if <I>v. g.</I> we $hould turn one of tho$e great wheels very fa$t about, wherein one or more men walking, crane up very great weights, as the huge ma$$ie $tone, u$ed by the Callander for pre$$ing of Cloaths; or the fraighted Barks which being haled on $hore, are hoi$ted out of one river into another; in ca$e the parts of that $ame Wheel $o $wiftly turn'd round, be not very well joyn'd and pin'd together, they would all be $hattered to pieces; and though many $tones or other ponderous $ub$tances, $hould be very fa$t bound to its outward Rimme, yet could they not re$i$t the impetuo$ity, which with great violence would hurl them every way far from the Wheel, and con$equently from its Centre. So that if the Earth did move with $uch and $o much greater velocity, what gravity, what tena- city of lime or plai$ter would keep together Stones, Buildings, and whole Cities, that they $hould not be to$t into the Air by $o pre- cipitous a motion? And both men and bea$ts, which are not fa- $tened to the Earth, how could they re$i$t $o great an <I>impetus</I>? Whereas, on the other $ide, we $ee both the$e, and far le$$e re- $i$tances of pebles, $ands, leaves re$t quietly on the Earth, and to return to it in falling, though with a very $low motion. See here, <I>Simplicius,</I> the mo$t potent arguments, taken, to $o $peak, from things Terre$trial; there remain tho$e of the other kind, namely, $uch as have relation to the appearances of Heaven, which rea$ons, to confe$$e the truth, tend more to prove the Earth to be in the centre of the Univer$e, and con$equently, to deprive it of the annual motion about the $ame, a$cribed unto it <foot>by</foot> <p n=>115</p> by <I>Copernicus.</I> Which arguments, as being of $omewhat a di$te- rent nature, may be produced, after we have examined the $trength of the$e already propounded.</P> <P>SAGR. What $ay you <I>Simplicius</I>? do you think that <I>Salviatus</I> is Ma$ter of, and knoweth how to unfold the <I>Ptolomean</I> and <I>Ari- $totelian</I> arguments? Or do you think that any <I>Peripatetick</I> is e- qually ver$t in the <I>Copernican</I> demon$trations?</P> <P>SIMPL. Were it not for the high e$teem, that the pa$t di$cour- $es have begot in me of the learning of <I>Salviatus,</I> and of the a- cutene$$e of <I>Sagredus,</I> I would by their good leave have gone my way without $taying for their an$wers; it $eeming to me a thing impo$$ible, that $o palpable experiments $hould be contradicted; and would, without hearing them farther, con$irm my $elf in my old per$wa$ion; for though I $hould be made to $ee that it was er- roneous, its being upheld by $o many probable rea$ons, would ren- der it excu$eable. And if the$e are fallacies, what true demon$tra- tions were ever $o fair?</P> <P>SAGR. Yet its good that we hear the re$pon$ions of <I>Salviatus</I>; which if they be true, mu$t of nece$$ity be more fair, and that by in$inite degrees; and tho$e mu$t be deformed, yea mo$t deformed, if the Metaphy $ical Axiome hold, That true and fair are one and <marg><I>True and fair are one and the $ame, as al$o fal$e and deformed.</I></marg> the $ame thing; as al$o fal$e and deformed. Therefore <I>Salviatus</I> let's no longer lo$e time.</P> <P>SALV. The fir$t Argument alledged by <I>Simplicius,</I> if I well re- member it, was this. The Earth cannot move circularly, becau$e $uch motion would be violent to the $ame, and therefore not per- petual: that it is violent, the rea$on was: Becau$e, that had it been natural, its parts would likewi$e naturally move round, which is impo$$ible, for that it is natural for the parts thereof to move with a right motion downwards. To this my reply is, that I could glad- ly wi$h, that <I>Ari$totle</I> had more cleerly expre$t him$elf, where he <marg><I>The an$wer to</I> Ari$totles <I>fir$t ar- gument.</I></marg> $aid; That its parts would likewi$e move circularly; for this mo- ving circularly is to be under$tood two wayes, one is, that every particle or atome $eparated from its <I>Whole</I> would move circularly about its particular centre, de$cribing its $mall Circulets; the other is, that the whole Globe moving about its centre in twenty four hours, the parts al$o would turn about the $ame centre in four and twenty hours. The fir$t would be no le$$e an impertinency, than if one $hould $ay, that every part of the circumference of a Circle ought to be a Circle; or becau$e that the Earth is Spherical, that therefore every part thereof be a Globe, for $o doth the <I>Axiome</I> require: <I>Eadem e$t ratio totius, & partium.</I> But if he took it in the other $en$e, to wit, that the parts in imitation of the <I>Whole</I> $hould move naturally round the Centre of the whole Globe in twenty four hours, I $ay, that they do $o; and it concerns you, <foot>P 2 in$tead</foot> <p n=>116</p> in$tead of <I>Ari$totle,</I> to prove that they do not.</P> <P>SIMPL. This is proved by <I>Ari$totle</I> in the $ame place, when he $aith, that the natural motion of the parts is the right motion downwards to the centre of the Univer$e; $o that the circular motion cannot naturally agree therewith.</P> <P>SALV. But do not you $ee, that tho$e very words carry in them a confutation of this $olution?</P> <P>SIMPL. How? and where?</P> <P>SALV. Doth not he $ay that the circular motion of the Earth would be violent? and therefore not eternal? and that this is ab- $urd, for that the order of the World is eternal?</P> <P>SIMPL. He $aith $o.</P> <P>SALV. But if that which is violent cannot be eternal, then by <marg><I>That which is violent, cannot be eternal, and that which cannot be e- ternal, cannot be natural.</I></marg> conver$ion, that which cannot be eternal, cannot be natural: but the motion of the Earth downwards cannot be otherwi$e eternal; therefore much le$$e can it be natural: nor can any other motion be natural to it, $ave onely that which is eternal. But if we make the Earth move with a circular motion, this may be eternal to it, and to its parts, and therefore natural.</P> <P>SIMPL. The right motion is mo$t natural to the parts of the Earth, and is to them eternal; nor $hall it ever happen that they move not with a right motion; alwayes provided that the impe- diments be removed.</P> <P>SALV. You equivocate <I>Simplicius</I>; and I will try to free you from the equivoke. Tell me, therefore, do you think that a Ship which $hould $ail from the Strait of <I>Gibralter</I> towards <I>Pale- $tina</I> can eternally move towards that Coa$t? keeping alwayes an equal cour$e?</P> <P>SIMPL. No doubtle$$e.</P> <P>SALV. And why not?</P> <P>SIMPL. Becau$e that Voyage is bounded and terminated be- tween the <I>Herculean</I> Pillars, and the $hore of the <I>Holy-land</I>; and the di$tance being limited, it is pa$t in a finite time, unle$$e one by returning back $hould with a contrary motion begin the $ame Voy- age anew; but this would be an interrupted and no continued motion.</P> <P>SALV. Very true. But the Navigation from the Strait of <I>Ma- galanes</I> by the <I>Pacifick</I> Ocean, the <I>Moluccha's,</I> the Cape <I>di buona Speranza,</I> and from thence by the $ame Strait, and then again by the <I>Pacifick</I> Ocean, &c. do you believe that it may be perpe- tuated?</P> <P>SIMPL. It may; for this being a circumgyration, which re- turneth about its $elf, with infinite replications, it may be perpetu- ated without any interruption.</P> <P>SALV. A Ship then may in this Voyage continue $ailing eter- nally.</P> <foot>SIMPL.</foot> <p n=>117</p> <P>SIMPL. It may, in ca$e the Ship were incorruptible, but the Ship decaying, the Navigation mu$t of nece$$ity come to an end.</P> <P>SALV. But in the Mediterrane, though the Ve$$el were incor- ruptible, yet could $he not $ail perpetually towards <I>Pale$tina,</I> that <marg><I>Two things re- qui$ite to the end a motion may per- petuate it $elf; an unlimited $pace, and an incorrupti- ble moveable.</I></marg> Voyage being determined. Two things then are required, to the end a moveable may without intermi$$ion move perpetually; the one is, that the motion may of its own nature be indeterminate and infinite; the other, that the moveable be likewi$e incorruptible and eternal.</P> <P>SIMPL. All this is nece$$ary.</P> <P>SALV. Therefore you may $ee how of your own accord you have confe$$ed it impo$$ible that any moveable $hould move eter- nally in a right line, in regard that right motion, whether it be up- <marg><I>Right motion cannot be eternal, and con$equently cannot be natural to the Earth.</I></marg> wards, or downwards, is by you your $elf bounded by the circum- ference and centre; $o that if a Moveable, as $uppo$e the Earth be eternal, yet fora$much as the right motion is not of its own na- ture eternall, but mo$t ^{*}terminate, it cannot naturally $uit with <marg>* Terminati$$imo.</marg> the Earth. Nay, as was $aid ^{*} ye$terday, <I>Ari$totle</I> him$elf is <marg>* By this expre$$i- on he every where means the prece- ding Dialogue, or <I>Giornata.</I></marg> con$trained to make the Terre$trial Globe eternally immoveable. When again you $ay, that the parts of the Earth evermore move downwards, all impediments being removed, you egregiou$ly equi- vocate; for then, on the other $ide they mu$t be impeded, contra- ried, and forced, if you would have them move; for, when they are once fallen to the ground, they mu$t be violently thrown up- wards, that they may a $econd time fall; and as to the impedi- ments, the$e only hinder its arrival at the centre; but if there were a <I>Well,</I> that did pa$$e thorow and beyond the centre, yet would not a clod of Earth pa$$e beyond it, unle$$e ina$much as being tran$- ported by its <I>impetus,</I> it $hould pa$$e the $ame to return thither a- gain, and in the end there to re$t. As therefore to the defending, that the motion by a right line doth or can agree naturally neither to the Earth, nor to any other moveable, whil'$t the Univer$e re- taineth its perfect order, I would have you take no further paines a- bout it, but (unle$$e you will grant them the circular motion) your be$t way will be to defend and maintain their immobility.</P> <P>SIMPL. As to their immoveablene$$e, the arguments of <I>Ari- $totle,</I> and moreover tho$e alledged by your $elf $eem in my opini- on nece$$arily to conclude the $ame, as yet; and I conceive it will be a hard matter to refute them.</P> <P>SALV. Come we therefore to the $econd Argument, which was, That tho$e bodies, which we are a$$ured do move circularly, have <marg><I>The an$wer to the $econd argu- ment.</I></marg> more than one motion, unle$$e it be the <I>Primum Mobile</I>; and therefore, if the Earth did move circularly, it ought to have two motions; from which alterations would follow in the ri$ing and $etting of the Fixed Stars: Which effect is not perceived to en$ue. <foot>There-</foot> <p n=>118</p> Therefore, &c. The mo$t proper and genuine an$wer to this Alle- gation is contained in the Argument it $elf; and even <I>Aristotle</I> puts it in our mouths, which it is impo$$ible, <I>Simplicius,</I> that you $hould not have $een.</P> <P>SIMPL. I neither have $een it, nor do I yet apprehend it.</P> <P>SALV. This cannot be, $ure, the thing is $o very plain.</P> <P>SIMPL. I will with your leave, ca$t an eye upon the <I>Text.</I></P> <P>SAGR. We will command the <I>Text</I> to be brought forthwith.</P> <P>SIMPL. I alwayes carry it about with me: See here it is, and I know the place perfectly well, which is in <I>lib. 2. De Cælo, cap.</I> 16. Here it is, <I>Text</I> 97. <I>Preterea omnia, quæ feruntur latione circulari $ubdeficere videntur, ac moveri pluribus una latione, præter primam Sphæram; quare & Terram nece$$ariam e$t, $ive circa medium, $ive in medio po$ita feratur, duabus moveri lationibus. Si autem hoc acciderit, nece$$ariam e$t fieri muta- tiones, ac conver$iones fixorum a$trorum. Hoc autem non vide- tur ficri, $ed $emper eadem, apud eadem loca ip$ius, & oriun- tur, & occidunt.</I> [In Engli$h thus:] Furthermore all that are <marg>* Subde$icere.</marg> carried with circular motion, $eem to ^{*} fore$low, and to move with more than one motion, except the fir$t Sphere; wherefore it is nece$$ary that the Earth move with two motions, whether <marg>* Or Centre.</marg> it be carried about the ^{*} middle, or placed in the middle. But if it be $o, there would of nece$$ity be alterations and conver$i- ons made among$t the fixed Stars. But no $uch thing is $een to be done, but the $ame Star doth alwayes ri$e and $et in the $ame place. In all this I find not any falacy, and my thinks the argu- ment is very forcible.</P> <P>SALV. And this new reading of the place hath confirmed me in the fallacy of the Sillogi$me, and moreover, di$covered ano- ther fal$ity. Therefore ob$erve. The Po$itions, or if you will, Conclu$ions, which <I>Ari$totle</I> endeavours to oppo$e, are two; one is that of tho$e, who placing the Earth in the mid$t of the World, do make it move in it $elf about its own centre. The other is of tho$e, who con$tituting it far from the middle, do make it re- volve with a circular motion about the middle of the Univer$e. And both the$e Po$itions he conjointly impugneth with one and the $ame argument. Now I affirm that he is out in both the one and the other impugnation; and that his error again$t the fir$t Po$ition is an Equivoke or Paralogi$me; and his mi$take touch- <marg>Ari$totles <I>argu- ment again$t the Earths motion, is defective in two things</I></marg> ing the $econd is a fal$e con$equence. Let us begin with the fir$t A$$ertion, which con$tituteth the Earth in the mid$t of the World, and maketh it move in it $elf about its own centre; and <marg>* The $ame word which a little above I tendred $tay <*>e- hind, as a bowle when it meets with ruls.</marg> let us confront it with the objection of <I>Ari$totle</I>; $aying, All moveables, that move circularly, $eem to ^{*} fore$low, and move with more than one Byas, except the fir$t Sphere (that is <I>the pri-</I> <foot><I>mum</I></foot> <p n=>119</p> <I>mum mobile</I>) therefore the Earth moving about its own centre, being placed in the middle, mu$t of nece$$ity have two bya$$es, and fore$low. But if this were $o, it would follow, that there $hould be a variation in the ri$ing and $etting of the fixed Stars, which we do not perceive to be done: Therefore the Earth doth not move, <I>&c.</I> Here is the Paralogi$me, and to di$cover it, I will argue with <I>Ari$totle</I> in this manner. Thou $ai$t, oh <I>Ari$totle,</I> that the Earth placed in the middle of the World, cannot move in it $elf (<I>i. e.</I> upon its own <I>axis</I>) for then it would be requi$ite to allow it two bya$$es; $o that, if it $hould not be nece$$ary to allow it more than one Byas onely, thou woulde$t not then hold it impo$$ible for it to move onely with that one; for thou would'$t unnece$$arily have con$ined the impo$$ibility to the plurality of bya$$es, if in ca$e it had no more but one, yet it could not move with that. And becau$e that of all the moveables in the World, thou make$t but one alone to move with one $ole byas; and all the re$t with more than one; and this $ame moveable thou af- firme$t to be the fir$t Sphere, namely, that by which all the fix- ed and erratick Stars $eem harmoniou$ly to move from Ea$t to We$t, if in ca$e the Earth may be that fir$t Sphere, that by mo- ving with one by as onely, may make the Stars appear to move from Ea$t to We$t, thou wilt not deny them it: But he that af- firmeth, that the Earth being placed in the mid$t of the World, moveth about its own Axis, a$cribes unto it no other motion, $ave that by which all the Stars appear to move from Ea$t to We$t; and $o it cometh to be that fir$t Sphere, which thou thy $elf ac- knowledge$t to move with but one by as onely. It is therefore ne- ce$$ary, oh <I>Ari$totle,</I> if thou wilt conclude any thing, that thou demon$trate, that the Earth being placed in the mid$t of the World, cannot move with $o much as one by as onely; or el$e, that much le$$e can the fir$t Sphere have one $ole motion; for o- therwi$e thou doe$t in thy very Sillogi$me both commit the falacy, and detect it, denying, and at that very time proving the $ame thing. I come now to the $econd Po$ition, namely, of tho$e who placing the Earth far from the mid$t of the Univer$e, make it moveable about the $ame; that is, make it a Planet and erra- tick Star; again$t which the argument is directed, and as to form is concludent, but faileth in matter. For it being granted, that the Earth doth in that manner move, and that with two by- a$$es, yet doth it not nece$$arily follow that though it were $o, it $hould make alterations in the ri$ings and $ettings of the fixed Stars, as I $hall in its proper place declare. And here I could gladly excu$e <I>Ari$totle</I>; rather I could highly applaud him for ha- ving light upon the mo$t $ubtil argument that could be produced again$t the <I>Copernican Hypothe$is</I>; and if the objection be inge- <foot>nious,</foot> <p n=>120</p> nious, and to outward appearance mo$t powerful, you may $ee how much more acute and ingenious the $olution mu$t be, and not to be found by a wit le$$e piercing than that of <I>Copernicus</I>; and again from the difficulty in under$tanding it, you may argue the $o much greater difficulty in finding it. But let us for the pre- $ent $u$pend our an$wer, which you $hall under$tand in due time and place, after we have repeated the objection of <I>Ari$totle,</I> and that in his favour, much $trengthened. Now pa$$e we to <I>Ari-</I> <marg><I>The an$wer to the third argu- ment.</I></marg> <I>$totles</I> third Argument, touching which we need give no farther reply, it having been $ufficiently an$wered betwixt the di$cour$es of ye$terday and to day: In as much as he urgeth, that the mo- tion of grave bodies is naturally by a right line to the centre; and then enquireth, whether to the centre of the Earth, or to that of the Univer$e, and concludeth that they tend naturally to the centre of the Univer$e, but accidentally to that of the Earth. <marg><I>The an$wer to the fourth argu- ment.</I></marg> Therefore we may proceed to the fourth, upon which its requi$ite that we $tay $ome time, by rea$on it is founded upon that expe- riment, from whence the greater part of the remaining argu- ments derive all their $trength. <I>Ari$totle</I> $aith therefore, that it is a mo$t convincing argument of the Earths immobility, to $ee that projections thrown or $hot upright, return perpendicularly by the $ame line unto the $ame place from whence they were $hot or thrown. And this holdeth true, although the motion be of a very great height; which could never come to pa$$e, did the Earth move: for in the time that the projected body is moving upwards and downwards in a $tate of $eparation from the Earth, the place from whence the motion of the projection began, would be pa$t, by means of the Earths revolution, a great way to- wards the Ea$t, and look how great that $pace was, $o far from that place would the projected body in its de$cent come to the ground. So that hither may be referred the argument taken from a bullet $hot from a Canon directly upwards; as al$o that other u$ed by <I>Ari$totle</I> and <I>Ptolomy,</I> of the grave bodies that falling from on high, are ob$erved to de$cend by a direct and perpendicu- lar line to the $urface of the Earth. Now that I may begin to untie the$e knots, I demand of <I>Simplicius</I> that in ca$e one $hould deny to <I>Ptolomy</I> and <I>Ari$totle</I> that weights in falling freely from on high, de$cend by a right and perpendicular line, that is, directly to the centre, what means he would u$e to prove it?</P> <P>SIMPL. The means of the $en$es; the which a$$ureth us, that that Tower or other altitude, is upright and perpendicular, and $heweth us that that $tone, or other grave body, doth $lide along the Wall, without inclining a hairs breadth to one $ide or ano- ther, and light at the foot thereof ju$t under the place from whence it was let fall.</P> <foot>SALV.</foot> <p n=>121</p> <P>SALV. But if it $hould happen that the Terre$trial Globe did move round, and con$equently carry the Tower al$o along with it, and that the $tone did then al$o grate and $lide along the $ide of the Tower, what mu$t its motion be then?</P> <P>SIMPL. In this ca$e we may rather $ay its motions: for it would have one wherewith to de$cend from the top of the Tower to the bottom, and $hould nece$$arily have another to follow the cour$e of the $aid Tower.</P> <P>SALV. So that its motion $hould be compounded of two, to wit, of that wherewith it mea$ureth the Tower, and of that o- ther wherewith it followeth the $ame: From which compo$ition would follow, that the $tone would no longer de$cribe that $imple right and perpendicular line, but one tran$ver$e, and perhaps not $treight.</P> <P>SIMPL. I can $ay nothing of its non-rectitude, but this I know very well, that it would of nece$$ity be tran$ver$e, and different from the other directly perpendicular, which it doth de$cribe, the Earth $tanding $till.</P> <P>SALV. You $ee then, that upon the meer ob$erving the falling $tone to glide along the Tower, you cannot certainly affirm that it de$cribeth a line which is $treight and perpendicular, unle$s you fir$t $uppo$e that the Earth $tandeth $till.</P> <P>SIMPL. True; for if the Earth $hould move, the $tones mo- tion would be tran$ver$e, and not perpendicular.</P> <P>SALV. Behold then the Paralogi$m of <I>Ari$totle</I> and <I>Ptolomey</I> <marg><I>The Paralogi$m of</I> Ari$totle <I>and</I> Ptolomey <I>in $up- po$ing that for known, which is in que$tion.</I></marg> to be evident and manife$t, and di$covered by you your $elf, wherein that is $uppo$ed for known, which is intended to be de- mon$trated.</P> <P>SIMPL. How can that be? To me it appeareth that the Syllogi$m is rightly demon$trated without <I>petitionem principii.</I></P> <P>SALV. You $hall $ee how it is; an$wer me a little. Doth he not lay down the conclu$ion as unknown?</P> <P>SIMPL. Unknown; why otherwi$e the demon$trating it would be $uperfluous.</P> <P>SALV. But the middle term, ought not that to be known?</P> <P>SIMPL. Its nece$$ary that it $hould; for otherwi$e it would be a proving <I>ignotum per æquè ignotum.</I></P> <P>SALV. Our conclu$ion which is to be proved, and which is un- known, is it not the $tability of the Earth?</P> <P>SIMPL. It is the $ame.</P> <P>SALV. The middle term, which ought to be known, is it not the $treight and perpendicular de$cent of the $tone?</P> <P>SIMPL. It is $o.</P> <P>SALV. But was it not ju$t now concluded, that we can have no certain knowledg whether that $ame $hall be direct and perpen- <foot>Q dicular,</foot> <p n=>122</p> dicular, unle$s we fir$t know that the Earth $tands $till? Therefore in your Syllogi$m the certainty of the middle term is a$$umed from the uncertainty of the conclu$ion. You may $ee then, what and how great the Paralogi$m is.</P> <P>SAGR. I would, in favour of <I>Simplicius,</I> defend <I>Ari$totle</I> if it were po$$ible, or at lea$t better $atisfie my $elf concerning the $trength of your illation. You $ay, that the $eeing the $tone rake along the Tower, is not $ufficient to a$$ure us, that its motion is perpendicular (which is the middle term of the Syllogi$m) unle$s it be pre$uppo$ed, that the Earth $tandeth $till, which is the con- clu$ion to be proved: For that if the Tower did move together with the Earth, and the $tone did $lide along the $ame, the motion of the $tone would be tran$ver$e, and not perpendicular. But I $hall an$wer, that $hould the Tower move, it would be impo$$ible that the $tone $hould fall gliding along the $ide of it; and there- fore from its falling in that manner the $tability of the Earth is in- ferred.</P> <P>SIMPL. It is $o; for if you would have the $tone in de$cend- ing to grate upon the Tower, though it were carried round by the Earth, you mu$t allow the $tone two natural motions, to wit, the $traight motion towards the Centre, and the circular about the Centre, the which is impo$$ible.</P> <P>SALV. <I>Ari$totles</I> defen$e then con$i$teth in the impo$$ibilitie, or at lea$t in his e$teeming it an impo$$ibility, that the $tone $hould move with a motion mixt of right and circular: for if he did not hold it impo$$ible that the $tone could move to the Centre, and about the Centre at once, he mu$t have under$tood, that it might come to pa$s that the cadent $tone might in its de$cent, race the Tower as well when it moved as when it $tood $till; and con- $equently he mu$t have perceived, that from this grating nothing could be inferred touching the mobility or immobility of the Earth. But this doth not any way excu$e <I>Aristotle</I>; a$well be- cau$e he ought to have expre$t it, if he had had $uch a conceit, it being $o material a part of his Argument; as al$o becau$e it can neither be $aid that $uch an effect is impo$$ible, nor that <I>Ari$totle</I> did e$teem it $o. The fir$t cannot be affirmed, for that by and by I $hall $hew that it is not onely po$$ible, but nece$$ary: nor <marg>Ari$totle <I>admit- teth that the Fire moveth directly upwards by na- ture, and round a- bout by participa- tion.</I></marg> much le$s can the $econd be averred, for that <I>Ari$totle</I> him$elf granteth fire to move naturally upwards in a right line, and to move about with the diurnal motion, imparted by Heaven to the whole Element of Fire, and the greater part of the Air: If there- fore he held it not impo$$ible to mix the right motion upwards, with the circular communicated to the Fire and Air from the con- cave of the Moon, much le$s ought he to account impo$$ible the mixture of the right motion downwards of the $tone, with the <foot>circular</foot> <p n=>123</p> circular which we pre$uppo$e natural to the whole Terre$trial Globe, of which the $tone is a part.</P> <P>SIMPL. I $ee no $uch thing: for if the element of Fire re- volve round together with the Air, it is a very ea$ie, yea a nece$$ary thing, that a $park of fire which from the Earth mounts upwards, in pa$$ing thorow the moving air, $hould receive the $ame motion, being a body $o thin, light, and ea$ie to be moved: but that a very heavy $tone, or a Canon bullet, that de$cendeth from on high, and that is at liberty to move whither it will, $hould $uffer it $elf to be tran$ported either by the air or any other thing, is altogether incredible. Be$ides that, we have the Experiment, which is $o proper to our purpo$e, of the $tone let fall from the round top of the Ma$t of a $hip, which when the $hip lyeth $till, falleth at the Partners of the Ma$t; but when the $hip $aileth, falls $o far di$tant from that place, by how far the $hip in the time of the $tones falling had run forward; which will not be a few fa- thoms, when the $hips cour$e is $wift.</P> <P>SALV. There is a great di$parity between the ca$e of the Ship <marg><I>The di$parity be- tween the fall of a $tone from the round top of a $hip, and from the top of a tower.</I></marg> and that of the Earth, if the Terre$trial Globe be $uppo$ed to have a diurnal motion. For it is a thing very manife$t, that the mo- tion of the Ship, as it is not natural to it, $o the motion of all tho$e things that are in it is accidental, whence it is no wonder that the $tone which was retained in the round top, being left at liberty, de$cendeth downwards without any obligation to follow the mo- tion of the Ship. But the diurnal conver$ion is a$cribed to the Terre$trial Globe for its proper and natural motion, and con$e- quently, it is $o to all the parts of the $aid Globe; and, as being impre$s'd by nature, is indelible in them; and therefore that $tone that is on the top of the Tower hath an intrin$ick inclination of revolving about the Centre of its <I>Whole</I> in twenty four hours, and this $ame natural in$tinct it exerci$eth eternally, be it placed in any $tate what$oever. And to be a$$ured of the truth of this, you have no more to do but to alter an antiquated impre$$ion made in your mind; and to $ay, Like as in that I hitherto holding it to be the property of the Terre$trial Globe to re$t immoveable about its Centre, did never doubt or que$tion but that all what$oever particles thereof do al$o naturally remain in the $ame $tate of re$t: So it is rea$on, in ca$e the Terre$trial Globe did move round by natural in$tinct in twenty four hours, that the intrin$ick and natu- ral inclination of all its parts $hould al$o be, not to $tand $till, but <marg>*That you may not $u$pect my tran$la- tion, or wonder what Oars have to do with a $hip, you are to know that the Author intends the Gallies u$ed in the Mediterrane.</marg> to follow the $ame revolution. And thus without running into any inconvenience, one may conclude, that in regard the motion conferred by the force of ^{*}Oars on the Ship, and by it on all the things that are contained within her, is not natural but forreign, it is very rea$onable that that $tone, it being $eparated from the $hip, <foot>Q 2 do</foot> <p n=>124</p> do reduce its $elf to its natural di$po$ure, and return to exerci$e <marg><I>The part of the Air inferiour to the higher moun- tains doth follow the motion of the Earth.</I></marg> its pure $imple in$tinct given it by nature. To this I add, that it's nece$$ary, that at lea$t that part of the Air which is beneath the greater heights of mountains, $hould be tran$ported and carried round by the roughne$s of the Earths $urface; or that, as being mixt with many Vapours, and terrene Exhalations, it do na- turally follow the diurnal motion, which occurreth not in the Air about the $hip rowed by Oars: So that your arguing from the $hip to the Tower hath not the force of an illation; becau$e that $tone which falls from the round top of the Ma$t, entereth into a <I>medium,</I> which is unconcern'd in the motion of the $hip: but that which departeth from the top of the Tower, finds a <I>medium</I> that hath a motion in common with the whole Ter- re$trial Globe; $o that without being hindred, rather being a$$i$ted by the motion of the air, it may follow the univer$al cour$e of the Earth.</P> <P>SIMPL. I cannot conceive that the air can imprint in a very <marg><I>The motion of the Air apt to carry with it light things but not heavy.</I></marg> great $tone, or in a gro$s Globe of Wood or Ball of Lead, as $uppo$e of two hundred weight, the motion wherewith its $elf is moved, and which it doth perhaps communicate to feathers, $now, and other very light things: nay, I $ee that a weight of that na- ture, being expo$ed to any the mo$t impetuous wind, is not there- by removed an inch from its place; now con$ider with your $elf whether the air will carry it along therewith.</P> <P>SALV. There is great difference between your experiment and our ca$e. You introduce the wind blowing again$t that $tone, $uppo$ed in a $tate of re$t, and we expo$e to the air, which already moveth, the $tone which doth al$o move with the $ame velocity; $o that the air is not to conferr a new motion upon it, but onely to maintain, or to $peak better, not to hinder the motion already acquired: you would drive the $tone with a $trange and preter- natural motion, and we de$ire to con$erve it in its natural. If you would produce a more pertinent experiment, you $hould $ay, that it is ob$erved, if not with the eye of the forehead, yet with that of the mind, what would evene, if an eagle that is carried by the cour$e of the wind, $hould let a $tone fall from its talons; which, in regard that at its being let go, it went along with the wind, and after it was let fall it entered into a <I>medium</I> that mo- ved with equal velocity, I am very confident that it would not be $een to de$cend in its fall perpendicularly, but that following the cour$e of the wind, and adding thereto that of its particular gra- vity, it would move with a tran$ver$e motion.</P> <P>SIMPI. But it would fir$t be known how $uch an experiment may be made; and then one might judg according to the event. In the mean time the effect of the $hip doth hitherto incline to fa- vour our opinion.</P> <foot>SALV.</foot> <p n=>125</p> <P>SALV. Well $aid you <I>hitherto,</I> for perhaps it may anon change countenance. And that I may no longer hold you in $u$pen$e, tell me, <I>Simplicius,</I> do you really believe, that the Experiment of the $hip $quares $o very well with our purpo$e, as that it ought to be believed, that that which we $ee happen in it, ought al$o to evene in the Terre$trial Globe?</P> <P>SIMPL. As yet I am of that opinion; and though you have alledged $ome $mall di$parities, I do not think them of $o great moment, as that they $hould make me change my judgment.</P> <P>SALV. I rather de$ire that you would continue therein, and hold for certain, that the effect of the Earth would exactly an$wer that of the $hip: provided, that when it $hall appear prejudicial to your cau$e, you would not be humorous and alter your thoughts. You may haply $ay, Fora$much as when the $hip $tands $till, the $tone falls at the foot of the Ma$t, and when $he is under $ail, it lights far from thence, that therefore by conver$ion, from the $tones falling at the foot is argued the $hips $tanding $till, and from its falling far from thence is argued her moving; and becau$e that which occurreth to the $hip, ought likewi$e to befall the Earth: that therefore from the falling of the $tone at the foot of the Tow- er is nece$$arily inferred the immobility of the Terre$trial Globe. Is not this your argumentation?</P> <P>SIMPL. It is; and reduced into that conci$ene$s, as that it is become mo$t ea$ie to be apprehended.</P> <P>SALV. Now tell me; if the $tone let fall from the Round- top, when the $hip is in a $wift cour$e, $hould fall exactly in the $ame place of the $hip, in which it falleth when the $hip is at anchor, what $ervice would the$e experiments do you, in order to the a$certaining whether the ve$$el doth $tand $till or move?</P> <P>SIMPL. Ju$t none: Like as, for exemple, from the beating of the pul$e one cannot know whether a per$on be a$leep or awake, $eeing that the pul$e beateth after the $ame manner in $leeping as in waking.</P> <P>SALV. Very well. Have you ever tryed the experiment of the Ship?</P> <P>SIMPL. I have not; but yet I believe that tho$e Authors which alledg the $ame, have accurately ob$erved it; be$ides that the cau$e of the di$parity is $o manife$tly known, that it admits of no que$tion.</P> <P>SALV. That it is po$$ible that tho$e Authors in$tance in it, without having made tryal of it, you your $elf are a good te$ti- mony, that without having examined it, alledg it as certain, and in a credulous way remit it to their authority; as it is now not onely po$$ible, but very probable that they likewi$e did; I mean, did remit the $ame to their Predece$$ors, without ever arriving at one <foot>that</foot> <p n=>126</p> that had made the experiment: for whoever $hall examine the $ame, $hall find the event $ucceed quite contrary to what hath been written of it: that is, he $hall $ee the $tone fall at all times in the $ame place of the Ship, whether it $tand $till, or move with any what$oever velocity. So that the $ame holding true in the <marg><I>The stone falling from the Mast of a $hip lights in the $ame place, whe- ther the $hip doth move or ly still.</I></marg> Earth, as in the Ship, one cannot from the $tones falling perpen- dicularly at the foot of the Tower, conclude any thing touching the motion or re$t of the Earth.</P> <P>SIMPL. If you $hould refer me to any other means than to experience, I verily believe our Di$putations would not come to an end in ha$te; for this $eemeth to me a thing $o remote from all humane rea$on, as that it leaveth not the lea$t place for credulity or probability.</P> <P>SALV. And yet it hath left place in me for both.</P> <P>SIMPL. How is this? You have not made an hundred, no nor one proof thereof, and do you $o confidently affirm it for true? I for my part will return to my incredulity, and to the confidence I had that the Experiment hath been tried by the principal Au- thors who made u$e thereof, and that the event $ucceeded as they affirm.</P> <P>SALV. I am a$$ured that the effect will en$ue as I tell you; for $o it is nece$$ary that it $hould: and I farther add, that you know your $elf that it cannot fall out otherwi$e, however you feign or $eem to feign that you know it not. Yet I am $o good at taming of wits, that I will make you confe$s the $ame whether you will or no. But <I>Sagredus</I> $tands very mute, and yet, if I mi$take not, I $aw him make an offer to $peak $omewhat.</P> <P>SAGR. I had an intent to $ay $omething, but to tell you true, I know not what it was; for the curio$ity that you have moved in me, by promi$ing that you would force <I>Simplicius</I> to di$cover the knowledg which he would conceal from us, hath made me to de- po$e all other thoughts: therefore I pray you to make good your vaunt.</P> <P>SALV. Provided that <I>Simplicius</I> do con$ent to reply to what I $hall ask him, I will not fail to do it.</P> <P>SIMPL. I will an$wer what I know, a$$ured that I $hall not be much put to it, for that of tho$e things which I hold to be fal$e, I think nothing can be known, in regard that Science re$pecteth truths and not fal$hoods.</P> <P>SALV. I de$ire not that you $hould $ay or reply, that you know any thing, $ave that which you mo$t a$$uredly know. Therefore tell me; If you had here a flat $uperficies as polite as a Looking- gla$s, and of a $ub$tance as hard as $teel, and that it were not pa- ralel to the Horizon, but $omewhat inclining, and that upon it you did put a Ball perfectly $pherical, and of a $ub$tance grave and <foot>hard,</foot> <p n=>127</p> hard, as $uppo$e of bra$s; what think you it would do being let go? do not you believe (as for my part I do) that it would lie $till?</P> <P>SIMPL. If that $uperficies were inclining?</P> <P>SALV. Yes; for $o I have already $uppo$ed.</P> <P>SIMPL. I cannot conceive how it $hould lie $till: nay, I am confident that it would move towards the declivity with much pro- pen$ne$s.</P> <P>SALV. Take good heed what you $ay, <I>Simplicius,</I> for I am confident that it would lie $till in what ever place you $hould lay it.</P> <P>SIMPL. So long as you make u$e of $uch $uppo$itions, <I>Sal- viatus,</I> I $hall cea$e to wonder if you inferr mo$t ab$urd con- clu$ions.</P> <P>SALV. Are you a$$ured, then, that it would freely move to- wards the declivity?</P> <P>SIMPL. Who doubts it?</P> <P>SALV. And this you verily believe, not becau$e I told you $o, (for I endeavoured to per$wade you to think the contrary) but of your $elf, and upon your natural judgment.</P> <P>SIMPL. Now I $ee what you would be at; you $poke not this as really believing the $ame; but to try me, and to wre$t matter out of my own mouth wherewith to condemn me.</P> <P>SALV. You are in the right. And how long would that Ball move, and with what velocity? But take notice that I in$tanced in a Ball exactly round, and a plain exqui$itely poli$hed, that all external and accidental impediments might be taken away. And $o would I have you remove all ob$tructions cau$ed by the Airs re- $i$tance to divi$ion, and all other ca$ual ob$tacles, if any other there can be.</P> <P>SIMPL. I very well under$tand your meaning, and as to your demand, I an$wer, that the Ball would continue to move <I>in in- finitum,</I> if the inclination of the plain $hould $o long la$t, and con- tinually with an accelerating motion; for $uch is the nature of ponderous moveables, that <I>vires acquirant eundo</I>: and the great- er the declivity was, the greater the velocity would be.</P> <P>SALV. But if one $hould require that that Ball $hould move upwards on that $ame $uperficies, do you believe that it would $o do?</P> <P>SIMPL. Not $pontaneou$ly; but being drawn, or violently thrown, it may.</P> <P>SALV. And in ca$e it were thru$t forward by the impre$$ion of $ome violent <I>impetus</I> from without, what and how great would its motion be?</P> <P>SIMPL. The motion would go continually decrea$ing and re- <foot>tarding,</foot> <p n=>128</p> tarding, as being contrary to nature; and would be longer or $horter, according to the greater or le$s impul$e, and according to the greater or le$s acclivity.</P> <P>SALV. It $eems, then, that hitherto you have explained to me the accidents of a moveable upon two different Planes; and that in the inclining plane, the grave moveable doth $pontaneou$ly de- $cend, and goeth continually accelerating, and that to retain it in re$t, force mu$t be u$ed therein: but that on the a$cending plane, there is required a force to thru$t it forward, and al$o to $tay it in re$t, and that the motion impre$$ed goeth continually dimini$hing, till that in the end it cometh to nothing. You $ay yet farther, that in both the one and the other ca$e, there do ari$e differences from the planes having a greater or le$s declivity or acclivity; $o that the greater inclination is attended with the greater velocity; and contrariwi$e, upon the a$cending plane, the $ame moveable thrown with the $ame force, moveth a greater di$tance, by how much the elevation is le$s. Now tell me, what would befall the $ame moveable upon a $uperficies that had neither acclivity nor declivity?</P> <P>SIMPL. Here you mu$t give me a little time to con$ider of an an$wer. There being no declivity, there can be no natural incli- nation to motion: and there being no acclivity, there can be no re$i$tance to being moved; $o that there would ari$e an indiffe- rence between propen$ion and re$i$tance of motion; therefore, methinks it ought naturally to $tand $till. But I had forgot my $elf: it was but even now that <I>Sagredus</I> gave me to under$tand that it would $o do.</P> <P>SALV. So I think, provided one did lay it down gently: but if it had an <I>impetus</I> given it towards any part, what would fol- low?</P> <P>SIMP. There would follow, that it $hould move towards that part.</P> <P>SALV. But with what kind of motion? with the continually accelerated, as in declining planes; or with the $ucce$$ively re- tarded, as in tho$e a$cending.</P> <P>SIMP. I cannot tell how to di$cover any cau$e of acceleration, or retardation, there being no declivity or acclivity.</P> <P>SALV. Well: but if there be no cau$e of retardation, much le$s ought there to be any cau$e of re$t. How long therefore would you have the moveable to move?</P> <P>SIMP. As long as that $uperficies, neither inclined nor decli- ned $hall la$t.</P> <P>SALV. Therefore if $uch a $pace were interminate, the motion upon the $ame would likewi$e have no termination, that is, would be perpetual.</P> <foot>SIMPL</foot> <p n=>129</p> <P>SIMP. I think $o, if $o be the moveable be of a matter durable.</P> <P>SALV. That hath been already $uppo$ed, when it was $aid, that all external and accidental impediments were removed, and the brittlene$$e of the moveable in this our ca$e, is one of tho$e impediments accidental. Tell me now, what do you think is the cau$e that that $ame Ball moveth $pontaneou$ly upon the inclining plane, and not without violence upon the erected?</P> <P>SIMP. Becau$e the inclination of grave bodies is to move to- wards the centre of the Earth, and onely by violence upwards to- wards the circumference; and the inclining $uperficies is that which acquireth vicinity to the centre, and the a$cending one, remotene$$e.</P> <P>SALV. Therefore a $uperficies, which $hould be neither de- clining nor a$cending, ought in all its parts to be equally di- $tant from the centre. But is there any $uch $uperficies in the World?</P> <P>SIMP. There is no want thereof: Such is our Terre$trial Globe, if it were more even, and not as it is rough and montai- nous; but you have that of the Water, at $uch time as it is calm and $till.</P> <P>SALV. Then a $hip which moveth in a calm at Sea, is one of tho$e moveables, which run along one of tho$e $uperficies that are neither declining nor a$cending, and therefore di$po$ed, in ca$e all ob$tacles external and accidental were removed, to move with the impul$e once imparted ince$$antly and uniformly.</P> <P>SIMPL. It $hould $eem to be $o.</P> <P>SALV. And that $tone which is on the round top, doth not it move, as being together with the $hip carried about by the cir- cumference of a Circle about the Centre; and therefore con$e- quently by a motion in it indelible, if all extern ob$tacles be removed? And is not this motion as $wift as that of the $hip.</P> <P>SIMPL. Hitherto all is well. But what followeth?</P> <P>SALV. Then in good time recant, I pray you, that your la$t conclu$ion, if you are $atisfied with the truth of all the pre- mi$es.</P> <P>SIMPL. By my la$t conclu$ion, you mean, That that $ame $tone moving with a motion indelibly impre$$ed upon it, is not to leave, nay rather is to follow the $hip, and in the end to light in the $elf $ame place, where it falleth when the $hip lyeth $till; and $o I al$o grant it would do, in ca$e there were no outward impe- diments that might di$turb the $tones motion, after its being let go, the which impediments are two, the one is the moveables inability to break through the air with its meer <I>impetus</I> onely, it being deprived of that of the $trength of Oars, of which it had <foot>R been</foot> <p n=>130</p> been partaker, as part of the $hip, at the time that it was upon the Ma$t; the other is the new motion of de$cent, which al$o mu$t needs be an hinderance of that other progre$$ive motion.</P> <P>SALV. As to the impediment of the Air, I do not deny it you; and if the thing falling were a light matter, as a feather, or a lock of wool, the retardation would be very great, but in an heavy $tone is very exceeding $mall. And you your $elf but even now did $ay, that the force of the mo$t impetuous wind $ufficeth not to $tir a great $tone from its place; now do but con- $ider what the calmer air is able to do, being encountred by a $tone no more $wift than the whole $hip. Neverthele$$e, as I $aid before, I do allow you this $mall effect, that may depend upon $uch an impediment; like as I know, that you will grant to me, that if the air $hould move with the $ame velocity that the $hip and $tone hath, then the impediment would be nothing at all. As to the other of the additional motion downwards; in the fir$t place it is manife$t, that the$e two, I mean the circular, about the centre, and the $treight, towards the centre, are not contra- ries, or de$tructive to one another, or incompatible. Becau$e that as to the moveable, it hath no repugnance at all to $uch motions, for you your $elf have already confe$t the repugnance to be a- gain$t the motion which removeth from the centre, and the incli- nation to be towards the motion which approacheth to the centre. Whence it doth of nece$$ity follow, that the moveable hath nei- ther repugnance, nor propen$ion to the motion which neither ap- proacheth, nor goeth from the centre, nor con$equently is there any cau$e for the dimini$hing in it the faculty impre$$ed. And for- a$much as the moving cau$e is not one alone, which it hath at- tained by the new operation of retardation; but that they are two, di$tinct from each other, of which, the gravity attends on- ly to the drawing of the moveable towards the centre, and the vertue impre$s't to the conducting it about the centre, there re- maineth no occa$ion of impediment.</P> <P>SIMPL. Your argumentation, to give you your due, is very probable; but in reality it is invelloped with certain intricacies, that are not ea$ie to be extricated. You have all along built upon <marg><I>The project ac- cording to</I> Ari$to- tle, <I>is not moved by vertue impre$$ed, but by the</I> medium.</marg> a $uppo$ition, which the <I>Peripatetick</I> Schools will not ea$ily grant you, as being directly contrary to <I>Aristotle,</I> and it is to take for known and manife$t, That the project $eparated from the proji- cient, continueth the motion by <I>vertue impre$$ed</I> on it by the $aid projicient, which <I>vertue impre$$ed</I> is a thing as much dete- $ted in <I>Peripatetick</I> Philo$ophy, as the pa$$age of any accident from one $ubject into another. Which doctrine doth hold, as I believe it is well known unto you, that the project is carried by the <I>medium,</I> which in our ca$e happeneth to be the Air. And <foot>there-</foot> <p n=>131</p> therefore if that $tone let fall from the round top, ought to fol- low the motion of the $hip, that effect $hould be a$cribed to the Air, and not to the vertue impre$$ed. But you pre$uppo$e that the Air doth not follow the motion of the $hip, but is tranquil. Moreover, he that letteth it fall, is not to throw it, or to give it <I>impetus</I> with his arm, but ought barely to open his hand and let it go; and by this means, the $tone, neither through the vertue impre$$ed by the projicient, nor through the help of the Air, $hall be able to follow the $hips motion, and therefore $hall be left behind.</P> <P>SALV. I think then that you would $ay, that if the $tone be not thrown by the arm of that per$on, it is no longer a pro- jection.</P> <P>SIMPL. It cannot be properly called a motion of projection.</P> <P>SALV. So then that which <I>Ari$totle</I> $peaks of the motion, the moveable, and the mover of the projects, hath nothing to do with the bu$ine$$e in hand; and if it concern not our purpo$e, why do you alledg the $ame?</P> <P>SIMP. I produce it on the ocea$ion of that impre$$ed vertue, named and introduced by you, which having no being in the World, can be of no force; for <I>non-entium nullæ $unt operatio- nes</I>; and therefore not onely of projected, but of all other pre- ternatural motions, the moving cau$e ought to be a$cribed to the <I>medium,</I> of which there hath been no due con$ideration had; and therefore all that hath been $aid hitherto is to no purpo$e.</P> <P>SALV. Go to now, in good time. But tell me, $eeing that your in$tance is wholly grounded upon the nullity of the vertue impre$$ed, if I $hall demon$trate to you, that the <I>medium</I> hath nothing to do in the continuation of projects, after they are $e- patated from the projicient, will you admit of the impre$$ed ver- tue, or will you make another attempt to overthrow it?</P> <P>SIMP. The operation of the <I>medium</I> being removed, I $ee not how one can have recour$e to any thing el$e $ave the faculty im- pre$$ed by the mover.</P> <P>SALV. It would be well, for the removing, as much as is po$$ible, the occa$ions of multiplying contentions, that you would explain with as much di$tinctne$$e as may be, what is that operation of the <I>medium</I> in continuing the motion of the project.</P> <marg><I>Operation of the</I> medium <I>in continu- ing the motion of the project.</I></marg> <P>SIMP. The projicient hath the $tone in his hand, and with force and violence throws his arm, with which jactation the $tone doth not move $o much as the circumambient Air; $o that when the $tone at its being for$aken by the hand, findeth it $elf in the Air, which at the $ame time moveth with impetou$ity, it is thereby born away; for, if the air did not operate, the $tone would fall at the foot of the projicient or thrower.</P> <foot>R2 SALV.</foot> <p n=>132</p> <marg><I>Many experi- ments, and rea- $ons again$t the cau$e of the moti- on of projects, a$- $igned by</I> Ari$totle.</marg> <P>SALV. And was you $o credulous, as to $uffer your $elf to be per$waded to believe the$e fopperies, $o long as you had your $en$es about you to confute them, and to under$tand the truth thereof? Therefore tell me, that great $tone, and that Canon bullet, which but onely laid upon a table, did continue immoveable again$t the mo$t impetuous winds, according as you a little before did affirm, if it had been a ball of cork or other light $tuffe, think you that the wind would have removed it from its place?</P> <P>SIMP. Yes, and I am a$$ured that it would have blown it quite away, and with $o much more velocity, by how much the matter was lighter, for upon this rea$on we $ee the clouds to be tran$ported with a velocity equal to that of the wind that drives them.</P> <P>SALV. And what is the Wind?</P> <P>SIMP. The Wind is defined to be nothing el$e but air moved.</P> <P>SALV. Then the moved air doth carry light things more $wiftly, and to a greater di$tance, then it doth heavy.</P> <P>SIMP. Yes certainly.</P> <P>SALV. But if you were to throw with your arm a $tone, and a lock of cotton wool, which would move $wi$te$t and farthe$t?</P> <P>SIMP. The $tone by much; nay the wool would fall at my feet.</P> <P>SALV. But, if that which moveth the projected $ub$tance, af- ter it is delivered from the hand, be no other than the air moved by the arm, and the moved air do more ea$ily bear away light than grave matters, how cometh it that the project of wool flieth not farther, and $wifter than that of $tone? Certainly it argu- eth that the $tone hath $ome other impul$e be$ides the motion of the air. Furthermore, if two $trings of equal length did hang at yonder beam, and at the end of one there was fa$tened a bul- let of lead, and a ball of cotton wool at the other, and both were carried to an equal di$tance from the perpendicular, and then let go; it is not to be doubted, but that both the one and the other would move towards the perpendicular, and that being carried by their own <I>impetus,</I> they would go a certain $pace be- yond it, and afterwards return thither again. But which of the$e two pendent Globes do you think, would continue longe$t in mo- tion, before that it would come to re$t in its perpendicularity?</P> <P>SIMP. The ball of lead would $wing to and again many times, and that of wool but two or three at the mo$t.</P> <P>SALV. So that that <I>impetus</I> and that <I>mobility</I> what$oever is the cau$e thereof, would con$erve its $elf longer in grave $ub- $tances, than light; I proceed now to another particular, and de- mand of you, why the air doth not carry away that Lemon which is upon that $ame Table?</P> <foot>SIMP.</foot> <p n=>133</p> <P>SIMP. Becau$e that the air it $elf is not moved</P> <P>SALV. It is requi$ite then, that the projicient do confer mo- tion on the Air, with which it afterward moveth the project. But if $uch a motion cannot be impre$$ed [<I>i. e. imparted</I>] it being im- po$$ible to make an accident pa$$e out of one $ubject into another, how can it pa$$e from the arm into the Air? Will you $ay that the Air is not a $ubject different from the arm?</P> <P>SIMP. To this it is an$wered that the Air, in regard it is nei- ther heavy nor light in its own Region, is di$po$ed with facility to receive every impul$e, and al$o to retain the $ame.</P> <P>SALV. But if tho$e <I>penduli</I> even now named, did prove unto us, that the moveable, the le$$e it had of gravity, the le$$e apt it was to con$erve its motion, how can it be that the Air which in the Air hath no gravity at all, doth of it $elf alone re- tain the motion acquired? I believe, and know that you by this time are of the $ame opinion, that the arm doth not $ooner re- turn to re$t, than doth the circumambient Air. Let's go into the Chamber, and with a towel let us agitate the Air as much as we can, and then holding the cloth $till, let a little candle be brought, that was lighted in the next room, or in the $ame place let a leaf of beaten Gold be left at liberty to flie any wav, and you $hall by the calm vagation of them be a$$ured that the Air is imme- diately reduced to tranquilty. I could alledg many other experi- ments to the $ame purpo$e, but if one of the$e $hould not $uf- fice, I $hould think your folly altogether incurable.</P> <P>SAGR. When an arrow is $hot again$t the Wind, how incredi- ble a thing is it, that that $ame $mall filament of air, impelled by the bow-$tring, $hould in de$pite of fate go along with the arrow? But I would willingly know another particular of <I>Ari$totle,</I> to which I intreat <I>Simplicius</I> would vouch$afe me an an$wer. Sup- po$ing that with the $ame Bow there were $hot two arrows, one ju$t after the u$ual manner, and the other $ide-wayes, placing it long-wayes upon the Bow-$tring, and then letting it flie, I would know which of them would go farthe$t. Favour me, I pray you with an an$wer, though the que$tion may $eem to you rather ridiculous than otherwi$e; and excu$e me, for that I, who am, as you $ee, rather blocki$h, than not, can reach no higher with my $peculative faculty.</P> <P>SIMPL. I have never $een an arrow $hot in that manner, yet neverthele$$e I believe, that it would not flie $ide-long, the twentieth part of the $pace that it goeth end-wayes.</P> <P>SAGR. And for that I am of the $ame opinion, hence it is, that I have a doubt ri$en in me, whether <I>Aristotle</I> doth not contradict experience. For as to experience, if I lay two arrows upon this Table, in a time when a $trong Wind bloweth, one towards <foot>the</foot> <p n=>134</p> the cour$e of the wind, and the other $idelong, the wind will quickly carry away this later, and leave the other where it was; and the $ame to my $eeming, ought to happen, if the Doctrine of <I>Ari$totle</I> were true, of tho$e two $hot out of a Bow: fora$much as the arrow $hot $ideways is driven by a great quantity of Air, moved by the bow$tring, to wit by as much as the $aid $tring is long, whereas the other arrow receiveth no greater a quantity of air, than the $mall circle of the $trings thickne$s. And I cannot imagine what may be the rea$on of $uch a difference, but would fain know the $ame.</P> <P>SIMP. The cau$e $eemeth to me $ufficiently manife$t; and it is, becau$e the arrow $hot endways, hath but a little quantity of air to penetrate, and the other is to make its way through a quan- tity as great as its whole length.</P> <P>SALV. Then it $eems the arrows $hot, are to penetrate the air? but if the air goeth along with them, yea, is that which carrieth them, what penetration can they make therein? Do you not $ee that, in this ca$e, the arrow would of nece$$ity move with greater velocity than the air? and this greater velocity, what doth confer it on the arrow? Will you $ay the air giveth them a velocity greater than its own? Know then, <I>Simplicius,</I> that the bu$ine$s proceeds quite contrary to that which <I>Ari$totle</I> $aith, and that the <marg><I>The</I> medium <I>doth impede and not cor- fer the motion of projects.</I></marg> <I>medium</I> conferreth the motion on the project, is as fal$e, as it is true, that it is the onely thing which procureth its ob$truction; and having known this, you $hall under$tand without finding any thing whereof to make que$tion, that if the air be really moved, it doth much better carry the dart along with it longways, than endways, for that the air which impelleth it in that po$ture, is much, and in this very little. But $hooting with the Bow, fora$much as the air $tands $till, the tran$ver$e arrow, being to force its pa$$age through much air, comes to be much impeded, and the other that was nock't ea$ily overcometh the ob$truction of the $mall quantity of air, which oppo$eth it $elf thereto.</P> <P>SALV. How many Propo$itions have I ob$erved in <I>Ari$totle,</I> (meaning $till in Natural Philo$ophy) that are not onely fal$e, but fal$e in $uch $ort, that its diametrical contrary is true, as it happens in this ca$e. But pur$uing the point in hand, I think that <I>Simplicius</I> is per$waded, that, from $eeing the $tone always to fall in the $ame place, he cannot conjecture either the motion or $ta- bility of the Ship: and if what hath been hitherto $poken, $hould not $uffice, there is the Experiment of the <I>medium</I> which may thorowly a$$ure us thereof; in which experiment, the mo$t that could be $een would be, that the cadent moveable might be left behind, if it were light, and that the air did not follow the motion of the $hip: but in ca$e the air $hould move with equal <foot>velocity,</foot> <p n=>135</p> velocity, no imaginable diver$ity could be found either in this, or any other experiment what$oever, as I am anon to tell you. Now if in this ca$e there appeareth no difference at all, what can be pretended to be $een in the $tone falling from the top of the Tower, where the motion in gyration is not adventitious, and ac- cidental, but natural and eternal; and where the air exactly fol- loweth the motion of the Tower, and the Tower that of the Ter- re$trial Globe? have you any thing el$e to $ay, <I>Simplicius,</I> upon this particular?</P> <P>SIMP. No more but this, that I $ee not the mobility of the Earth as yet proved.</P> <P>SALV. Nor have I any intention at this time, but onely to $hew, that nothing can be concluded from the experiments alledg- ed by our adver$aries for convincing Arguments: as I think I $hall prove the others to be.</P> <P>SAGR. I be$eech you, <I>Salviatus,</I> before you proceed any far- ther, to permit me to $tart certain que$tions, which have been rouling in my fancy all the while that you with $o much patience and equanimity, was minutely explaining to <I>Simplicius</I> the expe- riment of the Ship.</P> <P>SALV. We are here met with a purpo$e to di$pute, and it's fit that every one $hould move the difficulties that he meets withall, for this is the way to come to the knowledg of the truth. Therefore $peak freely.</P> <P>SAGR. If it be true, that the <I>impetus</I> wherewith the $hip moves, doth remain indelibly impre$$'d in the $tone, after it is let fall from the Ma$t; and if it be farther true, that this motion brings no im- pediment or retardment to the motion directly downwards, na- tural to the $tone: it's nece$$ary, that there do an effect en$ue of <marg><I>An admirable accident in the mo- tion of projects.</I></marg> a very wonderful nature. Let a Ship be $uppo$ed to $tand $till, and let the time of the falling of a $tone from the Ma$ts Round-top to the ground, be two beats of the pul$e; let the Ship afterwards be under $ail, and let the $ame $tone depart from the $ame place, and it, according to what hath been premi$ed, $hall $till take up the time of two pul$es in its fall, in which time the $hip will have run, $uppo$e, twenty yards; To that the true motion of the $tone will be a tran$ver$e line, con$iderably longer than the fir$t $traight and perpendicular line, which is the length of the ^{*} Ma$t, and yet <marg>*By the length of the ma$t he means the di$tance be- tween the upper- deck and Round- top.</marg> neverthele$s the ^{*} $tone will have pa$t it in the $ame time. Let it be farther $uppo$ed, that the Ships motion is much more accele- rated, $o that the $tone in falling $hall be to pa$s a tran$ver$e line much longer than the other; and in $um, increa$ing the Ships ve- <marg>* La palla.</marg> locity as much as you will, the falling $tone $hall de$cribe its tran$- ver$e lines $till longer and longer, and yet $hall pa$s them all in tho$e $elf $ame two pul$es. And in this fa$hion, if a Canon were <foot>level'd</foot> <p n=>136</p> level'd on the top of a Tower, and $hots were made therewith point blank, that is, paralel to the Horizon, let the Piece have a greater or le$s charge, $o as that the ball may fall $ometimes a thou$and yards di$tant, $ometimes four thou$and, $ometimes $ix, $ometimes ten, <I>&c.</I> and all the$e $hots $hall curry or fini$h their ranges in times equal to each other, and every one equal to the time which the ball would take to pa$s from the mouth of the Piece to the ground, being left, without other impul$e, to fall $imply downwards in a perpendicular line. Now it $eems a very admirable thing, that in the $ame $hort time of its falling perpen- dicularly down to the ground, from the height of, $uppo$e, an hundred yards, the $ame ball, being thru$t violently out of the Piece by the Fire, $hould be able to pa$s one while four hundred, another while a thou$and, another while four, another while ten thou$and yards, $o as that the $aid ball in all $hots made point blank, always continueth an equal time in the air.</P> <P>SALV. The con$ideration for its novelty is very pretty, and if the effect be true, very admirable: and of the truth thereof, I make no que$tion: and were it not for the accidental impediment of the air, I verily believe, that, if at the time of the balls going out of the Piece, another were let fall from the $ame height di- rectly downwards, they would both come to the ground at the $ame in$tant, though that $hould have curried ten thou$and miles in its range, and this but an hundred onely: pre$uppo$ing the $urface of the Earth to be equal, which to be a$$ured of, the experiment may be made upon $ome lake. As for the impediment which might come from the air, it would con$i$t in retarding the extreme $wift motion of the $hot. Now, if you think fit, we will proceed to the $olution of the other Objections, $eeing that <I>Sim- plicius</I> (as far as I can $ee) is convinc'd of the nullity of this fir$t, taken from things falling from on high downwards.</P> <P>SIMP. I find not all my $cruples removed, but it may be the fault is my own, as not being of $o ea$ie and quick an apprehen$ion as <I>Sagredus.</I> And it $eems to me, that if this motion, of which the $tone did partake whil$t it was on the Round-top of the Ships Ma$t, be, as you $ay, to con$erve it $elf indelibly in the $aid $tone, even after it is $eparated from the Ship, it would follow, that like- wi$e in ca$e any one, riding a hor$e that was upon his $peed, $hould let a bowl drop out of his hand, that bowl being fallen to the ground would continue its motion and follow the hor$es $teps, without tarrying behind him: the which effect, I believe, is not to be $een, unle$s when he that is upon the hor$e $hould throw it with violence that way towards which he runneth; but otherwi$e, I believe it will $tay on the ground in the $ame place where it fell.</P> <foot>SALV.</foot> <p n=>137</p> <P>SALV. I believe that you very much deceive your $elf, and am certain, that experience will $hew you the contrary, and that the ball being once arrived at the ground, will run together with the hor$e, not $taying behind him, unle$s $o far as the a$perity and uneven- ne$s of the Earth $hall hinder it. And the rea$on $eems to me very manife$t: for if you, $tanding $till, throw the $aid ball a- long the ground, do you think it would not continue its motion even after you had delivered it out of your hand? and that for $o much a greater $pace, by how much the $uperficies were more $mooth, $o that <I>v. g.</I> upon ice it would run a great way?</P> <P>SIMP. There is no doubt of it, if I give it <I>impetus</I> with my arm; but in the other ca$e it is $uppo$ed, that he who is upon the hor$e, onely drops it out of his hand:</P> <P>SALV. So I de$ire that it $hould be: but when you throw it with your arm, what other remaineth to the ball being once gone out of your hand, than the motion received from your arm, which motion being con$erved in the boul, it doth continue to carry it forward? Now, what doth it import, that that <I>impetus</I> be con- ferred on the ball rather from the arm than from the hor$e? Whil$t you were on hor$eback, did not your hand, and con$equently the ball run as fa$t as the hor$e it $elf? Doubtle$s it did: therefore in onely opening of the hand, the ball departs with the motion al- ready conceived, not from your arm, by your particular motion, but from the motion dependant on the $aid hor$e, which cometh to be communicated to you, to your arm, to your hand, and la$tly to the ball. Nay, I will tell you farther, that if the rider upon his $peed fling the ball with his arm to the part contrary to the cour$e, it $hall, after it is fallen to the ground, $ometimes (albeit thrown to the contrary part) follow the cour$e of the hor$e, and $ometimes lie $till on the ground; and $hall onely move contrary to the $aid cour$e, when the motion received from the arm, $hall exceed that of the carrier in velocity. And it is a vanity, that of $ome, who $ay that a hor$eman is able to ca$t a javelin thorow the air, that way which the hor$e runs, and with the hor$e to follow and over- take the $ame; and la$tly, to catch it again. It is, I $ay, a vanity, for that to make the project return into the hand, it is requi$ite to ca$t it upwards, in the $ame manner as if you $tood $till. For, let the carrier be never $o $wift, provided it be uniform, and the pro- ject not over-light, it $hall always fall back again into the hand of the projicient, though never $o high thrown.</P> <P>SAGR. By this Doctrine I come to know $ome Problems very <marg><I>Sundry curious Problems, touch- ing the motions of projects.</I></marg> curious upon this $ubject of projections; the fir$t of which mu$t $eem very $trange to <I>Simplicius.</I> And the Problem is this; I af- firm it to be po$$ible, that the ball being barely dropt or let fall, by one that any way runneth very $wiftly, being arrived at the <foot>S Earth,</foot> <p n=>138</p> Earth, doth not onely follow the cour$e of that per$on, but doth much out go him. Which Problem is connexed with this, that the moveable being thrown by the projicient above the plane of the Horizon, may acquire new velocity, greater by far than that confer'd upon it by the projicient. The which effect I have with admiration ob$erved, in looking upon tho$e who u$e the $port of tops, which, $o $oon as they are $et out of the hand, are $een to move in the air with a certain velocity, the which they afterwards much encrea$e at their coming to the ground; and if whipping them, they rub at any uneven place that makes them skip on high, they are $een to move very $lowly through the air, and falling a- gain to the Earth, they $till come to move with a greater velocity: But that which is yet more $trange, I have farther ob$erved, that they not onely turn always more $wiftly on the ground, than in the air, but of two $paces both upon the Earth, $ometimes a mo- tion in the $econd $pace is more $wift than in the fir$t. Now what would <I>Simplicius</I> $ay to this?</P> <P>SIMP. He would $ay in the fir$t place, that he had never made $uch an ob$ervation. Secondly, he would $ay, that he did not be- lieve the $ame. He would $ay again, in the third place, that if you could a$$ure him thereof, and demon$tratively convince him of the $ame, he would account you a great Dæmon.</P> <P>SAGR. I hope then that it is one of the Socratick, not infernal ones. But that I may make you under$tand this particular, you mu$t know, that if a per$on apprehend not a truth of him$elf, it is impo$$ible that others $hould make him under$tand it: I may in- deed in$truct you in tho$e things which are neither true nor fal$e; but the true, that is, the nece$$ary, namely, $uch as it is impo$$ible $hould be otherwi$e, every common capacity either comprehendeth them of him$elf, or el$e it is impo$$ible he $hould ever know them. And of this opinion I am confident is <I>Salviatus</I> al$o: and there- fore I tell you, that the rea$ons of the pre$ent Problems are known by you, but it may be, not apprehended.</P> <P>SIMP. Let us, for the pre$ent, pa$s by that controver$ie, and permit me to plead ignorance of the$e things you $peak of, and try whether you can make me capable of under$tanding the$e Pro- blems.</P> <P>SAGR. This fir$t dependeth upon another, which is, Whence cometh it, that $etting a top with the la$h, it runneth farther, and con$equently with greater force, than when its $et with the fin- gers?</P> <P>SIMP. <I>Ari$totle</I> al$o makes certain Problems about the$e kinds of projects.</P> <P>SALV. He doth $o; and very ingenious they are: particular- ly, That, Whence it cometh to pa$s that round tops run better than the $quare?</P> <foot>SAGR.</foot> <p n=>139</p> <P>SAGR. And cannot you, <I>Simplicius,</I> give a rea$on for this, without others prompting you?</P> <P>SIMP. Very good, I can $o; but leave your jeering.</P> <P>SAGR. In like manner you do know the rea$on of this other al$o. Tell me therefore; know you that a thing which moveth, being impeded $tands $till?</P> <P>SIMP. I know it doth, if the impediment be $o great as to $uffice.</P> <P>SAGR. Do you know, that moving upon the Earth is a greater impediment to the moveable, than moving in the air, the Earth be- ing rough and hard, and the air $oft and yielding?</P> <P>SIMP. And knowing this, I know that the top will turn fa$ter in the air, than on the ground, $o that my knowledg is quite con- trary to what you think it.</P> <P>SAGR. Fair and $oftly, <I>Simplicius.</I> You know that in the parts of a moveable, that turneth about its centre, there are found motions towards all $ides; $o that $ome a$cend, others de$cend; $ome go forwards, others backwards?</P> <P>SIMP. I know it, and <I>Aristotle</I> taught me the $ame.</P> <P>SAGR. And with what demon$tration, I pray you?</P> <P>SIMP. With that of $en$e.</P> <P>SAGR. <I>Ari$totle,</I> then, hath made you $ee that which without him you would not have $een? Did he ever lend you his eyes? You would $ay, that <I>Ari$totle</I> hath told, adverti$ed, remembered you of the $ame; and not taught you it. When then a top, with- out changing place, turns round, (or in the childrens phra$e, $leep- eth) not paralel, but erect to the Horizon, $ome of its parts a$cend, and the oppo$ite de$cend; the $uperiour go one way, the infe- riour another. Fancie now to your $elf, a top, that without chan- ging place, $wiftly turns round in that manner, and $tands $u$pen- ded in the air, and that in that manner turning, it be let fall to the Earth perpendicularly, do you believe, that when it is arrived at the ground, it will continue to turn round in the $ame manner, without changing place, as before?</P> <P>SIMP. No, Sir.</P> <P>SAGR. What will it do then?</P> <P>SIMP. It will run along the ground very fa$t.</P> <P>SAGR. And towards what part?</P> <P>SIMP. Towards that, whither its ^{*}reeling carrieth it.</P> <marg>* Vertigine.</marg> <P>SAGR. In its reeling there are parts, that is the uppermo$t, which do move contrary to the inferiour; therefore you mu$t in$tance which it $hall obey: for as to the parts a$cending and de$cending, the one kind will not yield to the other; nor will they all go downwards, being hindered by the Earth, nor upwards as being heavy.</P> <foot>2 3 SIMP.</foot> <p n=>140</p> <P>SIMP. The top will run reeling along the floor towards that part whither its upper parts encline it.</P> <P>SAGR. And why not whither the contrary parts tend, namely, tho$e which touch the ground?</P> <P>SIMP. Becau$e tho$e upon the ground happen to be impeded by the roughne$s of the touch, that is, by the floors unevenne$s; but the $uperiour, which are in the tenuous and flexible air, are hindred very little, if at all; and therefore the top will obey their inclination.</P> <P>SAGR. So that that taction, if I may $o $ay, of the neither parts on the floor, is the cau$e that they $tay, and onely the upper parts $pring the top forward.</P> <P>SALV. And therefore, if the top $hould fall upon the ice, or other very $mooth $uperficies, it would not $o well run forward, but might peradventure continue to revolve in it $elf, (or $leep) with- out acquiring any progre$$ive motion.</P> <P>SAGR. It is an ea$ie thing for it $o to do; but yet neverthe- le$s, it would not $o $peedily come to $leep, as when it falleth on a $uperficies $omewhat rugged. But tell me, <I>Simplicius,</I> when the top turning round about it $elf, in that manner, is let fall, why doth it not move forwards in the air, as it doth afterwards when it is upon the ground?</P> <P>SIMP. Becau$e having air above it, and beneath, neither tho$e parts, nor the$e have any where to touch, and not having more oc- ca$ion to go forward than backward, it falls perpendicularly.</P> <P>SAGR. So then the onely reeling about its $elf, without other <I>impetus,</I> can drive the top forward, being arrived at the ground, very nimbly. Now proceed we to what remains. That la$h, which the driver tyeth to his Top-$tick, and with which, winding it about the top, he $ets it (<I>i. e.</I> makes it go) what effect hath it on the $aid top?</P> <P>SIMP. It con$trains it to turn round upon its toe, that $o it may free it $elf from the Top-la$h.</P> <P>SAGR. So then, when the top arriveth at the ground, it cometh all the way turning about its $elf, by means of the la$h. Hath it not rea$on then to move in it $elf more $wiftly upon the ground, than it did whil$t it was in the air?</P> <P>SIMP. Yes doubtle$s; for in the air it had no other impul$e than that of the arm of the projicient; and if it had al$o the reel- ing, this (as hath been $aid) in the air drives it not forward at all: but arriving at the floor, to the motion of the arm is added the progre$$ion of the reeling, whereby the velocity is redoubled. And I know already very well, that the top skipping from the ground, its velocity will demini$h, becau$e the help of its circulation is wanting; and returning to the Earth will get it again, and by that <foot>means</foot> <p n=>141</p> means move again fa$ter, than in the air. It onely re$ts for me to under$tand, whether in this $econd motion on the Earth it move more $wiftly, than in the fir$t; for then it would move <I>in infini- tum,</I> alwayes accelerating.</P> <P>SAGR. I did not ab$olutely affirm, that this $econd motion is more $wift than the fir$t; but that it may happen $o to be $ome- times.</P> <P>SIMP. This is that, which I apprehend not, and which I de$ire to know.</P> <P>SAGR. And this al$o you know of your $elf. Therefore tell me: When you let the top fall out of your hand, without ma- king it turn round (<I>i. e.</I> $etting it) what will it do at its coming to the ground?</P> <P>SIMP. Nothing, but there lie $till.</P> <P>SAGR. May it not chance, that in its fall to the ground it may acquire a motion? Think better on it.</P> <P>SIMP. Unle$$e we let it fall upon $ome inclining $tone, as children do playing at ^{*} <I>Chio$a,</I> and that falling $ide-wayes upon <marg>* A Game in <I>Italy,</I> which is, to glide bullets down an inclining $tone, <I>&c.</I></marg> the $ame, it do acquire the motion of turning round upon its toe, wherewith it afterwards continueth to move progre$$ively on the floor, I know not in what other manner it can do any thing but lie $till where it falleth.</P> <P>SAGR. You $ee then that in $ome ca$e it may acquire a new revolution. When then the top jerked up from the ground, falleth down again, why may it not ca$ually hit upon the declivity of $ome $tone fixed in the floor, and that hath an inclination that way towards which it moveth, and acquiring by that $lip a new whirle over and above that conferred by the la$h, why may it not redouble its motion, and make it $wifter than it was at its fir$t lighting upon the ground?</P> <P>SIMP. Now I $ee that the $ame may ea$ily happen. And I am thinking that if the top $hould turn the contrary way, in ar- riving at the ground, it would work a contrary effect, that is, the motion of the accidental whirl would retard that of the pro- jicient.</P> <P>SAGR. And it would $ometimes wholly retard and $top it, in ca$e the revolution of the top were very $wift. And from hence a- ri$eth the re$olution of that $light, which the more skilful Tennis Players u$e to their advantage; that is, to gull their adver$ary by cutting (for $o is their Phra$e) the Ball; which is, to return it with a $ide Rachet, in $uch a manner, that it doth thereby ac- quire a motion by it $elf contrary to the projected motion, and $o by that means, at its coming to the ground, the rebound, which if the ball did not turn in that manner, would be towards the adver$ary, giving him the u$ual time to to$$e it back again, doth <foot>fail,</foot> <p n=>142</p> fail, and the ball runs tripping along the ground, or rebounds le$$e than u$ual, and breaketh the time of the return. Hence it is <marg>*A Game in <I>Italy,</I> wherein they $trive who $hall trundle or throw a wooden bowle neere$t to an a$$igned mark.</marg> that you $ee, tho$e who play at ^{*} Stool-ball, when they play in a $tony way, or a place full of. holes and rubs that make the ball trip an hundred $everal wayes, never $uffering it to come neer the mark, to avoid them all, they do not trundle the ball upon the ground, but throw it, as if they were to pitch a quait. But be- cau$e in throwing the ball, it i$$ueth out of the hand with $ome roling conferred by the fingers, when ever the hand is under the ball, as it is mo$t commonly held; whereupon the ball in its lighting on the ground neer to the mark, between the motion of the pro- jicient and that of the roling, would run a great way from the $ame: To make the ball $tay, they hold it artificially, with their hand uppermo$t, and it undermo$t, which in its delivery hath a contrary twirl or roling conferred upon it by the fingers, by means whereof in its coming to the ground neer the mark it $tays there, or runs very very little forwards. But to return to our principal problem which gave occa$ion for $tarting the$e others; I $ay it is po$$ible that a per$on carried very $wiftly, may let a ball drop out of his hand, that being come to the Earth, $hall not onely follow his motion, but al$o out-go it, moving with a great- er velocity. And to $ee $uch an effect, I de$ire that the cour$e may be that of a Chariot, to which on the out-$ide let a decli- ning board be fa$tened; $o as that the neither part may be towards the hor$es, and the upper towards the hind Wheel. Now, if in the Chariots full career, a man within it, let a ball fall gliding a- long the declivity of that board, it $hall in roling downward ac- quire a particular <I>vertigo</I> or turning, the which added to the motion impre$$ed by the Chariot, will carrie the ball along the ground much fa$ter than the Chariot. And if one accommodate another declining board over again$t it, the motion of the Cha- riot may be qualified $o, that the ball, gliding downwards along the board, in its coming to the ground $hall re$t immoveable, and al$o $hall $ometimes run the contrary way to the Chariot. But we are $trayed too far from the purpo$e, therefore if <I>Simplicius</I> be $atisfied with the re$olution of the fir$t argnment again$t the Earths mobility, taken from things falling perpendicularly, we may pa$$e to the re$t</P> <P>SALV. The digre$$ions made hitherto, are not $o alienated from the matter in hand, as that one can $ay they are wholly $trangers to it. Be$ides the$e argumentations depend on tho$e things that $tart up in the fancy not of one per$on, but of three, that we are: And moreover we di$cour$e for our plea$ure, nor are we obliged to that $trictne$$e of one who <I>ex profe$$o</I> treateth methodically of an argument, with an intent to publi$h the $ame. <foot>I</foot> <p n=>143</p> I will not con$ent that our Poem $hould be $o confined to that unity, as not to leave us fields open for Ep$ody's, which every $mall connection $hould $uffice to introduce; but with almo$t as much liberry as if we were met to tell $tories, it $hall be lawful for me to $peak, what ever your di$cour$e brings into my mind.</P> <P>SAGR. I like this motion very well; and $ince we are at this liberty, let me take leave, before we pa$$e any farther to ask of you <I>Salviatus,</I> whether you did ever con$ider what that line may be that is de$cribed by the grave moveable naturally falling down from the top of a Tower; and if you have reflected on it, be plea$ed to tell me what you think thereof.</P> <P>SALV. I have $ometimes con$idered of it, and make no que- $tion, that if one could be certain of the nature of that motion wherewith the grave body de$cendeth to approach the centre of the Terre$trial Globe, mixing it $elf afterwards with the common circular motion of the diurnal conver$ion; it might be exactly found what kind of line that is, that the centre of gravity of the moveable de$cribeth in tho$e two motions.</P> <P>SAGR. Touching the $imple motion towards the centre de- pendent on the gravity, I think that one may confidently, with- out error, believe that it is by a right line, as it would be, were the Earth immoveable.</P> <P>SALV. As to this particular, we may not onely believe it, but experience rendereth us certain of the $ame.</P> <P>SAGR. But how doth experience a$$ure us thereof, if we ne- ver $ee any motions but $uch as are compo$ed of the two, circular and de$cending.</P> <P>SALV. Nay rather <I>Sagredus</I> we onely $ee the $imple motion of de$cent; $ince that other circular one common to the Earth, the Tower and our $elves remains imperceptible, and as if it never were, and there remaineth perceptible to us that of the $tone, one- ly not participated by us, and for this, $en$e demon$trateth that it is by a right line, ever parallel to the $aid Tower, which is built upright and perpendicular upon the Terre$trial $urface.</P> <P>SAGR. You are in the right; and this was but too plainly de- mon$trated to me even now, $eeing that I could not remember $o ea$ie a thing; but this being $o manife$t, what more is it that you $ay you de$ire, for under$tanding the nature of this motion downwards?</P> <P>SALV. It $ufficeth not to know that it is $treight, but its requi- $ite to know whether it be uniform, or irregular; that is, whe- ther it maintain alwayes one and the $ame velocity, or el$e goeth retarding or accelerating.</P> <P>SAGR. It is already clear, that it goeth continually accelle- rating.</P> <foot>SALV.</foot> <p n=>144</p> <P>SALV. Neither doth this $uffice, but its requi$ite to know ac- cording to what proportion $uch accelleration is made; a Pro- blem, that I believe was never hitherto under$tood by any Phi- lo$opher or Mathematician; although Philo$ophers, and particu- larly the <I>Peripateticks,</I> have writ great and entire Volumes, touching motion.</P> <P>SIMP. Philo$ophers principally bu$ie them$elves about univer- $als; they find the definitions and more common $ymptomes, o- mitting certain $ubtilties and niceties, which are rather curio- $ities to the Mathematicians. And <I>Aristotle</I> did content him$elf to de$ine excellently what motion was in general; and of the lo- cal, to $hew the principal qualities, to wit, that one is natural, another violent; one is $imple, another compound; one is equal, another accellerate; and concerning the accelerate, con- tents him$elf to give the rea$on of acceleration, remitting the finding out of the proportion of $uch acceleration, and other particular accidents to the Mechanitian, or other inferiour Arti$t.</P> <P>SAGR. Very well <I>Simplicius.</I> But you <I>Salviatus,</I> when you de$cend $ometimes from the Throne of <I>Peripatetick</I> Maje$ty, have you ever thrown away any of your hours in $tudying to find this proportion of the acceleration of the motion of de$cending grave bodies?</P> <P>SALV. There was no need that I $hould $tudy for it, in regard that the Academick our common friend, heretofore $hewed me a Treati$e of his ^{*} <I>De Motu,</I> where this, and many other acci- <marg>This is that ex- cellent tract which we give the fir$t place in our $econd Volume.</marg> dents were demon$trated. But it would be too great a digre$$ion, if for this particular, we $hould interrupt our pre$ent di$cour$e, (which yet it $elf is al$o no better than a digre$$ion) and make as the Saying is, a Comedy within a Comedy.</P> <P>SAGR. I am content to excu$e you from this narration for the pre$ent, provided that this may be one of the Propo$itions re$er- ved to be examined among$t the re$t in another particular meeting, for that the knowledg thereof is by me very much de$ired; and in the mean time let us return to the line de$cribed by the grave body in its fall from the top of the Tower to its ba$e.</P> <P>SALV. If the right motion towards the centre of the Earth was uniforme, the circular towards the Ea$t being al$o uniforme, you would $ee compo$ed of them both a motion by a $piral line, of that kind with tho$e defined by <I>Archimedes</I> in his Book <I>Dc Spira- libus</I>; which are, when a point moveth uniformly upon a right line, while$t that line in the mean time turneth uniformly about one of its extreme points fixed, as the centre of his gyration. But becau$e the right motion of grave bodies falling, is continu- ally accelerated, it is nece$$ary, that the line re$ulting of the <foot>com-</foot> <p n=>145</p> compo$ition of the two motions do go alwayes receding with greater and greater proportion from the circumference of that cir- cle, which the centre of the $tones gravity would have de$igned, if it had alwayes $taid upon the Tower; it followeth of nece$$ity that this rece$$ion at the fir$t be but little, yea very $inall, yea, more, as $mall as can be imagined, $eeing that the de$cending grave body departing from re$t, that is, from the privation of motion, towards the bottom and entring into the right motion downwards, it mu$t needs pa$$e through all the degrees of tardi- ty, that are betwixt re$t, and any a$$igned velocity; the which degrees are infinite; as already hath been at large di$cour$ed and proved.</P> <P>It being $uppo$ed therefore, that the progre$$e of the accele- ration being after this manner, and it being moreover true, that the de$cending grave body goeth to terminate in the centre of the Earth, it is nece$$ary that the line of its mixt motion be $uch, that <marg><I>The line de$cri- bed by a moveable in its natural de- $cent, the motion of the Earth a- bout its own centre being pre$uppo$ed, would probably be the circumference of a circle.</I></marg> it go continually receding with greater and greater proportion from the top of the Tower, or to $peak more properly, from the circumference of the circle de$cribed by the top of the Tower, by means of the Earths conver$ion; but that $uch rece$$ions be le$$er and le$$er <I>in infinitum</I>; by how much the moveable finds it $elf to be le$$e and le$$e removed from the fir$t term where it re$ted. Moreover it is nece$$ary, that this line of the compound- ed motion do go to terminate in the centre of the Earth. Now having pre$uppo$ed the$e two things, I come to de$cribe about the centre A [<I>in Fig. 1. of this $econd Dialogue</I>;] with the $emi- diameter A B, the circle B I, repre$enting to me the Terre$trial Globe, and prolonging the $emidiameter A B to C, I have de- $cribed the height of the Tower B C; the which being carried about by the Earth along the circumference B I, de$cribeth with its top the arch C D: Dividing, in the next place, the line C A in the middle at E; upon the centre E, at the di$tance E C, I de- $cribe the $emicircle C I A: In which, I now affirm, that it is very probable that a $tone falling from the top of the Tower C, doth move, with a motion mixt of the circular, which is in common, and of its peculiar right motion. If therefore in the circumference C D, certain equal parts C F, F G, G H, H L, be marked, and from the points F, G, H, L, right lines be drawn towards the centre A, the parts of them intercepted between the two cir- cumferences C D and B I, $hall repre$ent unto us the $ame Tower C B, tran$ported by the Terre$trial Globe towards D I; in which lines the points where they come to be inter$ected by the arch of the $emicircle C I, are the places by whichfrom time to time the falling $tone doth pa$$e; which points go continually with greater and greater proportion receding from the top of the <foot>T Tower.</foot> <p n=>146</p> Tower. And this is the cau$e why the right motion made along the $ide of the Tower appeareth to us more and more accelerate. It appeareth al$o, how by rea$on of the infinite acutene$$e of the contact of tho$e two circles D C, C I, the rece$$ion of the cadent moveable from the circumference C F D; namely, from the top of the Tower, is towards the beginning extream $mall, which is as much as if one $aid its motion downwards is very $low, and more and more $low <I>in infinitum,</I> according to its vicinity to the term C, that is to the $tate of re$t. And la$tly it is $een how in the end this $ame motion goeth to terminate in the centre of the Earth A.</P> <P>SAGR. I under$tand all this very well, nor can I per$wade my $elf that the falling moveable doth de$cribe with the centre of its gravity any other line, but $uch an one as this.</P> <P>SALV. But $tay a little <I>Sagredus,</I> for I am to acquaint you al$o with three Ob$ervations of mine, that its po$$ible will not di$- <marg><I>A moveable fal- ting from the top of the Tower, moveth in the circumfe- rence of a circle.</I></marg> plea$e you. The fir$t of which is, that if we do well con$ider, the moveable moveth not really with any more than onely one motion $imply circular, as when being placed upon the Tower, it moved with one $ingle and circular motion. The $econd is yet more plea- <marg><I>It moveth neither more nor le$$e, than if it had $taid al- wayes there.</I></marg> $ant; for, it moveth neither more nor le$$e then if it had $taid con- tinually upon the Tower, being that to the arches C F, F G, G H, &c. that it would have pa$$ed continuing alwayes upon the Tower, the arches of the circumference C I are exactly equal, an$wering under the $ame C F, F G, G H, &c. Whence followeth the third <marg><I>It moveth with an uniform, not an accelerate mo- tion.</I></marg> wonder, That the true and real motion of the $tone is never acce- lerated, but alwayes even and uniforme, $ince that all the equal ar- ches noted in the circumference C D, and their re$pondent ones marked in the circumference C I, are pa$t in equal times; $o that we are left at liberty to $eek new cau$es of acceleration, or of o- ther motions, $eeing that the moveable, as well $tanding upon the Tower, as de$cending thence, alwayes moveth in the $ame fa$hion, that is, circularly, with the $ame velocity, and with the $ame uni- formity. Now tell me what you think of this my fanta$tical con- jecture.</P> <P>SAGR. I mu$t tell you, that I cannot with words $ufficiently expre$$e how admirable it $eemeth to me; and for what at pre- $ent offereth it $elf to my under$tanding, I cannot think that the bu$ine$s happeneth otherwi$e; and would to God that all the demon$trations of Philo$ophers were but half $o probable as this. However for my perfect $atisfaction I would gladly hear how you prove tho$e arches to be equal.</P> <P>SALV. The demon$tration is mo$t ea$ie. Suppo$e to your $elf a line drawn from I to E. And the Semidiameter of the circle CD, that is, the line C A, being double the Semidiameter C E of the <foot>cir-</foot> <p n=>147</p> circle C I, the circumference $hall be double to the circumference, and every arch of the greater circle double to every like arch of the le$$er; and con$equently, the half of the arch of the greater circle, equal to the whole arch of the le$$e. And becau$e the an- gle C E I made in the centre E of the le$$er circle, and which in$i- $teth upon the arch C I, is double the angle C A D, made in the centre A of the greater circle, to which the arch C D $ubtendeth; therefore the arch C D is half of the arch of the greater circle like to the arch C I, and therefore the two arches C D and C I are e- qual; and in the $ame manner we may demon$trate of all their parts. But that the bu$ine$s, as to the motion of de$cending grave bodies, proceedeth exactly thus, I will not at this time affirm; but this I will $ay, that if the line de$cribed by the cadent moveable be not exactly the $ame with this, it doth extream neerly re$emble the $ame.</P> <P>SAGR. But I, <I>Salviatus,</I> am ju$t now con$idering another par- <marg><I>Right motion $eemeth wholly ex- cluded in nature.</I></marg> ticular very admirable; and this it is; That admitting the$e con- $iderations, the right motion doth go wholly ^{*} mounting, and that <marg>* Vadia del tutto a monte, <I>rendered in the Latixe</I> omni- no pe$$um eat.</marg> Nature never makes u$e thereof, $ince that, even that that u$e, which was from the beginning granted to it, which was of redu- cing the parts of integral bodies to their place, when they were $eparated from their whole, and therefore con$tituted in a depra- ved di$po$ition, is taken from it, and a$$igned to the circular motion.</P> <P>SALV. This would nece$$arily follow, if it were concluded that the Terre$trial Globe moveth circularly; a thing, which I pretend not to be done, but have onely hitherto attempted, as I $hall $till, to examine the $trength of tho$e rea$ons, which have been alledged by Philo$ophers to prove the immobility of the Earth, of which this fir$t taken from things falling perpendicu- larly, hath begat the doubts, that have been mentioned; which I know not of what force they may have $eemed to <I>Simplicius</I>; and therefore before I pa$$e to the examination of the remaining arguments, it would be convenient that he produce what he hath to reply to the contrary.</P> <P>SIMP. As to this fir$t, I confe$$e indeed that I have heard $undry pretty notions, which I never thought upon before, and in regard they are new unto me, I cannot have an$wers $o ready for them, but this argument taken from things falling perpendi- cularly, I e$teem it not one of the $tronge$t proofs of the mobi- lity of the Earth; and I know not what may happen touching the $hots of great Guns, e$pecially tho$e aimed contrary to the diur- nal motion.</P> <P>SAGR. The flying of the birds as much puzzleth me as the objection of the Gun-$hot, and all the other experiments above <foot>T 2 al-</foot> <p n=>148</p> alledged. For the$e birds which at their plea$ure flie for- wards and backwards, and wind to and again in a thou$and fa$hions, and, which more importeth, lie whole hours upon the wing, the$e I $ay do not a little po$e me, nor do I $ee, how a- mong$t $o many circumgyrations, they $hould not lo$e the motion of the Earth, and how they $hould be able to keep pace with $o great a velocity as that which they $o far exceed with their flight.</P> <P>SALV. To $peak the truth, your $cruple is not without rea$on, and its po$$ible <I>Copernicus</I> him$elf could not find an an$wer for it, that was to him$elf entirely $atisfactory; and therefore haply pa$t it over in $ilence albeit he was, indeed, very brief in examining the other allegations of his adver$aries, I believe through his height of wit, placed on greater aud $ublimer contemplations, like as Lions are not much moved at the barking of little Dogs. We will therefore re$erve the in$tance of birds to the la$t place, and for the pre$ent, $ee if we can give <I>Simplicius</I> $atisfaction in the others, by $hewing him in our wonted manner, that he him- $elf hath their an$wers at hand, though upon fir$t thoughts he doth not di$cover them. And to begin with the $hots made at randome, with the $elf $ame piece, powder, and ball, the one towards the Ea$t, the other towards the We$t, let him tell me what it is that per$wades him to think that the Range towards the We$t (if the diurnal con- ver$ion belonged to the Earth) ought to be much longer than that towards the Ea$t.</P> <marg><I>The rea$on why a Gun $hould $iem to carry farther to- wards the We$t than towards the Ea$t.</I></marg> <P>SIMP. I am moved $o to think; becau$e in the $hot made to- wards the Ea$t, the ball whil'$t it is out of the piece, is follow- ed by the $aid piece, the which being carried round by the Earth, runneth al$o with much velocity towards the $ame part, where- upon the fall of the ball to the ground, cometh to be but little di$tant from the piece. On the contrary in the $hot towards the We$t, before that the ball falleth to the ground, the piece is re- tired very far towards the Ea$t, by which means the $pace be- tween the ball and the piece, that is Range, will appear longer than the other, by how much the piece, that is the Earth, had run in the time that both the bals were in the air.</P> <P>SALV. I could wi$h, that we did know $ome way to make an experiment corre$ponding to the motion of the$e projects, as that of the $hip doth to the motion of things perpendicularly falling from on high; and I am thinking how it may be done.</P> <marg><I>The experiment of a running cha- riot to find out the difference of Ran- ges.</I></marg> <P>SAGR. I believe, that it would be a very oppo$ite proof, to take an open Chariot, and to accomodate therein a ^{*}Stock-bow at half elevation, to the end the flight may prove the greate$t <marg>* Bale$trone da bol- zoni.</marg> that my be, and whil'$t the hor$es $hall run, to $hoot fir$t towards the part whither you drive, and then another backwards towards the contrary part, cau$ing $ome one to mark diligently where the Chariot was in that moment f time when the $haft came to <foot>the</foot> <p n=>149</p> the ground, as well in the one $hot as in the other: for thus you may $ee exactly how much one $haft flew farther than the other.</P> <P>SIMP. In my thoughts this experiment is very proper: and I do not doubt but that the flight, that is, the $pace between the $haft and the place where the chariot was at the $hafts fall, will be le$s by much when one $hooteth towards the chariots cour$e, than when one $hooteth the contrary way. For an example, let the flight of it $elf be three hundred yards, and the cour$e of the cha- riot in the time whil$t the $haft $tayeth in the air, an hundred yards, therefore $hooting towards the cour$e, of the three hundred yards of the flight, the chariot will have gone one hundred; $o then at the $hafts coming to the ground, the $pace between it and the chariot, $hall be but two hundred yards onely; but on the contrary, in the other $hoot, the chariot running contrary to the $haft, when the $haft $hall have pa$$ed its three hundred yards, and the chariot its other hundred the contrary way, the di$tance inter- po$ing $hall be found to be four hundred yards.</P> <P>SALV. Is there any way to $hoot $o that the$e flights may be equal?</P> <P>SIMP. I know no other way, unle$s by making the chariot to $tand $till.</P> <P>SALV. This we know; but I mean when the chariot runneth in full carreer.</P> <P>SIMP. In that ca$e you are to draw the Bow higher in $hoot- ing forwards, and to $lack it in $hooting the contrary way.</P> <P>SALV. Then you $ee that there is one way more. But how much is the bow to be drawn, and how much $lackened?</P> <P>SIMP. In our ca$e, where we have $uppo$ed that the bow car- ried three hundred yards, it would be requi$ite to draw it $o, as that it might carry four hundred, and in the other to $lacken it $o, as that it might carry no more than two hundred. For $o each of the flights would be but three hundred in relation to the chariot, the which, with its cour$e of an hundred yards which it $ub$tracts from the $hoot of four hundred, and addeth to that of two hun- dred, would reduce them both to three hundred.</P> <P>SALV. But what effect hath the greater or le$s inten$ne$s of the bow upon the $haft?</P> <P>SIMP. The $tiffer bow carrieth it with greater velocity, and the weaker with le$s; and the $ame $haft flieth $o much farther at one time than another, with how much greater velocity it goeth out of the tiller at one time, than another.</P> <P>SALV. So that to make the $haft $hot either way, to flie at e- qual di$tance from the running chariot, it is requi$ite, that if in the fir$t $hoot of the precedent example, it goeth out of the tiller with <I>v. g.</I> four degrees of velocity, that then in the other $hoot it de- <foot>part</foot> <p n=>150</p> part but with two onely: but if the $ame bow be u$ed, it always receiveth thence three degrees.</P> <P>SIMP. It doth $o; and for this rea$on, $hooting with the $ame bow in the chariots cour$e, the $hoots cannot be equal.</P> <P>SALV. I had forgot to ask, with what velocity it is $uppo$ed in this particular experiment, that the chariot runneth.</P> <P>SIMP. The velocity of the chariot mu$t be $uppo$ed to be one degree in compari$on to that of the bow, which is three,</P> <P>SALV. Very right, for $o computation gives it. But tell me, when the chariot moveth, doth not all things in the $ame move with the $ame velocity?</P> <P>SIMP. Yes doubtle$s.</P> <P>SALV. Then $o doth the $haft al$o, and the bow, and the $tring, upon which the $haft is nock't.</P> <P>SIMP. They do $o.</P> <P>SALV. Why then, in di$charging the $haft towards the cour$e of the chariot, the bow impre$$eth its three degrees of velocity on a $haft that had one degree of velocity before, by means of the chariot which tran$ported it $o fa$t towards that part; $o that in its going off it hath four degrees of velocity. On the contrary, in the other $hoot, the $ame bow conferreth its $ame three degrees of velocity on a $haft that moveth the contrary way, with one de- gree; $o that in its departing from the bow-$tring, it hath no more left but onely two degrees of velocity. But you your $elf have already $aid, that the way to make the $hoots equal, is to cau$e that the $haft be let flie the fir$t time with four degrees of velocity, and the $econd time with two. Therefore without changing the bow, the very cour$e of the chariot is that which adju$teth the <marg><I>The $olution of the argument ta- ken from great- Guns $hot towards the East & We$t.</I></marg> flights, and the experiment doth $o repre$ent them to any one who is not either wilfully or naturally incapable of rea$on. Now apply this di$cour$e to Gunnery, and you $hall find, that whether the Earth move or $tand $till, the $hots made with the $ame force, will always curry equal ranges, to what part $oever aimed. The error of <I>Ari$totle, Ptolomey, Iycho,</I> your $elf, and all the re$t, is ground- ed upon that fixed and $trong per$ua$ion, that the Earth $tandeth $till, which you have not judgment nor power to depo$e, no not when you have a de$ire to argue of that which would en$ue, pre- $uppo$ing the Earth to move. And thus, in the other argument, not con$idering that whil'$t the $tone is upon the Tower, it doth, as to moving or not moving, the $ame that the Terre$trial Globe doth, becau$e you have concluded with your $elf, that the Earth $tands $till, you always di$cour$e touching the fall of the $tone, as if it were to depart from re$t: whereas it behooveth to $ay, that if the Earth $tandeth $till, the $tone departeth from re$t, and de- $cendeth perpendicularly; but if the Earth do move, the $tone <foot>likewi$e</foot> <p n=>151</p> likewi$e moveth with like velocity, nor doth it depart from re$t, but from a motion equal to that of the Earth, wherewith it inter- mixeth the $upervenient motion of de$cent, and of tho$e two com- po$eth a third which is tran$ver$al or $ide-ways.</P> <P>SIMP. But for Gods $ake, if it move tran$ver$ly, how is it that I behold it to move directly and perpendicularly? This is no bet- ter than the denial of manife$t $en$e; and if we may not believe $en$e, at what other door $hall we enter into di$qui$itions of Philo- $ophy?</P> <P>SALV. In re$pect to the Earth, to the Tower, and to our $elves, which all as one piece move with the diurnal motion together with the $tone, the diurnal motion is as if it never had been, and becom- eth in$en$ible, imperceptible, and without any action at all; and the onely motion which we can perceive, is that of which we par- take not, that is the de$cent gliding along the $ide of the Tower: You are not the fir$t that hath felt great repugnance in apprehen- ding this non-operating of motion upon things to which it is com- mon.</P> <P>SAGR. Now I do remember a certain conceipt, that came one <marg><I>A notable ca$e of</I> Sagredus, <I>to $hew the non-operating of common motion.</I></marg> day into my fancy, whil$t I $ailed in my voyage to <I>Aleppo,</I> whither I went Con$ul for our Countrey, and po$$ibly it may be of $ome u$e, for explaining this nullity of operation of common motion, and being as if it never were to all the partakers thereof. And if it $tand with the good liking of <I>Simplicius,</I> I will rea$on with him upon that which then I thought of by my $elf alone.</P> <P>SIMP. The novelty of the things which I hear, makes me not $o much a patient, as a greedy and curious auditor: therefore go on.</P> <P>SAGR. If the neb of a writing pen, that I carried along with me in the $hip, through all my navigation from <I>Venice</I> to ^{*} <I>Scan-</I> <marg>* Ale$$andretta.</marg> <I>deron,</I> had had a facultie of leaving vi$ible marks of its whole voy- age, what $igns, what marks, what lines would it have left?</P> <P>SIMP. It would have left a line di$tended from <I>Venice</I> thither, not perfectly $treight, or to $ay better, di$tended in a perfect arch of a circle, but in $ome places more, in $ome le$s curved, according as the ve$$el had gone more or le$s fluctuating; but this its infle- cting in $ome places a fathom or two to the right hand or to the left, upwards or downwards, in a length of many hundred miles, would have brought but little alteration to the intire tract of the line, $o that it would have been hardly $en$ible; and without any con$iderable error, might have been called the part of a perfect arch.</P> <P>SAGR. So that the true and mo$t exact motion of the neb of my pen would have al$o been an arch of a perfect circle, if the ve$$els motion, the fluctuation of the billows cea$ing, had been <foot>calm</foot> <p n=>152</p> calm and tranquill. And if I had continually held that pen in my hand, and had onely moved it $ometimes an inch or two this way or that way, what alteration $hould I have made in that its principal, and very long tract or $troke?</P> <P>SIMP. Le$s than that which the declining in $everal places from ab$olute rectitude, but the quantity of a flea's eye makes in a right line of a thou$and yards long.</P> <P>SAGR. If a Painter, then, at our launching from the Port, had began to de$ign upon a paper with that pen, and continued his work till he came to <I>Scanderon,</I> he would have been able to have taken by its motion a perfect draught of all tho$e figures perfectly interwoven and $hadowed on $everal $ides with countreys, build- ings, living creatures, and other things; albeit all the true, real, and e$$ential motion traced out by the neb of that pen, would have been no other than a very long, but $imple line: and as to the proper operation of the Painter, he would have delineated the $ame to an hair, if the $hip had $tood $till. That therefore of the huge long motion of the pen there doth remain no other marks, than tho$e tracks drawn upon the paper, the rea$on thereof is be- cau$e the grand motion from <I>Venice</I> to <I>Scanderon,</I> was common to the paper, the pen, and all that which was in the $hip: but the petty motions forwards and backwards, to the right, to the left, com- municated by the fingers of the Painter unto the pen, and not to the paper, as being peculiar thereunto, might leave marks of it $elf upon the paper, which did not move with that motion. Thus it is likewi$e true, that the Earth moving, the motion of the $tone in de$cending downwards, was really a long tract of many hundreds and thou$ands of yards, and if it could have been able to have de- lineated in a calm air, or other $uperficies, the track of its cour$e, it would have left behind an huge long tran$ver$e line. But that part of all this motion which is common to the $tone, the Tower, and our $elves, is imperceptible to us, and as if it had never been, and that part onely remaineth ob$ervable, of which neither the Tower nor we are partakers, which is in fine, that wherewith the $tone falling mea$ureth the Tower.</P> <P>SALV. A mo$t witty conceipt to clear up this point, which was not a little difficult to many capacities. Now if <I>Simplicius</I> will make no farther reply, we may pa$s to the other experiments, the unfolding of which will receive no $mall facility from the things already declared.</P> <P>SIMP. I have nothing more to $ay: and I was well-nigh tran$- ported with that delineation, and with thinking how tho$e $trokes drawn $o many ways, hither, thither, upwards, downwards, for- wards, backwards, and interwoven with thou$ands of turnings, are not e$$entially or really other, than $mall pieces of one $ole line <foot>drawn</foot> <p n=>153</p> drawn all one way, and the $ame without any other alteration $ave the declining the direct rectitude, $ometimes a very in$en$ible mat- ter towards one $ide or another, and the pens moving its neb one while $ofter, another while $lower, but with very $mall inequality. And I think that it would in the $ame manner write a letter, and that tho$e frollike penmen, who to $hew their command of hand, without taking their pen from the paper in one $ole $troke, with infinite turnings draw a plea$ant knot, if they were in a boat that did tide it along $wiftly they would convert the whole motion of the pen, which in reality is but one $ole line, drawn all towards one and the $ame part, and very little curved, or declining from perfect rectitude, into a knot or flouri$h. And I am much plea$ed that <I>S agredus</I> hath helped me to this conceit: therefore let us go on, for the hope of meeting with more of them, will make me the $tricter in my attention.</P> <P>SAGR. If you have a curio$ity to hear $uch like $ubtilties, which <marg><I>Subtilties $uffici- ently in$ipid, ironi- cally, $poken and taken from a cer- tain</I> Encyclopædia.</marg> occurr not thus to every one, you will find no want of them, e$pe- cially in this particular of Navigation; and do you not think that a witty conceit which I met with likewi$e in the $ame voyage, when I ob$erved that the ma$t of the $hip, without either breaking or bend- ing, had made a greater voyage with its round-top, that is with its top-gallant, than with its foot; for the round top being more di$tant from the centre of the Earth than the foot is, it had de$cribed the arch of a circle bigger than the circle by which the foot had pa$$ed.</P> <P>SIMP. And thus when a man walketh he goeth farther with his head than with his feet.</P> <P>SAGR. You have found out the matter your $elf by help of your own mother-wit: But let us not interrupt <I>Salviatus.</I></P> <P>SALV. It plea$eth me to $ee <I>Simplicius</I> how he $ootheth up him$elf in this conceit, if happly it be his own, and that he hath not borrowed it from a certain little pamphlet of conclu$ions, where there are a great many more $uch fancies no le$s plea$ant & witty. It followeth that we $peak of the peice of Ordinance mounted per- <marg><I>An in$tance a- gainst the diurnal motion of the earth, taken from the $hot of a Peece of Ordi- nance perpendicu- larly.</I></marg> pendicular to the Horizon, that is, of a $hot towards our vertical point, and to conclude, of the return of the ball by the $ame line unto the $ame peice, though that in the long time which it is $e- parated from the peice, the earth hath tran$ported it many miles towards the Ea$t; now it $eemeth, that the ball ought to fall a like di$tance from the peice towards the We$t; the which doth not happen: therefore the peice without having been moved did $tay expecting the $ame. The an$wer is the $ame with that of the <marg><I>The an$wer to the objection, $hewing the equivoke.</I></marg> $tone falling from the Tower; and all the fallacy, and equivocati- on con$i$teth in $uppo$ing $till for true, that which is in que$tion; for the Opponent hath it $till fixed in his conceit that the ball departs from its re$t, being di$charged by the fire <foot>V from</foot> <p n=>154</p> from the piece; and the departing from the $tate of re$t, cannot be, unle$$e the immobility of the Terre$trial Globe be pre$uppo- $ed, which is the conclu$ion of that was in di$pute; Therefore, I reply, that tho$e who make the Earth moveable, an$wer, that the piece, and the ball that is in it, partake of the $ame motion with the Earth; nay that they have this together with her from nature; and that therefore the ball departs in no other manner from its quie$cence, but conjoyned with its motion about the cen- tre, the which by its projection upwards, is neither taken away, nor hindered; and in this manner following, the univer$al motion of the Earth towards the Ea$t, it alwayes keepeth perpendicular over the $aid piece, as well in its ri$e as in its return. And the $ame you $ee to en$ue, in making the experiment in a $hip with a bullet $hot upwards perpendicularly with a Cro$$e-bow, which returneth to the $ame place whether the $hip doth move, or $tand $till.</P> <marg><I>Another an$wer to the $ame objecti- on.</I></marg> <P>SAGR. This $atisfieth very well to all; but becau$e that I have $een that <I>Simplicius</I> taketh plea$ure with certain $ubtilties to puzzle his companions, I will demand of him whether, $uppo- $ing for this time that the Earth $tandeth $till, and the piece ere- cted upon it perpendicularly, directed to our Zenith, he do at all que$tion that to be the true perpendicular $hot, and that the ball in departing, and in its return is to go by the $ame right line, $till $uppo$ing all external and accidental impediments to be re- moved?</P> <P>SIMP. I under$tand that the matter ought to $ucceed exactly in that manner.</P> <P>SAGR. But if the piece were placed, not perpendicularly, but inclining towards $ome place, what would the motion of the ball be? Would it go haply, as in the other $hot, by the perpendi- cular line, and return again by the $ame?</P> <P>SIMP. It would not $o do; but i$$uing out of the piece, it would pur$ue its motion by a right line which prolongeth the e- rect perpendicularity of the concave cylinder of the piece, unle$$e $o far as its own weight would make it decline from that erection towards the Earth.</P> <P>SAGR. So that the mounture of the cylinder is the regulator of the motion of the ball, nor doth it, or would it move out of that line, if its own gravity did not make it decline downwards. And <marg><I>Projects conti- nue their motion by the right line that followeth the direction of the motion, made to- gether with the projicient, whil'$t they were conjoin'd therewith.</I></marg> therefore placing the cylinder perpendicularly, and $hooting the ball upwards, it returneth by the $ame right line downwards; be- cau$e the motion of the ball dependent on its gravity is down- ward, by the $ame perpendicular. The journey therefore of the ball out of the piece, continueth or prolongeth the rectitude or perpendicularity of that $mall part of the $aid journey, which it made within the $aid piece; is it not $o?</P> <foot>SIMP.</foot> <p n=>155</p> <P>SIMP. So it is, in my opinion.</P> <P>SAGR. Now imagine the cylinder to be erected, and that the Earth doth revolve about with a diurnal motion, carrying the piece along with it, tell me what $hall be the motion of the ball within the cylinder, having given fire?</P> <P>SIMP. It $hall be a $treight and perpendicular motion, the cylin- der being erected perpendicularly.</P> <P>SAGR. Con$ider well what you $ay: for I believe that it will not be perpendicular. It would indeed be perpendicular, if the Earth $tood $till, for $o the ball would have no other motion but that proceeding from the fire. But in ca$e the Earth turns round, <marg><I>The revolution of the Earth $up- po$ed, the ball in the piece erected perpendicularly, doth not move by a perpendicular, but an inclined line.</I></marg> the ball that is in the piece, hath likewi$e a diurnal motion, $o that there being added to the $ame the impul$e of the fire, it mo- veth from the breech of the piece to the muzzle with two motions, from the compo$ition whereof it cometh to pa$$e that the motion made by the centre of the balls gravity is an inclining line. And for your clearer under$tanding the $ame, let the piece A C [<I>in Fig.</I> 2.] be erected, and in it the ball B; it is manife$t, that the piece $tanding immoveable, and fire being given to it, the ball will make its way out by the mouth A, and with its centre, pa$- $ing thorow the the piece, $hall have de$cribed the perpendicular line B A, and it $hall pur$ue that rectitude when it is out of the piece, moving toward the Zenith. But in ca$e the Earth $hould move round, and con$equently carry the piece along with it, in the time that the ball driven out of the piece $hall move along the cylinder, the piece being carried by the Earth, $hall pa$$e in- to the $ituation D E, and the ball B, in going off, would be at the corni$h D, and the motion of the bals centre, would have been according to the line B D, no longer perpendicular, but in- clining towards the Ea$t; and the ball (as hath been concluded) being to continue its motion through the air, according to the direction of the motion made in the piece, the $aid motion $hall continue on according to the inclination of the line B D, and $o $hall no longer be perpendicular, but inclined towards the Ea$t, to which part the piece doth al$o move; whereupon the ball may follow the motion of the Eerth, and of the piece. Now <I>Simplicius,</I> you $ee it demon$trated, that the Range which you took to be perpendicular, is not $o.</P> <P>SIMP. I do not very well under$tand this bu$ine$s; do you, <I>Salviatus</I>?</P> <P>SALV. I apprehend it in part; but I have a certain kind of $cruple, which I wi$h I knew how to expre$s. It $eems to me, that according to what hath been $aid, if the Piece be erected perpen- dicular, and the Earth do move, the ball would not be to fall, as <I>Ari$totle</I> and <I>Tycho</I> will have it, far from the Piece towards the <foot>V 2 We$t,</foot> <p n=>156</p> We$t, nor as you would have it, upon the Piece, but rather far di$tant towards the Ea$t. For according to yo<*>r explanation, it would have two motions, the which would with one con$ent carry it thitherward, to wit, the common motion of the Earth, which carrieth the Piece and the ball from C A towards E D; and the fire which carrieth it by the inclined line B D, both motions to- wards the Ea$t, and therefore they are $uperiour to the motion of the Earth.</P> <P>SAGR. Not $o, Sir. The motion which carrieth the ball to- wards the Ea$t, cometh all from the Earth, and the fire hath no part at all therein: the motion which mounteth the ball upwards, is wholly of fire, wherewith the Earth hath nothing to do. And that it is $o, if you give not fire, the ball will never go out of the Piece, nor yet ri$e upwards a hairs breadth; as al$o if you make the Earth immoveable, and give fire, the ball without any incli- nation $hall go perpendicularly upwards. The ball therefore ha- ving two motions, one upwards, and the other in gyration, of both which the tran$ver$e line B D is compounded, the impul$e upward is wholly of fire, the circular cometh wholly from the Earth, and is equal to the Earths motion: and being equal to it, the ball maintaineth it $elf all the way directly over the mouth of the Piece, and at la$t falleth back into the $ame: and becau$e it al- ways ob$erveth the erection of the Piece, it appeareth al$o conti- nually over the head of him that is near the Piece, and therefore it appeareth to mount exactly perpendicular towards our Zenith, or vertical point.</P> <P>SIMP. I have yet one doubt more remaining, and it is, that in regard the motion of the ball is very $wift in the Piece, it $eems not po$$ible, that in that moment of time the tran$po$ition of the Piece from C A to A D $hould confer $uch an inclination upon the tran$ver$e line C D, that by means thereof, the ball when it cometh afterwards into the air $hould be able to follow the cour$e of the Earth.</P> <P>SAGR. You err upon many accounts; and fir$t, the inclination of the tran$ver$e line C D, I believe it is much greater than you take it to be, for I verily think that the velocity of the Earths mo- tion, not onely under the Equinoctial, but in our paralel al$o, is greater than that of the ball whil$t it moveth in the Piece; $o that the interval C E would be ab$olutely much bigger than the whole length of the Piece, and the inclination of the tran$ver$e line con- $equently bigger than half a right angle: but be the velocity of the Earth more, or be it le$s, in compari$on of the velocity of the fire, this imports nothing; for if the velocity of the Earth be $mall, and con$equently the inclination of the tran$ver$e line be little al$o; there is then al$o need but of little inclination to make the <foot>ball</foot> <p n=>157</p> ball $u$pend it $elf in its range directly over the Piece. And in a word, if you do but attentively con$ider, you will comprehend, that the motion of the Earth in transferring the Piece along with it from C A to E D, conferreth upon the tran$ver$e line C D, $o much of little or great inclination, as is required to adju$t the range to its perpendicularity. But you err, $econdly, in that you referr the faculty of carrying the ball along with the Earth to the impul$e of the fire, and you run into the $ame error, into which <I>Salviatus,</I> but even now $eemed to have fallen; for the faculty of following the motion of the Earth, is the primary and perpetual motion, indelibly and in$eparably imparted to the $aid ball, as to a thing terre$trial, and that of its own nature doth and ever $hall po$$e$s the $ame.</P> <P>SALV. Let us yield, <I>Simplicius,</I> for the bu$ine$s is ju$t as he <marg><I>The manner how Fowlers $hoot birds flying.</I></marg> $aith. And now from this di$cour$e let us come to under$tand the rea$on of a Venatorian Problem, of tho$e Fowlers who with their guns $hoot a bird flying; and becau$e I did imagine, that in regard the bird flieth a great pace, therefore they $hould aim their $hot far from the bird, anticipating its flight for a certain $pace, and more or le$s according to its velocity and the di$tance of the bird, that $o the bullet ha$ting directly to the mark aimed at, it might come to arrive at the $elf $ame time in the $ame point with its motion, and the bird with its flight, and by that means one to encounter the other: and asking one of them, if their practi$e was not $o to do; He told me, no; but that the $light was very ea$ie and certain, and that they took aim ju$t in the $ame manner as if they had $hot at a bird that did $it $till; that is, they made the flying bird their mark, and by moving their fowling-piece they followed her, keeping their aim $till full upon her, till $uch time as they let fly, and in this manner $hot her as they did others $itting $till. It is nece$$ary therefore that that motion, though $low, which the fowl- ing-piece maketh in turning and following after the flight of the bird do communicate it $elf to the bullet al$o, and that it be joyned with that of the fire; $o that the ball hath from the fire the mo- tion directly upwards, and from the concave Cylinder of the barrel the declination according to the flight of the Bird, ju$t as was $aid before of the $hot of a Canon; where the ball receiveth from the fire a virtue of mounting upwards towards the Zenith, and from the motion of the Earth its winding towards the Ea$t, and of both maketh a compound motion that followeth the cour$e of the Earth, and that to the beholder $eemeth onely to go directly up- wards, and return again downwards by the $ame line. The hold- ing therefore of the gun continually directed towards the mark, maketh the $hoot hit right, and that you may keep your gun di- rected to the mark, in ca$e the mark $tands $till, you mu$t al$o hold <foot>your</foot> <p n=>158</p> your gun $till; and if the mark $hall move, the gun mu$t be kept upon the mark by moving. And upon this dependeth the proper an$wer <marg><I>The an$wer to the objection tak n from the $hots of great Guns made towards the North and South.</I></marg> to the other argument taken from the $hot of a Canon, at the mark placed towards the South o<*> North: wherein is alledged, that if the Earth $hould move, the $hots would all range We$t- ward of the mark, becau$e that in the time whil$t the ball, being forc'd out of the Piece, goeth through the air to the mark, the $aid mark being carried toward the Ea$t, would leave the ball to the We$tward. I an$wer therefore, demanding whether if the Ca- non be aimed true at the mark, and permitted $o to continue, it will con$tantly hit the $aid mark, whether the Earth move or $tand $till? It mu$t be replied, that the aim altereth not at all, for if the mark doth $tand $till, the Piece al$o doth $tand $till, and if it, being tran$ported by the Earths motion, doth move, the Piece doth al$o move at the $ame rate, and, the aim maintained, the $hot proveth always true, as by what hath been $aid above, is mani- fe$t.</P> <P>SAGR. Stay a little, I entreat you, <I>Salviatus,</I> till I have pro- pounded a certain conceit touching the$e $hooters of birds flying, who$e proceeding I believe to be the $ame which you relate, and believe the effect of hitting the bird doth likewi$e follow: but yet I cannot think that act altogether conformable to this of $hooting in great Guns, which ought to hit as well when the piece and mark moveth, as when they both $tand $till; and the$e, in my opinion, are the particulars in which they di$agree. In $hooting with a great Gun both it and the mark move with equal velocity, being both tran$ported by the motion of the Terre$trial Globe: and al- beit $ometimes the piece being planted more towards the Pole, than the mark, and con$equently its motion being $omewhat flow- er than the motion of the mark, as being made in a le$$er circle, $uch a difference is in$en$ible, at that little di$tance of the piece from the mark: but in the $hot of the Fowler the motion of the Fowling-piece wherewith it goeth following the bird, is very $low in compari$on of the flight of the $aid bird; whence me thinks it $hould follow, that that $mall motion which the turning of the Birding-piece conferreth on the bullet that is within it, cannot, when it is once gone forth of it, multiply it $elf in the air, untill it come to equal the velocity of the birds flight, $o as that the $aid bullet $hould always keep direct upon it: nay, me thinketh the bird would anticipate it and leave it behind. Let me add, that in this act, the air through which the bullet is to pa$s, partaketh not of the motion of the bird: whereas in the ca$e of the Canon, both it, the mark, and the intermediate air, do equally partake of the com- mon diurnal motion. So that the true cau$e of the Marks-man his hitting the mark, as it $hould $eem, moreover and be$ides the <foot>following</foot> <p n=>159</p> following the birds flight with the piece, is his $omewhat anticipa- ting it, taking his aim before it; as al$o his $hooting (as I believe) not with one bullet, but with many $mall balls (called $hot) the which $cattering in the air po$$e$s a great $pace; and al$o the ex- treme velocity wherewith the$e $hot, being di$charged from the Gun, go towards the bird.</P> <P>SALV. See how far the winged wit of <I>Sagredus</I> anticipateth, and out-goeth the dulne$s of mine; which perhaps would have <marg><I>The an$wer to the Argument taken from the $hots at point blanck to- wards the Ea$t & We$t.</I></marg> light upon the$e di$parities, but not without long $tudie. Now turning to the matter in hand, there do remain to be con$idered by us the $hots at point blank, towards the Ea$t and towards the We$t; the fir$t of which, if the Earth did move, would always happen to be too high above the mark, and the $econd too low; fora$much as the parts of the Earth Ea$tward, by rea$on of the di- urnal motion, do continually de$cend beneath the tangent paralel to the Horizon, whereupon the Ea$tern $tars to us appear to a$cend; and on the contrary, the parts We$tward do more and more a$- cend, whereupon the We$tern $tars do in our $eeming de$cend: and therefore the ranges which are leveled according to the $aid tangent at the Oriental mark, (which whil$t the ball pa$$eth along by the tangent de$cendeth) $hould prove too high, and the Occidental too low by means of the elevation of the mark, whil$t the ball pa$$eth along the tangent. The an$wer is like to the re$t: for as the Ea$tern mark goeth continually de$cending, by rea$on of the Earths motion, under a tangent that continueth immove- able; $o likewi$e the piece for the $ame rea$on goeth continually inclining, and with its mounture pur$uing the $aid mark: by which means the $hot proveth true.</P> <P>But here I think it a convenient opportunity to give notice of <marg><I>The followers of Copernicus too freely admit cer- tain propo$itions for true, which are very doubtfull.</I></marg> certain conce$$ions, which are granted perhaps over liberally by the followers of <I>Copernicus</I> unto their Adver$aries: I mean of yielding to them certain experiments for $ure and certain, which yet the Adver$aries them$elves had never made tryal of: as for example, that of things falling from the round-top of a $hip whil$t it is in motion, and many others; among$t which I verily believe, that this of experimenting whether the $hot made by a Canon to- wards the Ea$t proveth too high, and the We$tern $hot too low, is one: and becau$e I believe that they have never made tryal thereof, I de$ire that they would tell me what difference they think ought to happen between the $aid $hots, $uppo$ing the Earth moveable, or $uppo$ing it moveable; and let <I>Simplieius</I> for this time an$wer for them.</P> <P>SIMP. I will not undertake to an$wer $o confidently as another more intelligent perhaps might do; but $hall $peak what thus upon the $udden I think they would reply; which is in effect the $ame <foot>with</foot> <p n=>160</p> with that which hath been $aid already, namely, that in ca$e the Earth $hould move, the $hots made Ea$tward would prove too high, &c. the ball, as it is probable, being to move along the tan- gent.</P> <P>SALV. But if I $hould $ay, that $o it falleth out upon triall, how would you cen$ure me?</P> <P>SIMP. It is nece$$ary to proceed to experiments for the pro- ving of it.</P> <P>SALV. But do you think, that there is to be found a Gunner $o skilful, as to hit the mark at every $hoot, in a di$tance of <I>v.g.</I> five hundred paces?</P> <P>SIMP. No Sir; nay I believe that there is no one, how good a marks-man $oever that would promi$e to come within a pace of the mark,</P> <P>SALV. How can we then, with $hots $o uncertain, a$$ure our $elves of that which is in di$pute?</P> <P>SIMP. We may be a$$ured thereof two wayes; one, by ma- king many $hots; the other, becau$e in re$pect of the great velo- city of the Earths motion, the deviation from the mark would in my opinion be very great.</P> <P>SALV. Very great, that is more than one pace; in regard that the varying $o much, yea and more, is granted to happen ordinarily even in the Earths mobility.</P> <P>SIMP. I verily believe the variation from the mark would be more than $o.</P> <marg><I>A Computation how much the ran- ges of great $hot ought to vary from the marke, the Earths motion be- ing granted.</I></marg> <P>SALV. Now I de$ire that for our $atisfaction we do make thus in gro$$e a $light calculation, if you con$ent thereto, which will $tand us in $tead likewi$e (if the computation $ucceed as I expect) for a warning how we do in other occurrences $uffer our $elves, as the $aying is, to be taken with the enemies $houts, and $urrender up our belief to what ever fir$t pre$ents it $elf to our fancy. And now to give all advantages to the <I>Peripateticks</I> and <I>Tychonicks,</I> let us $uppo$e our $elves to be under the Equinoctial, there to $hoot a piece of Ordinance point blank Ea$twards at a mark five hun- dred paces off. Fir$t, let us $ee thus (as I $aid) in a level, what time the $hot after it is gone out of the Piece taketh to arrive at the mark; which we know to be very little, and is certainly no more than that wherein a travailer walketh two $teps, which al$o is le$s than the $econd of a minute of an hour; for $uppo$ing that the travailer walketh three miles in an hour, which are nine thou$and paces, being that an hour containes three thou$and, $ix hundred $econd minutes, the travailer walketh two $teps and an half in a $econd, a $econd therefore is more than the time of the balls motion. And for that the diurnal revolution is twenty four hours, the We$tern horizon ri$eth fifteen degrees in an hour, that <foot>is,</foot> <p n=>161</p> is, fifteen fir$t minutes of a degree, in one fir$t minute of an hour; that is, fifteen $econds of a degree, in one $econd of an hour; and becau$e one $econd is the time of the $hot, therefore in this time the We$tern horizon ri$eth fifteen $econds of a degree, and $o much likewi$e the mark; and therefore fifteen $econds of that cir- cle, who$e $emidiameter is five hundred paces (for $o much the di- $tance of the mark from the Piece was $uppo$ed.) Now let us look in the table of Arches and Chords ($ee here is <I>Copernicus</I> his book) what part is the chord of fifteen $econds of the $emidiame- ter, that is, five hundred paces. Here you $ee the chord (or $ub- ten$e) of a fir$t minute to be le$s than thirty of tho$e parts, of which the $emidiameter is an hundred thou$and. Therefore the chord of a $econd minute $hall be le$s then half of one of tho$e parts, that is le$s than one of tho$e parts, of whichthe $emidiame- ter is two hundred thou$and; and therefore the chord of fifteen conds $hall be le$s than fifteen of tho$e $ame two hundred thou$and parts; but that which is le$s than <I>(a)</I> fifteen parts of two hun- <marg><I>(a)</I> That is, in plainer termes the fraction 15/200000, is more than the fra- ction 4/50000, for di- viding the denomi- nators by their no- minators, and the fir$t produceth 13333 1/3 the other but 12500.</marg> dred thou$and, is al$o more than that which is four cente$mes of five hundred; therefore the a$cent of the mark in the time of the balls motion is le$$e than four cente$mes, that is, than one twenty fifth part of a pace; it $hall be therefore <I>(b)</I> about two inches: And $o much con$equently $hall be the variation of each We$tern $hot, the Earth being $uppo$ed to have a diurnal motion. Now if I $hall tell you, that this variation (I mean of falling two inches $hort of what they would do in ca$e the Earth did not move) upon tri- <marg><I>(b)</I> It $hall be neer 2 2/5 inches, ac- counting the pace to be Geometrical, containing 5 foot.</marg> all doth happen in all $hots, how will you convince me <I>Simplicius,</I> $hewing me by an experiment that it is not $o? Do you not $ee that it is impo$$ible to confute me, unle$s you fir$t find out a way to $hoot at a mark with $o much exactne$$e, as never to mi$$e an hairs bredth? For whil$t the ranges of great $hot con$i$t of diffe- rent numbers of paces, as <I>de facto</I> they do, I will affirm that in each of tho$e variations there is contained that of two inches cau- $ed by the motion of the Earth.</P> <P>SAGR. Pardon me, <I>Salviatus,</I> you are too liberal. For I would <marg><I>It is demon$tra- ted with great $ub- tilty, that the Earths motion $up- po$ed, Canon $hot ought not to vary more than in re$t.</I></marg> tell the <I>Peripateticks,</I> that though every $hot $hould hit the very centre of the mark, that $hould not in the lea$t di$prove the motion of the Earth. For the Gunners are $o con$tantly imployed in le- velling the $ight and gun to the mark, as that they can hit the $ame, notwith$tanding the motion of the Earth. And I $ay, that if the Earth $hould $tand $till, the $hots would not prove true; but the Occidental would be too low, and the Oriental too high: now let <I>Simplicius</I> di$prove me if he can.</P> <P>SALV. This is a $ubtilty worthy of <I>Sagredus:</I> But whether this variation be to be ob$erved in the motion, or in the re$t of the Earth, it mu$t needs be very $mall, it mu$t needs be $wallowed up <foot>X in</foot> <p n=>162</p> in tho$e very great ones which $undry accidents continually pro- <marg><I>It is requi$ite to be very cautious in admitting experi- ments for true, to tho$e who never tried them.</I></marg> duce. And all this hath been $poken and granted on good grounds to <I>Simplicius,</I> and only with an intent to adverti$e him how much it importeth to be cautious in granting many experiments for true to tho$e who never had tried them, but only eagerly alledged them ju$t as they ought to be for the $erving their purpo$e: This is $po- ken, I $ay, by way of $urplu$$age and Corollary to <I>Simplicius,</I> for <marg><I>Experiments and arguments again$t the Earths motion $eem $o far con- cluding, as they lie hid under equi- vokes.</I></marg> the real truth is, that as concerning the$e $hots, the $ame ought ex- actly to befall a$well in the motion as in the re$t of the Terre$trial Globe; as likewi$e it will happen in all the other experiments that either have been or can be produced, which have at fir$t blu$h $o mnch $emblance of truth, as the antiquated opinion of the Earths motion hath of equivocation.</P> <P>SAGR. As for my part I am fully $atisfied, and very well un- der$tand that who $o $hall imprint in his fancy this general com- munity of the diurnal conver$ion among$t all things Terre$trial, to all which it naturally agreeth, a$well as in the old conceit of its re$t about the centre, $hall doubtle$$e di$cern the fallacy and equi- voke which made the arguments produced $eem eoncluding. There yet remains in me $ome hæ$itancy (as I have hinted be- fore) touching the flight of birds; the which having as it were an animate faculty of moving at their plea$ure with a thou$and mo- tions, and to $tay long in the Air $eparated from the Earth, and therein with mo$t irregular windings to go fluttering to and again, I cannot conceive how among$t $o great a confu$ion of motions, they $hould be able to retain the fir$t commune motion; and in what manner, having once made any $tay behind, they can get it up again, and overtake the $ame with flying, and kcep pace with the Towers and trees which hurry with $o precipitant a cour$e towards the Ea$t; I $ay $o precipitant, for in the great circle of the Globe it is little le$$e than a thou$and miles an hour, whereof the flight of the $wallow I believe makes not fifty.</P> <P>SALV. If the birds were to keep pace with the cour$e of the trees by help of their wings, they would o$ nece$$ity flie very fa$t; and if they were deprived of the univer$al conver$ion, they would lag as far behind; and their flight would $eem as furious towards the We$t, and to him that could di$cern the $ame, it would much exceed the flight of an arrow; but I think we could not be able to perceive it, no more than we $ee a Canon bullet, whil'$t driven by the fury of the fire, it flieth through the Air: But the truth is that the proper motion of birds, I mean of their flight, hath nothing to do with the univer$al motion, to which it is nei- ther an help, nor an hinderance; and that which maintaineth the $aid motion unaltered in the birds, is the Air it $elf, thorough which they flie, which naturally following the <I>Vertigo</I> of the <foot>Earth</foot> <p n=>163</p> Earth, like as it carrieth the clouds along with it, $o it tran$porteth birds and every thing el$e which is pendent in the $ame; in $o much that as to the bu$ine$$e of keeping pace with the Earth, the birds need take no care thereof, but for that work might $leep perpe- tually.</P> <P>SAGR. That the Air can carry the clouds along with it, as being matters ea$ie for their lightne$$e to be moved and deprived of all other contrary inclination, yea more, as being matters that partake al$o of the conditions and properties of the Earth; I com- prehend without any difficulty; but that birds, which as having life, may move with a motion quite contrary to the diurnal, once having $urcea$ed the $aid motion, the Air $hould re$tore them to it, $eems to me a little $trange, and the rather for that they are $olid and weighty bodies; and withal, we $ee; as hath been $aid, $tones and other grave bodies to lie unmoved again$t the <I>impetus</I> of the air; and when they $uffer them$elves to be overcome thereby, they never acquire $o much velocity as the wind which carrieth them.</P> <P>SALV. We a$cribe not $o little force, <I>Sagredus,</I> to the moved Air, which is able to move and bear before it $hips full fraught, to tear up trees by the roots, and overthrow Towers when it moveth $wiftly; and yet we cannot $ay that the motion of the Air in the$e violent operations is neer $o violent, as that of the diurnal revolution.</P> <P>SIMP. You $ee then that the moved Air may al$o cotinue the motion of projects, according to the Doctrine of <I>Ari$totle</I>; and it $eemed to me very $trange that he $hould have erred in this particular.</P> <P>SALV. It may without doubt, in ca$e it could continue it $elf, but lik as when the wind cea$ing neither $hips go on, nor trees are blown down, $o the motion in the Air not continuing after the $tone is gone out of the hand, and the Air cea$ing to move, it followeth that it mu$t be $omething el$e be$ides the Air that ma- keth the projects to move.</P> <P>SIMP. But how upon the winds being laid, doth the $hip cea$e to move? Nay you may $ee that when the wind is down, and the $ails furl'd, the ve$$el continueth to run whole miles.</P> <P>SALV. But this maketh again$t your $elf <I>Simplicius,</I> for that the wind being laid that filling the $ails drove on the $hip, yet ne- verthele$$e doth it without help of the <I>medium</I> continue its cour$e.</P> <P>SIMP. It might be $aid that the water was the <I>medium</I> which carried forward the $hip, and maintain'd it in motion.</P> <P>SALV. It might indeed be $o affirmed, if you would $peak quite contrary to truth; for the truth is, that the water, by rea- <foot>X 2 $on</foot> <p n=>164</p> $on of its great re$i$tance to the divi$ion made by the hull of the $hip, doth with great noi$e re$i$t the $ame; nor doth it permit it of a great while to acquire that velocity which the wind would confer upon it, were the ob$tacle of the water removed. Per- haps <I>Simplicius</I> you have never con$idered with what fury the water be$ets a bark, whil'$t it forceth its way through a $tanding water by help of Oars or Sails: for if you had ever minded that effect, you would not now have produced $uch an ab$urdity. And I am thinking that you have hitherto been one of tho$e who to find out how $uch things $ucceed, and to come to the know- ledg of natural effects, do not betake them$elves to a Ship, a Cro$$e-bow, or a piece of Ordinance, but retire into their $tu- dies, and turn over Indexes and Tables to $ee whether <I>Aristotle</I> hath $poken any thing thereof, and being a$$ured of the true $en$e of the Text, neither de$ire nor care for knowing any more.</P> <marg><I>The great feli- city for which they are much to be en- vied who per$wade them$elves that they know every thing.</I></marg> <P>SAGR. This is a great felicity, and they are to be much en- vied for it. For if knowledg be de$ired by all, and if to be wi$e, be to think ones $elf $o, they enjoy a very great happine$$e, for that they may per$wade them$elves that they know and under$tand all things, in $corn of tho$e who knowing, that they under$tand not what the$e think they under$tand, and con$equently $eeking that they know not the very lea$t particle of what is knowable, kill them$elves with waking and $tudying, and con$ume their days in experiments and ob$ervations. But pray you let us return to our birds; touching which you have $aid, that the Air being mo- ved with great velocity, might re$tore unto them that part of the diurnal motion which among$t the windings of their flight they might have lo$t; to which I reply, that the agitated Air $eemeth unable to confer on a $olid and grave body, $o great a velocity as its own: And becau$e that of the Air is as great as that of the Earth, I cannot think that the Air is able to make good the lo$$e of the birds retardation in flight.</P> <P>SALV. Your di$cour$e hath in it much of probability, and to $tick at trivial doubts is not for an acute wit; yet neverthele$$e the probability being removed, I believed that it hath not a jot more force than the others already con$idered and re$olved.</P> <P>SAGR. It is mo$t certain that if it be not nece$$atily conclu- dent, its efficacy mu$t needs be ju$t nothing at all, for it is onely when the conclu$ion is nece$$ary that the opponent hath no- thing to alledg on the contrary.</P> <P>SALV. Your making a greater $cruple of this than of the other in$tances dependeth, if I mi$take not, upon the birds being ani- mated, and thereby enabled to u$e their $trength at plea$ure a- gain$t the primary motion in-bred in terrene bodies: like as for <foot>example,</foot> <p n=>165</p> example, we $ee them whil'$t they are alive to fly upwards, a thing altogether impo$$ible for them to do as they are grave bodies; whereas being dead they can onely fall downwards; and there- fore you hold that the rea$ons that are of force in all the kinds of projects above named, cannot take place in birds: Now this is very true; and becau$e it is $o, <I>Sagredus,</I> that doth not appear to be done in tho$e projects, which we $ee the birds to do. For if <marg><I>The an$wer to the argument ta- ken from the flight of birds contrary to the motion of the Earth.</I></marg> from the top of a Tower you let fall a dead bird and a live one, the dead bird $hall do the $ame that a $tone doth, that is, it $hall fir$t follow the general motion diurnal, and then the motion of de$cent, as grave; but if the bird let fall, be a live, what $hall hinder it, (there ever remaining in it the diurnal motion) from $oaring by help of its wings to what place of the Horizon it $hall plea$e? and this new motion, as being peculiar to the bird, and not participated by us, mu$t of nece$$ity be vi$ible to us; and if it be moved by help of its wings towards the We$t, what $hall hinder it from returning with a like help of its wings unto the Tower. And, becau$e, in the la$t place, the birds wending its flight towards the We$t was no other than a withdrawing from the diurnal motion, (which hath, $upppo$e ten degrees of velocity) one degree onely, there did thereupon remain to the bird whil'$t it was in its flight nine degrees of velocity, and $o $oon as it did alight upon the the Earth, the ten common degrees returned to it, to which, by flying towards the Ea$t it might adde one, and with tho$e eleven overtake the Tower. And in $hort, if we well con- $ider, and more narrowly examine the effects of the flight of birds, they differ from the projects $hot or thrown to any part of the World in nothing, $ave onely that the projects are moved by an external projicient, and the birds by an internal principle. And <marg><I>An experiment with which alone is $hewn the nullity of all the objecti- ons produced a- gainst the motion of the Earth.</I></marg> here for a final proof of the nullity of all the experiments before alledged, I conceive it now a time and place convenient to demon$trate a way how to make an exact trial of them all. Shut your $elf up with $ome friend in the grand Cabbin between the decks of $ome large Ship, and there procure gnats, flies, and $uch other $mall winged creatures: get al$o a great tub (or other ve$$el) full of water, and within it put certain fi$hes; let al$o a certain bottle be hung up, which drop by drop letteth forth its water into another bottle placed underneath, having a narrow neck: and, the Ship lying $till, ob$erve diligently how tho$e $mall winged animals fly with like velocity towards all parts of the Ca- bin; how the fi$hes $wim indifferently towards all $ides; and how the di$tilling drops all fall into the bottle placed underneath. And ca$ting any thing towards your friend, you need not throw it with more force one way then another, provided the di$tances be equal: and leaping, as the $aying is, with your feet clo$ed, you will reach <foot>as</foot> <p n=>166</p> as far one way as another. Having ob$erved all the$e particulars, though no man doubteth that $o long as the ve$$el $tands $till, they ought to $ucceed in this manner; make the Ship to move with what velocity you plea$e; for ($o long as the motion is uniforme, and not fluctuating this way and that way) you $hall not di$cern any the lea$t alteration in all the forenamed effects; nor can you gather by any of them whether the Ship doth move or $tand $till. In leaping you $hall reach as far upon the floor, as before; nor for that the Ship moveth $hall you make a greater leap towards the poop than towards the prow; howbeit in the time that you $taid in the Air, the floor under your feet $hall have run the contrary way to that of your jump; and throwing any thing to your companion you $hall not need to ca$t it with more $trength that it may reach him, if he $hall be towards the prow, and you towards the poop, then if you $tood in a contrary $ituation; the drops $hall all di$till as before into the inferiour bottle and not $o much as one $hall fall towards the poop, albeit whil'$t the drop is in the Air, the Ship $hall have run many feet; the Fi$hes in their water $hall not $wim with more trouble towards the fore-part, than towards the hinder part of the tub; but $hall with equal velocity make to the bait placed on any $ide of the tub; and la$tly, the flies and gnats $hall continue their flight indifferently towards all parts; nor $hall they ever happen to be driven together towards the $ide of the Cabbin next the prow, as if they were wearied with fol- lowing the $wift cour$e of the Ship, from which through their $u$pen$ion in the Air, they had been long $eparated; and if burning a few graines of incen$e you make a little $moke, you $hall $ee it a$cend on high, and there in manner of a cloud $u$pend it $elf, and move indifferently, not inclining more to one $ide than another: and of this corre$pondence of effects the cau$e is for that the Ships motion is common to all the things contained in it, and to the Air al$o; I mean if tho$e things be $hut up in the Cabbin: but in ca$e tho$e things were above deck in the open Air, and not obliged to follow the cour$e of the Ship, differences more or le$$e notable would be ob$erved in $ome of the fore-named ef- fects, and there is no doubt but that the $moke would $tay behind as much as the Air it $elf; the flies al$o, and the gnats being hin- dered by the Air would not be able to follow the motion of the Ship, if they were $eparated at any di$tance from it. But keeping neer thereto, becau$e the Ship it $elf as being an unfractuous Fa- brick, carrieth along with it part of its neere$t Air, they would follow the $aid Ship without any pains or difficulty. And for the like rea$on we $ee $ometimes in riding po$t, that the trouble$ome flies and ^{*} hornets do follow the hor$es flying $ometimes to one, <marg>* Tafaris, <I>bor$e- flyes.</I></marg> $ometimes to another part of the body, but in the falling drops <foot>the</foot> <p n=>167</p> the difference would be very $mall; and in the $alts, and projecti- ons of grave bodies altogether imperceptible.</P> <P>SAGR. Though it came not into my thoughts to make triall of the$e ob$ervations, when I was at Sea, yet am I confident that they will $ucceed in the $ame manner, as you have related; in confirma- tion of which I remember that being in my Cabbin I have asked an hundred times whether the Ship moved or $tood $till; and $ometimes I have imagined that it moved one way, when it $teered quite another way. I am therefore as hitherto $atisfied and con- vinced of the nullity of all tho$e experiments that have been pro- duced in proof of the negative part. There now remains the ob- jection founded upon that which experience $hews us, namely, that a $wift <I>Vertigo</I> or whirling about hath a faculty to extrude and di$per$e the matters adherent to the machine that turns round; whereupon many were of opinion, and <I>Ptolomy</I> among$t the re$t, that if the Earth $hould turn round with $o great velocity, the $tones and creatures upon it $hould be to$t into the Skie, and that there could not be a morter $trong enough to fa$ten buildings $o to their foundations, but that they would likewi$e $uffer a like extru$ion.</P> <P>SALV. Before I come to an$wer this objection, I cannot but take notice of that which I have an hundred times ob$erved, and not without laughter, to come into the minds of mo$t men $o $oon as ever they hear mention made of this motion of the Earth, which is believed by them $o fixt and immoveable, that they not only ne- ver doubted of that re$t, but have ever $trongly believed that all other men a$well as they, have held it to be created immoveable, and $o to have continued through all $ucceeding ages: and being <marg><I>The $tupidity of $ome that think the Earth to have be- gun to move, when</I> Pythagoras <I>began to affirme that it did $o.</I></marg> $etled in this per$wa$ion, they $tand amazed to hear that any one $hould grant it motion, as if, after that he had held it to be immo- veable, he had fondly thought it to commence its motion then (and not till then) when <I>Pythagoras</I> (or whoever el$e was the fir$t hinter of its mobility) $aid that it did move. Now that $uch a foo- li$h conceit (I mean of thinking that tho$e who admit the motion of the Earth, have fir$t thought it to $tand $till from its creation, untill the time of <I>Pythagoras,</I> and have onely made it moveable after that <I>Pythagor as</I> e$teemed it $o) findeth a place in the mindes of the vulgar, and men of $hallow capacities, I do not much won- der; but that $uch per$ons as <I>Ari$totle</I> and <I>Ptolomy</I> $hould al$o run into this childi$h mi$take, is to my thinking a more admirable and unpardonable folly.</P> <P>SAGR. You believe then, <I>Salviatus,</I> that <I>Ptolomy</I> thought, that in his Di$putation he was to maintain the $tability of the Earth again$t $uch per$ons, as granting it to have been immoveable, un- till the time of <I>Pythagoras,</I> did affirm it to have been but then <foot>made</foot> <p n=>168</p> made moveable, when the $aid <I>Pythagoras</I> a$cribed unto it mo- tion.</P> <P>SALV. We can think no other, if we do but con$ider the way <marg>Ari$totle <I>and</I> Ptolomy <I>$eem to confute the mobili- ty of the Earth a- gain$t tho$e who thought that it ha- ving a long time $tood still, did be- gin to move in the time of</I> Pythagoras</marg> he taketh to confute their a$$ertion; the confutation of which con$i$ts in the demolition of buildings, and the to$$ing of $tones, living creatures and men them$elves up into the Air. And be- cau$e $uch overthrows and extru$ions cannot be made upon buil- dings and men, which were not before on the Earth, nor can men be placed, nor buildings erected upon the Earth, unle$$e when it $tandeth $till; hence therefore it is cleer, that <I>Ptolomy</I> argueth a- gain$t tho$e, who having granted the $tability of the Earth for $ome time, that is, $o long as living creatures, $tones, and Ma$ons were able to abide there, and to build Palaces and Cities, make it afterwards precipitately moveable to the overthrow and de$tructi- of Edifices, and living creatures, &c. For if he had undertook to di$pute again$t $uch as had a$cribed that revolution to the Earth from its fir$t creation, he would have confuted them by $aying, that if the Earth had alwayes moved, there could never have been placed upon it either men or $tones; much le$s could buildings have been erected, or Cities founded, &c.</P> <P>SIMP. I do not well conceive the$e <I>Ari$totelick</I> and <I>Ptolo- maick</I> inconveniences.</P> <P>SALV. <I>Ptolomey</I> either argueth again$t tho$e who have e$teem- ed the Earth always moveable; or again$t $uch as have held that it $tood for $ome time $till, and hath $ince been $et on moving. If again$t the fir$t, he ought to $ay, that the Earth did not always move, for that then there would never have been men, animals, or edifices on the Earth, its <I>vertigo</I> not permitting them to $tay thereon. But in that he arguing, $aith that the Earth doth not move, becau$e that bea$ts, men, and hou$es before plac'd on the Earth would precipitate, he $uppo$eth the Earth to have been once in $uch a $tate, as that it did admit men and bea$ts to $tay, and build thereon; the which draweth on the con$equence, that it did for $ome time $tand $till, to wit, was apt for the abode of a- nimals and erection of buildings. Do you now conceive what I would $ay?</P> <P>SIMP. I do, and I do not: but this little importeth to the merit of the cau$e; nor can a $mall mi$take of <I>Ptolomey,</I> com- mitted through inadvertencie be $ufficient to move the Earth, when it is immoveable. But omitting cavils, let us come to the $ub$tance of the argument, which to me $eems unan$werable.</P> <P>SALV. And I, <I>Simplicius,</I> will drive it home, and re-inforce it, by $hewing yet more $en$ibly, that it is true that grave bodies turn'd with velocity about a $ettled centre, do acquire an <I>impetus</I> of moving, and receding to a di$tance from that centre, even <foot>then</foot> <p n=>169</p> then when they are in a $tate of having a propen$ion of moving naturally to the $ame. Tie a bottle that hath water in it, to the end of a cord, and holding the other end fa$t in your hand, and making the cord and your arm the $emi-diameter, and the knitting of the $houlder the centre, $wing the bottle very fa$t a- bout, $o as that it may de$cribe the circumference of a circle, which, whether it be parallel to the Horizon, or perpendicular to it, or any way inclined, it $hall in all ca$es follow, that the wa- ter will not fall out of the bottle: nay, he that $hall $wing it, $hall find the cord always draw, and $trive to go farther from the $houlder. And if you bore a hole in the bottom of the bottle, you $hall $ee the water $pout forth no le$s upwards into the skie, than laterally, and downwards to the Earth; and if in$tead of wa- ter, you $hall put little pebble $tones into the bottle, and $wing it in the $ame manner, you $hall find that they will $trive in the like manner again$t the cord. And la$tly, we $ee boys throw $tones a great way, by $winging round a piece of a $tick, at the end of which the $tone is let into a $lit <I>(which $tick is called by them a $ling;)</I> all which are arguments of the truth of the conclu$ion, to wit, that the <I>vertigo</I> or $wing conferreth upon the moveable, a motion towards the circumference, in ca$e the motion be $wift: and therefore if the Earth revolve about its own centre, the mo- tion of the $uperficies, and e$pecially towards the great circle, as being incomparably more $wift than tho$e before named, ought to extrude all things up into the air.</P> <P>SIMP. The Argument $eemeth to me very well proved and inforced; and I believe it would be an hard matter to an$wer and overthrow it.</P> <P>SALV. Its $olution dependeth upon certain notions no le$s known and believed by you, than by my $elf: but becau$e they come not into your mind, therefore it is that you perceive not the an$wer; wherefore, without telling you it (for that you know the $ame already) I $hall with onely a$$i$ting your memory, make you to refute this argument.</P> <P>SIMP. I have often thought of your way of arguing, which hath made me almo$t think that you lean to that opinion of <I>Pla-</I> <marg><I>Our krowledg is a kind of remini$- cence according to</I> Plato.</marg> <I>to, Quòd no$trum $cire $it quoddam remini$ci</I>; therefore I intreat you to free me from this doubt, by letting me know your judg- ment.</P> <P>SALV. What I think of the opinion of <I>Plato,</I> you may gather from my words and actions. I have already in the precedent con- ferences expre$ly declared my $elf more than once; I will pur$ue the $ame $tyle in the pre$ent ca$e, which may hereafter $erve you for an example, thereby the more ea$ily to gather what my opi- nion is touching the attainment of knowledg, when a time $hall <foot>Y offer</foot> <p n=>170</p> offer upon $ome other day: but I would not have <I>Sagredus</I> of- fended at this digre$$ion.</P> <P>SAGR. I am rather very much plea$ed with it, for that I re- member that when I $tudied Logick, I could never comprehend that $o much cry'd up and mo$t potent demon$tration of <I>Ari$totle.</I></P> <P>SALV. Let us go on therefore; and let <I>Simplicius,</I> tell me what that motion is which the $tone maketh that is held fa$t in the $lit of the $ling, when the boy $wings it about to throw it a great way?</P> <P>SIMP. The motion of the $tone, $o long as it is in the $lit, is circular, that is, moveth by the arch of a circle, who$e $tedfa$t centre is the knitting of the $houlder, and its $emi-diameter the arm and $tick.</P> <P>SALV. And when the $tone leaveth the $ling, what is its mo- tion? Doth it continue to follow its former circle, or doth it go by another line?</P> <P>SIMP. It will continue no longer to $wing round, for then it would not go farther from the arm of the projicient, whereas we $ee it go a great way off.</P> <P>SALV. With what motion doth it move then?</P> <P>SIMP. Give me a little time to think thereof; For I have ne- ver con$idered it before.</P> <P>SALV. Hark hither, <I>Sagredus</I>; this is the <I>Quoddam remini$ci</I> in a $ubject well under$tood. You have pau$ed a great while, <I>Simplicius.</I></P> <P>SIMP. As far as I can $ee, the motion received in going out of the $ling, can be no other than by a right line; nay, it mu$t ne- ce$$arily be $o, if we $peak of the pure adventitious <I>impetus.</I> I was a little puzled to $ee it make an arch, but becau$e that arch bended all the way upwards, and no other way, I conceive that <marg><I>The motion im- pre$$ed by the pro- jicient is onely by a right line.</I></marg> that incurvation cometh from the gravity of the $tone, which na- turally draweth it downwards. The impre$$ed <I>impetus,</I> I $ay, without re$pecting the natural, is by a right line.</P> <P>SALV. But by what right line? Becau$e infinite, and towards every $ide may be produced from the $lit of the $ling, and from the point of the $tones $eparation from the $ling.</P> <P>SIMP. It moveth by that line which goeth directly from the motion which the $tone made in the $ling.</P> <P>SALV. The motion of the $tone whil$t it was in the $lit, you have affirmed already to be circular; now circularity oppo$eth directne$s, there not being in the circular line any part that is di- rect or $treight.</P> <P>SIMP I mean not that the projected motion is direct in re- $pect of the whole circle, but in reference to that ultimate point, where the circular motion determineth. I know what I would <foot>$ay,</foot> <p n=>171</p> $ay, but do not well know how to expre$s my $elf.</P> <P>SALV. And I al$o perceive that you under$tand the bu$ine$s, but that you have not the proper terms, wherewith to expre$s the $ame. Now the$e I can ea$ily teach you; teach you, that is, as to the words, but not as to the truths, which are things. And that you may plainly $ee that you know the thing I ask you, and onely want language to expre$s it, tell me, when you $hoot a bullet out of a gun, towards what part is it, that its acquired <I>impetus</I> carri- eth it?</P> <P>SIMP. Its acquired <I>impetus</I> carrieth it in a right line, which continueth the rectitude of the barrel, that is, which inclineth nei- ther to the right hand nor to the left, nor upwards not down- wards.</P> <P>SALV. Which in $hort is a$much as to $ay, it maketh no angle with the line of $treight motion made by the $ling.</P> <P>SIMP. So I would have $aid.</P> <P>SALV. If then the line of the projects motion be to continue without making an angle upon the circular line de$cribed by it, whil$t it was with the projicient; and if from this circular motion it ought to pa$s to the right motion, what ought this right line to be?</P> <P>SIMP. It mu$t needs be that which toucheth the circle in the point of $eparation, for that all others, in my opinion, being pro- longed would inter$ect the circumference, and by that means make $ome angle therewith.</P> <P>SALV. You have argued very well, and $hewn your $elf half a Geometrician. Keep in mind therefore, that your true opinion is expre$t in the$e words, namely, That the project acquireth an <I>impetus</I> of moving by the Tangent, the arch de$cribed by the motion of the projicient, in the point of the $aid projects $epara- tion from the projicient.</P> <P>SIMP. I under$tand you very well, and this is that which I would $ay.</P> <P>SALV. Of a right line which toucheth a circle, which of its points is the neare$t to the centre of that circle?</P> <P>SIMP. That of the contact without doubt: for that is in the circumference of a circle, and the re$t without: and the points of the circumference are all equidi$tant from the centre.</P> <P>SALV. Therefore a moveable departing from the contact, and moving by the $treight Tangent, goeth continually farther and farther from the contact, and al$o from the centre of the circle.</P> <P>SIMP. It doth $o doubtle$s.</P> <P>SALV. Now if you have kept in mind the propo$itions, which you have told me, lay them together, and tell me what you gather from them.</P> <P>SIMP. I think I am not $o forgetful, but that I do remember <foot>Y 2 them.</foot> <p n=>172</p> <marg><I>The project mo- veth by the Tan- gent of the circle of the motion prece- dent in the point of $eparation.</I></marg> them. From the things premi$ed I gather that the project $wiftly $winged round by the projicient, in its $eparating from it, doth re- tain an <I>impetus</I> of continuing its motion by the right line, which toucheth the circle de$cribed by the motion of the projicient in the point of $eparation, by which motion the project goeth con- tinually receding from the centre of the circle de$cribed by the motion of the projicient.</P> <P>SALV. You know then by this time the rea$on why grave bo- dies $ticking to the rim of a wheele, $wiftly moved, are extruded and thrown beyond the circumference to yet a farther di$tance from the centre.</P> <P>SIMP. I think I under$tand this very well; but this new know- ledg rather increa$eth than le$$eneth my incredulity that the Earth can turn round with $o great velocity, without extruding up into the sky, $tones, animals, <I>&c.</I></P> <P>SALV. In the $ame manner that you have under$tood all this, you $hall, nay you do under$tand the re$t: and with recollecting your $elf, you may remember the $ame without the help of o- thers: but that we may lo$e no time, I will help your memory therein. You do already know of your $elf, that the circular mo- tion of the projicient impre$$eth on the project an <I>impetus</I> of mo- ving (when they come to $eparate) by the right Tangent, the circle of the motion in the point of $eparation, and continuing a- long by the $ame the motion ever goeth receding farther and far- ther from the projicient: and you have $aid, that the project would continue to move along by that right line, if there were not by its proper weight an inclination of de$cent added unto it; from which the incurvation of the line of motion is derived. It $eems moreover that you knew of your $elf, that this incurvation al- ways bended towards the centre of the Earth, for thither do all grave bodies tend. Now I proceed a little farther, and ask you, whe- ther the moveable after its $eparation, in continuing the right mo- tion goeth always equally receding from the centre, or if you will, from the circumference of that circle, of which the precedent mo- tion was a part; which is as much as to $ay, Whether a moveable, that for$aking the point of a Tangent, and moving along by the $aid Tangent, doth equally recede from the point of contact, and from the circumference of the circle?</P> <P>SIMP. No, Sir: for the Tangent near to the point of contact, recedeth very little from the circumference, wherewith it keepeth a very narrow angle, but in its going farther and farther off, the di$tance always encrea$eth with a greater proportion; $o that in a circle that $hould have <I>v. g.</I> ten yards of diameter, a point of the Tangent that was di$tant from the contact but two palms, would be three or four times as far di$tant from the circumference <foot>of</foot> <p n=>173</p> of the circle, as a point that was di$tant from the contaction one palm, and the point that was di$tant half a palm, I likewi$e believe would $car$e recede the fourth part of the di$tance of the $econd: fo that within an inch or two of the contact, the $eparation of the Tangent from the circumference is $car$e di$cernable.</P> <P>SALV. So that the rece$$ion of the project from the circumfe- rence of the precedent circular motion is very $mall in the begin- ing?</P> <P>SIMP. Almo$t in$en$ible.</P> <P>SALV. Now tell me a little; the project, which from the mo- tion of the projicient receiveth an <I>impetus</I> of moving along the Tangent in a right line, and that would keep unto the $ame, did not its own weight depre$s it downwards, how long is it after the $eparation, ere it begin to decline downwards.</P> <P>SIMP. I believe that it beginneth pre$ently; for it not ha- ving any thing to uphold it, its proper gravity cannot but ope- rate.</P> <marg><I>A grave project, as $oon as it is $e- parated from the projicient begineth to decline.</I></marg> <P>SALV. So that, if that $ame $tone, which being extruded from that wheel turn'd about very fa$t, had as great a natural propen- $ion of moving towards the centre of the $aid wheel, as it hath to move towards the centre of the Earth, it would be an ea$ie mat- ter for it to return unto the wheel, or rather not to depart from it; in regard that upon the begining of the $eparation, the rece$$ion be- ing $o $inall, by rea$on of the infinite acutene$s of the angle of contact, every very little of inclination that draweth it back to- wards the centie of the wheel, would be $ufficient to retain it up- on the rim or circumference.</P> <P>SIMP. I que$tion not, but that if one $uppo$e that which nei- ther is, nor can be, to wit, that the inclination of tho$e grave bo- dies was to go towards the centre of the wheel, they would never come to be extruded or $haken off.</P> <P>SALV. But I neither do, nor need to $uppo$e that which is not; for I will not deny but that the $tones are extruded. Yet I $peak this by way of $uppo$ition, to the end that you might grant me the re$t. Now fancy to your $elf, that the Earth is that great wheel, which moved with $o great velocity is to extrude the $tones. You could tell me very well even now, that the motion of proje- ction ought to be by that right line which toucheth the Earth in the point of $eparation: and this Tangent, how doth it notably recede from the $uperficies of the Terre$trial Globe?</P> <P>SIMP. I believe, that in a thou$and yards, it will not recede from the Earth an inch.</P> <P>SALV. And did you not $ay, that the project being drawn by its own weight, declineth from the Tangent towards the centre of the Earth?</P> <foot>SIMP.</foot> <p n=>174</p> <P>SIMP. I $aid $o, and al$o confe$$e the re$t: and do now plainly under$tand that the $tone will not $eparate from the Earth, for that its rece$$ion in the beginning would be $uch, and $o $mall, that it is a thou$and times exceeded by the inclination which the $tone hath to move towards the centre of the Earth, which cen- tre in this ca$e is al$o the centre of the wheel. And indeed it mu$t be confe$$ed that the $tones, the living creatures, and the other grave bodies cannot be extruded; but here again the lighter things beget in me a new doubt, they having but a very weak propen$ion of de$cent towards the centre; $o that there being wanting in them that faculty of withdrawing from the $uperficies, I $ee not, but that they may be extruded; and you know the rule, that <I>ad de$truendum $ufficit unum.</I></P> <P>SAVL. We will al$o give you $atisfaction in this. Tell me therefore in the fir$t place, what you under$tand by light matters, that is, whether you thereby mean things really $o light, as that they go upwards, or el$e not ab$olutely light, but of $o $mall gra- vity, that though they de$cend downwards, it is but very $lowly; for if you mean the ab$olutely light, I will be readier than your $elf to admit their extru$ion.</P> <P>SIMP. I $peak of the other $ort, $uch as are feathers, wool, cot- ton, and the like; to lift up which every $mall force $ufficeth: yet neverthele$$e we $ee they re$t on the Earth very quietly.</P> <P>SALV. This pen, as it hath a natural propen$ion to de$cend to- wards the $uperficies of the Earth, though it be very $mall, yet I mu$t tell you that it $ufficeth to keep it from mounting upwards: and this again is not unknown to you your $elf; therefore tell me if the pen were extruded by the <I>Vertigo</I> of the Earth, by what line would it move?</P> <P>SIMP. By the tangent in the point of $eparation.</P> <P>SALV. And when it $hould be to return, and re-unite it $elf to the Earth, by what line would it then move?</P> <P>SIMP. By that which goeth from it to the centre of the Earth.</P> <P>SALV. So then here falls under our con$ideration two moti- ons; one the motion of projection, which beginneth from the point of contact, and proceedeth along the tangent; and the o- ther the motion of inclination downwards, which beginneth from the project it $elf, and goeth by the $ecant towards the centre; and if you de$ire that the projection follow, it is nece$$ary that the <I>im- petus</I> by the tangent overcome the inclination by the $ecant: is it not $o?</P> <P>SIMP. So it $eemeth to me.</P> <P>SALV. But what is it that you think nece$$ary in the motion of the projicient, to make that it may prevail over that inclina- <foot>tion,</foot> <p n=>175</p> tion, from which en$ueth the $eparation and elongation of the pen from the Earth?</P> <P>SIMP. I cannot tell.</P> <P>SALV. How, do you not know that? The moveable is here the $ame, that is, the $ame pen; now how can the $ame moveable $uperate and exceed it $elf in motion?</P> <P>SIMP. I do not $ee how it can overcome or yield to it $elf in motion, unle$$e by moving one while fa$ter, and another while $lower.</P> <P>SALV. You $ee then, that you do know it. If therefore the projection of the pen ought to follow, and its motion by the tan- gent be to overcome its motion by the $ecant, what is it requi$ite that their velocities $hould be?</P> <P>SIMP. It is requi$ite that the motion by the tangent be greater than that other by the $ecant. But wretch that I am! Is it not only many thou$and times greater than the de$cending motion of the pen, but than that of the $tone? And yet like a $imple fellow I had $uffered my $elf to be per$waded, that $tones could not be extruded by the revolution of the Earth. I do therefore revoke my former $entence, and $ay, that if the Earth $hould move, $tones, Elephants, Towers, and whole Cities would of nece$$ity be to$t up into the Air; and becau$e that that doth not evene, I con- clude that the Earth doth not move.</P> <P>SALV. Softly <I>Simplicius,</I> you go on $o fa$t, that I begin to be more afraid for you, than for the pen. Re$t a little, and ob$erve what I am going to $peap. If for the reteining of the $tone or pen an- nexed to the Earths $urface it were nece$$ary that its motion of de$cent were greater, or as much as the motion made by the tan- gent; you would have had rea$on to $ay, that it ought of nece$$ity to move as fa$t, or fa$ter by the $ecant downwards, than by the tangent Ea$twards: But did not you tell me even now, that a thou$and yards of di$tance by the tangent from the contact, do remove hardly an inch from the circumference? It is not $uffici- ent therefore that the motion by the tangent, which is the $ame with that of the diurnall <I>Vertigo,</I> (or ha$ty revolution) be fimply more $wift than the motion by the $ecant, which is the $ame with that of the pen in de$cending; but it is requi$ite that the $ame be $o much more $wift as that the time which $ufficeth for the pen to move <I>v.g.</I> a thou$and yards by the tangent, be in$ufficient for it to move one $ole inch by the $ecant. The which I tell you $hall never be, though you $hould make that motion never $o $wift, and this never $o $low.</P> <P>SIMP. And why might not that by the tangent be $o $wift, as not to give the pen time to return to the $urface of the Earth?</P> <P>SALV. Try whether you can $tate the ca$e in proper termes, <foot>and</foot> <p n=>176</p> and I will give you an an$wer. Tell me therefore, how much do you think $ufficeth to make that motion $wifter than this?</P> <P>SIMP. I will $ay for example, that if that motion by the tan- gent were a million of times $wifter than this by the $ecant, the pen, yea, and the $tone al$o would come to be extruded.</P> <P>SALV. You $ay $o, and $ay that which is fal$e, onely for want, not of Logick, Phy$icks, or Metaphy$icks, but of Geome- try; for if you did but under$tand its fir$t elements, you would know, that from the centre of a circle a right line may be drawn to meet the tangent, which inter$ecteth it in $uch a manner, that the part of the tangent between the contact and the $ecant, may be one, two, or three millions of times greater than that part of the $ecant which lieth between the tangent and the circumference, and that the neerer and neerer the $ecant $hall be to the contact, this proportion $hall grow greater and greater <I>in infinitum</I>; $o that it need not be feared, though the <I>vertigo</I> be $wift, and the motion downwards $low, that the pen or other lighter matter can begin to ri$e upwards, for that the inclination downwards always exceedeth the velocity of the projection.</P> <P>SAGR. I do not perfectly apprehend this bu$ine$$e.</P> <P>SALV. I will give you a mo$t univer$al yet very ea$ie demon- <marg><I>A geometrical demon$tration to prove the impo$$i- bility of <*>tru$ion by means of the terre$trial</I> vertigo.</marg> $tration thereof. Let a proportion be given between B A [<I>in Fig.</I> 3.] and C: And let B A be greater than C at plea$ure. And let there be de$cribed a circle, who$e centre is D. From which it is required to draw a $ecant, in $uch manner, that the tangent may be in proportion to the $aid $ecant, as B A to C. Let A I be $uppo$ed a third proportional to B A and C. And as B I is to I A, $o let the diameter F E be to E G; and from the point G, let there be drawn the tangent G H. I $ay that all this is done as was required; and as B A is to C, $o is H G to G E. And in re- gard that as B I is to I A, $o is F E to E G; therefore by compo- $ition, as B A is to A I; $o $hall F G be to G E. And becau$e C is the middle proportion between <I>B</I> A and A I; and G H is a middle term between F G and G E; therefore, as B A is to C, $o $hall F G be to G H; that is H G to G E, which was to be demon$trated.</P> <P>SAGR. I apprehend this demon$tration; yet neverthele$$e, I am not left wholly without hæ$itation; for I find certain confu- $ed $cruples role to and again in my mind, which like thick and dark clouds, permit me not to di$cern the cleerne$$e and nece$$ity of the conclu$ion with that per$picuity, which is u$ual in Mathe- matical Demon$trations. And that which I $tick at is this. It is true that the $paces between the tangent and the circumference do gradually dimini$h <I>in infinitum</I> towards the contact; but it is al$o true on the contrary, that the propen$ion of the moveable to <foot>de-</foot> <p n=>177</p> de$cending groweth le$s & le$s in it, the nearer it is to the fir$t term of its de$cent; that is, to the $tate of re$t; as is manife$t from that which you declare unto us, demon$trating that the de$cending grave body departing from re$t, ought to pa$$e thorow all the degrees of tardity comprehended between the $aid re$t, & any a$$igned degree of velocity, the which grow le$s and le$s <I>in infinitum.</I> To which may be added, that the $aid velocity and propen$ion to motion, doth for another rea$on dimini$h to infinity; and it is becau$e the gravity of the $aid moveable may infinitely dimini$h. So that the cau$es which dimini$h the propen$ion of a$cending, and con$equently favour the projection, are two; that is, the levity of the moveable, and its vicinity to the $tate of re$t; both which are augmentable <I>in infinit.</I> and the$e two on the contrary being to contract but with one $ole cau$e of making the projection, I cannot conceive how it alone, al- though it al$o do admit of infinite augmentation, $hould be able to remain invincible again$t the union & confederacy of the others, w^{ch} are two, and are in like manner capable of infinite augmentation.</P> <P>SALV. This is a doubt worthy of <I>Sagredus</I>; and to explain it $o as that we may more cleerly apprehend it, for that you $ay that you your $elf have but a confu$ed <I>Idea</I> of it, we will di$tingui$h of the $ame by reducing it into figure; which may al$o perhaps afford us $ome ca$e in re$olving the $ame. Let us therefore [<I>in Fig.</I> 4.] draw a perpendicular line towards the centre, and let it be AC, and to it at right angles let there be drawn the Horizontal line A <I>B,</I> upon which the motion of the projection ought to be made; now the pro- ject would continue to move along the $ame with an even motion, if $o be its gravity did not incline it downwards. Let us $uppo$e from the point A a right line to be drawn, that may make any angle at plea$ure with the line A B; which let be A E, and upon A<I>B</I> let us mark $ome equal $paces AF, FH, HK, and from them let us let fall the perpendiculars FG, HI, K L, as far as AE. And becau$e, as al ready hath been $aid, the de$cending grave body departing from re$t, goeth from time to time acquiring a greater degree of velocity, according as the $aid time doth $ucce$$ively encrea$e; we may con- ceive the $paces AF, FH, HK, to repre$ent unto us equal times; and the perpendiculars FG, HI, KL, degrees of velocity acquired in the $aid times; $o that the degree of velocity acquired in the whole time A K, is as the line K L, in re$pect to the degree H I, acquired in the time AH, and the degree FG in the time AF; the which degrees KL, HI, FG, are (as is manife$t) the $ame in proportion, as the times K A, HA, F A, and if other perpendiculars were drawn from the points marked at plea$ure in the line F A, one might $ucce$$ively find de- grees le$$e and le$$e <I>in infinitum,</I> proceeding towards the point A, repre$enting the fir$t in$tant of time, and the fir$t $tate of re$t. And this retreat towards A, repre$enteth the fir$t propen$ion to the <foot>Z mo-</foot> <p n=>178</p> motion of de$cent, dimini$hed <I>in infinitum</I> by the approach of the moveable to the fir$t $tate of re$t, which approximation is augmentable <I>in infinitum.</I> Now let us find the other diminution of velocity, which likewi$e may proceed to infinity, by the di- minution of the gravity of the moveable, and this $hall be repre- $ented by drawing other lines from the point A, which contein angles le$$e than the angle B A E, which would be this line A D, the which inter$ecting the parallels K L, H I, F G, in the points M, N, and O, repre$ent unto us the degrees F O, H N, K M, acquired in the times A F, A H, A K, le$$e than the other de- grees F G, H I, K L, acquired in the $ame times; but the$e latter by a moveable more ponderous, and tho$e other by a moveable more <I>light.</I> And it is manife$t, that by the retreat of the line E A towards A B, contracting the angle E A B (the which may be done <I>in infinitum,</I> like as the gravity may <I>in infi- nitum</I> be dimini$hed) the velocity of the cadent moveable may in like manner be dimini$hed <I>in infinitum,</I> and $o con$equently the cau$e that impeded the projection; and therefore my thinks that the union of the$e two rea$ons again$t the projection, dimi- ni$hed to infinity, cannot be any impediment to the $aid proje- ction. And couching the whole argument in its $horte$t terms, we will $ay, that by contracting the angle E A B, the degrees of ve- locity L K, I H, G F, are dimini$hed; and moreover by the re- treat of the parallels K L, H I, F G, towards the angle A, the fame degrees are again dimini$hed; and both the$e diminutions extend to infinity: Therefore the velocity of the motion of de- $cent may very well dimini$h $o much, (it admitting of a two$old diminution <I>in infinitum</I>) as that it may not $uffice to re$tore the moveable to the circumference of the wheel, and thereupon may occa$ion the projection to be hindered and wholly obviated.</P> <P>Again on the contrary, to impede the projection, it is nece$- $ary that the $paces by which the project is to de$cend for the reuniting it $elf to the Wheel, be made $o $hort and clo$e toge- ther, that though the de$cent of the moveable be retarded, yea more, dimini$hed <I>in infinitum,</I> yet it $ufficeth to reconduct it thither: and therefore it would be requi$ite, that you find out a diminuti- on of the $aid $paces, not only produced to infinity, but to $uch an infinity, as that it may $uperate the double infinity that is made in the diminution of the velocity of the de$cending moveable. But how can a magnitude be dimini$hed more than another, which hath a twofold diminution <I>in infinitum</I>? Now let <I>Simplicius</I> ob- $erve how hard it is to philo$ophate well in nature, without <I>Geo- metry.</I> The degrees of velocity dimini$hed <I>in infinitum,</I> as well by the diminution of the gravity of the moveable, as by the ap- proxination to the fir$t term of the motion, that is, to the $tate <foot>of</foot> <p n=>179</p> of re$t, are alwayes determinate, and an$wer in proportion to the parallels comprehended between two right lines that concur in an angle, like to the angle B A E, or B A D, or any other infinitely more acute, alwayes provided it be rectilineall- But the diminution of the $paces thorow which the moveable is to be conducted along the circumference of the wheel, is propor- tionate to another kind of diminution, comprehended between lines that contain an angle infinitely more narrow and acute, than any rectilineal angle, how acute $oever, which is that in our pre- $ent ca$e. Let any point be taken in the perpendicular A C, and making it the centre, de$cribe at the di$tance C A, an arch A M P, the which $hall inter$ect the parallels that determine the degrees of velocity, though they be very minute, and comprehended within a mo$t acute rectilineal angle; of which parallels the parts that lie between the arch and the tangent A B, are the quantities of the $paces, and of the returns upon the wheel, alwayes le$$er (and with greater proportion le$$er, by how much neerer they approach to the contact) than the $aid parallels of which they are parts. The parallels comprehended between the right lines in retiring to- wards the angle dimini$h alwayes at the $ame rate, as <I>v.g.</I> A H be- ing divided in two equal parts in F, the parallel H I $hall be dou- ble to F G, and $ub-dividing F A, in two equal parts, the paral- lel produced from the point of the divi$ion $hall be the half of F G; and continuing the $ub-divi$ion <I>in infinitum,</I> the $ub$equent parallels $hall be alwayes half of the next preceding; but it doth not $o fall out in the lines intercepted between the tangent and the circumference of the circle: For if the $ame $ub-divi$ion be made in F A; and $uppo$ing for example, that the parallel which cometh from the point H, were double unto that which commeth from F, this $hall be more then double to the next following, and continually the neerer we come towards the contact A, we $hall find the precedent lines contein the next following three, four, ten, an hundred, a thou$and, an hundred thou$and, an hundred millions of times, and more <I>in infinitum.</I> The brevity therefore of $uch lines is $o reduced, that it far exceeds what is requi$ite to make the project, though never $o light, return, nay more, continue unremoveable upon the circumference.</P> <P>SAGR. I very well comprehend the whole di$cour$e, and upon what it layeth all its $tre$$e, yet neverthele$$e methinks that he that would take pains to pur$ue it, might yet $tart $ome further que$tions, by $aying, that of tho$e two cau$es which render the de$cent of the moveable $lower and $lower <I>in infinitum,</I> it is mani- fe$t, that that which dependeth on the vicinity to the fir$t term of the de$cent, increa$eth alwayes in the $ame proportion, like as the parallels alwayes retain the $ame proportion to each other, &c. <foot>Z 2 but</foot> <p n=>180</p> but that the diminution of the $ame velocity, dependent on the diminution of the gravity of the moveable (which vvas the $econd cau$e) doth al$o ob$erve the $ame proportion, doth not $o plainly appear, And vvho $hall a$$ure us that it doth not proceed accor- ding to the proportion of the lines intercepted between the $ecant, and the circumference; or vvhether vvith a greater proportion?</P> <P>SALV. I have a$$umed for a truth, that the velocities of movea- bles de$cending naturally, vvill follovv the proportion of their gra- vities, with the favour of <I>Simplicius,</I> and of <I>Ari$totle,</I> who doth in many places affirm the $ame, as a propo$ition manife$t: You, in favour of my adver$ary, bring the $ame into que$tion, and $ay that its po$$ible that the velocity increa$eth with greater propor- tion, yea and greater <I>in infinitum</I> than that of the gravity; $o that all that hath been $aid falleth to the ground: For maintaining whereof, I $ay, that the proportion of the velocities is much le$$e than that of the gravities; and thereby I do not onely $upport but confirme the premi$es. And for proof of this I appeal unto experience, which will $hew us, that a grave body, howbeit thirty or fourty times bigger then another; as for example, a ball of lead, and another of $ugar, will not move much more than twice as fa$t. Now if the projection would not be made, albeit the ve- locity of the cadent body $hould dimini$h according to the pro- portion of the gravity, much le$$e would it be made $o long as the velocity is but little dimini$hed, by abating much from the gravi- ty. But yet $uppo$ing that the velocity dimini$heth with a propor- tion much greater than that wherewith the gravity decrea$eth, nay though it were the $elf-$ame wherewith tho$e parallels conteined between the tangent and circumference do decrea$e, yet cannot I $ee any nece$$ity why I $hould grant the projection of matters of never $o great levity; yea I farther averre, that there could no $uch projection follow, meaning alwayes of matters not properly and ab$olutely light, that is, void of all gravity, and that of their own natures move upwards, but that de$cend very $lowly, and have very $mall gravity. And that which moveth me $o to think is, that the diminution of gravity, made according to the propor- tion of the parallels between the tangent and the circumference, hath for its ultimate and highe$t term the nullity of weight, as tho$e parallels have for their la$t term of their diminution the contact it $elf, which is an indivi$ible point: Now gravity never dimini$heth $o far as to its la$t term, for then the moveable would cea$e to be grave; but yet the $pace of the rever$ion of the project to the circumference is reduced to the ultimate minuity, which is when the moveable re$teth upon the circumference in the very point of contact; $o as that to return thither it hath no need of $pace: and therefore let the propen$ion to the motion of de$cent be ne- <foot>ver</foot> <p n=>181</p> ver $o $mall, yet is it alwayes more than $ufficient to reconduct the moveable to the circumference, from which it is di$tant but its lea$t $pace, that is, nothing at all.</P> <P>SAGR. Your di$cour$e, I mu$t confe$s, is very accurate; and yet no le$s concluding than it is ingenuous; and it mu$t be gran- ted that to go about to handle natural que$tions, without <I>Geome- try,</I> is to attempt an impo$$ibility.</P> <P>SALV. But <I>Simplicius</I> will not $ay $o; and yet I do not think that he is one of tho$e <I>Peripateticks</I> that di$$wade their Di$ciples from $tudying the <I>Mathematicks,</I> as Sciences that vitiate the rea- $on, and render it le$$e apt for contemplation.</P> <P>SIMP. I would not do $o much wrong to <I>Plato,</I> but yet I may truly $ay with <I>Aristotle,</I> that he too much lo$t him$elf in, and too much doted upon that his <I>Geometry</I>: for that in conclu$ion the$e Mathematical $ubtilties <I>Salviatus</I> are true in ab$tract, but applied to $en$ible and Phy$ical matter, they hold not good. For the Mathematicians will very well demon$trate for example, that <I>Sphæratangit planum in puncto</I>; a po$ition like to that in di$pute, but when one cometh to the matter, things $ucceed quite another way. And $o I may $ay of the$e angles of contact, and the$e proportions; which all evaporate into Air, when they are applied to things material and $en$ible.</P> <P>SALV. You do not think then, that the tangent toucheth the $uperficies of the terre$trial Globe in one point only?</P> <P>SIMP. No, not in one $ole point; but I believe that a right line goeth many tens and hundreds of yards touching the $urface not onely of the Earth, but of the water, before it $eparate from the $ame.</P> <P>SALV. But if I grant you this, do not you perceive that it ma- keth $o much the more again$t your cau$e? For if it be $uppo$ed that the tangent was $eparated from the terre$trial $uperficies, yet it hath been however demon$trated that by rea$on of the great a- cuity of the angle of contingence (if happily it may be call'd an angle) the project would not $eparate from the $ame; how much le$$e cau$e of $eparation would it have, if that angle $hould be wholly clo$ed, and the $uperficies and the tangent become all one? <marg><I>The truth $ometimes gaines $trength by con- tradiction.</I></marg> Perceive you not that the Projection would do the $ame thing up- on the $urface of the Earth, which is a$much as to $ay, it would do ju$t nothing at all? You $ee then the power of truth, which while you $trive to oppo$e it, your own a$$aults them$elves uphold and defend it. But in regard that you have retracted this errour, I would be loth to leave you in that other which you hold, namely, that a material Sphere doth not touch a plain in one $ole point: and I could wi$h $ome few hours conver$ation with $ome per$ons conver$ant in <I>Geometry,</I> might make you a little more intelligent <foot>among$t</foot> <p n=>182</p> among$t tho$e who know nothing thereof. Now to $hew you how great their errour is who $ay, that a Sphere <I>v.g.</I> of bra$$e, doth not touch a plain <I>v.g.</I> of $teel in one $ole point, Tell me what con- ceipt you would entertain of one that $hould con$tantly aver, that the Sphere is not truly a Sphere.</P> <P>SIMP. I would e$teem him wholly devoid of rea$on.</P> <P>SALV. He is in the $ame ca$e who $aith that the material Sphere <marg><I>The sphere al- though material, toucheth the mate- rial plane but in one point onely.</I></marg> doth not touch a plain, al$o material, in one onely point; for to $ay this is the $ame, as to affirm that the Sphere is not a Sphere. And that this is true, tell me in what it is that you con$titute the Sphere to con$i$t, that is, what it is that maketh the Sphere differ from all other $olid bodies.</P> <P>SIMP. I believe that the e$$ence of a Sphere con$i$teth in ha- <marg><I>The definition of the $phere.</I></marg> ving all the right lines produced from its centre to the circumfe- rence, equal.</P> <P>SALV. So that, if tho$e lines $hould not be equal, there $ame $olidity would be no longer a $phere?</P> <P>SIMP. True.</P> <P>SALV. Go to; tell me whether you believe that among$t the many lines that may be drawn between two points, that may be more than one right line onely.</P> <P>SIMP. There can be but one.</P> <P>SALV. But yet you under$tand that this onely right line $hall again of nece$$ity be the $horte$t of them all?</P> <P>SIMP. I know it, and al$o have a demon$tration thereof, pro- duced by a great <I>Peripatetick</I> Philo$opher, and as I take it, if my memory do not deceive me, he alledgeth it by way of reprehending <I>Archimedes,</I> that $uppo$eth it as known, when it may be demon- $trated.</P> <P>SALV. This mu$t needs be a great Mathematician, that knew how to demon$trate that which <I>Archimedes</I> neither did, nor could demon$trate. And if you remember his demon$tration, I would gladly hear it: for I remember very well, that <I>Archimedes</I> in his Books, <I>de Sphærà & Cylindro,</I> placeth this Propo$ition among$t the <I>Po$tulata</I>; and I verily believe that he thought it demon$trated.</P> <P>SIMP. I think I $hall remember it, for it is very ea$ie and $hort.</P> <P>SALV. The di$grace of <I>Archimedes,</I> and the honour of this Phi- lo$opher $hall be $o much the greater.</P> <P>SIMP. I will de$cribe the Figure of it. Between the points <marg><I>The demon$tra- tion of a Peripate- tick, to prove the right line to be the $horte$t of all lines.</I></marg> A and B, [<I>in Fig.</I> 5.] draw the right line A B, and the curve line A C B, of which we will prove the right to be the $horter: and the proof is this; take a point in the curve-line, which let be C, and draw two other lines, A C and C B, which two lines together; are longer than the $ole line A B, for $o demon$trateth <I>Euelid.</I> <foot>But</foot> <p n=>183</p> But the curve-line A C B, is greater than the two right-lines A C, and C B; therefore, <I>à fortiori,</I> the curve-line A C B, is much greater than the right line A B, which was to be demon$trated.</P> <marg><I>The Paralogi$m of the $ame Peripa- tetick, which pro- veth</I> ignotum per ignotius.</marg> <P>SALV. I do not think that if one $hould ran$ack all the Para- logi$ms of the world, there could be found one more commodious than this, to give an example of the mo$t $olemn fallacy of all fallacies, namely, than that which proveth <I>ignotum per ignotius.</I></P> <P>SIMP. How $o?</P> <P>SALV. Do you ask me how $o? The unknown conclu$ion which you de$ire to prove, is it not, that the curved line A C B, is longer than the right line A B; the middle term which is taken for known, is that the curve-line A C B, is greater than the two lines A C and C B, the which are known to be greater than A B; And if it be unknown whether the curve-line be greater than the $ingle right-line A B, $hall it not be much more unknown whether it be greater than the two right lines A C & C B, which are known to be greater than the $ole line A B, & yet you a$$ume it as known?</P> <P>SIMP. I do not yet very well perceive wherein lyeth the fal- lacy.</P> <P>SALV. As the two right lines are greater than A B, (as may be known by <I>Euclid</I>) and in as much as the curve line is longer than the two right lines A C and B C, $hall it not not be much greater than the $ole right line A B?</P> <P>SIMP. It $hall $o.</P> <P>SALV. That the curve-line A C B, is greater than the right line A B, is the conclu$ion more known than the middle term, which is, that the $ame curve-line is greater than the two right- lines A C and C B. Now when the middle term is le$s known than the conclu$ion, it is called a proving <I>ignotum per ignotius.</I> But to return to our purpo$e, it is $ufficient that you know the right line to be the $horte$t of all the lines that can be drawn be- tween two points. And as to the principal conclu$ion, you $ay, that the material $phere doth not touch the $phere in one $ole point. What then is its contact?</P> <P>SIMP. It $hall be a part of its $uperficies.</P> <P>SALV. And the contact likewi$e of another $phere equal to the fir$t, $hall be al$o a like particle of its $uperficies?</P> <P>SIMP. There is no rea$on vvhy it $hould be othervvi$e.</P> <P>SALV. Then the tvvo $pheres vvhich touch each other, $hall touch vvith the tvvo $ame particles of a $uperficies, for each of them agreeing to one and the $ame plane, they mu$t of nece$$ity agree in like manner to each other. Imagine now that the two $pheres <marg><I>A demon $tration that the $phere tou- cheth the plane but in one point.</I></marg> [<I>in Fig.</I> 6.] who$e centres are A and B, do touch one another: and let their centres be conjoyned by the right line A B, which pa$$eth through the contact. It pa$$eth thorow the point C, and <foot>another</foot> <p n=>184</p> another point in the contact being taken as D, conjoyn the two right lines A D and B D, $o as that they make the triangle A D B; of which the two $ides A D and D B $hall be equal to the other one A C B, both tho$e and this containing two $emidiameters, which by the definition of the $phere are all equal: and thus the right line A B, drawn between the two centres A and B, $hall not be the $horte$t of all, the two lines A D and D B being equal to it: which by your own conce$$ion is ab$urd.</P> <P>SIMP. This demon$tration holdeth in the ab$tracted, but not in the material $pheres.</P> <P>SALV. In$tance then wherein the fallacy of my argument con- $i$teth, if as you $ay it is not concluding in the material $pheres, but holdeth good in the immaterial and ab$tracted.</P> <marg><I>Why the $phere in ab$tract, toucheth the plane onely in one point, and not the material in conerete.</I></marg> <P>SIMP. The material $pheres are $ubject to many accidents, which the immaterial are free from. And becau$e it cannot be, that a $phere of metal pa$$ing along a plane, its own weight $hould not $o depre$s it, as that the plain $hould yield $omewhat, or that the $phere it $elf $hould not in the contact admit of $ome impre$$i- on. Moreover, it is very hard for that plane to be perfect, if for nothing el$e, yet at lea$t for that its matter is porous: and per- haps it will be no le$s difficult to find a $phere $o perfect, as that it hath all the lines from the centre to the $uperficies, exactly equal.</P> <P>SALV. I very readily grant you all this that you have $aid; but it is very much be$ide our purpo$e: for whil$t you go about to $hew me that a material $phere toucheth not a material plane in one point alone, you make u$e of a $phere that is not a $phere, and of a plane that is not a plane; for that, according to what you $ay, either the$e things cannot be found in the world, or if they may be found, they are $poiled in applying them to work the effect. It had been therefore a le$s evil, for you to have granted the con- clu$ion, but conditionally, to wit, that if there could be made of matter a $phere and a plane that were and could continue perfect, they would touch in one $ole point, and then to have denied that any $uch could be made.</P> <P>SIMP. I believe that the propo$ition of Philo$ophers is to be under$tood in this $en$e; for it is not to be doubted, but that the imperfection of the matter, maketh the matters taken in con- crete, to di$agree with tho$e taken in ab$tract.</P> <P>SALV. What, do they not agree? Why, that which you your $elf $ay at this in$tant, proveth that they punctually agree.</P> <P>SIMP. How can that be?</P> <P>SALV. Do you not $ay, that through the imperfection of the matter, that body which ought to be perfectly $pherical, and that plane which ought to be perfectly level, do not prove to be the <foot>$ame</foot> <p n=>185</p> $ame in concrete, as they are imagined to be in ab$tract?</P> <P>SIMP. This I do affirm.</P> <P>SALV. Then when ever in concrete you do apply a material Sphere <marg><I>Things are ex- actly the $ame in ab$tract as in con- crete.</I></marg> to a material plane, youapply an imperfect Sphere to an imperfect plane, & the$e you $ay do not touch only in one point. But I mu$t tell you, that even in ab$tract an immaterial Sphere, that is, not a perfect Sphere, may touch an immaterial plane, that is, not a per- fect plane, not in one point, but with part of its $uperficies, $o that hitherto that which falleth out in concrete, doth in like manner hold true in ab$tract. And it would be a new thing that the com- putations and rates made in ab$tract numbers, $hould not after- wards an$wer to the Coines of Gold and Silver, and to the mer- chandizes in concrete. But do you know <I>Simplicius,</I> how this commeth to pa$$e? Like as to make that the computations agree with the Sugars, the Silks, the Wools, it is nece$$ary that the accomptant reckon his tares of che$ts, bags, and $uch other things: So when the <I>Geometricall Philo$opher</I> would ob$erve in concrete the effects demon$trated in ab$tract, he mu$t defalke the impedi- ments of the matter, and if he know how to do that, I do a$$ure you, the things $hall jump no le$$e exactly, than <I>Arithmstical</I> computations. The errours therefore lyeth neither in ab$tract, nor in concrete, nor in <I>Geometry,</I> nor in <I>Phy$icks,</I> but in the Calcula- tor, that knoweth not how to adju$t his accompts. Therefore if you had a perfect Sphere and plane, though they were material, you need not doubt but that they would touch onely in one point. And if $uch a Sphere was and is impo$$ible to be procured, it was much be$ides the purpo$e to $ay, <I>Quod Sphæra ænea non tangit in puncto.</I> Furthermore, if I grant you <I>Simplicius,</I> that in matter a figure cannot be procured that is perfectly $pherical, or perfectly level: Do you think there may be had two materiall bodies, who$e $uperficies in $ome part, and in $ome $ort are incurvated as irregularly as can be de$ired?</P> <P>SIMP. Of the$e I believe that there is no want.</P> <P>SALV. If $uch there be, then they al$o will touch in one $ole <marg><I>Contact in a $in- gle point is not pe- culiar to the per- fect Spheres onely? but belongeth to all curved figures.</I></marg> point; for this contact in but one point alone is not the $ole and peculiar priviledge of the perfect Sphere and perfect plane. Nay, he that $hould pro$ecute this point with more $ubtil contemplations would finde that it is much harder to procure two bodies that <marg><I>It is more diffi- cult to find Figures that touch with a part of their $ur- face, than in one $ole point.</I></marg> touch with part of their $nper$icies, than with one point onely. For if two $uperficies be required to combine well together, it is nece$$ary either, that they be both exactly plane, or that if one be convex, the other be concave; but in $uch a manner concave, that the concavity do exactly an$wer to the convexity of the other: the which conditions are much harder to be found, in regard of their too narrow determination, than tho$e others, which in their ca$uall latitude are infinite.</P> <foot>Aa SIMP.</foot> <p n=>186</p> <P>SIMP. You believe then, that two $tones, or two pieces of I- ron taken at chance, and put together, do for the mo$t part touch in one $ole point?</P> <P>SALV. In ca$ual encounters, I do not think they do; as well becau$e for the mo$t part there will be $ome $mall yielding filth upon them, as becau$e that no diligence is u$ed in applying them without $triking one another; and every $mall matter $ufficeth to make the one $uperficies yield $omewhat to the other; $o that they interchangeably, at lea$t in $ome $mall particle, receive $igure from the impre$$ion of each other. But in ca$e their $uperficies were very ter$e and polite, and that they were both laid upon a table, that $o one might not pre$$e upon the other, and gently put towards one another, I que$tion not, but that they might be brought to the $imple contact in one onely point.</P> <P>SAGR. It is requi$ite, with your permi$$ion, that I propound a certain $cruple of mine, which came into my minde, whil'$t I heard propo$ed by <I>Simplicius,</I> the impo$$ibility of finding a materiall and $olid body, that is, perfectly of a Spherical figure, and whil'$t J law <I>Salviatus</I> in a certain manner, not gain$aying, to give his con$ent thereto; therefore I would know, whether there would be the $ame difficulty in forming a $olid of $ome other figure, that is, to expre$$e my $elf better, whether there is more difficulty in reducing a piece of Marble into the figure of a perfect Sphere, than into a perfect Pyramid, or into a perfect Hor$e, or into a perfect Gra$$e-hopper?</P> <P>SALV. To this I will make you the fir$t an$wer: and in the fir$t place, I will acquit my $elf of the a$$ent which you think I gave to <I>Simplicius,</I> which was only for a time; for I had it al$o in my thoughts, betore I intended to enter upon any other matter, to $peak that, which, it may be, is the $ame, or very like to that which you are about to $ay, And an$wering to your fir$t que$tion, I $ay, <marg><I>The Sphericall Figure is ea$ier to be made than any other.</I></marg> that if any figure can be given to a Solid, the Spherical is the ea$i- e$t of all others, as it is likewi$e the mo$t $imple, and holdeth the $ame place among$t $olid figures, as the Circle holdeth among$t <marg><I>The circular Fi- gure only is placed amongst the</I> po$tu- lata <I>of Mathema- ticians.</I></marg> the $uperficial. The de$cription of which Circle, as being more ea- $ie than all the re$t, hath alone been judged by <I>Mathematicians</I> worthy to be put among$t the ^{*} <I>po$tulata</I> belonging to the de$cri- <marg>* Demands or Petitions.</marg> ption of all other figures. And the formation of the Sphere is $o very ea$ie, that if in a plain plate of hard metal you take an empty or hollow circle, within which any Solid goeth ca$ually re- volving that was before but gro$ly rounded, it $hall, without any other artifice be reduced to a Spherical figure, as perfect as is po$- $ible for it to be; provided, that that $ame Solid be not le$$e than the Sphere that would pa$$e thorow that Circle. And that which is yet more worthy of our con$ideration is, that within the $elf-$ame <foot>incavity</foot> <p n=>187</p> incavity one may form Spheres of $everal magnitudes. But what <marg><I>Sphericall Fi- gures of $undry magnitudes may be made with one onely in$trument.</I></marg> is required to the making of an Hor$e, or (as you $ay) of a Gra$s- hopper, I leave to you to judge, who know that there are but few $tatuaries in the world able to undertake $uch a piece of work. And I think that herein <I>Simplicius</I> will not di$$ent from me.</P> <P>SIMP. I know not whether I do at all diffent from you; my opinion is this, that none of the afore-named figures can be per- fectly obteined; but for the approaching as neer as is po$$ible to the mo$t perfect degree, I believe that it is incomparably more ea- $ie to reduce the Solid into a Spherical figure, than into the $hape of an Hor$e, or Gra$$e-hopper?</P> <P>SAGR. And this greater difficulty, wherein think you doth it depend?</P> <P>SIMP. Like as the great facility in forming the Sphere ari$eth <marg><I>Irregular forms difficult to be in- troduced.</I></marg> from its ab$olute $implicity and uniformity $o the great irregu- larity rendereth the con$truction of all other figures difficult.</P> <P>SAGR. Therefore the irregularity being the cau$e of the diffi- culty, than the figure of a $tone broken with an hammer by chance, $hall be one of the figures that are difficult to be introdu- ced, it being perhaps more irregular than that of the hor$e?</P> <P>SIMP. So it $hould be.</P> <P>SAGR. But tell me; that figure what ever it is which the $tone hath, hath it the $ame in perfection, or no?</P> <P>SIMP. What it hath, it hath $o perfectly, that nothing can be more exact.</P> <P>SAGR. Then, if of figures that are irregular, and con$equent- ly hard to be procured, there are yet infinite which are mo$t per- fectly obteined, with what rea$on can it be $aid, that the mo$t $imple, and con$equently the mo$t ea$ie of all, is impo$$ible to be procured?</P> <P>SALV. Gentlemen, with your favour, I may $ay that we have $allied out into a di$pute not much more worth than the wool of a goat; and whereas our argumentations $hould continually be con- ver$ant about $erious and weighty points, we con$ume our time in <marg><I>The con$titution of the Univer$e is one of the mo$t no- ble Problems.</I></marg> frivolous and impertinent wranglings. Let us call to minde, I pray you, that the $earch of the worlds con$titution, is one of the grea- te$t and noble$t Problems that are in nature; and $o much the greater, ina$much as it is directed to the re$olving of that other; to wit, of the cau$e of the Seas ebbing and flowing, enquired in- to by all the famous men, that have hitherto been in the world, and po$$ibly found out by none of them. Therefore if we have nothing more remaining for the full confutation of the argument taken from the Earths <I>vertigo,</I> which was the la$t, alledged to prove its immobility upon its own centre, let us pa$$e to the ex- amination of tho$e things that are alledged for, and again$t the <I>Annual Motion.</I></P> <foot>Aa2 SAGR.</foot> <p n=>188</p> <P>SAGR. I would not have you, <I>Salviatus,</I> mea$ure our wits by the $cale of yours: you, who u$e to be continually bu$ied about the $ublime$t contemplations, e$teem tho$e notions frivolous and below you, which we think matters worthy of our profounde$t thoughts: yet $ometimes for our $atisfaction do not di$dain to $toop $o low as to give way a little to our curio$ity. As to the refutation of the la$t argument, taken from the extru$ions of the diurnal <I>vertigo,</I> far le$s than what hath been $aid, would have given me $atisfaction: and yet the things $uperfluou$ly $poken, $eemed to me $o ingenious, that they have been $o far from wea- rying my fancy, as that they have, by rea$on of their novelty, en- tertained me all along with $o great delight, that I know not how to de$ire greater: Therefore, if you have any other $peculation to add, produce it, for I, as to my own particular, $hall gladly hearken to it.</P> <P>SALV. I have always taken great delight in tho$e things which I have had the fortune to di$cover, and next to that, which is my chief content, I find great plea$ure in imparting them to $ome friends, that apprehendeth and $eemeth to like them: Now, in re- gard you are one of the$e, $lacking a little the reins of my ambi- tion, which is much plea$ed when I $hew my $elf more per$pi- cacious, than $ome other that hath the reputation of a $harp $ight, I will for a full and true mea$ure of the pa$t di$pute, pro- duce another fallacy of the Sectators of <I>Ptolomey</I> and <I>Ari$totle,</I> which I take from the argument alledged.</P> <P>SAGR. See how greedily I wait to hear it.</P> <P>SALV. We have hitherto over-pa$$ed, and granted to <I>Ptolomey,</I> as an effect indubitable, that the extru$ion of the $tone proceed- ing from the velocity of the wheel turn'd round upon its centre, the cau$e of the $aid extru$ion encrea$eth in proportion, as the ve- locity of the <I>vertigo</I> (or whirling) is augmented: from whence it was inferred, that the velocity of the Earth's <I>vertigo</I> being very much greater than that of any machin what$oever, that we can make to turn round artificially; the extru$ion of $tones, of animals, &c. would con$equently be far more violent. Now, I ob$erve that there is a great fallacy in this di$cour$e, in that we do compare the$e velocities indifferently and ab$olutely to one ano- ther. It's true, that if I compare the velocities of the $ame wheel, or of two wheels equal to each other, that which $hall be more $wiftly turn'd round, $hall extrude the $tone with greater vio- lence; and the velocity encrea$ing, the cau$e of the projection $hall likewi$e encrea$e: but when the velocity is augmented, not by encrea$ing the velocity in the $ame wheel, which would be by cau$ing it to make a greater number of revolutions in equal times; but by encrea$ing the diameter, and making the wheel greater, $o as that the conver$ion taking up the $ame time in the le$$er wheel, <foot>as</foot> <p n=>189</p> as in the greater, the velocity is greater onely in the bigger wheel, <marg><I>The cau$e of the projection increa$- eth not according to the proportion of the velocity, in- crea$ed by making the wheel bigger.</I></marg> for that its circumference is bigger; there is no man that thinketh that the cau$e of the extru$ion in the great wheel will encrea$e ac- cording to the proportion of the velocity of its circumference, to the velocity of the circumference of the other le$$er wheel; for that this is mo$t fal$e, as by a mo$t expeditious experiment I $hall thus gro$ly declare: We may $ling a $tone with a $tick of a yard long, farther than we can do with a $tick $ix yards long, though the motion of the end of the long $tick, that is of the $tone placed in the $lit thereof, were more than double as $wift as the mo- tion of the end of the other $horter $tick, as it would be if the velocities were $uch that the le$$er $tick $hould turn thrice round in the time whil$t the greater is making one onely con- ver$ion.</P> <P>SAGR. This which you tell me, <I>Salviatus,</I> mu$t, I $ee, needs $ucceed in this very manner; but I do not $o readily apprehend the cau$e why equal velocities $hould not operate equally in extruding projects, but that of the le$$er wheel much more than the other of the greater wheel; therefore I intreat you to tell me how this cometh to pa$s.</P> <P>SIMP. Herein, <I>Sagredus,</I> you $eem to differ much from your $elf, for that you were wont to penetrate all things in an in$tant, and now you have overlook'd a fallacy couched in the experiment of the $tick, which I my $elf have been able to di$cover: and this is the different manner of operating, in making the projection one while with the $hort $ling and another while with the long one, for if you will have the $tone fly out of the $lit, you need not con- tinue its motion uniformly, but at $uch time as it is at the $wifte$t, you are to $tay your arm, and $top the velocity of the $tick; where- upon the $tone which was in its $wifte$t motion, flyeth out, and moveth with impetuo$ity: but now that $top cannot be made in the great $tick, which by rea$on of its length and flexibility, doth not entirely obey the check of the arm, but continueth to accom- pany the $tone for $ome $pace, and holdeth it in with $o much le$s force, and not as if you had with a $tiff $ling $ent it going with a jerk: for if both the $ticks or $lings $hould be check'd by one and the $ame ob$tacle, I do believe they would fly a$well out of the one, as out of the other, howbeit their motions were equally $wift.</P> <P>SAGR. With the permi$$ion of <I>Salviatus,</I> I will an$wer $ome- thing to <I>Simplicius,</I> in regard he hath addre$$ed him$elf to me; and I $ay, that in his di$cour$e there is $omewhat good and $omewhat bad: good, becau$e it is almo$t all true; bad, becau$e it doth not agree with our ca$e: Truth is, that when that which carrieth the $tones with velocity, $hall meet with a <foot>check</foot> <p n=>190</p> check that is immoveable, they $hall fly out with great impetuo- $ity: the $ame effect following in that ca$e, which we $ee dayly to fall out in a boat that running a $wift cour$e, runs a-ground, or meets with $ome $udden $top, for all tho$e in the boat, being $ur- <marg><I>Graming the di- urnal</I> vertigo <I>of the Earth, & that by $ome $udden $top or ob$tacle it were arre$ted, hou$es, mountains them- $elves, and perhaps the whole Globe would be $haken n pieces.</I></marg> prized, $tumble forwards, and fall towards the part whither the boat $teered. And in ca$e the Earth $hould meet with $uch a check, as $hould be able to re$i$t and arre$t its <I>vertigo,</I> then indeed I do believe that not onely bea$ts, buildings and cities, but moun- tains, lakes and $eas would overturn, and the globe it $elf would go near to $hake in pieces; but nothing of all this concerns our pre$ent purpo$e, for we $peak of what may follow to the motion of the Earth, it being turn'd round uniformly, and quietly about its own centre, howbeit with a great velocity. That likewi$e which you $ay of the $lings, is true in part; but was not alledged by <I>Salviatus,</I> as a thing that punctually agreed with the matter whereof we treat, but onely, as an example, for $o in gro$s it may prompt us in the more accurate con$ideration of that point, whe- ther, the velocity increa$ing at any rate, the cau$e of the proje- ction doth increa$e at the $ame rate: $o that <I>v. g.</I> if a wheel of ten yards diameter, moving in $uch a manner that a point of its circumference will pa$s an hundred yards in a minute of an hour, and $o hath an <I>impetus</I> able to extrude a $tone, that $ame <I>impetus</I> $hall be increa$ed an hundred thou$and times in a wheel of a million of yards diameter; the which <I>Salviatus</I> denieth, and I incline to his opinion; but not knowing the rea$on thereof, I have reque$ted it of him, and $tand impatiently expecting it.</P> <P>SALV. I am ready to give you the be$t $atisfaction, that my abilities will give leave: And though in my fir$t di$cour$e you thought that I had enquired into things e$tranged from our pur- po$e, yet neverthele$$e I believe that in the $equel of the di$pute, you will find that they do not prove $o. Therefore let <I>Sagredus</I> tell me wherein he hath ob$erved that the re$i$tance of any move- able to motion doth con$i$t.</P> <P>SAGR. I $ee not for the pre$ent that the moveable hath any internal re$i$tance to motion, unle$$e it be its natural inclination and propen$ion to the contrary motion, as in grave bodies, that have a propen$ion to the motion downwards, the re$i$tance is to the motion upwards; and I $aid an internal re$i$tance, becau$e of this, I think, it is you intend to $peak, and not of the external re$i$tances, which are many and accidental.</P> <P>SALV. It is that indeed I mean, and your nimblene$$e of wit hath been too hard for my craftine$$e, but if I have been too $hort in asking the que$tion, I doubt whether <I>Sagredus</I> hath been full enough in his an$wer to $atis$ie the demand; and whether there be not in the moveable, be$ides the natural inclination to the <foot>contrary</foot> <p n=>191</p> contrary term, another intrin$ick and natural quality, which ma- <marg><I>The inclination of grave bodies to the motion downwards, is equal to their re$i$tance to the motion upwards.</I></marg> keth it aver$e to motion. Therefore tell me again; do you not think that the inclination <I>v. g.</I> of grave bodies to move down- wards, is equal to the re$i$tance of the $ame to the motion of pro- jection upwards?</P> <P>SAGR. I believe that it is exactly the $ame. And for this rea$on I $ee that two equal weights being put into a ballance, they do $tand $till in <I>equilibrium,</I> the gravity of the one re$i$ting its be- ing rai$ed by the gravity wherewith the other pre$$ing down- wards would rai$e it.</P> <P>SALV. Very well; $o that if you would have one rai$e up the other, you mu$t encrea$e the weight of that which depre$$eth, or le$$en the weight of the other. But if the re$i$tance to a$cend- ing motion cun$i$t onely in gravity, how cometh it to pa$$e, that <marg>* A portable bal- lance wherewith market-people weigh their com- modities, giving it gravity by remo- ving the weight farther from the cock: call'd by the Latines, <I>Campana trutina.</I></marg> in ballances of unequal arms, to wit in the ^{*} <I>Stiliard,</I> a weight $ometimes of an hundred pounds, with its pre$$ion downwards, doth not $uffice to rai$e up on of four pounds; that $hall counter- poi$e with it, nay this of four, de$cending $hall rai$e up that of an hundred; for $uch is the effect of the pendant weight upon the weight which we would weigh? If the re$i$tance to motion re$ideth onely in the gravity, how can the arm with its weight of four pounds onely, re$i$t the weight of a $ack of wool, or bale of $ilk, which $hall be eight hundred, or a thou$and weight; yea more, how can it overcome the $ack with its moment, and rai$e it up? It mu$t therefore be confe$t <I>Sagredus,</I> that here it maketh u$e of $ome other re$i$tance, and other force, be$ides that of $imple gravity.</P> <P>SAGR. It mu$t needs be $o; therefore tell me what this $e- cond virtue $hould be.</P> <P>SALV. It is that which was not in the ballance of equal arms; you $ee then what variety there is in the Stiliard; and up- on this doubtle$$e dependeth the cau$e of the new effect.</P> <P>SAGR. I think that your putting me to it a $econd time, hath made me remember $omething that may be to the purpo$e. In both the$e beams the bu$ine$s is done by the weight, and by the motion; in the ballance, the motions are equal, and therefore the one weight mu$t exceed it in gravity before it can move it; in the $tiliard, the le$$er weight will not move the greater, unle$s when this latter moveth little, as being $lung at a le$$er di$tance, and the other much, as hanging at a greater di$tance from the lacquet or cock. It is nece$$ary therefore to conclude, that the le$$er weight overcometh the re$i$tance of the greater, by moving much, whil$t the other is moved but little.</P> <P>SALV. Which is as much as to $ay, that the velocity of the moveable le$s grave, compen$ateth the gravity of the moveable more grave and le$s $wift.</P> <foot>SAGR.</foot> <p n=>192</p> <marg><I>The greater velo- city exactly com- pen$ates thegreater gravity.</I></marg> <P>SAGR. But do you think that the velocity doth fully make good the gravity? that is, that the moment and force of a move- able of <I>v. g.</I> four pounds weight, is as great as that of one of an hundred weight, when$oever that the fir$t hath an hundred degrees of velocity, and the later but four onely?</P> <P>SALV. Yes doubtle$s, as I am able by many experiments to demon$trate: but for the pre$ent, let this onely of the $tiliard $uffice: in which you $ee that the light end of the beam is then able to $u$tain and equilibrate the great Wool $ack, when its di- $tance from the centre, upon which the $tiliard re$teth and turn- eth, $hall $o much exceed the le$$er di$tance, by how much the ab- $olute gravity of the Wool-$ack exceedeth that of the pendent weight. And we $ee nothing that can cau$e this in$ufficiencie in the great $ack of Wool, to rai$e with its weight the pendent weight $o much le$s grave, $ave the di$parity of the motions which the one and the other $hould make, whil$t that the Wool $ack by de$cending but one inch onely, will rai$e the pendent weight an hundred inclies: ($uppo$ing that the $ack did weigh an hundred times as much, and that the di$tance of the $mall weight from the centre of the beam were an hundred times greater, than the di- $tance between the $aid centre and the point of the $acks $u$pen$i- on.) And again, the pendent weight its moving the $pace of an hundred inches, in the time that the $ack moveth but one inch onely, is the $ame as to $ay, that the velocity of the motion of the little pendent weight, is an hundred times greater than the velo- city of the motion of the $ack. Now fix it in your belief, as a true and manife$t axiom, that the re$i$tance which proceedeth from the velocity of motion, compen$ateth that which dependeth on the gravity of another moveable: So that con$equently, a move- able of one pound, that moveth with an hundred degrees of ve- locity, doth as much re$i$t all ob$truction, as another moveable of an hundred weight, who$e velocity is but one degree onely. And two equal moveables will equally re$i$t their being moved, if that they $hall be moved with equal velocity: but if one be to be moved more $wiftly than the other, it $hall make greater re- $i$tance, according to the greater velocity that $hall be conferred on it. The$e things being premi$ed, let us proceed to the expla- nation of our Problem; and for the better under$tanding of things, let us make a $hort Scheme thereof. Let two unequal wheels be de$cribed about this centre A, [<I>in Fig.</I> 7.] and let the circumference of the le$$er be B G, and of the greater C E H, and let the $emidiameter A B C, be perpendicular to the Horizon; and by the points B and C, let us draw the right lined Tangents B F and C D; and in the arches B G and C E, take two equal parts B G and C E: and let the two wheels be $uppo$ed to be turn'd <foot>round</foot> <p n=>193</p> round upon their centres with equal velocities, $o as that two mo- veables, which $uppo$e for example to be two $tones placed in the points B and C, come to be carried along the circumferences B G and C E, with equal velocities; $o that in the $ame time that the $tone B $hall have run the arch B G, the $tone C will have pa$t the arch C E. I $ay now, that the whirl or <I>vertigo</I> of the le$$er wheel is much more potent to make the projection of the $tone B, than the <I>vertigo</I> of the bigger wheel to make that of the $tone C. Therefore the projection, as we have already declared, being to be made along the tangent, when the $tones B and C are to $eparate from their wheels, and to begin the motion of projection from the points B and C, then $hall they be extruded by the <I>impetus</I> con- ceived from the <I>vertigo</I> by (or along) the tangents B F and C D. The two $tones therefore have equal impetuo$ities of running a- long the tangents B F and C D, and would run along the $ame, if they were not turn'd a$ide by $ome other force: is it not $o <I>Sa- gredus</I>?</P> <P>SAGR. In my opinion the bu$ine$$e is as you $ay.</P> <P>SALV. But what force, think you, $hould that be which averts the $tones from moving by the tangents, along which they are cer- tainly driven by the <I>impetus</I> of the <I>vertigo.</I></P> <P>SAGR. It is either their own gravity, or el$e $ome glutinous matter that holdeth them fa$t and clo$e to the wheels.</P> <P>SALV. But for the diverting of a moveable from the motion to which nature inciteth it, is there not required greater or le$$er force, according as the deviation is intended to be greater or le$- $er? that is, according as the $aid moveable in its deviation hath a greater or le$$er $pace to move in the $ame time?</P> <P>SAGR. Yes certainly: for it was concluded even now, that to make a moveable to move; the movent vertue mu$t be increa$ed in proportion to the velocity wherewith it is to move.</P> <P>SALV. Now con$ider, that for the deviating the $tone upon the le$$e wheel from the motion of projection, which it would make by the tangent B F, and for the holding of it fa$t to the wheel, it is required, that its own gravity draw it back the whole length of the $ecant F G, or of the perpendicular rai$ed from the point G, to the line B F, whereas in the greater wheel the retracti- on needs to be no more than the $ecant D E, or the perpendicu- lar let fall from the tangent D G to the point E, le$$e by much than F G, and alwayes le$$er and le$$er according as the wheel is made bigger. And fora$much as the$e retractions (as I may call them) are required to be made in equal times, that is, whil'$t the wheels pa$$e the two equal arches B G and C E, that of the $tone B, that is, the retraction F G ought to be more $wift than the o- ther D E; and therefore much greater force will be required for <foot>Bb holding</foot> <p n=>194</p> holding fa$t the $tone B to its little wheel, than for the holding the $tone C to its great one, which is as much as to $ay, that $uch a $mall thing will impede the extru$ion in the great wheel, as will not at all hinder it in the little one. It is manife$t therefore that the more the wheel augmenteth, the more the cau$e of the pro- jection dimini$heth.</P> <P>SAGR. From this which I now under$tand, by help of your mi- nute di$$ertation, I am induced to think, that I am able to $atisfie my judgment in a very few words. For equal <I>impetus</I> being im- pre$$ed on both the $tones that move along the tangents, by the equal velocity of the two wheels, we $ee the great circumference, by means of its $mall deviation from the tangent, to go $econding, as it were, and in a fair way refraining in the $tone the appetite, if I may $o $ay, of $eparating from the circumference; $o that any $mall retention, either of its own inclination, or of $ome glutina- tion $ufficeth to hold it fa$t to the wheel. Which, again, is not a- ble to work the like effect in the little wheel, which but little pro- $ecuting the direction of the tangent, $eeketh with too much ea- gerne$$e to hold fa$t the $tone; and the re$triction and glutination not being $tronger than that which holdeth the other $tone fa$t to <marg>* Strappar la ca- vezza, <I>is to break the bridle.</I></marg> the greater wheel, it ^{*} breaks loo$e, and runneth along the tan- gent. Therefore I do not only finde that all tho$e have erred, who have believed the cau$e of the projection to increa$e accor- ding to the augmentation of the <I>vertigo's</I> velocity; but I am further thinking, that the projection dimini$hing in the inlarging of the wheel, $o long as the $ame velocity is reteined in tho$e wheels; it may po$$ibly be true, that he that would make the great wheel extrude things like the little one, would be forced to increa$e them as much in velocity, as they increa$e in diameter, which he might do, by making them to fini$h their conver$ions in equal times; and thus we may conclude, that the Earths revolution or <I>vertigo</I> would be no more able to extrude $tones, than any little wheel that goeth $o $lowly, as that it maketh but one turn in twen- ty four hours.</P> <P>SALV. We will enquire no further into this point for the pre- $ent: let it $uffice that we have abundantly (if I deceive not my $elf) demon$trated the invalidity of the argument, which at fir$t $ight $eemed very concluding, and was $o held by very famous men: and I $hall think my time and words well be$towed, if I have but gained $ome belief in the opinion of <I>Simplicius,</I> I will not $ay or the Earths mobility, but only that the opinion of tho$e that believe it, is not $o ridiculous and fond, as the rout of vulgar Philo$ophers e$teem it.</P> <P>SIMP. The an$wers hitherto produced again$t the arguments brought again$t this <I>Diurnal Revolution</I> of the Earth taken from <foot>grave</foot> <p n=>195</p> grave bodies falling from the top of a Tower, and from proje- ctions made perpendicularly upwards, or according to any inclina- tion $idewayes towards the Ea$t, We$t, North, South, &c. have $omewhat abated in me the antiquated incredulity I had conceived again$t that opinion: but other greater doubts run in my mind at this very in$tant, which I know not in the lea$t how to free my $elf of, and haply you your $elf will not be able to re$olve them; nay, its po$$ible you may not have heard them, for they are very modern. And the$e are the objections of two Authours, that <I>ex profe$$o</I> write again$t <I>Copernicus.</I> Some of which are read in a <marg><I>Other objections of two modern Au- thors against</I> Co- pernicus.</marg> little Tract of natural conclu$ions; The re$t are by a great both Philo$opher and Mathematician, in$erted in a Treati$e which he hath written in favour of <I>Aristotle,</I> and his opinion touching the inalterability of the Heavens, where he proveth, that not onely the Comets, but al$o the new $tars, namely, that <I>anno</I> 1572. in <I>Ca$$iopeia,</I> and that <I>anno</I> 1604. in <I>Sagittarius</I> were not above the Spheres of the Planets, but ab$olutely beneath the concave of the Moon in the Elementary Sphere, and this he demon$trateth a- gain$t <I>Tycho, Kepler,</I> and many other Aftronomical Ob$ervators, and beateth them at their own weapon; to wit, the Doctrine of Parallaxes. If you like thereof, I will give you the rea$ons of both the$e Authours, for I have read them more than once, with attention; and you may examine their $trength, and give your opinion thereon.</P> <P>SALV. In regard that our principal end is to bring upon the $tage, and to con$ider what ever hath been $aid for, or again$t the two Sy$temes, <I>Ptolomaick,</I> and <I>Copernican,</I> it is not good to omit any thing that hath been written on this $ubject.</P> <P>SIMP. I will begin therefore with the objections which I finde in the Treati$e of Conclu$ions, and afterwards proceed to the <marg><I>The fir$t obje- ction of the mo- dern Author of the little tract of</I> Conclu$ions.</marg> re$t. In the fir$t place then, he be$toweth much paines in calcu- lating exactly how many miles an hour a point of the terre$trial Globe $ituate under the Equinoctial, goeth, and how many miles are pa$t by other points $ituate in other parallels: and not being content with finding out $uch motions in horary times, he findeth them al$o in a minute of an hour; and not contenting him$elf with a minute, he findes them al$o in a $econd minute; yea more, he goeth on to $hew plainly, how many miles a Cannon bullet would go in the $ame time, being placed in the concave of the Lu- <marg><I>A Cannon bul- let would $pend more than $ix days in falling from the Concave of the Moon to the cen- tre of the Earth, according to the o- pinion of that mo- dern Author of the</I> Conclu$ions.</marg> nar Orb, $uppo$ing it al$o as big as <I>Copernicus</I> him$elf repre$enteth it, to take away all $ubterfuges from his adver$ary. And having made this mo$t ingenious and exqui$ite $upputation, he $heweth, that a grave body falling from thence above would con$ume more than $ix dayes in attaining to the centre of the Earth, to which all grave bodies naturally move. Now if by the ab$olute Divine <foot>Bb 2 Power</foot> <p n=>196</p> Power, or by $ome Angel, a very great Cannon bullet were carri- ed up thither, and placed in our Zenith or vertical point, and from thence let go at liberty, it is in his, and al$o in my opinion, a mo$t incredible thing that it, in de$cending downwards, $hould all the way maintain it $elf in our vertical line, continuing to turn round with the Earth, about its centre, for $o many dayes, de$cribing under the Equinoctial a Spiral line in the plain of the great circle it $elf: and under other Parallels, Spiral lines about Cones, and under the Poles falling by a $imple right line. He, in the next place, $tabli$heth and confirmeth this great improbability by pro- ving, in the way of interrogations, many difficulties impo$$ible to be removed by the followers of <I>Copernicus</I>; and they are, if I do well remember-----.</P> <P>SALV. Take up a little, good <I>Simplicius,</I> and do not load me with $o many novelties at once: I have but a bad memory, and therefore I mu$t not go too fa$t. And in regard it cometh into my minde, that I once undertook to calculate how long time $uch a grave body falling from the concave of the Moon, would be in pa$$ing to the centre of the Earth, and that I think I remember that the time would not be $o long; it would be fit that you $hew us by what rule this Author made his calculation.</P> <P>SIMP. He hath done it by proving his intent <I>à fortiori,</I> a $uffi- cient advantage for his adver$aries, $uppo$ing that the velocity of the body falling along the vertical line, towards the centre of the Earth, were equal to the velocity of its circular motion, which it made in the grand circle of the concave of the Lunar Orb. Which by equation would come to pa$$e in an hour, twelve thou- $and $ix hundred German miles, a thing which indeed $avours of impo$$ibility: Yet neverthele$$e, to $hew his abundant caution, and to give all advantages to his adver$aries, he $uppo$eth it for true, and concludeth, that the time o$ the fall ought however to be more than $ix dayes.</P> <P>SALV. And is this the $um of his method? And doth he by this demon$tration prove the time of the fall to be above $ix dayes?</P> <P>SAGR. Me thinks that he hath behaved him$elf too mode$tly, for that having it in the power of his will to give what velocity he plea$ed to $uch a de$cending body, and might a$well have made it $ix moneths, nay, $ix years in falling to the Earth, he is content with $ix dayes. But, good <I>Salviatus,</I> $harpen my appetite a lit- tle, by telling me in what manner you made your computation, in regard you $ay, that you have heretofore ca$t it up: for I am con- fident that if the que$tion had not required $ome ingenuity in working it, you would never have applied your minde unto it.</P> <foot>SALV.</foot> <p n=>197</p> <P>SALV. It is not enough, <I>Sagredus,</I> that the $ubjects be noble and great, but the bu$ine$$e con$i$ts in handling it nobly. And who knoweth not, that in the di$$ection of the members of a bea$t, there may be di$covered infinite wonders of provident and prudent Nature; and yet for one, that the Anatomi$t di$- $ects, the butcher cuts up a thou$and. Thus I, who am now $eeking how to $atisfie your demand, cannot tell with which of the two $hapes I had be$t to appear on the Stage; but yet, taking heart from the example of <I>Simplicius,</I> his Authour, I will, with- out more delays, give you an account (if I have not forgot) how I proceeded. But before I go any further, I mu$t not omit to tell you, that I much fear that <I>Simplicius</I> hath not faithfully related the manner how this his Authour found, that the Cannon bul- let in coming from the concave of the Moon to the centre of the Earth, would $pend more than fix dayes: for if he had $uppo- $ed that its velocity in de$cending was equal to that of the concave (as <I>Simplicius</I> $aith he doth $uppo$e) he would have $hewn him$elf ignorant of the fir$t, and more $imple principles of <I>Geometry</I>; yea, I admire that <I>Simplicius,</I> in admitting the $uppo$ition which he $peaketh of, doth not $ee the mon$trous ab- $urdity that is couched in it.</P> <P>SIMP. Its po$$ible that I may have erred in relating it; but that I $ee any fallacy in it, I am $ure is not true.</P> <P>SALV. Perhaps I did not rightly apprehend that which you $aid, Do you not $ay, that this Authour maketh the velocity of the bullet in de$cending equall to that which it had in tur- ning round, being in the concave of the Moon, and that com- ming down with the $ame velocity, it would reach to the centre in $ix dayes?</P> <P>SIMP. So, as I think, he writeth.</P> <P>SALV. And do not you perceive a $hamefull errour therein? But que$tionle$$e you di$$emble it: For it cannot be, but that you $hould know that the $emidiameter of the Circle is le$$e than <marg><I>A $hamefull errour in the Ar- gument taken from the bullets falling out of the Moons concave.</I></marg> the $ixth part of the circumference; and that con$equently, the time in which the moveable $hall pa$$e the $emidiameter, $hall be le$$e than the $ixth part of the time; in which, being moved with the $ame velocity, it would pa$$e the circumference; and that therefore the bullet de$cending with the velocity, where- with it moved in the concave, will arrive in le$$e than four hours at the centre, $uppo$ing that in the concave one revolution $hould be con$ummate in twenty four hours, as he mu$t of ne- ce$$ity have $uppo$ed it, for to keep it all the way in the $ame vertical line.</P> <P>SIMP. Now I thorowly perceive the mi$take: but yet I would not lay it upon him unde$ervedly, for it's po$$ible that I <foot>may</foot> <p n=>198</p> may have erred in rehear$ing his Argument, and to avoid running into the $ame mi$takes for the future, I could wi$h I had his Book; and if you had any body to $end for it, I would take it for a great favour.</P> <P>SAGR. You $hall not want a Lacquey that will runne for it with all $peed: and he $hall do it pre$ently, without lo$ing any time; in the mean time <I>Salviatus</I> may plea$e to oblige us with his computation.</P> <P>SIMP. If he go, he $hall finde it lie open upon my Desk, together with that of the other Author, who al$o argueth a- gain$t <I>Copernicus.</I></P> <P>SAGR. We will make him bring that al$o for the more cer- tainty: and in the interim <I>Salviatus</I> $hall make his calculation: I have di$patch't away a me$$enger.</P> <P>SALV. Above all things it mu$t be con$idered, that the motion of de$cending grave bodies is not uniform, but departing from <marg><I>An exact com- pute of the time of the fall of the Ca- non bullet from the Moons concave to the Earths centre.</I></marg> re$t they go continually accelerating: An effect known and ob- $erved by all men, unle$$e it be by the forementioned modern Au- thour, who not $peaking of acceleration, maketh it even and u- niforme. But this general notion is of no avail, if it be not known according to what proportion this increa$e of velocity is made; a conclu$ion that hath been until our times unknown to all <I>Philo$o- phers</I>; and was fir$t found out & demon$trated by the ^{*} <I>Academick,</I> <marg>* The Author.</marg> our common friend, who in $ome of his ^{*} writings not yet publi$h- <marg>* By the$e <I>Wri- tings,</I> he every where meanes his Dialogues, <I>De mo- tu,</I> which I promi$e to give you in my $econd Volume.</marg> ed, but in familiarity $hewn to me, and $ome others of his ac- quaintance he proveth, how that the acceleration of the right mo- tion of grave bodies, is made according to the numbers uneven beginning <I>ab unitate,</I> that is, any number of equal times being a$- $igned, if in the fir$t time the moveable departing from re$t $hall <marg><I>Acceleration of the natural motion of grave bodies is made according to the odde numbers beginning at unity.</I></marg> have pa$$ed $uch a certain $pace, as for example, an ell, in the $e- cond time it $hall have pa$$ed three ells, in the third five, in the fourth $even, and $o progre$$ively, according to the following odd numbers; which in $hort is the $ame, as if I $hould $ay, that the $paces pa$$ed by the moveable departing from its re$t, are unto <marg><I>The $paces pa$t by the falling grave body are as the $quares of their times.</I></marg> each other in proportion double to the proportion of the times, in which tho$e $paces are mea$ured; or we will $ay, that the $paces pa$$ed are to each other, as the $quares of their times.</P> <P>SAGR. This is truly admirable: and do you $ay that there is a Mathematical demon$tration for it?</P> <P>SALV. Yes, purely Mathematical; and not onely for this, but for many other very admirable pa$$ions, pertaining to natural mo- tions, and to projects al$o, all invented, and demon$trated by <I>Our</I> <marg><I>An intire and new Science of the</I> Academick <I>concer- ning local motion.</I></marg> <I>Friend,</I> and I have $een and con$idered them all to my very great content and admiration, $eeing a new compleat Doctrine to $pring up touching a $ubject, upon which have been written hundreds of <foot>Volumes;</foot> <p n=>199</p> Volumes; and yet not $o much as one of the infinite admirable conclu$ions that tho$e his writings contain, hath ever been ob- $erved, or under$tood by any one, before <I>Our Friend</I> made them out.</P> <P>SAGR. You make me lo$e the de$ire I had to under$tand more in our di$putes in hand, onely that I may hear $ome of tho$e demon$trations which you $peak of; therefore either give them me pre$ently, or at lea$t promi$e me upon your word, to appoint a particular conference concerning them, at which <I>Sim- plicius</I> al$o may be pre$ent, if he $hall have a mind to hear the pa$$ions and accidents of the primary effect in Nature.</P> <P>SIMP. I $hall undoubtedly be much plea$ed therewith, though indeed, as to what concerneth Natural Philo$ophy, I do not think that it is nece$$ary to de$cend unto minute particularities, a gene- ral knowledg of the definition of motion, and of the di$tin- ction of natural and violent, even and accelerate, and the like, $ufficing: For if this were not $ufficient, I do not think that <I>Ari- $totle</I> would have omitted to have taught us what ever more was nece$$ary.</P> <P>SALV. It may be $o. But let us not lo$e more time about this, which I promi$e to $pend half a day apart in, for your $atis- faction; nay, now I remember, I did promi$e you once before to $atisfie you herein. Returning therefore to our begun calcula- tion of the time, wherein the grave cadent body would pa$s from the concave of the Moon to the centre of the Earth, that we may not proceed arbitrarily and at randon, but with a Logical method, we will fir$t attempt to a$certain our $elves by experiments often repeated, in how long time a ball <I>v. g.</I> of Iron de$cendeth to the Earth from an altitude of an hundred yards.</P> <P>SAGR. Let us therefore take a ball of $uch a determinate weight, and let it be the $ame wherewith we intend to make the computation of the time of de$cent from the Moon.</P> <P>SALV. This is not material, for that a ball of one, of ten, of an hundred, of a thou$and pounds, will all mea$ure the $ame hundred yards in the $ame time.</P> <P>SIMP. But this I cannot believe, nor much le$s doth <I>Ari$totle</I> think $o, who writeth, that the velocities of de$cending grave bodies, are in the $ame proportion to one another, as their gra- vities.</P> <P>SALV. If you will admit this for true, <I>Simplicius,</I> you mu$t be- <marg><I>The error of</I> Ari- $totle <I>in affirming, falling grave bo- dies to move accor- ding to the propor- tion of their gravi- ties.</I></marg> lieve al$o, that two balls of the $ame matter, being let fall in the $ame moment, one of an hundred pounds, and another of one, from an altitude of an hundred yards, the great one arriveth at the ground, before the other is de$cended but one yard onely: Now bring your fancy, if you can, to imagine, that you $ee the great <foot>ball</foot> <p n=>200</p> ball got to the ground, when the little one is $till within le$s than a yard of the top of the Tower.</P> <P>SAGR. That this propo$ition is mo$t fal$e, I make no doubt in the world; but yet that yours is ab$olutely true, I cannot well a$$ure my $elf: neverthele$s, I believe it, $eeing that you $o re- $olutely affirm it; which I am $ure you would not do, if you had not certain experience, or $ome clear demon$tration thereof.</P> <P>SALV. I have both: and when we $hall handle the bu$ine$s of motions apart, I will communicate them: in the interim, that we may have no more occa$ions of interrupting our di$cour$e, we will $uppo$e, that we are to make our computation upon a ball of <marg><I>(a) (b)</I> Note that the$e Calculations are made in <I>Itali- an</I> weights and mea$ures. And 100 pounds <I>Haverdu- poi$e</I> make 131 <I>l. Florentine.</I> And 100 Engli$h yards makes 150 2/5 Braces <I>Florent.</I> $o that the brace or yard of our <I>Author</I> is 3/4 of cur yard.</marg> Iron of an hundred <I>(a)</I> pounds, the which by reiterated experi- ments de$cendeth from the altitude of an hundred <I>(b)</I> yards, in five $econd-minutes of an hour. And becau$e, as we have $aid, the $paces that are mea$ured by the cadent moveable, increa$e in double proportion; that is, according to the $quares of the times, being that the time of one fir$t-minute is duodecuple to the time of five $econds, if we multiply the hundred yards by the $quare of 12, that is by 144, we $hall have 14400, which $hall be the num- ber of yards that the $ame moveable $hall pa$s in one fir$t-minute of an hour: and following the $ame rule becau$e one hour is 60 minutes, multiplying 14400, the number of yards pa$t in one mi- nute, by the $quare of 60, that is, by 3600, there $hall come forth 51840000, the number of yards to be pa$$ed in an hour, which make 17280 miles. And de$iring to know the $pace that the $aid ball would pa$s in 4 hours, let us multiply 17280 by 16, (which is the $quare of 4) and the product will be 276480 miles: which number is much greater than the di$tance from the Lunar concave to the centre of the Earth, which is but 196000 miles, making the di$tance of the concave 56 $emidiameters of the Earth, as that mo- dern Author doth; and the $emidiameter of the Earth 3500 miles, <marg>* The <I>Italian</I> mea- $ure which I com- monly tran$l te yards.</marg> of 3000 ^{*}<I>Braces</I> to a †mile, which are our <I>Italian</I> miles.</P> <P>Therefore, <I>Simplicius,</I> that $pace from the concave of the Moon to the centre of the Earth, which your Accomptant $aid could <marg>† The <I>Italian</I> mile is 1000/1056 of our mile.</marg> not be pa$$ed under more than $ix days, you $ee that (computing by experience, and not upon the fingers ends) that it $hall be pa$- $ed in much le$s than four hours; and making the computation exact, it $hall be pa$$ed by the moveable in 3 hours, 22 <I>min. prim.</I> and 4 $econds.</P> <P>SAGR. I be$eech you, dear Sir, do not defraud me of this ex- act calculation, for it mu$t needs be very excellent.</P> <P>SALV. So indeed it is: therefore having (as I have $aid) by diligent tryal ob$erved, that $uch a moveable pa$$eth in its de$cent, the height of 100 yards in 5 $econds of an hour, we will $ay, if 100 yards are pa$$ed in 5 $econds; in how many $econds $hall <foot>588000000</foot> <p n=>201</p> 588000000 yards (for $o many are in 56 diameters of the Earth) be pa$$ed? The rule for this work is, that the third number mu$t be multiplied by the $quare of the $econd, of which doth come 14700000000, which ought to be divided by the fir$t, that is, by 100, and the root $quare of the quotient, that is, 12124 is the number $ought, namely 12124 <I>min. $ecun.</I> of an hour, which are 3 hours, 22 <I>min. prim.</I> and 4 $econds.</P> <P>SAGR. I have $een the working, but I know nothing of the rea$on for $o working, nor do I now think it a time to ask it.</P> <P>SALV. Yet I will give it, though you do not ask it, becau$e it is very ea$ie. Let us mark the$e three numbers with the Letters A fir$t, B $econd, C <fig> third. A and C are the numbers of the $paces, B is the number of the time; the fourth number is $ought, of the time al$o. And becau$e we know, that look what proportion the $pace A, hath to the $puace C, the $ame proportion $hall the $quare of the time B have to the $qare of the time, which is $ought. Therefore by the Golden Rule, let the number C be multi- plied by the $quare of the number B, and let the product be di- vided by the number A, and the quotient $hall be the $quare of the number $ought, and its $quare root $hall be the number it $elf that is $ought. Now you $ee how ea$ie it is to be under$tood.</P> <P>SAGR. So are all truths, when once they are found out, but the difficulty lyeth in finding them. I very well apprehend it, and kindly thank you. And if there remain any other curio$ity touching this point, I pray you let us hear it; for if I may $peak my mind, I will with the favour of <I>Simplicius,</I> that from your di$cour$es I al- wayes learn $ome new motion, but from tho$e of his Philo$o- phers, I do not remember that I have learn't any thing of mo- ment.</P> <P>SALV. There might be much more $aid touching the$e local motions; but according to agreement, we will re$erve it to a par- ticular conference, and for the pre$ent I will $peak $omething touching the Author named by <I>Simplicius,</I> who thinketh he hath given a great advantage to the adver$e party in granting that, that Canon bullet in falling from the concave of the Moon may de- $cend with a velocity equal to the velocity wherewith it would <foot>Cc turn</foot> <p n=>202</p> turn round, $taying there above, and moving along with the di- urnal conver$ion. Now I tell him, that that $ame ball falling from the concave unto the centre, will acquire a degree of velocity much more than double the velocity of the diurnal motion of the Lunar concave; and this I will make out by $olid and not imper- <marg><I>The falling move- able if it move with a degree of veloci- ty acquired in a like time with an uniform motion, it $hall paß a $pace double to that pa$- $ed with the acce- leratedmotion.</I></marg> tinent $uppo$itions. You mu$t know therefore that the grave body falling and acquiring all the way new velocity according to the proportion already mentioned, hath in any what$oever place of the line of its motion $uch a degree of velocity, that if it $hould continue to move therewith, uniformly without farther encrea$ing it; in another time like to that of its de$cent, it would pa$$e a $pace double to that pa$$ed in the line of the precedent motion of de$cent. And thus for example, if that ball in coming from the concave of the Moon to its centre hath $pent three hours, 22 min. <I>prim.</I> and 4 $econds, I $ay, that being arrived at the cen- tre, it $hall find it $elf con$tituted in $uch a degree of velocity, that if with that, without farther encrea$ing it, it $hould continue to move uniformly, it would in other 3 hours, 22 min. <I>prim.</I> and 4 $econds, pa$$e double that $pace, namely as much as the whole diameter of the Lunar Orb; and becau$e from the Moons con- cave to the centre are 196000 miles, which the ball pa$$eth in 3 hours 22 <I>prim.</I> min. and 4 $econds, therefore (according to what hath been $aid) the ball continuing to move with the velocity which it is found to have in its arrival at the centre, it would pa$$e in other 3 hours 22 min. prim. and 4 $econds, a $pace dou- ble to that, namely 392000 miles; but the $ame continuing in the concave of the Moon, which is in circuit 1232000 miles, and moving therewith in a diurnal motion, it would make in the $ame time, that is in 3 hours 22 min. prim. and 4 $econds, 172880 miles, which are fewer by many than the half of the 392000 miles. You $ee then that the motion in the concave is not as the modern Author $aith, that is, of a velocity impo$$ible for the fall- ing ball to partake of, <I>&c.</I></P> <P>SAGR. The di$cour$e would pa$s for current, and would give me full $atisfaction, if that particular was but $alved, of the mo- ving of the moveable by a double $pace to that pa$$ed in falling in another time equal to that of the de$cent, in ca$e it doth continue to move uniformly with the greate$t degree of velocity acquired in de$cending. A propo$ition which you al$o once before $uppo- $ed as true, but never demon$trated.</P> <P>SALV. This is one of the demon$trations of <I>Our Friend,</I> and you $hall $ee it in due time; but for the pre$ent, I will with $ome conjectures (not teach you any thing that is new, but) remember you of a certain contrary opinion, and $hew you, that it may haply $o be. A bullet of lead hanging in a long and fine thread fa$tened to the <foot>roof,</foot> <p n=>203</p> roof, if we remove it far from perpendicularity, and then let it go, have you not ob$erved that, it declining, will pa$s freely, and well near as far to the other $ide of the perpendicular?</P> <P>SAGR. I have ob$erved it very well, and find (e$pecially if the plummet be of any con$iderable weight) that it ri$eth $o little le$s than it de$cended, $o that I have $ometimes thought, that the a- $cending arch is equal to that de$cending, and thereupon made it a que$tion whether the vibrations might not perpetuate them$elves; and I believe that they might, if that it were po$$ible to remove <marg><I>The motion of grave</I> penduli <I>might be perpetua- ted, impediments being removed.</I></marg> the impediment of the Air, which re$i$ting penetration, doth $ome $mall matter retard and impede the motion of the <I>pendulum,</I> though indeed that impediment is but $mall: in favour of which opinion the great number of vibrations that are made before the moveable wholly cea$eth to move, $eems to plead.</P> <P>SALV. The motion would not be perpetual, <I>Sagredus,</I> al- though the impediment of the Air were totally removed, becau$e there is another much more ab$tru$e.</P> <P>SAGR. And what is that? as for my part I can think of no other?</P> <P>SALV. You will be plea$ed when you hear it, but I $hall not tell it you till anon: in the mean time, let us proceed. I have propo$ed the ob$ervation of this <I>Pendulum,</I> to the intent, that you $hould under$tand, that the <I>impetus</I> acquired in the de$cending arch, where the motion is natural, is of it $elf able to drive the $aid ball with a violent motion, as far on the other $ide in the like a$cending arch; if $o, I $ay, of it $elf, all external impediments being removed: I believe al$o that every one takes it for granted, that as in the de$cending arch the velocity all the way increa$eth, till it come to the lowe$t point, or its perpendicularity; $o from this point, by the other a$cending arch, it all the wav dimini$heth, untill it come to its extreme and highe$t point: and dimini$hing with the $ame proportions, where with it did before increa$e, $o that the dgrees of the velocities in the points equidi$tant from the point of perpendicularity, are equal to each other. Hence it $eemeth to me (arguing with all due mode$ty) that I might ea$ily be induced to believe, that if the Terre$trial Globe were bored thorow the <marg><I>If the Terre$trial Globe were perfo- rated, a grave bo- dy de$cending by that bore, would paß and a$cend as far beyond the cen- tre, as it did de- $cend.</I></marg> centre, a Canon bullet de$cending through that Well, would ac- quire by that time it came to the centre, $uch an impul$e of velo- city, that, it having pa$$ed beyond the centre, would $pring it up- wards the other way, as great a $pace, as that was wherewith it had de$cended, all the way beyond the centre dimini$hing the velocity with decrea$ements like to the increa$ements acquired in the de- $cent: and the time $pent in this $econd motion of a$cent, I be- lieve, would be equal to the time of de$cent. Now if the move- able by dimini$hing that its greate$t degree of velocity which it <foot>Cc 2 had</foot> <p n=>204</p> had in the centre, $ucce$$ively until it come to total extinction, do carry the moveable in $uch a time $uch a certain $pace, as it had gone in $uch a like quantity of time, by the acqui$t of velocity from the total privation of it until it came to that its greate$t degree; it $eemeth very rea$onable, that if it $hould move always with the $aid greate$t degree of velocity it would pa$s, in $uch another quantity of time, both tho$e $paces: For if we do but in our mind $ucce$$ively divide tho$e velocities into ri$ing and falling degrees, as <I>v. g.</I> the$e numbers in the margine; $o that the fir$t $ort unto 10 be $uppo$ed the increa$ing velocities, and the others unto 1, be the decrea$ing; and let tho$e of the time of the de$cent, and the others of the time of the a$cent being added all together, make as many, as if one of the two $ums of them had been all of the greate$t degrees, and therefore the whole $pace pa$$ed by all the degrees of the increa$ing veloci- ties, and decrea$ing, (which put together is the whole diame- ter) ought to be equal to the $pace pa$$ed by the greate$t velo- cities, that are in number half the aggregate of the increa$ing and decrea$ing velocities. I know that I have but ob$curely expre$$ed my $elf, and I wi$h I may be under$tood.</P> <P>SAGR. I think I under$tand you very well; and al$o that I can in a few words $hew, that I do under$tand you. You had a mind to $ay, that the motion begining from re$t, and all the way increa$ing the velocity with equal augmentations, $uch as are tho$e of continuate numbers begining at 1, rather at 0, which repre$enteth the $tate of re$t, di$po$ed as in the margine: and continued at plea$ure, $o as that the lea$t degree may be 0, and the greate$t <I>v. g.</I> 5, all the$e degrees of velocity wherewith the moveable is moved, make the $um of 15; but if the moveable $hould move with as many degrees in number as the$e are, and each of them equal to the bigge$t, which is 5, the aggregate of all the$e la$t velocities would be double to the others, namely 30. And therefore the moveable moving with a like time, but with uniform velocity, which is that of the highe$t degree 5, ought to pa$s a $pace double to that which it pa$$eth in the accelerate time, which beginneth at the $tate of re$t.</P> <P>SALV. According to your quick and piercing way of appre- hending things, you have explained the whole bu$ine$s with more plainne$s than I my $elf; and put me al$o in mind of adding $ome- thing more: for in the accelerate motion, the augmentation be- ing continual, you cannot divide the degrees of velocity, which continually increa$e, into any determinate number, becau$e chan- ging every moment, they are evermore infinite. Therefore we $hall be the better able to exemplifie our intentions by de$cribing a Triangle, which let be this A B C, [<I>in Fig.</I> 8.] taking in the <foot>$ide</foot> <p n=>205</p> $ide A C, as many equal parts as we plea$e, A D, D E, E F, F G, and drawing by the points D, E, F, G, right lines parallel to the ba$e B C. Now let us imagine the parts marked in the line A C, to be equal times, and let the parallels drawn by the points D, E, F, G, repre$ent unto us the degrees of velocity accelerated, and increa$- ing equally in equal times; and let the point A be the $tate of re$t, from which the moveable departing, hath <I>v. g.</I> in the time A D, acquired the degree of velocity D H, in the $econd time we will $uppo$e, that it hath increa$ed the velocity from D H, as far as to E I, and $o $uppo$ing it to have grown greater in the $ucceeding times, according to the increa$e of the lines F K, G L, <I>&c.</I> but <marg><I>The acceleration of grave bodies na- turally de$cendent, increa$eth from moment to moment.</I></marg> becau$e the acceleration is made continually from moment to mo- ment, and not disjunctly from one certain part of time to another; the point A being put for the lowe$t moment of velocity, that is, for the $tate of re$t, and A D for the fir$t in$tant of time follow- ing; it is manife$t, that before the acqui$t of the degree of velocity D H, made in the time A D, the moveable mu$t have pa$t by infinite other le$$er and le$$er degrees gained in the infinite in$tants that are in the time D A, an$wering the infinite points that are in the line D A; therefore to repre$ent unto us the infinite degrees of velocity that precede the degree D H, it is nece$$ary to imagine infinite lines $ucce$$ively le$$er and le$$er, which are $uppo$ed to be drawn by the infinite points of the line D A, and parallels to D H, the which infinite lines repre$ent unto us the $uperficies of the Triangle A H D, and thus we may imagine any $pace pa$$ed by the moveable, with a motion which begining at re$t, goeth uni- formly accelerating, to have $pent and made u$e of infinite degrees of velocity, increa$ing according to the infinite lines that begin- ing from the point A, are $uppo$ed to be drawn parallel to the line H D, and to the re$t I E, K F, L G, the motion continuing as far as one will.</P> <P>Now let us compleat the whole Parallelogram A M B C, and let us prolong as far as to the $ide thereof B M, not onely the Parallels marked in the Triangle, but tho$e infinite others imagined to be drawn from all the points of the $ide A C; and like as B C, was the greate$t of tho$e infinite parallels of the Triangle, repre$ent- ing unto us the greate$t degree of velocity acquired by the move- able in the accelerate motion, and the whole $uperficies of the $aid Triangle, was the ma$s and $um of the whole velocity, wherewith in the time A C it pa$$ed $uch a certain $pace, $o the parallelogram is now a ma$s and aggregate of a like number of degrees of ve- locity, but each equal to the greate$t B C, the which ma$s of ve- locities will be double to the ma$s of the increa$ing velocities in the Triangle, like as the $aid Parallelogram is double to the Tri- angle: and therefore if the moveable, that falling did make u$e <foot>of</foot> <p n=>206</p> of the accelerated degrees of velocity, an$wering to the triangle A B C, hath pa$$ed in $uch a time $uch a $pace, it is very rea$onable and probable, that making u$e of the uniform velocities an$wering to the parallelogram, it $hall pa$$e with an even motion in the $ame time a $pace double to that pa$$ed by the accelerate mo- tion.</P> <P>SAGR. I am entirely $atisfied. And if you call this a probable Di$cour$e, what $hall the nece$$ary demon$trations be? I wi$h that in the whole body of common Philo$ophy, I could find one that was but thus concludent.</P> <marg><I>In natural Sci- ences it is not ne- ce$$ary to $eek Ma- thematicall evi- dence.</I></marg> <P>SIMP. It is not nece$$ary in natural Philo$ophy to $eek exqui- $ite Mathematical evidence.</P> <P>SAGR. But this point of motion, is it not a natural que$tion? and yet I cannot find that <I>Ari$totle</I> hath demon$trated any the lea$t accident of it. But let us no longer divert our intended Theme, nor do you fail, I pray you <I>Salviatus,</I> to tell me that which you hinted to me to be the cau$e of the <I>Pendulum's</I> qui- e$cence, be$ides the re$i$tance of the <I>Medium</I> ro penetration.</P> <P>SALV. Tell me; of two <I>penduli</I> hanging at unequal di$tan- ces, doth not that which is fa$tned to the longer threed make its vibrations more $eldome?</P> <marg><I>The</I> pendulum <I>hanging at a long- er threed, maketh its vibrations more $eldome than the</I> pendulum <I>hanging at a $horter threed.</I></marg> <P>SAGR. Yes, if they be moved to equall di$tances from their perpendicularity.</P> <P>SALV. This greater or le$$e elongation importeth nothing at all, for the $ame <I>pendulum</I> alwayes maketh its reciprocations in e- quall times, be they longer or $horter, that is, though the <I>pendulum</I> <marg><I>The vibrations of the $ame</I> pen- dulum <I>are made with the $ame fre- quency, whether they be $mall or great.</I></marg> be little or much removed from its perpendicularity, and if they are not ab$olutely equal, they are in$en$ibly different, as expe- rience may $hew you: and though they were very unequal, yet would they not di$countenance, but favour our cau$e. There- fore let us draw the perpendicular A B [<I>in Fig.</I> 9.] and hang from the point A, upon the threed A C, a plummet C, and another up- on the $ame threed al$o, which let be E, and the threed A C, being removed from its perpendicularity, and then letting go the plum- mets C and E, they $hall move by the arches C B D, E G F, and the plummet E, as hanging at a le$$er di$tance, and withall, as (by what you $aid) le$$e removed, will return back again fa$ter, and make its vibrations more frequent than the plummet C, and therefore $hall hinder the $aid plummet C, from running $o much farther towards the term D, as it would do, if it were free: and thus the plummet E bringing unto it in every vibration continuall <marg><I>The cau$e which impedeth the</I> pen- dulum, <I>and redu- ceth it to re$t.</I></marg> impediment, it $hall finally reduce it to quie$cence. Now the $ame threed, (taking away the middle plummet) is a compo$ition of many grave <I>penduli,</I> that is, each of its parts is $uch a <I>pendu- lum</I> fa$tned neerer and neerer to the point A, and therefore di$po- <foot>$ed</foot> <p n=>207</p> $ed to make its vibrations $ucce$$ively more and more frequent; and con$equently is able to bring a continual impediment to the plummet C; and for a proof that this is $o, if we do but ob$erve the thread A C, we $hall $ee it di$tended not directly, but in an arch; and if in$tead of the thread we take a chain, we $hall di$- cern the effect more per$ectly; and e$pecially removing the gra- <marg><I>The thread or chain to which a</I> pendulum <I>is fa$t- ned, maketh an arch, and doth not $tretch it $elfe $treight out in its vibrations.</I></marg> vity C, to a con$iderable di$tance from the perpendicular A B, for that the chain being compo$ed of many loo$e particles, and each of them of $ome weight, the arches A E C, and A F D, will appear notably incurvated. By rea$on therefore, that the parts of the chain, according as they are neerer to the point A, de$ire to make their vibrations more frequent, they permit not the lower parts of the $aid chain to $wing $o far as naturally they would: and by continual detracting from the vibrations of the plummet C, they finally make it cea$e to move, although the impediment of the air might be removed.</P> <P>SAGR. The books are now come; here take them <I>Simplicius,</I> and find the place you are in doubt of.</P> <P>SIMP. See, here it is where he beginneth to argue again$t the diurnal motion of the Earth, he having fir$t confuted the annual. <I>Motus terræ annuus a$$errere</I> Copernicanos <I>cogit conver$ionem e- ju$dem quotidianam; alias idem terræ Hemi$phærium continenter ad Solem e$$et conver$um obumbrato $emper aver$o. [In Engli$h thus:]</I> The annual motion of the Earth doth compell the <I>Co- pernicans</I> to a$$ert the daily conver$ion thereof; otherwi$e the $ame Hemi$phere of the Earth would be continually turned to- wards the Sun, the $hady $ide being always aver$e. And $o one half of the Earth would never come to $ee the Sun.</P> <P>SALV. I find at the very $ir$t $ight, that this man hath not rightly apprehended the <I>Copernican Hypothe$is,</I> for if he had but taken notice how he alwayes makes the Axis of the terre$trial Globe perpetually parallel to it $elf, he would not have $aid, that one half of the Earth would never $ee the Sun, but that the year would be one entire natural day, that is, that thorow all parts of the Earth there would be $ix moneths day, and $ix moneths night, as it now befalleth to the inhabitants under the Pole, but let this mi$take be forgiven him, and let us come to what remai- neth.</P> <P>SIMP. It followeth, <I>Hanc autem gyrationem Terræ im- po$$ibilem e$$e $ic demon$tramus.</I> Which $peaks in Engli$h thus: That this gyration of the Earth is impo$$ible we thus demon$trate. That which en$ueth is the declaration of the following figure, wherein is delineated many de$cending grave bodies, and a$cend- ing light bodies, and birds that fly too and again in the air, &c.</P> <P>SAGR. Let us $ee them, I pray you. Oh! what fine figures, <foot>what</foot> <p n=>208</p> what birds, what balls, and what other pretty things are here?</P> <P>SIMP. The$e are balls which come from the concave of the Moon.</P> <P>SAGR. And what is this?</P> <P>SIMP. This is a kind of Shell-fi$h, which here at <I>Venice</I> they call <I>buovoli</I>; and this al$o came from the Moons concave.</P> <P>SAGR. Indeed, it $eems then, that the Moon hath a great pow- <marg>* Pe$ci armai, <I>or</I> armati.</marg> er over the$e Oy$ter-fi$hes, which we call ^{*} <I>armed $i$bes.</I></P> <P>SIMP. And this is that calculation, which I mentioned, of this Journey in a natural day, in an hour, in a fir$t minute, and in a $econd, which a point of the Earth would make placed under the Equinoctial, and al$o in the parallel of 48 <I>gr.</I> And then followeth this, which I doubted I had committed $ome mi$take in reciting, therefore let us read it. <I>His po$itis, nece$$e est, terra circulariter mota, omnia ex aëre eidem, &c. Quod $i ha$ce pilas æquales po- nemus pondere, magnitudine, gravitate, & in concavo Sphæræ Lu- naris po$itas libero de$cen$ui permittamus, $i motum deor$um æque- mus celeritate motui circum, (quod tamen $ecus e$t, cum pila A, &c.) elabentur minimum (ut multum cedamus adver$ariis) dies $ex: quo tempore $exies circa terram, &c. [In Engli$b thus.]</I> The$e things being $uppo$ed, it is nece$$ary, the Earth being cir- cularly moved, that all things from the air to the $ame, &c. So that if we $uppo$e the$e balls to be equal in magnitude and gra- vity, and being placed in the concave of the Lunar Sphere, we permit them a free de$cent, and if we make the motion down- wards equal in velocity to the motion about, (which neverthele$s is otherwi$e, if the ball A, &c.) they $hall be falling at lea$t (that we may grant much to our adver$aries) $ix dayes; in which time they $hall be turned $ix times about the Earth, &c.</P> <P>SALV. You have but too faithfully cited the argument of this per$on. From hence you may collect <I>Simplicius,</I> with what cau- tion they ought to proceed, who would give them$elves up to be- lieve others in tho$e things, which perhaps they do not believe them$elves. For me thinks it a thing impo$$ible, but that this Au- thor was advi$ed, that he did de$ign to him$elf a circle, who$e dia- meter (which among$t Mathematicians, is le$$e than one third part of the circumference) is above 72 times bigger than it $elf: an errour that affirmeth that to be con$iderably more than 200, which is le$$e than one.</P> <P>SAGR. It may be, that the$e Mathematical proportions, which are true in ab$tract, being once applied in concrete to Phy$ical and Elementary circles, do not $o exactly agree: And yet, I think, that the Cooper, to find the $emidiameter of the bottom, which he is to fit to the Cask, doth make u$e of the rule of Mathematicians in ab$tract, although $uch bottomes be things meerly material, <foot>and</foot> <p n=>209</p> and concrete: therefore let <I>Simplicius</I> plead in excu$e of this Author; and whether he chinks that the Phy$icks can differ $o very much from the Mathematicks.</P> <P>SIMP. The $ub$tractions are in my opinion in$ufficient to $alve this difference, which is $o extreamly too great to be reconciled: and in this ca$e I have no more to $ay but that, <I>Quandoque bonus dormitet Homerus.</I> But $uppo$ing the calculation of ^{*} <I>Salviatus</I> <marg>* Not <I>Sagre- dus,</I> as the Latine ha hit.</marg> to be more exact, and that the time of the de$cent of the ball were no more than three hours; yet me thinks, that coming from the concave of the Moon, which is $o great a di$tance off, it would be an admirable thing, that it $hould have an in$tinct of maintain- ing it $elf all the way over the $elf-$ame point of the Earth, over which it did hang in its departure thence and not rather be left a very great way behind.</P> <P>SALV. The effect may be admirable, and not admirable, but natural and ordinary, according as the things precedent may fall out. For if the ball (according to the Authors $uppo$itions) whil$t it $taid in the concave of the Moon, had the circular motion of twenty four hours together with the Earth, and with the re$t of the things contained within the $aid Concave; that very vertue which made it turn round before its de$cent, will continue it in the $ame motion in its de$cending. And $o far it is from not keep- ing pace with the motion of the Earth, and from $taying behind, that it is more likely to out-go it; being that in its approaches to the Earth, the motion of gyration is to be made with circles con- tinually le$$er and le$$er; $o that the ball retaining in it $elf that $elf-$ame velocity which it had in the concave, it ought to antici- pate, as I have $aid, the <I>vertigo</I> or conver$ion of the Earth. But if the ball in the concave did want that circulation, it is not obli- ged in de$cending to maintain it $elf perpendicularly over that point of the Earth, which was ju$t under it when the de$cent be- gan. Nor will <I>Copernicus,</I> or any of his followers affirm the $ame.</P> <P>SIMP. But the Author maketh an objection, as you $ee, de- manding on what principle this circular motion of grave and light bodies, doth depend: that is, whether upon an internal or an ex- ternal principle.</P> <P>SALV. Keeping to the Probleme of which we $peak, I $ay, that that very principle which made the ball turn round, whil'$t it was in the Lunar concave, is the $ame that maintaineth al$o the circulation in the de$cent: yet I leave the Author at liberty to make it internal or external at his plea$ure.</P> <P>SIMP. The Author proveth, that it can neither be inward nor outward.</P> <P>SALV. And I will $ay then, that the ball in the concave did <foot>Dd not</foot> <p n=>210</p> not move, and $o he $hall not be bound to $hew how that in de$- cending it continueth all the way vertically over one point, for that it will not do any $uch thing.</P> <P>SIMP. Very well; But if grave bodies, and light can have no principle, either internal or external of moving circularly, than neither can the terre$trial Globe move with a circular motion: and thus you have the intent of the Author.</P> <P>SALV. I did not $ay, that the Earth had no principle, either interne, or externe to the motion of gyration, but I $ay, that I do not know which of the two it hath; and yet my not knowing it hath not a power to deprive it of the $ame; but if this Author can tell by what principle other mundane bodies are moved round, of who$e motion there is no doubt; I $ay, that that which ma- keth the Earth to move, is a vertue, like to that, by which <I>Mars</I> and <I>Jupiter</I> are moved, and wherewith he believes that the $tarry Sphere it $elf al$o doth move; and if he will but a$$ure me, who is the mover of one of the$e moveables, I will undertake to be able to tell him who maketh the Earth to move. Nay more; I will undertake to do the $ame, if he can but tell me, who moveth the parts of the Earth downwards.</P> <P>SIMP. The cau$e of this is mo$t manife$t, and every one knows that it is gravity.</P> <P>SALV. You are out, <I>Simplicius,</I> you $hould $ay, that every one knowes, that it is called Gravity: but I do not que$tion you about the name, but the e$$ence of the thing, of which e$$ence you know not a tittle more than you know the e$$ence of the mover of the $tars in gyration; unle$$e it be the name that hath been put to this, and made familiar, and dome$tical, by the many <marg><I>We know no more who moveth grave bodies downwards; than who moveth the Stars round, nor know we any thing of the$e cau- $es, more than the names impo$ed on them by us.</I></marg> experiences which we $ee thereof every hour in the day,: but not as if we really under$tand any more, what principle or vertue that is which moveth a $tone downwards, than we know who moveth it upwards, when it is $eparated from the projicient, or who mo- veth the Moon round, except (as I have $aid) onely the name, which more particularly and properly we have a$$igned to the mo- tion of de$cent, namely, Gravity; whereas for the cau$e of cir- cular motion, in more general termes, we a$$ign the <I>Vertue impre$- $ed,</I> and call the $ame an <I>Intelligence,</I> either a$$i$ting, or informing; and to infinite other motions we a$cribe Nature for their cau$e.</P> <P>SIMP. It is my opinion, that this Author asketh far le$$e than that, to which you deny to make an$wer; for he doth not ask what is nominally and particularly the principle that moveth grave and light bodies circularly, but what$oever it be, he de$i- reth to know, whether you think it intrin$ecal, or extrin$ecal: For howbeit, <I>v. gr.</I> I do not know, what kind of thing that gravity is, by which the Earth de$cendeth; yet I know that it is an intern <foot>princi-</foot> <p n=>211</p> principle, $eeing that if it be not hindered, it moveth $pontane- ou$ly: and on the contrary, I know that the principle which mo- veth it upwards, is external, although that I do not know, what thing that vertue is, impre$$ed on it by the projicient.</P> <P>SALV. Into how many que$tions mu$t we excurre, if we would decide all the difficulties, which $ucce$$ively have dependance one upon another! You call that an external (and you al$o call it a preternatural and violent) principle, which moveth the grave pro- ject upwards; but its po$$ible that it may be no le$$e interne and natural, than that which moveth it downwards; it may peradven- <marg><I>The vertue which carrieth grave pro- jects upwards, is no le$$e natural to them, than the gravity which mo- veth them down- wards.</I></marg> ture be called external and violent, $o long as the moveable is joy- ned to the projicient; but being $eparated, what external thing remaineth for a mover of the arrow, or ball? In $umme, it mu$t nece$$arliy be granted, that that vertue which carrieth $uch a move- able upwards, is no le$$e interne, than that which moveth it down- wards; and I think the motion of grave bodies a$cending by the <I>impetus</I> conceived, to be altogether as natural, as the motion of de$cent depending on gravity.</P> <P>SIMP. I will never grant this; for the motion of de$cent hath its principle internal, natural, and perpetual, and the motion of a$cent hath its principle externe, violent, and finite.</P> <P>SALV. If you refu$e to grant me, that the principles of the motions of grave bodies downwards and upwards, are equally in- <marg><I>Contrary prin- ciples cannot natu- rally re$ide in the $ame $ubject.</I></marg> ternal and natural; what would you do, if I $hould $ay, that they may al$o be the $ame in number?</P> <P>SIMP. I leave it to you to judge.</P> <P>SALV. But I de$ire you your $elf to be the Judge: Therefore tell me, Do you believe that in the $ame natural body, there may re$ide interne principles, that are contrary to one another?</P> <P>SIMP. I do verily believe there cannot.</P> <P>SALV. What do you think to be the natural inclination of Earth, of Lead, of Gold, and in $um, of the mo$t ponderous mat- ters; that is, to what motion do you believe that their interne principle draweth them?</P> <P>SIMP. To that towards the centre of things grave, that is, to the centre of the Univer$e, and of the Earth, whither, if they be not hindered, it will carry them.</P> <P>SALV. So that, if the Terre$trial Globe were bored thorow, and a Well made that $hould pa$$e through the centre of it, a Cannon bullet being let fall into the $ame, as being moved by a natural and intrin$ick principle, would pa$$e to the centre; and it would make all this motion $pontaneou$ly, and by intrin$ick prin- ciple, is it not $o?</P> <P>SIMP. So I verily believe.</P> <P>SALV. But when it is arrived at the centre, do you think that <foot>Dd 2 it</foot> <p n=>212</p> it will pa$$e any further, or el$e that there it would immediately $tand $till, and move no further?</P> <P>SIMP. I believe that it would continue to move a great way further.</P> <P>SALV. But this motion beyond the centre, would it not be up- wards, and according to your a$$ertion preternatural, and violent? And yet on what other principle do you make it to depend, but only upon the $elf $ame, which did carry the ball to the centre, and which you called intrin$ecal, and natural? Finde, if you can, another external projicient, that overtaketh it again to drive it upwards. And this that hath been $aid of the motion thorow the centre, is al$o $een by us here above; for the interne <I>impetus</I> <marg><I>The natural mo- tion changeth it $elfe into that which is called pre- ternatural and vi- olent.</I></marg> of a grave body falling along a declining $uperficies, if the $aid $uperficies be reflected the other way, it $hall carry it, without a jot interrupting the motion, al$o upwards. A ball of lead that hangeth by a thread, being removed from its perpendicularity, de- $cendeth $pontaneou$ly, as being drawn by its internal inclination, and without any interpo$ure of re$t, pa$$eth beyond the lowe$t point of perpendicularity: and without any additional mover, moveth upwards. I know that you will not deny, but that the principle of grave bodies that moveth them downwards, is no le$s natural, and intrin$ecal, than that principle of light bodies, which moveth them upwards: $o that I propo$e to your con$ideration a ball of lead, which de$cending through the Air from a great al- titude, and $o moving by an intern principle, and comming to a depth of water, continueth its de$cent, and without any other ex- terne mover, $ubmergeth a great way; and yet the motion of de$cent in the water is preternatural unto it; but yet neverthele$s dependeth on a principle that is internal, and not external to the ball. You $ee it demon$trated then, that a moveable may be moved by one and the $ame internal principle, with contrary mo- tions.</P> <P>SIMP. I believe there are $olutions to all the$e objections, though for the pre$ent I do not remember them; but however it be, the Author continueth to demand, on what principle this cir- cular motion of grave and light bodies dependeth; that is, whe- ther on a principle internal, or external; and proceeding for- wards, $heweth, that it can be neither on the one, nor on the other, $aying; <I>Si ab externo; Deu$ne illum excitat per continuum mira- culum? an verò Angelus, an aër? Et hunc quidem multi a$$ig- nant. Sed contra---- [In Engli$h thus]</I> If from an externe prin- ciple; Whether God doth not excite it by a continued Miracle? or an Angel, or the Air? And indeed many do a$$ign this. But on the contrary-----.</P> <P>SALV. Trouble not your $elf to read his argument; for I am <foot>none</foot> <p n=>213</p> none of tho$e who a$cribe that principle to the ambient air. As to the Miracle, or an Angel, I $hould rather incline to this $ide; for that which taketh beginning from a Divine Miracle, or from an Angelical operation; as for in$tance, the tran$portation of a Can- non ball or bullet into the concave of the Moon, doth in all pro- bability depend on the vertue of the $ame principle for perform- ing the re$t. But, as to the Air, it $erveth my turn, that it doth not hinder the circular motion of the moveables, which we did $uppo$e to move thorow it. And to prove that, it $ufficeth (nor is more required) that it moveth with the $ame motion, and fini$h- eth its circulations with the $ame velocity, that the Terre$trial Globe doth.</P> <P>SIMP. And he likewi$e makes his oppo$ition to this al$o; demanding who carrieth the air about, Nature, or Violence? And proveth, that it cannot be Nature, alledging that that is con- trary to truth, experience, and to <I>Copernicus</I> him$elf.</P> <P>SALV. It is not contrary to <I>Copernicus</I> in the lea$t, who writeth no $uch thing; and this Author a$cribes the$e things to him with two exce$$ive courte$ie. It's true, he $aith, and for my part I think he $aith well, that the part of the air neer to the Earth, be- ing rather a terre$trial evaporation, may have the $ame nature, and naturally follow its motion; or, as being contiguous to it, may follow it in the $ame manner, as the Peripateticks $ay, that the $uperiour part of it, and the Element of fire, follow the mo- tion of the Lunar Concave, $o that it lyeth upon them to declare, whether that motion be natural, or violent.</P> <P>SIMP. The Author will reply, that if <I>Copernicus</I> maketh only the inferiour part of the Air to move, and $uppo$eth the upper part thereof to want the $aid motion, he cannot give a rea$on, how that quiet air can be able to carry tho$e grave bodies along with it, and make them keep pace with the motion of the Earth.</P> <P>SALV. <I>Copernicus</I> will $ay, that this natural propen$ion of the <marg><I>The propen$ion of elementary bo- dies to follow the Earth, hath a li- mited Sphere of activity.</I></marg> elementary bodies to $ollow the motion of the Earth, hath a li- mited Sphere, out of which $uch a natural inclination would cea$e; be$ides that, as I have $aid, the Air is not that which carrieth the moveables along with it; which being $eparated from the Earth, do follow its motion; $o that all the objections come to nothing, which this Author produceth to prove, that the Air cannot cau$e $uch effects.</P> <P>SIMP. To $hew therefore, that that cannot be, it will be nece$- $ary to $ay, that $uch like effects depend on an interne principle, again$t which po$ition, <I>oboriuntur difficillimæ, immò inextricabiles quæ$tiones $ecundæ,</I> of which $ort are the$e that follow. <I>Princi- pium illud internum vel e$t accidens, vel $ub$tantia. Si primum; quale nam illud? nam qualitas locomotiva circum, hactenus nulla</I> <foot><I>vide-</I></foot> <p n=>214</p> <I>videtur agnita. (In Engli$h thus:)</I> Contrary to which po$ition there do ari$e mo$t difficult, yea inextricable $econd que$tions, $uch as the$e; That intern principle is either an accident, or a $ub$tance. If the fir$t; what manner of accident is it? For a locomotive quality about the centre, $eemeth to be hitherto ac- knowledged by none.</P> <P>SALV. How, is there no $uch thing acknowledged? Is it not known to us, that all the$e elementary matters move round, to- gether with the Earth? You $ee how this Author $uppo$eth for true, that which is in que$tion.</P> <P>SIMP. He $aith, that we do not $ee the $ame; and me thinks, he hath therein rea$on on his $ide.</P> <P>SALV. We $ee it not, becau$e we turn round together with them.</P> <P>SIMP. Hear his other Argument. <I>Quæ etiam $i e$$et, quo- modo tamen inveniretur in rebus tam contrariis? in igne, ut in a- quâ; in aëre, ut in terra; in viventibus, ut in anima carentibus? [in Engli$h thus:]</I> Which although it were, yet how could it be found in things $o contrary? in the fire, as in the water? in the air, as in the earth? in living creatures, as in things wanting life?</P> <P>SALV. Suppo$ing for this time, that water and fire are contra- ries; as al$o the air and earth; (of which yet much may be $aid) the mo$t that could follow from thence would be, that tho$e mo- tions cannot be common to them, that are contrary to one ano- ther: $o that <I>v. g.</I> the motion upwards, which naturally agreeth to fire, cannot agree to water; but that, like as it is by nature con- trary to fire: $o to it that motion $uiteth, which is contrary to the motion of fire, which $hall be the motion <I>deor$ùm</I>; but the cir- cular motion, which is not contrary either to the motion <I>$ur$ùm,</I> or to the motion <I>deor$ùm,</I> but may mix with both, as <I>Aristotle</I> him$elf affirmeth, why may it not equally $uit with grave bodies and with light? The motions in the next place, which cannot be common to things alive, and dead, are tho$e which depend on the $oul: but tho$e which belong to the body, in as much as it is ele- mentary, and con$equently participateth of the qualities of the e- lements, why may not they be common as well to the dead corps, as to the living body? And therefore, if the circular motion be proper to the elements, it ought to be common to the mixt bodies al$o.</P> <P>SAGR. It mu$t needs be, that this Author holdeth, that a dead cat, falling from a window, it is not po$$ible that a live cat al$o could fall; it not being a thing convenient, that a carca$e $hould partake of the qualities which $uit with things alive.</P> <P>SALV. Therefore the di$cour$e of this Author concludeth <foot>nothing</foot> <p n=>215</p> nothing again$t one that $hould affirm, that the principle of the cir- cular motions of grave and light bodies is an intern accident: I know not how he may prove, that it cannot be a $ub$tance.</P> <P>SIMP. He brings many Arguments again$t this. The fir$t of which is in the$e words: <I>Si $ecundum (nempè, $i dieas tale princi- pium e$$e $ub$tantiam) illud e$t aut materia, aut forma, aut compo- $itum. Sed repugnant iterum tot diver$æ rerum naturæ, quales $unt aves, limaces, $axa, $agittæ, nives, fumi, grandines, pi$ces, &c. quæ tamen omnia $pecie & genere differentia, moverentur à naturâ $uâ circulariter, ip$a naturis diver$i$$ima, &c. [In Engli$h thus]</I> If the $econd, (that is, if you $hall $ay that this principle is a $ub$tance) it is either matter, or form, or a compound of both. But $uch diver$e natures of things are again repugnant, $uch as are birds, $nails, $tones, darts, $nows, $moaks, hails, fi$hes, &c. all which notwith$tanding their differences in $pecies and kind, are moved of their own nature circularly, they being of their natures mo$t different, &c.</P> <P>SALV. If the$e things before named are of diver$e natures, and things of diver$e natures cannot have a motion in common, it mu$t follow, if you would give $atisfaction to all, that you are to think of, more than two motions onely of upwards and downwards: and if there mu$t be one for the arrows, another for the $nails, another for the $tones, and another for fi$hes; then are you to bethink your $elf of worms, topazes and mu$hrums, which are not le$s different in nature from one another, than $now and hail.</P> <P>SIMP. It $eems that you make a je$t of the$e Arguments.</P> <P>SALV. No indeed, <I>Simplicius,</I> but it hath been already an- $wered above, to wit, that if one motion, whether downwards or upwards, can agree with all tho$e things afore named, a circular motion may no le$s agree with them: and as you are a <I>Peripate- tick,</I> do not you put a greater difference between an elementary comet and a celeftial $tar, than between a fi$h and a bird? and yet both tho$e move circularly. Now propo$e your $econd Ar- gument.</P> <P>SIMP. <I>Si terra $taret per voluntatem Dei, rotaréntne cætera, an non? $i hoc, fal$um e$t à naturâ gyrare; $i illud, redeunt priores quæ$tiones. Et $anè mirum e$$et, quòd Gavia pi$ciculo, Alauda nidulo $uo, & corvus limaci, petraque, etiam volans, imminere non po$$et. [Which I thus render</I>:] If the Earth be $uppo$ed to $tand $till by the will of God, $hould the re$t of bodies turn round or no? If not, then it's fal$e that they are revolved by nature; if the other, the former que$tions will return upon us. And truly it would be $trange that the Sea-pie $hould not be able to hover over the $mall fi$h, the Lark over her ne$t, and the Crow o- ver the $nail and rock, though flying.</P> <foot>SALV.</foot> <p n=>216</p> <P>SALV. I would an$wer for my $elf in general terms, that if it were appointed by the will of God, that the Earth $hould cea$e from its diurnal revolution, tho$e birds would do what ever $hould plea$e the $ame Divine will. But if this Author de$ire a more particular an$wer, I $hould tell him, that they would do quite con- trary to what they do now, if whil$t they, being $eparated from the Earth, do bear them$elves up in the air, the Terre$trial Globe by the will of God, $hould all on a $udden be put upon a precipi- tate motion; it concerneth this Author now to a$certain us what would in this ca$e $ucceed.</P> <P>SAGR. I pray you, <I>Salviatus,</I> at my reque$t to grant to this Author, that the Earth $tanding $till by the will of God, the other things, $eparated from it, would continue to turn round of their own natural motion, and let us hear what impo$$ibilities or incon- veniences would follow: for I, as to my own particular, do not $ee how there can be greater di$orders, than the$e produced by the Author him$elf, that is, that Larks, though they $hould flie, could not be able to hover over their ne$ts, nor Crows over $nails, or rocks: from whence would follow, that Crows mu$t $uffer for want of $nails, and young Larks mu$t die of hunger, and cold, not being able to be fed or $heltered by the wings of the old ones. This is all the ruine that I can conceive would follow, $uppo$ing the Authors $peech to be true. Do you $ee, <I>Simplicius,</I> if grea- ter inconveniences would happen?</P> <P>SIMP. I know not how to di$cover greater; but it is very cre- dible, that the Author be$ides the$e, di$covered other di$orders in Nature, which perhaps in reverend re$pect of her, he was not will- ing to in$tance in. Therefore let us proceed to the third Obje- ction. <I>In$uper quî fit, ut istæ res tam variæ tantùm moveantur ab Occa$u in Ortum, parallelæ ad Æquatorem? ut $emper movean- tur, nunquam quie$cant? [which $peaks to this $en$e:]</I> Moreover, how comes it to pa$s that the$e things, $o diver$e, are onely moved from the We$t towards the Ea$t, parallel to the Æquinoctial? that they always move, and never re$t?</P> <P>SALV. They move from We$t to Ea$t parallel to the Æqui- noctial without cea$ing, in the $ame manner as you believe the fixed $tars to move from Ea$t to We$t, parallel to the Æquinocti- al, without ever re$ting.</P> <P>SIMP. <I>Quarè, quò $unt altiores, celeriùs; quò humiliores, tar- diùs? (i. e.)</I> Why are the higher the $wifter, and the lower the $lower?</P> <P>SALV. Becau$e that in a Sphere or circle, that turns about up- on its own centre, the remoter parts de$cribe greater circuits, and the parts nearer at hand de$cribe le$$er in the $ame time.</P> <P>SIMP. <I>Quare, quæ Æquinoctiali propriores, in majori; quæ</I> <foot><I>remotiores,</I></foot> <p n=>217</p> <I>remotiores, in minori circulo feruntur? [$cilicet:]</I> Why are tho$e near the Æquinoctial carried about in a greater circle, and tho$e which are remote in a le$$er?</P> <P>SALV. To imitate the $tarry Sphere, in which tho$e neare$t to the Æquinoctial, move in greater circles, than the more re- mote.</P> <P>SIMP. <I>Quarè Pila eadem $ub Æquinoctiali tota circa centrum terr æ, ambitu maximo, celeritate incredibili; $ub Polo verò circa centrum proprium, gyro nullo, tarditate $upremâ volveretur? [That is:]</I> Why is the $ame ball under the Æquinoctial wholly turned round the centre of the Earth in the greate$t circumfe- rence, with an incredible celerity; but under the Pole about its own centre, in no circuite, but with the ultimate degree of tar- dity?</P> <P>SALV. To imitate the $tars of the Firmament, that would do the like if they had the diurnal motion.</P> <P>SIMP. <I>Quare eadem res, pila v. g. plumbea, $i $emel terram circuivit, de$cripto circulo maximo, eandem ubique non circum- migret $ecundùm circulum maximum, $ed tran$lata extra Æquino- ctialem in circulis minoribus agetur? [Which $peaketh thus:]</I> Why doth not the $ame thing, as for example, a ball of lead turn round every where according to the $ame great circle, if once de$cribing a great circle, it hath incompa$$ed the Earth, but being removed from the Æquinoctial, doth move in le$$er circles?</P> <P>SALV. Becau$e $o would, nay, according to the doctrine of <I>Ptolomey,</I> $o have $ome fixed $tars done, which once were very near the Æquinoctial, and de$cribed very va$t circles, and now that they are farther off, de$cribe le$$er.</P> <P>SAGR. If I could now but keep in mind all the$e fine no- tions, I $hould think that I had made a great purcha$e; I mu$t needs intreat you, <I>Simplicius,</I> to lend me this Book, for there can- not chu$e but be a $ea of rare and ingenious matters contained in it.</P> <P>SIMP. I will pre$ent you with it.</P> <P>SAGR. Not $o, Sir; I would not deprive you of it: but are the Queries yet at an end?</P> <P>SIMP. No Sir; hearken therefore. <I>Si latio circularis gra- vibus & levibus e$t naturalis, qualis e$t ea quæ fit $ecundùm line- am rectam? Nam $i naturalis, quomodo & is motus qui circum est, naturalis e$t, cùm $pecie differat à recto? Si violentus, quî fit, ut mi$$ile ignitum $ur$ùm evolans $cintillo$um caput $ur$ùm à terrâ, non autem circum volvatur, &c. [Which take in our idiom:]</I> If a circular lation is natural to heavy and light things, what is that which is made according to a right line? For if it be natural, how then is that motion which is about the centre natural, $eeing it <foot>Ee differs</foot> <p n=>218</p> differs in $pecies from a right motion? If it be violent, how is it that a fiery dart flying upwards, $parkling over our heads at a di- $tance from the Earth, but not turning about, <I>&c.</I></P> <marg><I>Of the mixt mo- <*>ion we $ee not the <*>art that is circu- <*>ar, becau$e we <*>artake thereof.</I></marg> <P>SALV. It hath been $aid already very often, that the circular motion is natural to the whole, and to its parts, whil$t they are in perfect di$po$ure, and the right is to reduce to order the parts di$ordered; though indeed it is better to $ay, that neither the parts ordered or di$ordered ever move with a right motion, but with one mixed, which might as well be averred meerly circular: but to us but one part onely of this motion is vi$ible and ob$er- vable, that is, the part of the right, the other part of the circular being imperceptible to us, becau$e we partake thereof. And this an$wers to the rays which move upwards, and round about, but we cannot di$tingui$h their circular motion, for that, with that we our $elves move al$o. But I believe that this Author never thought of this mixture; for you may $ee that he re$olutely $aith, that the rays go directly upwards, and not at all in gyration.</P> <P>SIMP. <I>Quare centrum $phære delap$æ $ub Æquatore $piram de- $cribit in ejus plano: $ub aliis parallelis $piram de$cribit in cono? $ub Polo de$cendit in axe lineam gyralem, decurrens in $uperficie cylindricâ con$ignatam</I>? (In Engli$h to this purpo$e:) Why doth the centre of a falling Globe under the Æquinoctial de$cribe a $piral line in the plane of the Æquator; and in other parallels a $piral about a Cone; and under the Pole de$cend in the axis de$cribing a gyral line, running in a Cylindrical Super$i- cies?</P> <P>SALV. Becau$e of the lines drawn from the Centre to the cir- cumference of the $phere, which are tho$e by which <I>graves</I> de- fcend, that which terminates in the Æquinoctial de$igneth a cir- cle, and tho$e that terminate in other parallels de$cribe conical $uperficies; now the axis de$cribeth nothing at all, but continueth in its own being. And if I may give you my judgment freely, I will $ay, that I cannot draw from all the$e Queries, any $en$e that interfereth with the motion of the Earth; for if I demand of this Author, (granting him that the Earth doth not move) what would follow in all the$e particulars, $uppo$ing that it do move, as <I>Co- pernicus</I> will have it; I am very confident, that he would $ay that all the$e effects would happen, that he hath objected, as inconve- niences to di$prove its mobility: $o that in this mans opinion ne- ce$$ary con$equences are accounted ab$urdities: but I be$eech you, if there be any more, di$patch them, and free us $peedily from this weari$om task.</P> <P>SIMP. In this which follows he oppo$es <I>Copernicus</I> & his Sectators, who affirm, that the motion of the parts $eparated from their whole, is onely to unite them$elves to their whole; but that the moving <foot>circularly</foot> <p n=>219</p> circularly along with the vertigenous diurnal revolution is ab$o- lutely natural: again$t which he objecteth, $aying, that according to the$e mens opinion; <I>Si tota terra, unà cum aquâ in nihilum redigeretur, nulla grando aut pluvia è nube decideret, $ed natu- raliter tantùm circumferetur, neque ignis ullus, aut igneum a$cen- deret, cùm illorum non improbabili $ententià ignis nullus $it $uprà.</I> [Which I tran$late to this $en$e:] If the whole Earth, together with the Water were reduced into nothing, no hail or rain would fall from the clouds, but would be onely naturally carried round; neither any fire or fiery thing would a$cend, $eeing to the$e that men it is no improbable opinion that there is no fire above.</P> <P>SALV. The providence of this Philo$opher is admirable, and worthy of great applau$e, for he is not content to provide for things that might happen, the cour$e of Nature continuing, but will $hew hic care in what may follow from tho$e things that he very well knows $hall never come to pa$s. I will grant him there- fore, (that I may get $om pretty pa$$ages out of him) that if the Earth and Water $hould be reduced to nothing, there would be no more hails or rains, nor would igneal matters a$cend any longer upwards, but would continually turn round: what will follow? what will the Philo$opher $ay then?</P> <P>SIMP. The objection is in the words which immediately fol- low; here they are: <I>Quibus tamen experientia & ratio adver- $atur.</I> Which neverthele$s ($aith he) is contrary to experience and rea$on.</P> <P>SALV. Now I mu$t yield, $eeing he hath $o great an advan- tage of me as experience, of which I am unprovided. For as yet I never had the fortune to $ee the Terre$trial Globe and the ele- ment of Water turn'd to nothing, $o as to have been able to ob- $erve what the hail and water did in that little Chaos. But he perhaps tells us for our in$truction what they did.</P> <P>SIMP. No, he doth not.</P> <P>SALV. I would give any thing to change a word or two with this per$on, to ask him, whether when this Globe vani$hed, it car- ried away with it the common centre of gravity, as I believe it did; in which ca$e, I think that the hail and water would remain in$en- $ate and $tupid among$t the clouds, without knowing what to do with them$elves. It might be al$o, that attracted by that great void <I>Vacuum,</I> left by the Earths ab$enting, all the ambients would be rarified, and particularly, the air, which is extreme ea$ily drawn, and would run thither with very great ha$te to fill it up. And perhaps the more $olid and material bodies, as birds, (for there would in all probability be many of them $cattered up and down in the air) would retire more towards the centre of the great va- cant $phere; (for it $eemeth very rea$onable, that $ub$tances that <foot>Ee 2 under</foot> <p n=>220</p> under $mall bulk contain much matter, $hould have narrower pla- ces a$$igned them, leaving the more $pacious to the more rarified) and there being dead of hunger, and re$olved into Earth, would form a new little Globe, with that little water, which at that time was among the clouds. It might be al$o, that tho$e matters as not beholding the light, would not perceive the Earths departure, but like blind things, would de$cend according to their u$ual cu$tom to the centre, whither they would now go, if that globe did not hinder them. And la$tly, that I may give this Philo$opher a le$s irre$olute an$wer, I do tell him, that I know as much of what would follow upon the annihilation of the Terre$trial Globe, as he would have done that was to have followed in and about the $ame, before it was created. And becau$e I am certain he will $ay, that he would never have been able to have known any of all tho$e things which experience alone hath made him knowing in, he ought not to deny me pardon, and to excu$e me if I know not that which he knows, touching what would en$ue upon the annihilation of the $aid Globe: for that I want that experience which he hath. Let us hear if he have any thing el$e to $ay.</P> <P>SIMP. There remains this figure, which repre$ents the Terre- $trial Globe with a great cavity about its centre, full of air; and to $hew that <I>Graves</I> move not downwards to unite with the Ter- re$trial Globe, as <I>Copernicus</I> $aith, he con$tituteth this $tone in the centre; and demandeth, it being left at liberty, what it would do; and he placeth another in the $pace of this great vacuum, and asketh the $ame que$tion. Saying, as to the fir$t: <I>Lapis in centro con$titutus, aut a$cendet ad terram in punctum aliquod, aut non. Si $ecundum; fal$um est, partes ob $olam $ejunctionem à toto, ad il- lud moveri. Si primum; omnis ratio & experientia renititur, neque gravia in $uœ gravitatis centro conquie$cent. Item $i $u- $pen$us lapis, liberatus decidat in centrum, $eparabit $e à toto, con- tra</I> Copernicum<I>: $i pendeat, refragatur omnis experientia, cùm videamus integros fornices corruere.</I> (Wherein he $aith:) The $tone placed in the centre, either a$cendeth to the Earth in $ome point, or no. If the $econd, it is fal$e that the parts $eparated from the whole, move unto it. If the fir$t; it contradicteth all rea$on and experience, nor doth the grave body re$t in the centre of its gravity. And if the $tone being $u$pended in the air, be let go, do de$cend to the centre, it will $eparate from its whole, con- trary to <I>Copernicus:</I> if it do hang in the air, it contradicteth all experience: $ince we $ee whole Vaults to fall down.</P> <P>SALV. I will an$wer, though with great di$advantage to my $elf, $eeing I have to do with one who hath $een by experience, what the$e $tones do in this great Cave: a thing, which for my part I have not $een; and will $ay, that things grave have an exi- <foot>$tence</foot> <p n=>221</p> $tence before the common centre of gravity: $o that it is not one <marg><I>Things grave are before the centre of gravity.</I></marg> centre alone, which is no other than indivi$ible point, and therefore of no efficacie, that can attract unto it grave matters; but that tho$e matters con$piring naturally to unite, form to them$elves a com- mon centre, which is that about which parts of equal moment con$i$t: $o that I hold, that if the great aggregate of grave bo- <marg><I>The great ma$s of grave bodies be- ing transferred out of their place, the $eparated parts would follow that maß.</I></marg> dies were gathered all into any one place, the $mall parts that were $eparated from their whole, would follow the $ame, and if they were not hindered, would penetrate wherever they $hould find parts le$s grave than them$elves: but coming where they $hould meet with matters more grave, they would de$cend no farther. And therefore I hold, that in the Cave full of air, the whole Vault would pre$s, and violently re$t it $elf onely upon that air, in ca$e its hardne$s could not be overcome and broken by its gravity; but loo$e $tones, I believe, would de$cend to the centre, and not $wim above in the air: nor may it be $aid, that they move not to their whole, though they move whither all the parts of the whole would transfer them$elves, if all impediments were removed.</P> <P>SIMP. That which remaineth, is a certain Errour which he ob- $erveth in a Di$ciple of <I>Copernicus,</I> who making the Earth to move with an annual motion, and a diurnal, in the $ame manner as the Cart-wheel moveth upon the circle of the Earth, and in it $elf, did con$titute the Terre$trial Globe too great, or the great Orb too little; for that 365 revolutions of the Æquinoctial, are le$s by far than the circumference of the great Orb.</P> <P>SALV. Take notice that you mi$take, and tell us the direct contrary to what mu$t needs be written in that Book; for you $hould $ay, that that $ame <I>Copernican</I> Author did con$titute the Terre$trial Globe too little, and the great Orb too big; and not the Terre$trial Globe too big, and the annual too little.</P> <P>SIMP. The mi$take is not mine; $ee here the words of the Book. <I>Non videt, quòd vel circulum annuum æquo minorem, vel orbem terreum ju$to multò fabricet majorem.</I> (In Engli$h thus:) He $eeth not, that he either maketh the annual circle equal to the le$s, or the Terre$trial Orb much too big.</P> <P>SALV. I cannot tell whether the fir$t Author erred or no, $ince the Author of this Tractate doth not name him; but the error of this Book is certain and unpardonable, whether that follower of <I>Copernicus</I> erred or not erred; for that your Author pa$$eth by $o material an error, without either detecting or correcting it. But let him be forgiven this fault, as an error rather of inadvertencie, than of any thing el$e: Farthermore, were it not, that I am al- ready wearied and tired with talking and $pending $o mnch time with very little profit, in the$e frivolous janglings and alterca- tions, I could $hew, that it is not impo$$ible for a circle, though <foot>no</foot> <p n=>222</p> <marg><I>It is not impo$$i- ble with the cir- cumference of a $mall circle few times revolved to mea$ure and de- $cribe a line bigger than any great cir- cle what $oever.</I></marg> no bigger than a Cart-wheel, with making not 365, but le$$e than 20 revolutions, to de$cribe and mea$ure the circumference, not onely of the grand Orb, but of one a thou$and times greater; and this I $ y to $hew, that there do not want far greater $ubtil- ties, than this wherewith your Author goeth about to detect the errour of <I>Copernicus</I>: but I pray you, let us breath a little, that $o we may proceed to the other Philo$opher, that oppo$eth of the $ame <I>Copernicus.</I></P> <P>SAGR. To confe$$e the truth, I $tand as much in need of re- $pite as either of you; though I have onely wearied my eares: and were it not that I hope to hear more ingenious things from this other Author, I que$tion whether I $hould not go my ways, to <marg>Gondola.</marg> take the air in my ^{*} Plea$ure-boat.</P> <P>SIMP. I believe that you will hear things of greater moment; for this is a mo$t accompli$hed Philo$opher, and a great Mathema- tician, and hath confuted <I>Tycho</I> in the bu$ine$$e of the Comets, and new Stars.</P> <marg>* The name of the <I>Author</I> is <I>Sci- pie Claramontius.</I></marg> <P>SALV. Perhaps he is the $ame with the Author of the Book, called <I>Anti-Tycho</I>?</P> <P>SIMP. He is the very $ame: but the confutation of the new Stars is not in his <I>Anti-Tycho,</I> onely $o far as he proveth, that they were not prejudicial to the inalterability and ingenerability of the Heavens, as I told you before; but after he had publi$hed his <I>Anti-Tycho,</I> having found out, by help of the Parallaxes, a way to demon$trate, that they al$o are things elementary, and contained within the concave of the Moon, he hath writ this other Book, <I>de tribus uovis Stellis, &c.</I> and therein al$o in$erted the Argu- ments again$t <I>Copernicus</I>: I have already $hewn you what he harh written touching the$e new Stars in his <I>Anti-Tycho,</I> where he denied not, but that they were in the Heavens; but he proved, that their production altered not the inalterability of the Heavens, and that he did, with a Di$cour$e purely philo$ophical, in the $ame man ner as you have already heard. And I then forgot to tell you, how that he afterwards did finde out a way to remove them out of the Heavens; for he proceeding in this confutation, by way of com- putations and parallaxes, matters little or nothing at all under- $tood by me, I did not mention them to you, but have bent all my $tudies upon the$e arguments again$t the motion of the Earth, which are purely natural.</P> <P>SALV. I under$tand you very well: and it will be convenient after we have heard what he hath to $ay again$t <I>Copernicus,</I> that we hear, or $ee at lea$t the manner wherewith he, by way of Pa- rallaxes, proveth tho$e new $tars to be elementary, which $o many famous A$tronomers con$titute to be all very high, and among$t the $tars of the Firmament; and as this Author accompli$heth $uch <foot>an</foot> <p n=>223</p> an enterprize of pulling the new $tars out of heaven, and placing them in the elementary Sphere, he $hall be worthy to be highly exalted, and transferred him$elf among$t the $tars, or at lea$t, that his name be by fame eternized among$t them. Yet before we enter upon this, let us hear what he alledgeth again$t the opinion of <I>Copernicus,</I> and do you begin to recite his Arguments.</P> <P>SIMP. It will not be nece$$ary that we read them <I>ad verbum,</I> becau$e they are very prolix; but I, as you may $ee, in reading them $everal times attentively, have marked in the margine tho$e words, wherein the $trength of his arguments lie, and it will $uffice to read them. The $ir$t Argument beginneth here. <I>Et</I> <marg><I>The opinion of</I> Copernicus <I>over- throws the</I> Crite- rium <I>of Philo$ophy</I></marg> <I>primo, $i opinio Copernici recipiatur, Criterium naturalis Philo- $ophiæ, ni pror$us tollatur, vehementer $altem labefactari videtur.</I> [In our Idiom thus] And fir$t, if <I>Copernicus</I> his opinion be imbraced, the <I>Criterium</I> of natural Philo$ophy will be, if not wholly $ubverted, yet at lea$t extreamly $haken.</P> <P>Which, according to the opinion of all the $ects of Philo$ophers requireth, that Sen$e and Experience be our guides in philo$opha- ting: But in the <I>Copernican</I> po$ition the Sen$es are greatly delu- ded, whil'$t that they vi$ibly di$cover neer at hand in a pure <I>Medi- um,</I> the grave$t bodies to de$cend perpendicularly downwards, ne- ver deviating a $ingle hairs breadth from rectitude; and yet accor- ding to the opinion of <I>Copernicus,</I> the $ight in $o manife$t a thing is deceived, and that motion is not reall $traight, but mixt of right and circular.</P> <P>SALV. This is the fir$t argument, that <I>Ari$totle, Ptolomy,</I> and all their followers do produce; to which we have abundant- ly an$wered, and $hewn the Paralogi$me, and with $ufficient plainne$$e proved, that the motion in common to us and other mo- veables, is, as if there were no $uch thing; but becau$e true con- clu$ions meet with a thou$and accidents, that confirme them, I <marg><I>Common motion is, as if it never were.</I></marg> will, with the favour of this Philo$opher, adde $omething more; and you <I>Simplicius</I> per$onating him, an$wer me to what I $hall ask you: And fir$t tell me, what effect hath that $tone upon you, <marg><I>The argument taken from things falling perpendicu- larly, another way confuted.</I></marg> which falling from the top of the Tower, is the cau$e that you per- ceive that motion; for if its fall doth operate upon you neither more nor le$$e, than its $tanding $till on the Towers top, you doubtle$$e could not di$cern its de$cent, or di$tingui$h its moving from its lying $till.</P> <P>SIMP. I comprehend its moving, in relation to the Tower, for that I $ee it one while ju$t again$t $uch a mark in the $aid Tower, and another while again$t another lower, and $o $ucce$- $ively, till that at la$t I perceive it arrived at the ground.</P> <P>SALV. Then if that $tone were let fall from the tallons of an Eagle flying, and $hould de$cend thorow the $imple invi$ible Air, <foot>and</foot> <p n=>224</p> and you had no other object vi$ible and $table, wherewith to make compari$ons to that, you could not perceive its motion?</P> <P>SIMP. No, nor the $tone it $elf; for if I would $ee it, when <marg><I>Whence the mo- tion of a cadent bo- dy is collected.</I></marg> it is at the highe$t, I mu$t rai$e up my head, and as it de$cendeth I mu$t hold it lower and lower, and in a word, mu$t continually move either that, or my eyes, following the motion of the $aid $tone.</P> <P>SALV. You have now rightly an$wered: you know then that <marg><I>The motion of the eye argueth the motion of the object looked on.</I></marg> the $tone lyeth $till, when without moving your eye, you alwayes $ee it before you; and you know that it moveth, when for the keeping it in $ight, you mu$t move the organ of $ight, the eye. So then when ever without moving your eye, you continually be- hold an object in the $elf $ame a$pect, you do always judge it immoveable.</P> <P>SIMP. I think it mu$t needs be $o.</P> <P>SALV. Now fancy your $elf to be in a $hip, and to have fixed your eye on the point of the Sail-yard: Do you think, that be- cau$e the $hip moveth very fa$t, you mu$t move your eye, to keep your $ight alwayes upon the point of the Sail-yard, and to fol- low its motion?</P> <P>SIMP. I am certain, that I $hould need to make no change at all; and that not only in the $ight; but if I had aimed a Musket at it, I $hould never have need, let the $hip move how it will, to $tir it an hairs breadth to keep it full upon the $ame.</P> <P>SALV. And this happens becau$e the motion, which the Ship conferreth on the Sail-yard, it conferreth al$o upon you, and upon your eye; $o that you need not $tir it a jot to behold the top of the Sail-yard: and con$equently, it will $eem to you immovea- able. Now this Di$cour$e being applied to the revolution of the Earth, and to the $tone placed in the top of the Tower, in which you cannot di$cern any motion, becau$e that you have that mo- tion which is nece$$ary for the following of it, in common with it from the Earth; $o that you need not move your eye. When a- gain there is conferred upon it the motion of de$cent, which is its particular motion, and not yours, and that it is intermixed with the circular, that part of the circular which is common to the $tone, and to the eye, continueth to be imperceptible, and the right one- ly is perceived, for that to the perception of it, you mu$t follow it with your eye, looking lower and lower. I wi$h for the undecei- ving of this Philo$opher, that I could advi$e him, that $ome time <marg><I>An experiment that $heweth how the common motion is imperceptible.</I></marg> or other going by water, he would carry along with him a Ve$$el of rea$onable depth full of water, and prepare a ball of wax, or other matter that would de$cend very $lowly to the bottome, $o that in a minute of an hour, it would $carce $ink a yard; and that rowing the boat as fa$t as could be, $o that in a minute of an hour <foot>it</foot> <p n=>225</p> it $hould run above an hundred yards, he would let the ball $ub- merge into the water, & freely de$cend, & diligently ob$erve its mo- tion. If he would but do thus, he $hould $ee, fir$t, that it would go in a direct line towards that point of the bottom of the ve$$el, whither it would tend, if the boat $hould $tand $till; & to his eye, and in rela- tion to the ve$$el, that motion would appear mo$t $traight and per- pendicular, and yet he could not $ay, but that it would be compo$ed of the right motion downwards, and of the circular about the ele- ment of water. And if the$e things befall in matters not natural, and in things that we may experiment in their $tate of re$t; & then again in the contrary $tate of motion, and yet as to appearance no diver$ity at all is di$covered, & that they $eem to deceive our $en$e what can we di$tingui$h touching the Earth, which hath been per- petually in the $ame con$titution, as to motion and re$t? And in what time can we experiment whether any difference is di$cernable among$t the$e accidents of local motion, in its diver$e $tates of mo- tion and re$t, if it eternally indureth in but one onely of them?</P> <P>SAGR. The$e Di$cour$es have $omewhat whetted my $tomack, which tho$e fi$hes, and $nails had in part nau$eated; and the former made me call to minde the correction of an errour, that hath $o much appearance of truth, that I know not whether one of a thou$and would refu$e to admit it as unque$tionable. And it was this, that $ailing into <I>Syria,</I> and carrying with me a very good <I>Tele$cope,</I> that had been be$towed on me by our <I>Common Friend,</I> who not many dayes before had invented, I propo$ed to the Ma- riners, that it would be of great benefit in Navigation to make u$e of it upon the round top of a $hip, to di$cover and kenne Ve$$els afar off. The benefit was approved, but there was objected the <marg><I>An ingenuous con$ideration a- bout the po$$ibility of u$ing the</I> Tele$- cope <I>with as much facility on the round top of the Ma$t of a $hip, as on the Deck.</I></marg> difficulty of u$ing it, by rea$on of the Ships continual fluctuation; and e$pecially on the round top, where the agitation is $o much greater, and that it would be better for any one that would make u$e thereof to $tand at the Partners upon the upper Deck, where the to$$ing is le$$e than in any other place of the Ship. I (for I will not conceal my errour) concurred in the $ame opinion, and for that time $aid no more: nor can I tell you by what hints I was moved to return to ruminate with my $elf upon this bu$ine$$e, and in the end came to di$cover my $implicity (although excu$able) in admitting that for true, which is mo$t fal$e; fal$e I $ay, that the great agitation of the basket or round top, in compari$on of the $mall one below, at the partners of the Ma$t, $hould render the u$e of the <I>Tele$cope</I> more difficult in finding out the object.</P> <P>SALV. I $hould have accompanied the Mariners, and your $elf at the beginning.</P> <P>SIMP. And $o $hould I have done, and $till do: nor can I be- lieve, if I $hould think of it an hundred years, that I could under- $tand it otherwi$e.</P> <foot>Ff SAGR.</foot> <p n=>226</p> <P>SAGR. I may then, it $eems, for once prove a Ma$ter to you both. And becau$e the proceeding by interrogatories doth in my opinion much dilucidate things, be$ides the plea$ure which it affords of con- founding our companion, forcing from him that which he thought he knew not, I will make u$e of that artifice. And fir$t, I $uppo$e that the Ship, Gally, or other Ve$$el, which we would di$cover, is a great way off, that is, four, $ix, ten, or twenty ^{*} miles, for that to kenne tho$e <marg>* I deviate here from the $trict Sea Diallect, which denominatesall di- $tances by Leagues.</marg> neer at hand there is no need of the$e Gla$$es: & con$equently, the <I>Tele$cope</I> may at $uch a di$tance of four or $ix miles conveniently di$cover the whole Ve$$el, & a muchgreater bulk. Now I demand what for $pecies, & how many for number are the motions that are made upon the round top, depending on the fluctuation of the Ship.</P> <P>SALV. We will $uppo$e that the Ship goeth towards the Ea$t. Fir$t, in a calme Sea, it would have no other motion than <marg><I>Different moti- ons depending on the fluctuation of the Ship.</I></marg> this of progre$$ion, but adding the undulation of the Waves, there $hall re$ult thence one, which alternately hoy$ting and low- ering the poop and prow, maketh the round top, to lean forwards and backwards; other waves driving the ve$$el $idewayes, bow the Ma$t to the Starboard and Larboard; others, may bring the $hip $omewhat abovt, and bear her away by the Mi$ne from Ea$t, one <marg>* <I>Greco,</I> which the Latine Tran- $lator according to his u$ual carele$$e- ne$$e (to call it no wor$e) tran$lates <I>Corum Ventum,</I> the Northwe$t Wind, for <I>Ventum Libanotum.</I></marg> while towards the ^{*} Northea$t; another while toward the South- ea$t; others bearing her up by the Carine may make her onely to ri$e, and fall; and in $um, the$e motions are for $pecies two, one that changeth the direction of the <I>Tele$cope</I> angularly, the other lineally, without changing angle, that is, alwayes keeping the tube of the In$trument parallel to its $elf.</P> <P>SAGR. Tell me, in the next place, if we, having fir$t directed <marg><I>Two mutations made in the Tele- $cope, depending on the agitation of the Ship.</I></marg> the <I>Tele$cope</I> yonder away towards the Tower of ^{*} <I>Burano,</I> $ix miles from hence, do turn it angularly to the right hand, or to the left, or el$e upwards or downwards, but a ^{*}$traws breadth, what ef- <marg>* This is a Ca$tle $ix Italian miles from <I>Venice</I> Northwards.</marg> fect $hall it have upon us touching the finding out of the $aid tower?</P> <P>SALV. It would make us immediately lo$e $ight of it, for $uch a declination, though $mall here, may import there hundreds and <marg>* <I>Vnnerod' ug- na,</I> the black or paring of a nail.</marg> thou$ands of yards.</P> <P>SAGR. But if without changing the angle, keeping the tube alwayes parallel to it $elf, we $hould transfer it ten or twelve yards farther off to the right or left hand, upwards or downwards, what alteration would it make as to the Tower?</P> <P>SALV. The change would be ab$olutely undi$cernable; for that the $paces here and there being contained between parallel rayes, the mutations made here and there, ought to be equal, and becau$e the $pace which the In$trument di$covers yonder, is capa- ble of many of tho$e Towers; therefore we $hall not lo$e $ight of it.</P> <P>SAGR. Returning now to the Ship, we may undoubtedly af- firm, that the <I>Tele$cope</I> moving to the right or left, upwards, or <foot>down-</foot> <p n=>227</p> downwards, and al$o forwards or backwards ten or fifteen fathom, keeping it all the while parallel to its $elf, the vi$ive ray cannot $tray from the point ob$erved in the object, more than tho$e fif- teen fathom; and becau$e in a di$tance of eight or ten miles, the In$trument takes in a much greater $pace than the Gally or other Ve$$el kenn'd; therefore that $mall mutation $hall not make me lo$e $ight of her. The impediment therefore, and the cau$e of lo$ing the object cannot befall us, unle$$e upon the mutation made angularly; $ince that <I>Tele$copes</I> tran$portation higher or lower, to the right, or to the left, by the agitation of the $hip, cannot import any great number of fathomes. Now $uppo$e that you had two <I>Tele$copes</I> fixed, one at the Partners clo$e by the Deck, and the o- ther at the round top, nay at the main top, or main top-gallant top, where you hang forth the <I>Pennon</I> or $treamer, and that they be both directed to the Ve$$el that is ten miles off, tell me, whe- ther you believe that any agitation of the $hip, & inclination of the Ma$t, can make greater changes, as to the angle, in the higher tube, than in the lower? One wave a<*>i$ing, the prow will make the main top give back fifteen or twenty fathom more than the foot of the Ma$t, and it $hall carry the upper tube along with it $o greata $pace, & the lower it may be not a palm; but the angle $hall change in one In$trument a$well as in the other; and likewi$e a $ide-billow $hall bear the higher tube an hundred times as far to the Larboard or Starboard, as it will the other below; but the angles change not at all, or el$e alter both alike. But the mutation to the right hand or left, forwards or backwards, upwards or downwards, bringeth no $en$ible impediment in the kenning of objects remote, though the alteration of the angle maketh great change therein; Therefore it mu$t of nece$$ity be confe$$ed, that the u$e of the <I>Tele$cope</I> on the round top is no more difficult than upon the Deck at the Partners; $eeing that the angular mutations are alike in both places.</P> <P>SALV. How much circum$pection is there to be u$ed in affirming or denying a propo$ition? I $ay again, thar hearing it re$olutely affir- med, that there is a greater motion made on the Ma$ts top, than at its partners, every one will per$wade him$elf, that the u$e of the <I>Te- le$cope</I> is much more difficult above than below. And thus al$o I w ill excu$e tho$e Philo$ophers, who grow impatient and fly out into pa$$ion again$t $uch as will not grant them, that that Cannon bullet which they cleerly $ee to fall in a right line perpendicularly, doth ab$olutely move in that manner; but will have its motion to be by an arch, and al$o very much inclined and tran$ver$al: but let us leave them in the$e labyrinths, and let us hear the other objections, that our Author in hand brings again$t <I>Copernicus.</I></P> <P>SIMP. The Author goeth on to demon$trate that in the Do- ctrine of <I>Copernicus,</I> it is requi$ite to deny the Sen$es, and the <foot>Ff 2 greate$t</foot> <p n=>228</p> greate$t Sen$ations, as for in$tance it would be, if we that feel the <marg><I>The annual mo- tion of the Earth mu$t cau$e a per- petual and $trong winde.</I></marg> re$pirations of a gentle gale, $hould not feel the impul$e of a per- petual winde that beateth upon us with a velocity that runs more than 2529 miles an hour, for $o much is the $pace that the centre of the Earth in its annual motion pa$$eth in an hour upon the cir- cumference of the grand Orb, as he diligently calculates; and becau$e, as he $aith, by the judgment of <I>Copernicus, Cum terra movetur circumpo$itus aër, motus tamen ejus, velocior licet ac ra- pidior celerrimo quocunque vento, à nohis non $entiretur, $ed $um- ma tum tranquilitas reputaretur, ni$i alius motus accederet. Quid e$t verò decipi $en$um, ni$i hæc e$$et deceptio</I>? [<I>Which I make to $peak to this $en$e.</I>] The circumpo$ed air is moved with the Earth, yet its motion, although more $peedy and rapid than the $wifte$t wind what$oever, would not be perceived by us, but then would be thought a great tranquillity, unle$$e $ome other motion $hould happen; what then is the deception of the $en$e, if this be not?</P> <P>SALV. It mu$t needs be that this Philo$opher thinketh, that that Earth which <I>Copernicus</I> maketh to turn round, together with the ambient air along the circumference of the great Orb, is not that whereon we inhabit, but $ome other $eparated from this; for that this of ours carrieth us al$o along with it with the $ame velocity, as al- <marg><I>The air alwayes touching us with the $ame part of it cannot make us <*>eel it.</I></marg> $o the circumjacent air: And what beating of the air can we feel, when we fly with equal $peed from that which $hould acco$t us? This Gentleman forgot, that we no le$s than the Earth and air are carried about, and that con$equently we are always touch'd by one and the $ame part of the air, which yet doth not make us feel it.</P> <P>SIMP. But I rather think that he did not $o think; hear the words which immediately follow. <I>Præterea nos quoque rotamur ex circumductione terræ &c.</I></P> <P>SALV. Now I can no longer help nor excu$e him; do you plead for him and bring him off, <I>Simplicius.</I></P> <P>SIMP. I cannot thus upon the $udden think of an excu$e that plea$eth me.</P> <P>SALV. Go to; take this whole night to think on it, and de- fend him to morrow; in the mean time let us hear $ome other of his objections.</P> <P>SIMP. He pro$ecuteth the $ame Objection, $hewing, that in the <marg><I>He that will fol- low</I> Copernicus, <I>must deny his $er- $es.</I></marg> way of <I>Copernicus,</I> a man mu$t deny his own $en$es. For that this principle whereby we turn round with the Earth, either is intrin$ick to us, or external; that is, a rapture of that Earth; and if it be this $econd, we not feeling any $uch rapture, it mu$t be confe$$ed that the $en$e of feeling, doth not feel its own object touching it, nor its impre$$ion on the $en$ible part: but if the prin- <foot>ciple</foot> <p n=>229</p> ciple be intrin$ecal, we $hall not perceive a local motion that is de- rived from our $elves, and we $hall never di$cover a propen$ion per- petually annexed to our $elves.</P> <P>SALV. So that the in$tance of this Philo$opher lays its $tre$s up- on this, that whether the principle by which we move round with the Earth be either extern, or intern, yet however we mu$t per- ceive it, and not perceiving it, it is neither the one nor the other, and therefore we move not, nor con$equently the Earth. Now I <marg><I>Our motion may be either interne or externe, and yet we never perceive or feel it.</I></marg> $ay, that it may be both ways, and yet we not perceive the $ame. And that it may be external, the experiment of the boat $upera- bundantly $atis$ieth me; I $ay, $uperabundantly, becau$e it being in our power at all times to make it move, and al$o to make it $tand $till, and with great exactne$s to make ob$ervation, whether by $ome diver$ity that may be comprehended by the $en$e of feel- ing, we can come to know whether it moveth or no, $eeing that as yet no $uch $cience is obtained: Will it then be any matter of wonder, if the $ame accident is unknown to us on the Earth, the which may have carried us about perpetually, and we, without our <marg><I>The motion of a Boat in$en$ible to tho$e that are with in it, as to the $en$e of feeling.</I></marg> being ever able to experiment its re$t? You, <I>Simplicius,</I> as I be- lieve, have gone by boat many times to <I>Padoua,</I> and if you will confe$s the truth, you never felt in your $elf the participation of that motion, unle$s when the boat running a-ground, or encoun- tring $ome ob$tacle, did $top, and that you with the other Pa$$en- gers being taken on a $udden, were with danger over-$et. It would be nece$$ary that the Terre$trial Globe $hould meet with $ome rub that might arre$t it, for I a$$ure you, that then you would di$cern the impul$e re$iding in you, when it $hould to$s you up towards the Stars. It's true, that by the other $en$es, but yet <marg><I>The boats moti- on is perceptible to the $ight joyn'd with rea$on.</I></marg> a$$i$ted by Rea$on, you may perceive the motion of the boat, that is, with the $ight, in that you $ee the trees and buildings placed on the $hoar, which being $eparated from the boat, $eem to move the <marg><I>The terre$trial motion collected from the $tars.</I></marg> contrary way. But if you would by $uch an experiment receive intire $atisfaction in this bu$ine$s of the Terre$trial motion, look on the $tars, which upon this rea$on $eem to move the contrary way. As to the wondering that we $hould not feel $uch a prin- ciple, $uppo$ing it to be internal, is a le$s rea$onable conceit; for if we do not feel $uch a one, that cometh to us from without, and that frequently goeth away, with what rea$on can we expect to feel it, if it immutably and continually re$ides in us? Now let us $ee what you have farther to allege on this argument.</P> <P>SIMP. Take this $hort exclamation. <I>Ex hac itaque opinione nece$$e est diffidere no$tris $en$ibus, ut penitùs fall acibus vel $tupidis in $en$ilibus, etiam conjuncti$$imis, dijudicandis. Quam ergò ve- ritatem $perare po$$umus à facultate adeò fallaci ortum trabentem</I>? [Which I render thus:] From this opinion likewi$e, we mu$t of <foot>nece$$ity</foot> <p n=>230</p> nece$$ity $u$pect our own $en$es, as wholly fallible, or $tupid in judging of $en$ible things even very near at hand. What truth therefore can we hope for, to be derived from $o deceiveable a fa- culty?</P> <P>SALV. But I de$ire not to deduce precepts more profitable, or more certain, learning to be more circum$pect and le$s confident about that which at fir$t blu$h is repre$ented to us by the $en$es, which may ea$ily deceive us. And I would not have this Author trouble him$elf in attemptiug to make us comprehend by $en$e, that this motion of de$cending Graves is $imply right, and of no other kind; nor let him exclaim that a thing $o clear, manife$t, and obvious $hould be brought in que$tion; for in $o doing, he maketh others believe, that he thinketh tho$e that deny that mo- tion to be ab$olutely $treight, but rather circular, the $tone did $en$ibly $ee it to move in an arch, $eeing that he inviteth their $en$es more than their Rea$on, to judg of that effect: which is not true, <I>Simplicius,</I> for like as I, that am indifferent in all the$e opini- ons, and onely in the manner of a Comedian, per$onate <I>Coperni- cus</I> in the$e our repre$entations, have never $een, nor thought that I have $een that $tone fall otherwi$e than perpendicularly, $o I believe, that to the eyes of all others it $eemed to do the $ame. Better it is therefore, that depo$ing that appearance in which all agree, we make u$e of our Rea$on, either to confirm the reality of that, or to di$cover its fallacy.</P> <P>SAGR. If I could any time meet with this Philo$opher, who yet me thinks is more $ublime than the re$t of the followers of the $ame doctrines, I would in token of my affection put him in mind of an accident which he hath doubtle$s very often beheld; from which, with great conformity to that which we now di$cour$e of, it may be collected how ea$ily one may be deceived by the bare appearance, or, if you will, repre$entation of the $en$e. And the accident is, the Moons $eeming to follow tho$e that walk the $treets in the night, with a pace equal to theirs, whil$t they $ee it go gli- ding along the Roofs of hou$es, upon which it $heweth ju$t like a cat, that really running along the ridges of hou$es, leaveth them behind. An appearance that, did not rea$on interpo$e, would but too manife$tly delude the $ight.</P> <P>SIMP. Indeed there want not experiments that render us cer- <marg><I>Arguments a- gain$t the Earths motion taken,</I> ex rerum natura.</marg> tain of the fallacy of the meer $en$es; therefore $u$pending $uch $en$ations for the pre$ent, let us hear the Arguments that follow which are taken, as he $aith, <I>ex rerum naturâ.</I> The fir$t of which is, that the Earth cannot of its own nature move with three moti- ons very different; or otherwi$e we mu$t deny many manife$t <marg><I>Three Axioms that are $uppo$ed manife$t.</I></marg> Axioms. The fir$t whereof is, that <I>Omnïs effectus dependeat ab aliquâ cau$â; [i. e.]</I> that every effect dependeth on $ome cau$e. <foot>The</foot> <p n=>231</p> The $econd, that <I>Nulla res $eip$am producat; [i. e.]</I> that nothing produceth it $elf: from whence it follows, that it is not po$$i- ble that the mover and moved $hould be totally the $ame thing: And this is manife$t, not onely in things that are moved by an ex- trin$ick mover; but it is gathered al$o from the principles pro- pounded, that the $ame holdeth true in the natural motion depen- dent on an intrin$ick principle; otherwi$e, being that the mover, as a mover, is the cau$e, and the thing moved, as moved, is the effect, the $ame thing would totally be both the cau$e and effect. Therefore a body doth not move its whole $elf, that is, $o as that all moveth, and all is moved; but its nece$$ary in the thing moved to di$tingui$h in $ome manner the efficient principle of the motion, and that which with that motion is moved. The third Axiom is, that <I>in rebus quæ $en$ui $ubjiciuntur, unum, quatenus unum, unam $olam rem producat; i. e.</I> That in things $ubject to the $en$es, one, as it is one, produceth but onely one thing: That is, the $oul in animals produceth its true divers operations, as the $ight, the hearing, the $mell, generation, <I>&c.</I> but all the$e with $everal in$truments. And in $hort, in things $en$ible, the diver$i- ty of operations, is ob$erved to derive it $elf from the diver$ity that is in the cau$e. Now if we put all the$e Axioms together, it <marg><I>A $imple body as the Earth, can- not move with three $everal moti- ons.</I></marg> will be a thing very manife$t, that one $imple body, as is the Earth, cannot of its own nature move at the $ame time with three motions, very divers: For by the foregoing $uppo$itions, all moveth not its $elf all; it is nece$fary therefore to di$tingui$h in it three principles of its three motions; otherwi$e one and the $ame principle would produce many motions; but if it contein in it three principles of natural motions, be$ides the part moved, it $hall not be a $imple body, but compounded of three principle movers, and of the part moved. If therefore the Earth be a $im- <marg><I>The Earth can- not move with any of the motions a$$i- gned it by</I> Coperni- cus.</marg> ple body, it $hall not move with three motions; nay more, it will not move with any of tho$e which <I>Copernicus</I> a$cribeth to it, it being to move but with one alone, for that it is manife$t, by the rea$ons of <I>Ari$totle,</I> that it moveth to its centre, as its parts do $hew, which de$cend at right angles to the Earths Spherical Surface.</P> <P>SALV. Many things might be $aid, and con$idered touching the connection of this argument; but in regard that we can re- <marg><I>An$wers to the arguments contra- ry to the Earths motion, taken</I> ex rerum natura.</marg> $olve it in few words, I will not at this time without need inlarge upon it; and $o much the rather, becau$e the $ame Author hath furni$hed me with an an$wer, when he $aith that from one $ole prin- ple in animals, there are produced divers operations; $o that for the pre$ent my an$wer $hall be, that in the $ame manner the Earth from one onely principle deriveth $everal operations.</P> <P>SIMP. But this an$wer will not at all $atisfie the Author who <foot>makes</foot> <p n=>232</p> makes the objection, yea, it is totally overthrown by that which immediately after he addeth for a greater confirmation of his argu- ment, as you $hall hear. He re-inforceth his argument, I $ay, with <marg><I>A fourth Ax- iome again$t the motion of the Earth</I></marg> another Axiome, which is this; That <I>natura in rebus nece$$ari is nec deficiat, nec abundat: i.e.</I> That nature in things nece$$ary is neither defective, nor $uperfluous. This is obvious to the ob$er- <marg><I>Flexures nece$- $ary in animals for the diver$ity of their motions.</I></marg> vers of natural things, and chiefly of animals, in which, becau$e they are to move with many motions, Nature hath made many flexures, and hath thereunto commodiou$ly knitted the parts for motion, as to the knees, to the hips, for the inabling of living creatures to go, and run at their plea$ure. Moreover in man he hath framed many flexions, and joynts, in the elbow, and hand, to enable them to perform many motions. From the$e things the ar- <marg><I>Another argu- ment again$t the three fold motion of the Earth.</I></marg> gument is taken again$t the threefold motion of the Earth. [<I>Ei- ther the Body, that is one, and continuate, without any manner of knittings or flexions, can exerci$e divers motions, or cannot: If it can without them, then in vain hath nature framed the flexures in animals; which is contrary to the Axiome: but if it cannot with- out them, then the Earth, one body, and continuate, and deprived of flexures, and joynts, cannot of its own nature move with plurali- ty of motions.</I>] You $ee now how craftily he falls upon your an- $wer, as if he had fore$een it.</P> <P>SALV. Are you $erious, or do you je$t?</P> <P>SIMP. I $peak it with the be$t judgment I have.</P> <P>SALV. You mu$t therefore $ee that you have as fortunate an hand in defending the reply of this Philo$opher, again$t $ome o- ther rejoynders made to him; therefore an$wer for him, I pray you, $eeing we cannot have him here. You fir$t admit it for true, that Nature hath made the joynts, flexures, and knuckles of li- ving creatures, to the intent that they might move with $nndry and divers motions; and I deny this propo$ition; and $ay, that the$e flexions are made, that the animal may move one, or more <marg><I>The Flexures in animals are not made for the di- ver$ity of motions.</I></marg> of its parts, the re$t remaining immoved: and I $ay, that as to the $pecies and differences of motions tho$e are of one kind alone, to wit, all circular, and for this cau$e you $ee all the ends of the mo- <marg><I>The motions of animals are of one $ort.</I></marg> veable bones to be convex or concave, and of the$e $ome are $phe- rical, as are tho$e that are to move every way, as in the $houlder- <marg><I>The ends of the bones are all ro- tund.</I></marg> joynt, the arme of the En$igne doth, in di$playing the Colours, and that of the Falconer in bringing his Hawk to the lure; and $uch is the flexure of the elbow, upon which the hand turns round, in boring with an augure: others are circular onely one way, and as it were cylindrical, which $erve for the members that bend one- <marg><I>It is demon$tra- ted, that the ends of the bones are of nece$$ity to be ro- tund.</I></marg> ly in one fa$hion, as the joynts of the fingers one above another, &c. But without more particular inductions, one only general di$- cour$e may make this truth under$tood; and this is, that of a $olid <foot>body</foot> <p n=>233</p> body that moveth, one of its extreams $tanding $till without chan- ching place, the motion mu$t needs be circular, and no other: and <marg><I>The motions of animals are all circular.</I></marg> becau$e in the living creatures moving, one of its members doth not $eparate from the other its conterminal, therefore that motion is of nece$$ity circular.</P> <P>SIMP. How can this be? For I $ee the animal move with an hundred motions that are not circular, and very different from one another, as to run, to skip, to climbe, to de$cend, to $wim, and many others.</P> <marg><I>Secondary moti- ons of animals de- pendent on the fir$t</I></marg> <P>SALV. Tis well: but the$e are $econdary motions, depending on the preceding motions of the joynts and flexures. Upon the plying of the legs to the knees, and the thighs to the hips, which are circular motions of the parts, is produced, as con$equents, the skip, or running, which are motions of the whole body, and the$e may po$$ibly not be circular. Now becau$e one part of the ter- <marg><I>The</I> Terre$triall Globe <I>hath noe need of flexures.</I></marg> re$trial Globe is not required to move upon another part immove- able, but that the motion is to be of the whole body, there is no need in it of flexures.</P> <P>SIMP. This (will the aduer$ary rejoyn) might be, if the moti- on were but one alone, but they being three, and tho$e very dif- ferent from each other, it is not po$$ible that they $hould concur in <marg>* Without joynts</marg> an ^{*} articulate body.</P> <P>SALV. I verily believe that this would be the an$wer of the Philo$opher. Again$t which I make oppo$ition another way; and ask you, whether you think that by way of joynts and flexures one may adapt the terre$trial Globe to the participation of three diffe- rent circular motions? Do you not an$wer me? Seeing you are $peechle$$e, I will undertake to an$wer for the Philo$opher, who would ab$olutely reply that they might; for that otherwi$e it would have been $uperfluous, and be$ides the purpo$e to have pro- po$ed to con$ideration, that nature maketh the flexions, to the end, the moveable may move with different motions; and that therefore the terre$trial Globe having no flexures, it cannot have tho$e three motions which are a$cribed to it. For if he had thought, that neither by help of flexures, it could be rendered apt for $uch motions, he would have freely affirmed, that the Globe could not move with three motions. Now granting this, I intreat <marg><I>It is de$ired to know, by means of what flexures and joynts the</I> Terre- $trial Globe <I>might move with three diver$e motions.</I></marg> you, and by you, if it were po$$ible, that Philo$opher, Au- thor of the Argument, to be $o courteous as to teach me in what manner tho$e flexures $hould be accommodated, $o that tho$e three motions might commodiou$ly be excerci$ed; and I grant you four or $ix moneths time to think of an an$wer. As to me, it $eem- eth that one principle onely may cau$e a plurality of motions in <marg><I>One only princi- ple may cau$e a plurality of moti- ons in the Earth.</I></marg> the Terre$trial Globe, ju$t in the $ame manner that, as I told you before, one onely principle with the help of various in$truments <foot>Gg pro-</foot> <p n=>234</p> produceth $undry and divers motions in living creatures. And as to the flexures there is no need of them, the motions being of the whole, and not of $ome particular parts; and becau$e they are to be circular, the meer $pherical figure is the mo$t perfect articu- lation or flection that can be de$ired.</P> <P>SIMP. The mo$t that ought to be granted upon this, would be, that it may hold true in one $ingle motion, but in three different motions, in my opinion, and that of the Author, it is impo$$i- ble; as he going on, pro$ecuting the objection, writes in the fol- lowing words. <I>Let us $uppo$e, with</I> Copernicus, <I>that the Earth moveth of its own faculty, and upon an intrin$ick principle from We$t to Ea$t in the plane of the Ecliptick; and again, that it al$o by an intrin$ick principle revolveth about its centre, from Ea$t to We$t; and for a third motion, that it of its own inclination defle- cteth from North to South, and $o back again.</I> It being a conti- nuate body, and not knit together with joints and flections, our fancy and our judgment will never be able to comprehend, that one and the $ame natural and indi$tinct principle, that is, that one and the $ame propen$ion, $hould actuate it at the $ame in$tant with different, and as it were of contrary motions. I cannot be- lieve that any one would $ay $uch a thing, unle$$e he had under- took to maintain this po$ition right or wrong.</P> <P>SALV. Stay a little; and find me out this place in the Book. <I>Fingamus modo cum Copernico terram aliqua $uâ vi, & ab indito principio impelli ab Occa$u ad Ortum in Eclipticæ plano; tum rur- $us revolvi ab indito etiam principio, circa $uimet centrum, ab</I> <marg><I>A gro$$e error of the oppo$er of</I> Copernicus.</marg> <I>Ortu in Occa$um; tertio de$lecti rur$us $u opte nutu à $eptentrio- ne in Au$trum, & vici$$im.</I> I had thought, <I>Simplicius,</I> that that you might have erred in reciting the words of the Au- thor, but now I $ee that he, and that very gro$$ely, decei- veth him$elf; and to my grief, I find that he hath $et him$elf to oppo$e a po$ition, which he hath not well under$tood; for the$e are not the motions which <I>Copernicus</I> a$$ignes to the Earth. Where doth he find that <I>Copernicus</I> maketh the annual motion by the Ecliptick contrary to the motion about its own centre? It mu$t needs be that he never read his Book, which in an hundred places, and in the very fir$t Chapters affirmeth tho$e motions to be both towards the $ame parts, that is from We$t to Ea$t. But without others telling him, ought he not of him$elf to com- prehend, that attributing to the Earth the motions that are ta ken, one of them from the Sun, and the other from the <I>pri- mum wobile,</I> they mu$t of nece$$ity both move one and the $ame way.</P> <marg><I>A $ubtil and withal $imple ar- gument again$t</I> Copernicus.</marg> <P>SIMP. Take heed that you do not erre your $elf, and <I>Coperni- cus</I> al$o. The Diurnal motion of the <I>primum mobile,</I> is it not from <foot>Ea$t</foot> <p n=>235</p> Ea$t to We$t? And the annual motion of the Sun through the Ecliptick, is it not on the contrary from We$t to Ea$t? How then can you make the$e motions being conferred on the Earth, of contraries to become con$i$tents?</P> <P>SAGR. Certainly, <I>Simplicius</I> hath di$covered to us the original cau$e of error of this Philo$opher; and in all probability he would have $aid the very $ame.</P> <P>SALV. Now if it be in our power, let us at lea$t recover <I>Simplicius</I> from this errour, who $eeing the Stars in their ri$ing to appear above the Oriental Horizon, will make it no difficult thing to under$tand, that in ca$e that motion $hould not belong <marg><I>The error of the Antagoni$t is ma- nife$t, by decla- ring that the an- nual and diurnal motions belonging to the Earth are both one way, and not contrary.</I></marg> to the Stars, it would be nece$$ary to confe$$e, that the Horizon, with a contrary motion would go down; and that con$equently the Earth would reoolve in it $elf a contrary way to that where- with the Stars $eem to move, that is from We$t to Ea$t, which is according to the order of the Signes of the Zodiack. As, in the next place, to the other motion, the Sun being fixed in the cen- tre of the Zodiack, and the Earth moveable about its circumfe- rence, to make the Sun $eem unto us to move about the $aid Zo- diack, according to the order of the Signes, it is nece$$ary, that the E arth move according to the $ame order, to the end that the Sun may $eem to us to po$$e$$e alwayes that degree in the Zodiack, that is oppo$ite to the degree in which we find the Earth; and thus the Earth running, <I>verbi gratia,</I> through <I>Aries,</I> the Sun will appear to run thorow <I>Libra</I>; and the Earth pa$$ing thorow the $igne <I>Taurus,</I> the Sun will pa$$e thorow <I>Scorpio,</I> and $o the Earth going thorow <I>Gemini,</I> the Sun $eemeth to go thorow <I>Sa- gittarius</I>; but this is moving both the $ame way, that is accord- ing to the order of the $ignes; as al$o was the revolution of the Earth about its own centre.</P> <P>SIMP. I under$tand you very well, and know not what to al- ledge in excu$e of $o gro$$e an error.</P> <P>SALV. And yet, <I>Simplicius,</I> there is one yet wor$e then this; and it is, that he makes the Earth move by the diurnal motion about its own centre from Ea$t to We$t; and perceives not that if this were $o, the motion of twenty four hours appropriated by him to the Univer$e, would, in our $eeming, proceed from We$t to Ea$t; the quite contrary to that which we behold.</P> <P>SIMP. Oh $trange! Why I, that have $carce $een the fir$t elements of the Sphere, would not, I am confident, have erred $o horribly.</P> <P>SALV. Judg now what pains this Antagoni$t may be thought to have taken in the Books of <I>Copernicus,</I> if he ab$olutely invert <marg><I>By another gro$<*> error it is $een that the Antagoni$t had but little $tudied</I> Copernicus.</marg> the $en$e of this grand and principal Hypothe$is, upon which is founded the whole $umme of tho$e things wherein <I>Copernicus</I> <foot>Gg 2 di$$enteth</foot> <p n=>236</p> di$$enteth from the doctrine of <I>Ari$totle</I> and <I>Ptolomy.</I> As again, <marg><I>It is que$tioned, whether the oppo- nent under$tood the third motion a$$igned to the Earth by</I> Coperni- cus.</marg> to this third motion, which the Author a$$ignes to the Terre$trial Globe, as the judgment of <I>Copernicus,</I> I know not which he would mean thereby: it is not that que$tionle$$e, which <I>Copernicus</I> a$- cribes unto it conjunctly with the other two, annual and diurnal, which hath nothing to do with declining towards the South and North; but onely $erveth to keep the axis of the diurnal revoluti- on continually parallel to it $elf; $o that it mu$t be confe$t, that either the Authour did not under$tand this, or that el$e he di$$em- bled it. But although this great mi$take $ufficeth to free us from any obligation of a farther enquiry into his objections; yet ne- verthele$$e I $hall have them in e$teem; as indeed they de$erve to be valued much before the many others of impertinent Antago- ni$ts. Returning therefore to his objection, I $ay, that the two motions, annual and diurnal, are not in the lea$t contrary, nay are towards the $ame way, and therefore may depend on one and the $ame principle. The third is of it $elf, and voluntarily $o con$equen- tial to the annual, that we need not trouble our $elves (as I $hall $hew in its place) to $tudy for principles either internal or external, from which, as from its cau$e, to make it produced.</P> <P>SAGR. I $hall al$o, as being induced thereto by natural rea$on, $ay $omething to this Antagoni$t. He will condemn <I>Copernicus,</I> unle$$e I be able to an$wer him to all objections, and to $atisfie him in all que$tions he $hall ask; as if my ignorance were a nece$- $ary argument of the fal$hood of his Doctrine. But if this way of condemning Writers be in his judgment legal, he ought not to think it unrea$onable, if I $hould not approve of <I>Arî$totle</I> and <I>Pto- lomy,</I> when he cannot re$olve, better than my $elf, tho$e doubts which I propound to him, touching their Doctrine. He asketh me, what are the principles by which the Terre$trial Globe is moved <marg><I>The $ame argu- ment an$wered by examples of the like motions in o- ther cœle$tial bo- dies.</I></marg> with the Annual motion through the Zodiack, and with the Diur- nal through the Equinoctial about its own axis. I an$wer, that they are like to tho$e by which <I>Saturn</I> is moved about the Zodi- ack in thirty years, and about its own centre in a much $horter time along the Equinoctial, as the collateral apparition and oc- cultation of its Globes doth evince. They are principles like to tho$e, whereby he $crupleth not to grant, that the Sun runneth tho- row the Ecliptick in a year, and revolveth about its own centre parallel to the Equinoctial in le$$e than a moneth, as its $pots doth $en$ibly demon$trate. They are things like to tho$e whereby the Medicean Stars run through the Zodiack in twelve years, and all the while revolve in $mall circles, and $hort periods of time a- bout <I>Jupiter.</I></P> <P>SIMP. This Author will deny all the$e things, as delu$ions of the fight, cau$ed by the cry$tals of the <I>Tele$cope.</I></P> <foot>SAGR.</foot> <p n=>237</p> <P>SAGR. But this would be to draw a further inconvenience up- on him$elf, in that he holdeth, that the bare eye cannot be decei- ved in judging of the right motion of de$cending graves, and yet holds that it is deceived in beholding the$e other motions at $uch time as its vi$ive vertue is perfected, and augmented to thirty times as much as it was before. We tell him therefore, that the Earth in like manner partaketh of the plurality of motions: and it is per- haps the $ame, whereby the Load$tone hath its motion down- wards, as grave, and two circular motions, one Horizontal, and the other Vertical under the Meridian. But what more; tell me, <I>Sim- plicius,</I> between which do you think this Author would put a greater difference, 'twixt right and circular motion, or 'twixt moti- on and re$t?</P> <P>SIMP. 'Twixt motion and re$t, certainly. And this is mani- <marg><I>Motion and re$t are more different than right motion and circular.</I></marg> fe$t, for that circular motion is not contrary to the right, according to <I>Aristotle</I>; nay, he granteth that they may mix with each o- ther; which it is impo$$ible for motion and re$t to do.</P> <P>SAGR. Therefore its a propo$ition le$$e improbable to place in one natural body two internal principles, one to right motion, and the other to circular, than two $uch interne principles one to motion, and the other to re$t. Now both the$e po$itions agree to <marg><I>One may more rationally a$cribe to the Earth two internal principles to the right, and circular motion, than two to motion and re$t.</I></marg> the natural inclination that re$ideth in the parts of the Earth to re- turn to their whole, when by violence they are divided from it; and they onely di$$ent in the operation of the whole: for the lat- ter of them will have it by an interne principle to $tand $till, and the former a$cribeth to it the circular motion. But by your con- ce$$ion, and the confe$$ion of this Philo$opher, two principles, one to motion, and the other to re$t, are incompatible together, like as their effects are incompatible: but now this evenes not in the two motions, right, and circular, which have no repugnance to each other.</P> <P>SALV. Adde this more, that in all probability it may be that <marg><I>The motion of the parts of the Earth returning to their whole may be circular.</I></marg> the motion, that the part of the Earth $eparated doth make whil$t it returneth towards its whole, is al$o circular, as hath been alrea- dy declared; $o that in all re$pects, as far as concernes the pre$ent ca$e, Mobility $eemeth more likely than Re$t. Now proceed, <I>Simplicius,</I> to what remains.</P> <P>SIMP. The Authour backs his Argument with producing ano- ther ab$urdity, that is, that the $ame motions agree to Natures ex- treamly different; but experience $heweth, that the operations <marg><I>The diver$ity of motions helpeth us in knowing the di- ver$ity of natures.</I></marg> and motions of different natures, are different; and Rea$on con- firmeth the $ame: for otherwi$e we $hould have no way left to know and di$tingui$h of natures, if they $hould not have their particular motions and operations, that might guide us to the knowledge of their $ub$tances.</P> <foot>SAGR.</foot> <p n=>238</p> <P>SAGR. I have twice or thrice ob$erved in the di$cour$es of this Authour, that to prove that a thing is $o, or $o, he $till alledgeth, that in that manner it is conformable with our under$tanding; or that otherwi$e we $hould never be able to conceive of it; or that the <I>Criterium</I> of Philo$ophy would be overthrown. As if that na- <marg><I>Nature fir$t made things as $he plea$ed, and after- wards capacitated mens under$tand- ings for conceiving of them.</I></marg> ture had fir$t made mens brains, and then di$po$ed all things in conformity to the capacity of their intellects. But I incline rather to think that Nature fir$t made the things them$elves, as $he be$t liked, and afterwards framed the rea$on of men capable of con- ceiving (though not without great pains) $ome part of her $e- crets.</P> <P>SALV. I am of the $ame opinion. But tell me, <I>Simplicius,</I> which are the$e different natures, to which, contrary to expe- rience and rea$on, <I>Copernicus</I> a$$ignes the $ame motions and ope- rations.</P> <P>SIMP. They are the$e. The Water, the Air, (which doubt- le$$e are Natures different from the Earth) and all things that are in tho$e elements compri$ed, $hall each of them have tho$e three motions, which <I>Copernicus</I> pretends to be in the Terre$triall Globe; and my Authour proceedeth to demon$trate Geometri- <marg>Copernicus <I>er- roneou$ly a$$igneth the $ame operations to different natures</I></marg> cally, that, according to the <I>Copernican</I> Doctrine, a cloud that is $u$pended in the Air, and that hangeth a long time over our heads without changing place, mu$t of nece$$ity have all tho$e three motions that belong to the Terre$trial Globe. The demon$tra- tion is this, which you may read your $elf, for I cannot repeat it without book.</P> <P>SALV. I $hall not $tand reading of it, nay I think it an imper- tinency in him to have in$erted it, for I am certain, that no <I>Copernican</I> will deny the $ame. Therefore admitting him what he would demon$trate, let us $peak to the objection, which in my judgment hath no great $trength to conclude any thing contrary to the <I>Copernican Hypothe$is,</I> $eeing that it derogates nothing from tho$e motions, and tho$e operations, whereby we come to the knowledge of the natures, &c. An$wer me, I pray you, <I>Simplici- us:</I> Tho$e accidents wherein $ome things exactly concur, can they $erve to inform us of the different natures of tho$e things?</P> <marg><I>From commune accidents one can- not know different natures.</I></marg> <P>SIMP. No Sir: nay rather the contrary, for from the idendity of operations and of accidents nothing can be inferred, but an idendity of natures.</P> <P>SALV. So that the different natures of the Water, Earth, Air, and other things conteined in the$e Elements, is not by you argu- ed from tho$e operations, wherein all the$e Elements and their af- fixes agree, but from other operations; is it $o?</P> <P>SIMP. The very $ame.</P> <P>SALV. So that he who $hould leave in the Elements all tho$e <foot>motions,</foot> <p n=>239</p> motions, operations, and other accidents, by which their natures are di$tingui$hed, would not deprive us of the power of coming to the knowledge of them; although he $hould remove tho$e o- perations, in which they unitedly concur, and which for that rea$on are of no u$e for the di$tingui$hing of tho$e natures.</P> <P>SIMP. I think your di$$ertation to be very good.</P> <P>SALV. But that the Earth, Water, Air, are of a nature equally con$tituted immoveable about the centre, is it not the opinion of your $elf, <I>Ari$totle, Prolomy,</I> and all their $ectators?</P> <P>SIMP. Its on all hands granted as an undeniable truth.</P> <P>SALV. Then from this common natural condition of quie$- cence about the centre, there is no argument drawn of the different natures of the$e Elements, and things elementary, but that knowledge mu$t be collected from other qualities not common; and therefore who$o $hould deprive the Elements of this common re$t only, and $hould leave unto them all their other operations, would not in the lea$t block up the way that leadeth to the know- ledge of their e$$ences. But <I>Copernicus</I> depriveth them onely of this common re$t, and changeth the $ame into a common motion, leaving them gravity, levity, the motions upwards, downwards, <marg><I>The concurrence of the Elements in a common motion importeth no more or le$$e, than their concurrence in a common re$t.</I></marg> $lower, fa$ter, rarity, den$ity, the qualities of hot, cold, dry, moi$t, and in a word, all things be$ides. Therefore $uch an ab$urdity, as this Authour imagineth to him$elf, is no <I>Copernican</I> po$ition; nor doth the concurrence in an identity of motion import any more or le$s, than the concurrence in an identity of re$t about the diver$i- fying, or not diver$ifying of natures. Now tell us, if there be any argument to the contrary.</P> <P>SIMP. There followeth a fourth objection, taken from a natu- <marg><I>A fourth argu- ment again$t</I> Co- pernicus.</marg> ral ob$ervation, which is, <I>That bodies of the $ame kind, have mo- tions that agree in kinde, or el$e they agree in re$t. But by the</I> Co- pernican Hypothe$is, <I>bodies that agree in kinde, and are mo$t $em-</I> <marg><I>Bodies of the $ame kinde have motions that agree in kinde.</I></marg> <I>blable to one another, would be very di$crepant, yea diametrically repugnant as to motion; for that Stars $o like to one another, would be neverthele$$e $o unlike in motion, as that $ix Planets would perpe- tually turn round; but the Sun and all the fixeed Stars would $tand perpetually immoveable.</I></P> <P>SALV. The forme of the argument appeareth good; but yet I believe that the application or matter is defective: and if the Authour will but per$i$t in his a$$umption, the con$equence $hall make directly again$t him. The Argument runs thus; Among$t mundane bodies, $ix there are that do perpetually move, and they <marg><I>From the Earths ob$curity, and the $plendour of the Sun, and fixed Stars, is argued, that it is movea- ble, and they im- moveable.</I></marg> are the $ix Planets; of the re$t, that is, of the Earth, Sun, and fixed Stars, it is di$putable which of them moveth, and which $tands $till, it being nece$$ary, that if the Earth $tand $till, the Sun and $ixed Stars do move; and it being al$o po$$ible, that the Sun <foot>and</foot> <p n=>240</p> and fixed Stars may $tand immoveable, in ca$e the Earth $hould move: the matter of fact in di$pute is, to which of them we may with mo$t convenience a$cribe motion, and to which re$t. Natural rea$on dictates, that motion ought to be a$$igned to the bodies, which in kind and e$$ence mo$t agree with tho$e bodies which do undoubtedly move, and re$t to tho$e which mo$t di$$ent from them; and in regard that an eternal re$t and perpetual motion are mo$t different, it is manife$t, that the nature of the body always move- able ought to be mo$t different from the body alwayes $table. Therefore, in regard that we are dubious of motion and re$t, let us enquire, whether by the help of $ome other eminent affecti- on, we may di$cover, which mo$t agreeth with the bodies certain- ly moveable, either the Earth, or the Sun and fixed Stars. But $ee how Nature, in favour of our nece$$ity and de$ire, pre$ents us with two eminent qualities, and no le$s different than motion and re$t, and they are light and darkne$s, to wit, the being by nature mo$t bright, and the being ob$cure, and wholly deprived of light: the bodies therefore adorned with an internal and eternal $plen- dour, are mo$t different in e$$ence from tho$e deprived of light: The Earth is deprived of light, the Sun is mo$t $plendid in it $elf, and $o are the fixed Stars. The $ix Planets do ab$olutely want light, as the Earth; therefore their e$$ence agreeth with the Earth, and differeth from the Sun and fixed Stars. There- fore is the Earth moveable, immoveable the Sunne and Starry Sphere.</P> <P>SIMP. But the Authour will not grant, that the $ix Planets are tenebro$e, and by that negative will he abide. Or he will argue the great conformity of nature between the $ix Planets, and the Sun, and Fixed Stars; and the di$parity between them and the Earth from other conditions than from tenebro$ity and light; yea, now I remember in the fifth objection, which followeth, he layeth down the va$t difference between the Earth and the Cœle$tial <marg><I>A fifih argu- ment again$t</I> Co- pernicus.</marg> Bodies, in which he writeth, <I>That the</I> Copernican Hypothe$is <I>would make great confu$ion and perturbation in the Sy$teme of the Vniver$e, and amongst its parts:</I> As for in$tance, among$t Cœ- <marg><I>Another diffe- rence between the Earth and the Cœ- le$tial bodies, ta- ken from purity & impurity.</I></marg> bodies that are immutable and incorruptible, according to <I>Ari$to- tle, Tycho,</I> and others; among$t bodies, I $ay, of $uch nobility, by the confe$$ion of every one, and <I>Copernicus</I> him$elf, who affirmeth them to be ordinate, and di$po$ed in a perfect con$titution, and removeth from them all incon$tancy of vertue among$t, the$e bo- dies, I $ay once more, $o pure, that is to $ay, among$t <I>Venus, Mars, &c.</I> to place the very $ink of all corruptible matters, to wit, the Earth, Water, Air, and all mixt bodies.</P> <P>But how much properer a di$tribution, and more with nature, yea with God him$elf, the Architect, is it, to $eque$ter the pure <foot>from</foot> <p n=>241</p> from the impure, the mortal from the immortal, as other Schools teach; which tell us that the$e impure and frail matters are con- teined within the angu$t concave of the Lunar Orb, above which with uninterrupted Series the things Cele$tial di$tend them$elves.</P> <P>SALV. It's true that the Copernican Sy$teme introduceth di- <marg>Copernicus <I>in troduceth confu$ion in the Univer$e of</I> Ari$totle.</marg> $traction in the univer$e of <I>Aristotle</I>; but we $peak of our own Univer$e, that is true and real. Again if this Author will infer the di$parity of e$$ence between the Earth and Cele$tial bodies from the incorruptibility of them, and the corruptibility of it in the method of <I>Ari$totle,</I> from which di$parity he concludeth mo- tion to belong to the Sun and fixed Stars, and the immobility of the Earth, he will flatter him$elf with a Paralogi$me, $uppo$ing <marg><I>The Paralogi$me of the Author of Anti-Tycho.</I></marg> that which is in que$tion; for <I>Ari$totle</I> inferreth the incorruptibi- lity of Cele$tial bodies from motion, which is in di$pute, whe- ther it belongeth to them or to the Earth. Of the vanity of the$e Rhetorical Illations enough hath been $poken. And what can be <marg><I>It $eemeth a folly to affirm the Earth to be with- out the Heavens.</I></marg> more fond, than to $ay, that the Earth and Elements are bani- $hed and $eque$tred from the Cele$tial Spheres, and confined within the Lunar Orb? Is, not then the Moons Orb one of the Cele$tial Spheres, and according to con$ent compri$ed in the middle of all the re$t? Its a new way to $eparate the pure from the impure, and the $ick from the $ound, to a$$igne the infected quarters in the heart of the City: I had thought that the ^{*} Pe$t- <marg>* Lazeretto</marg> hou$e ought to have been removed as far off as was po$$ible. <I>Copernicus</I> admireth the di$po$ition of the parts of the Univer$e, for that God hath con$tituted the grand Lamp, which is to give light all over his Temple in the centre of it, and not on one $ide. And as to the Earths being betwixt <I>Venus</I> and <I>Mars,</I> we will but hint the $ame; and do you, in favour of this Author, trie to remove it thence. But let us not ^{*} mix the$e Rhetorical <marg>* <I>Intrecciare,</I> to twine flowers in a garland.</marg> Flowers with $olid Demon$trations, rather let us leave them to the Orators, or if you will to the Poets, who know how in their drolling way to exalt by their pray$es things mo$t $ordid, yea and $ometimes mo$t pernicious. And if any thing el$e remain, let us di$patch it, as we have done the re$t.</P> <P>SIMP. There is the $ixth and la$t argument, wherein he ma- <marg><I>A $ixth argu- ment again$t</I> Co- pernicus, <I>taken from animals, who have need of re$t<*> though their moli- on be natural.</I></marg> keth it a very improbale thing. [<I>That a corruptible and di$$ipable body $hould move with a perpetual and regular motion; and this he confirmeth with the example of living creatures, which moving with a motion natural to them, yet grow weary, and have need of repo$e to re$tore their $trength.</I>] But what hath this motion to do with that of the Earth, that in compari$on to theirs is immen$e? Be$ides, to make it move with three motions that run and draw $everal wayes: Who would ever a$$ert $uch Paradoxes, unle$$e he had $worn to be their defender? Nor doth that avail in this <foot>H h ca$e,</foot> <p n=>242</p> ca$e, which <I>Copernicus</I> alledgeth, that by rea$on this motion is natural to the Earth and not violent, it worketh contrary effects to violent motions; and that tho$e things di$$olve and cannot long $ub$i$t, to which impul$e is conferred, but tho$e $o made by nature do continue in their perfect di$po$ure; this an$wer $uf- ficeth not, I $ay, for it is overthrown by that of ours. For the a- nimal is a natural body, and not made by art, and its motion is natural, deriving it $elf from the $oul, that is, from an intrin$ick principle; and that motion is violent, who$e beginning is with- out, and on which the thing moved conferreth nothing; how- ever, if the animal continueth its motion any long time, it grows weary, and al$o dyeth, if it ob$tinately $trive to per$i$t therein. You $ee then that in nature we meet on all $ides with notions con- trary to the <I>Copernican Hypothe$is,</I> and none in favour of it. And for that I have nothing more wherein to take the part of this Op- ponent, hear what he produceth again$t <I>Keplerus</I> (with whom he di$puteth) upon that argument, which the $aid <I>Kepler</I> bringeth again$t tho$e who think it an inconvenient, nay impo$$ible thing, to augment the Starry Sphere immen$ely, as the <I>Copernican</I> Hy- pothe$is requireth. <I>Kepler</I> therefore in$tanceth, $aying: <I>Difficili- us ect, accidens præter modulum $ubjecti intendere, quàm $ub-</I> <marg><I>An argument from</I> Kepler <I>in fa- vour of</I> Coperni- cus.</marg> <I>jectum $ine accidente augere. Copernicus ergo veri$imilius facit, qui auget Orbem Stellarum fixarum ab$que motu, quam Ptolomæus, qui auget motum fixarum immen$à velocitate.</I> [Which makes this Engli$h.] Its harder to $tretch the accident beyond the model of the $ubject than to augment the $ubject without the accident. <I>Coperni-</I> hath more probability on his $ide, who encrea$eth the Orb of the fixed Stars without motion, than <I>Ptolomy</I> who augmenteth the motion of the fixed Stars to an immen$e degree of velocity. <marg><I>The Author of the Anti Tycho op- po$eth</I> Kepler.</marg> Which objection the Author an$wereth, wondering how much <I>Kepler</I> deceived him$elf, in $aying, that in the Ptolomaick Hypothe- $is the motion encrea$eth beyond the model of the $ubject, for in his judgment it doth not encrea$e, $ave onely in conformity to the model, and that according to its encrea$ement, the velocity of <marg><I>The velocity of the circular moti- on increa$eth, ac- cording to the en- crea$e of the dia- meter of the circle.</I></marg> the motion is augmented. Which he proveth by $uppo$ing a ma- chine to be framed, that maketh one revolution in twenty four hours, which motion $hall be called mo$t $low; afterwards $up- po$ing its $emidiameter to be prolonged, as far as to the di$tance of the Sun, its extreme will equal the velocity of the Sun; and it being cantinued out unto the Starry Sphere, it will equal the velocity of the fixed Stars, though in the circumferrnce of the machine it be very $low. Now applying this con$ideration of the machine to the Starry Sphere, let us imagine any point in its $e- midiameter, as neer to the centre as is the $emidiameter of the ma- chine; the $ame motion that in the Starry Sphere is exceeding <foot>$wift,</foot> <p n=>243</p> $wift, $hall in that point be exceeding $low; But the great mag- nitude of the body is that which maketh it of exceeding $low, to become exceeding $wift, although it continueth $till the $ame, and thus the velocity encrea$eth, not beyond the model of the $ub- ject, but rather according to it, and to its magnitude; very dif- ferently from the imagination of <I>Kepler.</I></P> <P>SALV. I do not believe that this Author hath entertained $o mean and poor a conceit of <I>Kepler,</I> as to per$wade him$elf that he did not under$tand, that the highe$t term of a line drawn from the centre unro the Starry Sphere, moveth more $wiftly than a point of the $ame line taken within a yard or two of the centre. And therefore of nece$$ity he mu$t have conceived and comprehend- <marg><I>An explanation of the true $en$e of</I> Kepler <I>and his de- fence.</I></marg> ed that the mind and intention of <I>Kepler</I> was to have $aid, that it is le$$e inconvenient to encrea$e an immoveable body to an ex- traordinary magnitude, than to a$cribe an extraordinary velocity to a body, though very bigge, having regard to the model, that is to the gauge, and to the example of other natural bodies; in which we $ee, that the di$tance from the centre encrea$ing, the velocity dimini$heth; that is, that the periods of their circulati- ons take up longer times. But in re$t which is not capable of aug- <marg><I>The greatne$$e and $malne$$e of the body make a difference in moti- on and not in re$t.</I></marg> mentation or diminution, the grandure or $malne$$e of the body maketh no differeuce. So that if the an$wer of the Author would be directed again$t the argument of <I>Kepler,</I> it is nece$$ary, that that Author doth hold, that to the movent principle its one and the $ame to move in the $ame time a body very $mall, or very im- men$e, in regard that the augmentation of velocity in$eparably attends the augmentation of the ma$$e. But this again is contrary <marg><I>The order of na- ture is to make the le$$er Orbs to cir- culate in $horter times, and the big- ger in longer times.</I></marg> to the Architectonical rule of nature, which doth in the le$$er Spheres, as we $ee in the Planets, and mo$t $en$ibly in the Medi- cean Stars, ob$erve to make the le$$er Orbs to circulate in $horter times: Whence the time of <I>Saturns</I> revolution is longer than all the times of the other le$$er Spheres, it being thirty years; now the pa$$ing from this to a Sphere very much bigger, and to make it move in 24. hours, may very well be $aid to exceed the rules of the model. So that if we would but attentively con$ider it, the Authors an$wer oppo$eth not the intent and $en$e of the argument, but the expre$$ing and manner of delivering of it; where again the Author is injurious, and cannot deny but that he artificially di$$embled his under$tanding of the words, that he might charge <I>Kepler</I> with gro$$e ignorance; but the impo$ture was $o very dull and obvions, that he could not with all his craft alter the opini- on which <I>Kepler</I> hath begot of his Doctrine in the minds of all the Learned. As in the next place, to the in$tance again$t the perpetual motion of the Earth, taken from the impo$$ibility of its moving long without wearine$$e, in regard that living crea- <foot>Hh 2 tures</foot> <p n=>244</p> tures them$elves, which yet move naturally, and from an intern principle, do grow weary, and have need of re$t to relax and re- fre$h their members --------</P> <P>SAGR. Methinks I hear <I>Kepler</I> an$wer him to that, that there are $ome kinde of animals which refre$h them$elves after wearine$$e, by rowling on the Earth; and that therefore there <marg><I>The feigned an- $wer of</I> Kepler <I>co- vered with an ar- tificial Irony.</I></marg> is no need to fear that the Terre$trial Globe $hould tire, nay it may be rea$onably affirmed, that it enjoyeth a perpetual & mo$t tranquil repo$e, keeping it $elf in an eternal rowling.</P> <P>SALV. You are too tart and Satyrical, <I>Sagredus:</I> but let us lay a$ide je$ts, whil$t we are treating of $erious things.</P> <P>SAGR. Excu$e me, <I>Salviatus,</I> this that I $ay is not $o ab$o- lutely be$ides the bu$ine$s, as you perhaps make it; for a motion that $erveth in$tead of re$t, and removeth wearine$s from a body tired with travail, may much more ea$ily $erve to prevent the co- <marg><I>Animals would not grow weary of their motion, pro- ceeding as that which is a$$igned to the terre$trial Globe.</I></marg> ming of that wearine$s, like as preventive remedies are more ea$ie than curative. And I hold for certain, that if the motion of ani- mals $hould proceed in the $ame manner as this that is a$cribed to the Earth, they would never grow weary; Seeing that the weari- ne$s of the living creature, deriveth it $elf, in my opinion, from <marg><I>The cau$e of the wearine$$e of ani- mals.</I></marg> the imployment of but one part alone in the moving of its $elf, and all the re$t of the body; as <I>v. g.</I> in walking, the thighs and the legs onely are imployed for carrying them$elves and all the re$t: on the contrary, you $ee the motion of the heart to be as it were indefatigable, becau$e it moveth it $elf alone. Be$ides, I <marg><I>The motion of an animal is rather to be called violent than natural.</I></marg> know not how true it may be, that the motion of the animal is na- tural, and not rather violent: nay, I believe that one may truly $ay, that the $oul naturally moveth the members of an animal with a motion preternatural, for if the motion upwards is preternatu- ral to grave bodies, the lifting up of the legs, and the thighs, which are grave bodies, in walking, cannot be done without vio- lence, and therefore not without labour to the mover. The climbing upwards by a ladder carrieth the grave body contrary to its natural inclination upwards, from whence followeth wearine$s, by rea$on of the bodies natural aver$ne$s to that motion: but in moving a moveable with a motion, to which it hath no aver$ion, <marg><I>The $trength di- mini$heth not, where it is not im- ployed.</I></marg> what la$$itude, what diminution of vertue and $trength need we fear in the mover? and how $hould the forces wa$te, where they are not at all imployed?</P> <P>SIMP. They are the contrary motions wherewith the Earth is pretended to move, again$t which the Authour produceth his ar- gument.</P> <P>SAGR. It hath been $aid already, that they are no wi$e con- traries, and that herein the Authour is extteamly deceived, $o that the whole $trength of the argument recoileth upon the Op- <foot>ponent</foot> <p n=>245</p> ponent him$elf, whil$t he will make the <I>Fir$t Mover</I> to hurry <marg><I>The argument of</I> Cla<*>ntius, <I>recoileth upon him- $elf.</I></marg> along with it all the inferiour Spheres, contrary to the motion which they them$elves at the $ame time exerci$e. It belongs there- fore to the <I>Primum Mobile</I> to grow weary, which be$ides the moving of its $elf is made to carry $o many other Spheres, and which al$o $trive again$t it with a contrary motion. So that the ultimate conclu$ion that the Authour inferred, $aying, that di$cour$ing of the effects of Nature, a man alwayes meets with things that favour the opinion of <I>Ari$toile</I> and <I>Ptolomy,</I> and ne- ver any one that doth not interfer with <I>Copernicus,</I> $tands in need of great con$ideration; and it is better to $ay, that one of the$e two <I>Hypothe$es</I> being true, and the other nece$$arily fal$e, it is impo$$ible that a man $hould ever be able to finde any argu- ment, experience, or right rea$on, in favour of that which is <marg><I>True Propo$iti- ons meet with ma- ny conclu$ive ar- guments, $o do not the fal$e.</I></marg> fal$e, like as to the truth none of the$e things can be repugnant. Va$t difference, therefore, mu$t needs be found between the rea- $ons and arguments produced by the one and other party, for and again$t the$e two opinions, the force of which I leave you your $elf to judge of, <I>Simplicius.</I></P> <P>SALV. But you, <I>Sagredus,</I> being tran$ported by the velocity of your wit, have taken my words out of my mouth, whil$t I was about to $ay $omething, touching this la$t argument of the Author; and although you have more then $ufficiently refuted him, yet neverthele$$e I will adde $omewhat, which then ran in my minde. He propo$eth it as a thing very unlikely, that a body di$$ipable and corruptible, as the Earth, $hould perpetually move with a re- gular motion, c$pecially for that we $ee living creatures in the end to grow weary, and to $tand in need of re$t: and the improbability is increa$ed, in that the $aid motion is required to be of velocity incomparable and immen$e, in re$pect to that of animals. Now, I cannot $ee why the velocity of the Earth $hould, at pre$ent, trou- ble it; $o long as that of the $tarry Sphere $o very much bigger doth not occa$ion in it any di$turbance more con$iderable, than that which the velocity of a machine, that in 24 hours maketh but one $ole revolution, produceth in the $ame. If the being of the velo- city of the Earths conver$ion, according to the model of that ma- chine, inferreth things of no greater moment than that, let the Au- thor cea$e to fear the Earths growing weary; for that not one of the mo$t feeble and $low-pac't animals, no not a Chamæleon would <marg>* Cinque ò $ei braccia Fiorentini.</marg> tire in moving no more than ^{*} four or five yards in 24 hours; but if he plea$e to con$ider the velocity to be no longer, in relation to <marg><I>Wearineß more to be feared in the $tarry Sphere than in the terre$triall Globe.</I></marg> the model of the machine, but ab$olutely, and ina$much as the moveable in 24 hours is to pa$s a very great $pace, he ought to $hew him$elf $o much more re$erved in granting it to the $tarry Sphere, which with a velocity incomparably greater than that of the <foot>Earth</foot> <p n=>246</p> Earth is to carry along with it a thou$and bodies, each much big- ger than the Terre$trial Globe.</P> <P>Here it remains for us to $ee the proofs, whereby the Authour concludes the new $tars <I>Anno</I> 1572. and <I>Anno</I> 1604. to be $ublu- nary, and not cœle$tial, as the <I>Astronomers</I> of tho$e times were generally per$waded; an enterprize very great certainly; but I have con$idered, that it will be better, in regard the Book is new and long, by rea$on of its many calculations, that between this e- vening and to morrow morning I make them as plain as I can, and $o meeting you again to morrow to continue our wonted confe- rences, give you a brief of what I $hall ob$erve therein; and if we have time left, we will $ay $omething of the <I>Annual motion</I> a$cri- bed to the Earth. In the mean time, if either of you, and <I>Simpli- cius</I> in particular, hath any thing to $ay more, touching what relates to the <I>Diurnal motion,</I> at large examined by me, we have a little time $till left to treat thereof.</P> <P>SIMP. I have no more to $ay, unle$$e it be this, that the di$cour- $es that this day have falne under our debate, have appeared to me fraught with very acute and ingenious notions, alledged on <I>Coper- nicus</I> his $ide, in confirmation of the motion of the Earth, but yet I find not my $elf per$waded to believe it; for in $hort, the things that have been $aid conclude no more but this, that the rea$ons for the $tability of the Earth are not nece$$ary; but all the while no demon$tration hath been produced on the other $ide, that doth nece$$arily convince and prove its mobility.</P> <P>SALV. I never undertook, <I>Simplicius,</I> to remove you from that your opinion; much le$s dare I pre$ume to determine definitively in this controver$ie: it onely was, and $till $hall be in the en$uing di$putations my intent, to make it appear to you, that tho$e who have thought that mo$t $wift motion of 24 hours doth belong to the Earth alone, and not to the Univer$e, the Earth onely exclu- ded, were not induced to believe, that $o it might and ought to do out of any blind per$wa$ion; but that they did very well $ee, try, and examine the rea$ons on the contrary $ide, and al$o not $light- ly an$wer them. With the $ame intention, if it $tand with your liking, and that of <I>Sagredus,</I> we may pa$$e to the con$ideration of that other motion; fir$t, by <I>Aristarchus Samius,</I> and afterwards by <I>Nicholaus Copernicus</I> a$cribed to the $aid Terre$trial Globe, which is, as, I believe, you have heretofore heard, made under the Zodiack within the $pace of a year about the Sun, immoveably placed in the centre of the $aid Zodiack.</P> <P>SIMP. The di$qui$ition is $o great, and $o noble, that I $hall gladly hearken to the di$cu$$ion thereof, per$wading my $elf that I $hall hear what ever can be $aid of that matter. And I will after- <foot>wards</foot> <pb> <fig> <cap><I>Place this Plate at the end of the Second</I></cap> <cap>Dialogue.</cap> <p n=>247</p> wards by my $elf, according to my u$ual cu$tome, make more de- liberate reflexions upon what hath been, and is to be $poken; and if I $hould gain no more but this, it will be no $mall benefit that I $hall be able to di$cour$e more Logically.</P> <P>SAGR. Therefore, that we may no further weary <I>Salviatus,</I> we will put a period to the di$putations of this day, and re- a$$ume our conference to morrow in the u$ual manner, with hope to hear very plea$ing novelties.</P> <P>SIMP. I will leave with you the Book <I>De $tellis novis,</I> and car- ry back this of the Conclu$ions, to $ee what is written therein a- gain$t the Annual motion, which are to be the arguments of our di$cour$e to morrow.</P> <fig> <p n=>249</p> <head>GALILÆUS Galilæus Lyncæus, HIS SYSTEME OF THE WORLD.</head> <head>The Third Dialogue.</head> <head><I>INTERLOCVTORS.</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <P>SAGR. The great de$ire wherewith I have expected your coming, that I might hear the novel conceits touching the annual conver$i- on of this our Globe, hath made me think the houres of the la$t night, and tho$e of this morning very tedious, al- though I $pent them not idly, but lying awake I imployed a good part thereof in ruminating upon our ye$terdayes di$cour- $es, weighing the rea$ons alledged by both parties, in favour of the two contrary Hypothe$es, that of <I>Ari$totle</I> and <I>Ptolomy,</I> and this of <I>Ari$tarchus,</I> and <I>Copernicus.</I> And really methinks, that which ever of the$e parties have been deceived, they are worthy of excu$e, $o $pecious and valid in appearance are the rea$ons that may have per$waded them either way; though neverthele$$e we <foot>Ii do</foot> <p n=>250</p> do for the mo$t part clo$e with tho$e produced by the grave Au- thours fir$t above mentioned. But albeit that the <I>Peripatetick Hy- pothe$is,</I> by rea$on of its antiquity, hath had many followers and fautors, and the other very few; fir$t, for its ob$curity, and next, for its novelty, yet methinks I di$cover among$t tho$e many, and particularly among$t the modernes $ome, who for the $up- port of the opinion by them e$teemed true, have introduced other rea$ons $ufficiently childi$h, I could $ay ridiculous.</P> <P>SALV. I have met with the like, and $o much wor$e than <marg><I>Some in arguing fir$t fix in their minds the conclu- $ion beleeved by them, and then a- dapt their rea$ons to that.</I></marg> yours, that I blu$h to rehear$e them, not $o much to $pare the fame of their Authours, the names of whom might be perpetually con- cealed, as becau$e I am a$hamed $o much to $tain the honour of mankinde. In ob$erving of the$e men, I have found that $ome there are who prepo$terou$ly rea$oning, fir$t $tabli$h the conclu- $ion in their fancy, and (either becau$e it is their own, or el$e be- longs to a per$on whom they much confide in) $o firmly imprint it in their opinions, that it is altogether impo$$ible ever wholly to efface it: and tho$e rea$ons which they them$elves $tumble upon, or which they hear others to alledge in confirmation of the con- ceit entertained, though never $o $imple and in$ipid, in$tantly find credit and applau$e with them: but on the contrary, tho$e which are brought again$t their opinion, though ingenuous and conclu- ding, they receive not only with nau$eating, but with di$dain and bitter indignation, yea, you $hall have one of the$e $o inraged, as that he will not be backward to try all wayes to $uppre$s and $ilence their adver$aries: and of this I my $elf have had $ome experience.</P> <P>SAGR. Indeed the$e men deduce not the conclu$ion from the premi$es, nor confirme them with rea$ons, but accomodate, or to $ay better, di$commodate and di$tort the premi$es and arguments to make them $peak in favour of their pre-a$$umed and pertinaci- ous conclu$ions. It is not good therefore to contract familiarity with the$e men; and the rather, for that their conver$ation is not only unplea$ant, but al$o dangerous. Yet let us continue our con- ference with <I>Simplicius</I> however, whom I have known this long while for a man of great ingenuity, and altogether void of malice: be$ides he is well ver$t in the Peripatetick Doctrine; $o that I may a$$ure my $elf, that what $hall not fall within the reach of his rea- $on for the $upport of the <I>Ari$totelian</I> Hypothe$is, will not ea$ily be found out by others. But $ee yonder he comes, quite out of winde, who$e company we have $o long de$ired: we were ju$t now $peaking again$t the $mall ha$t you made to come to us.</P> <P>SIMP. You mu$t not blame me, but <I>Neptune,</I> for this my long $tay; which in the ebbe of this mornings tide hath in a manner drain'd away the waters, for the <I>Gondola</I> that brought me, being entered not far from hence into a certain Channel, wanting depth, <foot>where</foot> <p n=>251</p> where I was $tranded, and forced to $tay there more than a full hour, in expecting the return of the tide: and there waiting in this manner, without being able to get out of the boat, which on a $udden ran a ground, I ob$erved a certain accident, which to me <marg><I>The motion of the water in ebbing and flowing not in- terrupted by re$t.</I></marg> $eemed very $trange; and it was this, that in the waters ebbing I $aw it retreat very fa$t by $everal $mall rivolets, the ouze being in many places di$covered, and whil$t I $tood looking upon this ef- fect, I $aw this motion in an in$tant to cea$e, and without a mi- nutes interval the $ame water to begin to return back again, and the tide from ebbing to become young flood, without $tanding $till a moment: an effect that as long as I have dwelt in <I>Venice,</I> I never took notice of before.</P> <P>SAGR. It is very much, that you $hould be left thus on ground, among$t $mall Channels; in which rivolets, as having very little declivity, the ri$ing or falling of the main $ea, the thickne$s onely of a paper is $ufficient to make the water to ebbe and flow for good long $paces of time: like as in $ome creeks of the Sea, its flowing four or $ix ^{*} yards onely, maketh the water to overflow the adja- cent Mar$hes for $ome hundreds and thou$ands of ^{*} acres.</P> <marg>* Pertiche vene- tiani.</marg> <P>SIMP. This I know very well, but I $hould have thought, that between the ultimate terme of ebbing, and the fir$t beginnng to flow, there $hould have interpo$ed $ome con$iderable interval of re$t.</P> <P>SAGR. This will appear unto you, if you ca$t your eye upon the bank or piles, where the$e mutations are made perpendicular- ly, but not that there is any real time of ce$$ation.</P> <P>SIMP. I did think, that becau$e the$e two motions were con- trary, there ought to be in the mid$t between them $ome kind of re$t; conformable to the Doctrine of <I>Ari$totle,</I> which demon$trates. that <I>in puncto regre$$us mediat quies.</I></P> <P>SAGR. I very well remember this place: but I bear in minde al$o, that when I read Philo$ophy, I was not thorowly $atisfied with <I>Ari$totles</I> demon$tration; but that I had many experiments on the contrary, which I could $till rehear$e unto you, but I am unwilling to $ally out into any other digre$$ions, we being met here to di$cour$e of the propo$ed mattes, if it be po$$ible, without the$e excur$ions wherewith we have interrupted our di$putes in tho$e dayes that are pa$t.</P> <P>SIMP. And yet we may with convenience, if not interrupt them, at lea$t prolong them very much, for returning ye$ter- day home, I $et my $el$ to read the Tractate of Conclu$ions, where I find Demon$trations again$t this annual motion a$cribed to the Earth, very $olid; and becau$e I would not tru$t my memory with the punctual relation of them, I have brought back the Book a- long with me.</P> <foot>Ii 2 SAGR</foot> <p n=>252</p> <P>SAGR. You have done very well; but if we would re-a$$ume our Di$putations according to ye$terdayes appointment, it is re- qui$ite that we fir$t hear what account <I>Salviatus</I> hath to give us of the Book, <I>De $tellis novis,</I> and then without interruption we may proceed to the Annual motion. Now what $ay you, <I>Salvia- tus</I> touching tho$e $tars? Are they really pull'd down from Hea- ven to the$e lower regions, by vertue of that Authours calculati- ons, whom <I>Simplicius</I> mentioneth?</P> <P>SALV. I $et my $elf la$t night to peru$e his proceedings, and I have this morning had another view of him, to $ee whether that which he $eemed over night to affirm, were really his $en$e, or my dreams and phanta$tical nocturnal imaginations; and in the clo$e found to my great grief that tho$e things were really written and printed, which for the reputation-$ake of this Philo$opher I was unwilling to believe. It is in my judgment impo$$ible, but that he $hould perceive the vanity of his undertaking, a$well becau$e it is too apert, as becau$e I remember, that I have heard him mentio- ned with applau$e by the <I>Academick our Friend</I>: it $eemeth to me al$o to be a thing very unlikely, that in complacency to others, he $hould be induced to $et $o low a value upon his reputation, as to give con$ent to the publication of a work, for which he could expect no other than the cen$ure of the Learned.</P> <P>SAGR. Yea, but you know, that tho$e will be much fewer than one for an hundred, compared to tho$e that $hall celebrate and extoll him above the greate$t wits that are, or ever have been in the world: He is one that hath mentioned the Peripate- tick inalterability of Heaven again$t a troop of <I>A$tronomers,</I> and that to their greater di$grace hath foiled them at their own wea- pons; and what do you think four or five in a Countrey that di$- cern his triflings, can do again$t the innumerable multitude, that, not being able to di$cover or comprehend them, $uffer them$elves to be taken with words, and $o much more applaud him, by how much the le$$e they under$tand him? You may adde al$o, that tho$e few who under$tand, $corn to give an an$wer to papers $o trivial and unconcludent; and that upon very good rea$ons, be- cau$e to the intelligent there is no need thereof, and to tho$e that do not under$tand, it is but labour lo$t.</P> <P>SALV. The mo$t de$erved puni$hment of their demerits would certainly be $ilence, if there were not other rea$ons, for which it is haply no le$$e than nece$$ary to re$ent their timerity: one of which is, that we <I>Italians</I> thereby incur the cen$ure of Illiterates, and attract the laughter of Forreigners; and e$pecially to $uch who are $eparated from our Religion; and I could $hew you ma- ny of tho$e of no $mall eminency, who $coff at our <I>Academick,</I> and the many Mathematicians that are in <I>Italie,</I> for $uffering the <foot>follies</foot> <p n=>253</p> follies of $uch a ^{*} Fabler again$t <I>A$tronomers</I> to come into the <marg>* Lorenzini.</marg> light, and to be openly maintained without contradiction; but this al$o might be di$pen$ed with, in re$pect of the other greater occa$ions of laughter, wherewith we may confront them depend- ing on the di$$imulation of the intelligent, touching the follies of the$e opponents of the Doctrines that they well enough under- $tand.</P> <P>SAGR. I de$ire not a greater proof of tho$e mens petulancy, and the infelicity of a <I>Copernican,</I> $ubject to be oppo$ed by $uch as under$tand not $o much as the very fir$t po$itions, upon which he undertakes the quarrel.</P> <P>SALV. You will be no le$$e a$toni$hed at their method in con- futing the <I>Astronomers,</I> who affirm the new Stars to be $uperiour to the Orbs of the Planets; and peradventure in the ^{†} Firmament <marg>† He taketh the Firmament for the Starry Sphere, and as we vulgarly re- ceive the word.</marg> it $elf.</P> <P>SAGR. But how could you in $o $hort a time examine all this Book, which is $o great a Volume, and mu$t needs contain very many demon$trations.?</P> <P>SALV. I have confined my $elf to the$e his fir$t confutations, in which with twelve demon$trations founded upon the ob$ervations of twelve <I>A$tronomers,</I> (who all held, that the Star, <I>Anno</I> 1572. which appeared in <I>Ca$$iopeia,</I> was in the Firmament) he proveth it on the contrary, to be beneath the Moon, conferring, two by two, the meridian altitudes, proceeding in the method that you $hall under$tand by and by. And becau$e, I think, that in the exami- nation of this his fir$t progre$$ion, I have di$covered in this Au- thour a great unlikelihood of his ability to conclude any thing a- gain$t the <I>A$tronomers,</I> in favour of the <I>Peripatetick Philo$ophers,</I> and that their opinion is more and more concludently confirmed, I could not apply my $elf with the like patience in examining his other methods, but have given a very $light glance upon them, and am certain, that the defect that is in the$e fir$t impugnations, is likewi$e in the re$t. And as you $hall $ee, by experience, very few words will $uffice to confute this whole Book, though compi- led with $o great a number of laborious calculations, as here you <marg><I>The method ob- $erved by</I> Clar. <I>in confuting the A- $tronomers, and by</I> Salviatus <I>in confu- ting him.</I></marg> $ee. Therefore ob$erve my proceedings. This Authour under- taketh, as I $ay, to wound his adver$aries with their own weapons, <I>i.e.</I> a great number of ob$ervations made by them$elves, to wit, by twelve or thirteen Authours in number, and upon part of them he makes his $upputations, and concludeth tho$e $tars to have been below the Moon. Now becau$e the proceeding by interrogato- ries very much plea$eth me, in regard the Authour him$elf is not here, let <I>Simplicius</I> an$wer me to the que$tions that I $hall ask him, as he thinks he him$elf would, if he were pre$ent. And pre- $uppo$ing that we $peak of the fore$aid Star, of <I>Anno</I> 1572. ap- <foot>pearing</foot> <p n=>254</p> pearing in <I>Ca$$iopeia,</I> tell me, <I>Simplicius,</I> whether you believe that it might be in the $ame time placed in divers places, that is, a- mong$t the Elements, aud al$o among$t the planetary Orbs, and al$o above the$e among$t the fixed Stars, and yet again infinitely more high.</P> <P>SIMP. There is no doubt, but that it ought to be confe$$ed that it is but in one only place, and at one $ole and determinate di$tance from the Earth.</P> <P>SALV. Therefore if the ob$ervations made by the A$trono- mers were exact, and the calculations made by this Author were not erroneous, it were ea$ie from all tho$e and all the$e to re- collect the $ame di$tances alwayes to an hair, is not this true?</P> <P>SIMP. My rea$on hitherto tells me that $o it mu$t needs be; nor do I believe that the Author would contradict it</P> <P>SALV. But when of many and many computations that have been made, there $hould not be $o much as two onely that prove true, what would you think of them?</P> <P>SIMP. I would think that they were all fal$e, either through the fault of the computi$t, or through the defect of the ob$er- vators, and at the mo$t that could be $aid, I would $ay, that but onely one of them and no more was true; but as yet I know not which to choo$e.</P> <P>SALV. Would you then from fal$e fundamentals deduce and e$tabli$h a doubtful conclu$ion for ttue? Certainly no. Now the calculations of this Author are $uch, that no one of them agrees with another, you may $ee then what credit is to be given to them.</P> <P>SIMP. Indeed, if it be $o, this is a notable failing.</P> <P>SAGR. But by the way I have a mind to help <I>Simplicius,</I> and the Author by telling <I>Salviatus,</I> that his arguments would hold good if the Author had undertook to go about to find out exact- ly the di$tance of the Star from the Earth, which I do not think to be his intention; but onely to demon$trate that from tho$e ob$ervations he collected that the Star was $ublunary. So that if from tho$e ob$ervations, and from all the computations made thereon, the height of the Star be alwayes collected to be le$$e than that of the Moon, it $erves the Authors turn to con- vince all tho$e A$tronomers of mo$t impardonable ignorance, that through the defect either of Geometry or Arithmetick, have not known how to draw true conclu$ions from their own ob$erva- tions them$elves.</P> <P>SALV. It will be convenient therefore that I turn my $elf to you, <I>Sagredus,</I> who $o cunningly aphold the Doctrine of this Author. And to $ee whether I can make <I>Simplicius,</I> though not very expert in calcnlations, and demon$trations to apprehend the <foot>in-</foot> <p n=>255</p> inconclu$ivene$$e at lea$t of the demon$trations of this Author, fir$t propo$ed to con$ideration, and how both he, and all the A$tronomers with whom he contendeth, do agree that the new Star had not any motion of its own, and onely went round with the diurnal motion of the <I>primum mobile</I>; but di$$ent about the placing of it, the one party putting it in the Cele$tial Region, that is above the Moon, and haply above the fixed Stars, and the other judging it to be neer to the Earth, that is, under the concave of the Lunar Orb. And becau$e the $ituation of the new $tar, of which we $peak, was towards the North, and at no very great di$tance from the Pole, $o that to us <I>Septentrionals,</I> it did never $et, it was an ea$ie matter with A$tronomical in$truments to have taken its $everal meridian altitudes, as well its $malle$t under the Pole, as its greate$t above the $ame; from the compa- ring of which altitudes, made in $everal places of the Earth, $ituate at different di$tances from the North, that is, different from one another in relation to polar altitudes, the $tars di$tance might be inferred: For if it was in the Firmament among$t the <marg><I>The greate$t and lea$t elevations of the new $tar differ not from each o- ther more than the polar allitudes, the $aid $tar being in the Firmnment.</I></marg> other fixed $tars, its meridian altitudes taken in divers elevations of the pole, ought nece$$arily to differ from each other with the $ame variations that are found among$t tho$e elevations them- $elves; that is, for example, if the elevation of the $tar above the horizon was 30 degrees, taken in the place where the polar altitude was <I>v. gr.</I> 45 degrees, the elevation of the $ame $tar ought to have been encrea$ed 4 or 5 degrees in tho$e more Nor- thernly places where the pole was higher by the $aid 4 or 5 de- grees. But if the $tars di$tance from the Earth was but very little, in compari$on of that of the Firmament; its meridian altitudes ought approaching to the North to encrea$e con$iderably more than the polar altitudes; and by that greater encrea$e, that is, by the exce$$e of the encrea$e of the $tars elevation, above the encrea$e of the polar elevation (which is called the difference of Parallaxes) is readily calculated with a cleer and $ure method, the $tars di$tance from the centre of the Earth. Now this Author taketh the ob$ervations made by thirteen A$tronomers in $undry elevations of the pole, and conferring a part of them at his plea- $ure, he computeth by twelve collations the new $tars height to have been alwayes beneath the Moon; but this he adventures to do in hopes to find $o gro$$e ignorance in all tho$e, into who$e hands his book might come, that to $peak the truth, it hath turn'd my $tomack; and I wait to $ee how tho$e other A$tronomers, and particularly <I>Kepler,</I> again$t whom this Author principally in- veigheth, can contein them$elves in $ilence, for he doth not u$e to hold his tongue on $uch occa$ions; unle$$e he did po$$ibly think the enterprize too much below him. Now to give you to <foot>un-</foot> <p n=>256</p> under$tand the $ame, I have upon this paper tran$cribed the con- clu$ions that he inferreth from his twelve indagations; the fir$t of which is upon the two ob$ervations:</P> <table> <row><col>Of <I>Maurolicus</I> and <I>Hainzelius,</I> from which the Star is collected to have been di$tant from the centre le$$e than 3 $emidiameters of the Earth, the difference of Parallaxes being 4 <I>gr. 42 m.</I> 30 <I>$ec.</I></col><col>3 <I>$emid.</I></col></row> <row><col>2. And is calculated on the ob$ervations of <I>Hain- zelius,</I> with Parall. of 8. <I>m. 30 $ec.</I> and its di- $tance from the centre is computed to be more than</col><col>25 <I>$emid.</I></col></row> <row><col>3. And upon the ob$ervations of <I>Tycho</I> and <I>Hain- zelius,</I> with Parall. of 10 <I>m.</I> and the di$tance of the centre is collected to be little le$$e than</col><col>19 <I>$emid.</I></col></row> <row><col>4. And upon the ob$ervations of <I>Tycho</I> and the <I>Landgrave,</I> with Parall. of 14 <I>m.</I> the di$tance from the centre is made to be about</col><col>10 <I>$emid.</I></col></row> <row><col>5. And upon the ob$ervations of <I>Hainzelius</I> and <I>Gemma,</I> with Parall. of 42 <I>m. 30 $ec.</I> whereby the di$tance is gathered to be about</col><col>4 <I>$emid.</I></col></row> <row><col>6. And upon the ob$ervations of the <I>Landgrave</I> and <I>Camerarius,</I> with Parall. of 8 <I>m.</I> the di- $tance is concluded to be about</col><col>4 <I>$emid.</I></col></row> <row><col>7. And upon the ob$ervations of <I>Tycho</I> and <I>Hage- cius,</I> with Parall. of 6 <I>m.</I> and the di$tance is made</col><col>31 <I>$emid.</I></col></row> <row><col>8. And upon the ob$ervations of <I>Hagecius</I> and <I>Vr- $inus</I> with Parall. of 43 <I>m.</I> and the $tars di$tance from the $uperficies of the Earth is rendred</col><col>1/2 <I>$emid.</I></col></row> <row><col>9. And upon the ob$ervations of <I>Landgravius</I> and <I>Bu$chius,</I> with Parall. of 15 <I>m.</I> and the di- $tance from the $uperficies of the Earth is by $upputation</col><col>1/48 <I>$emid.</I></col></row> <row><col>10. And upon the ob$ervations of <I>Maurolice</I> and <I>Munocius,</I> with Parall. of 4 <I>m. 30 $ec.</I> and the compnted di$tance from the Earths $urface is</col><col>1/5 <I>$emid.</I></col></row> <row><col>11. And upon the ob$ervations of <I>Munocius</I> and <I>Gemma,</I> with Parall. of 55 <I>m.</I> and the di$tance from the centre is rendred</col><col>13 <I>$emid.</I></col></row> <foot>12. And</foot> <p n=>257</p> <row><col>12. And upon the ob$ervations of <I>Muno$ius</I> and <I>Vr$inus</I> with Parall. of 1 <I>gr. 36 m.</I> and the di- $tance from the centre cometh forth le$$e than</col><col>7 <I>$emid.</I></col></row> </table> <P>The$e are twelve indagations made by the Author at his electi- on, among$t many which, as he $aith, might be made by combi- ning the ob$ervations of the$e thirteen ob$ervators. The which twelve we may believe to be the mo$t favourable to prove his intention.</P> <P>SAGR. I would know whether among$t the $o many other in- dagations pretermitted by the Author, there were not $ome that made again$t him, that is, from which calculating one might find the new $tar to have been above the Moon, as at the very fir$t $ight I think we may rea$onably que$tion; in regard I $ee the$e already produced to be $o different from one another, that $ome of them give me the di$tance of the $aid $tar from the Earth, 4, 6, 10, 100, a thou$and, and an hundred thou$and times bigger one than another; $o that I may well $u$pect that among$t tho$e that he did not calculate, there was $ome one in fauour of the adver$e party. And I gue$$e this to be the more probable, for that I can- not conceive that tho$e A$tronomers the ob$ervators could want the knowledg and practice of the$e computations, which I think do not depend upon the ab$truce$t things in the World. And in- deed it will $eem to me a thing more than miraculous, if whil$t in the$e twelve inve$tigations onely, there are $ome that make the $tar to be di$tant from the Earth but a few miles, and others that make it to be but a very fmall matter below the Moon, there are none to be found that in favour of the contrary part do make it $o much as twenty yards above the Lunar Orb. And that which $hall be yet again more extravagant, that all tho$e A$tronomers $hould have been $o blind as not to have di$covered that their $o apparent mi$take.</P> <P>SALV. Begin now to prepare your ears to hear with infinite admiration to what exce$$es of confidence of ones own authority and others folly, the de$ire of contradicting and $hewing ones $elf wi$er than others, tran$ports a man. Among$t the indaga- tions omitted by the Author, there are $uch to be found as make the new $tar not onely above the Moon, but above the fixed $tars al$o. And the$e are not a few, but the greater part, as you $hall $ee in this other paper, where I have $et them down.</P> <P>SAGR. But what $aith the Author to the$e? It may be he did not think of them?</P> <P>SALV. He hath thought of them but too much: but $aith, that the ob$ervations upon which the calculations make the $tar to be infinitely remote, are erroneous, and that they cannot be com- bined to one another.</P> <foot>Kk SIMP.</foot> <p n=>258</p> <P>SIMP. But this $eemeth to me a very lame eva$ion; for the ad- ver$e party may with as much rea$on reply, that tho$e are errone- ous wherewith he collecteth the $tar to have been in the Elemen- tary Region.</P> <P>SALV. Oh <I>Simplicius,</I> if I could but make you comprehend the craft, though no great craftine$$e of this Author, I $hould make you to wonder, and al$o to be angry to $ee how that he palliating his $agacity with the vail of the $implicity of your $elf; and the re$t of meer Philo$ophers, would in$inuate him$elf into your good opinion, by tickling your cars, and $welling your am- bition, pretending to have convinced and $ilenced the$e petty A$tronomers, who went about to a$$ault the impregnable inalte- rability of the <I>Peripatetick</I> Heaven, and which is more, to have foild and conquered them with their own arms. I will try with all my ability to do the $ame; and in the mean time let <I>Sagredus</I> take it in good part, if <I>Simplicius</I> and I try his patience, perhaps a little too much, whil$t that with a $uperfluous circumlocution ($uperfluous I $ay to his mo$t nimble apprehen$ion) I go about to make out a thing, which it is not convenient $hould be hid and unknown unto him.</P> <P>SAGR. I $hall not onely without wearine$$e, but al$o with much delight hearken to your di$cour$es; and $o ought all <I>Peripa- tetick</I> Philo$ophers, to the end they may know how much they are oblieged to this their Protector.</P> <P>SALV. Tell me, <I>Simplicius,</I> whether you do well comprehend, how, the new $tar being placed in the meridian circle yonder to- wards the North, the $ame to one that from the South $hould go towards the North, would $eem to ri$e higher and higher a- bove the Horizon, as much as the Pole, although it $hould have been $cituate among$t the fixed $tars; but, that in ca$e it were con$iderably lower, that is nearer to the Earth, it would appear to a$cend more than the $aid pole, and $till more by how much its vicinity was greater?</P> <P>SIMP. I think that I do very well conceive the $ame; in to- ken whereof I will try if I can make a mathematical Scheme of it, and in this great circle <I>[in Fig. 1. of this Dialogue.]</I> I will marke the pole P; and in the$e two lower circles I will note two $tars beheld from one place on the Earth, which let be A; and let the two $tars be the$e B and C, beheld in the $ame line A B C, which line I prolong till it meet with a fixed $tar in D. And then walking along the Earth, till I come to the term E, the two $tars will appear to me $eparated from the fixed $tar D, and ad- vanced neerer to the pole P, and the lower $tar B more, which will appear to me in G, and the $tar C le$$e, which will ap pear to me in F, but the fixed $tar D will have kept the $ame di$tance from the Pole.</P> <foot>SALV.</foot> <p n=>259</p> <P>SALV. I $ee that you under$tand the bu$ine$$e very well. I be- lieve that you do likewi$e comprehend, that, in regard the $tar B is lower than C, the angle which is made by the rayes of the $ight, which departing from the two places A and E, meet in C, to wit, this angle A C E, is more narrow, or if we will $ay more acute than the angle con$tituted in B, by the rayes A B and E <I>B.</I></P> <P>SIMP. This I likewi$e under$tand very well.</P> <P>SALV. And al$o, the Earth beine very little and almo$t in$en- $ible, in re$pect of the Firmament <I>(or Starry Sphere</I>;) and con- $equently the $pace A E, paced on the Earth, being very $mall in compari$on of the immen$e length of the lines E G and E F, pa$- $ing from the Earth unto the Firmament, you thereby collect that the $tar C might ri$e and a$cend $o much and $o much above the Earth, that the angle therein made by the rayes which depart from the $aid $tationary points A and E, might become mo$t a- cute, and as it were ab$olutely null and in$en$ible.</P> <P>SIMP. And this al$o is mo$t manife$t to $en$e.</P> <P>SALV. Now you know <I>Simplicius</I> that A$tronomers and Ma- thematicians have found infallible rules by way of Geometry and Arithmetick, to be able by help of the quantity of the$e angles <I>B</I> and C, and of their differences, with the additional knowledg of the di$tance of the two places A and E, to find to a foot the remotene$$e of $ublime bodies; provided alwayes that the afore- $aid di$tance, and angles be exactly taken.</P> <P>SIMP. So that if the Rules dependent on <I>Geometry</I> and <I>A$tro- nomy</I> be true, all the fallacies and errours that might be met with in attempting to inve$tigate tho$e altitudes of new Stars or Co- mets, or other things mu$t of nece$$ity depend on the di$tance A E, and on the angles B and C, not well mea$ured. And thus all tho$e differences which are found in the$e twelve workings depend, not on the de$ects of the rules of the Calculations, but on the errours committed in finding out tho$e angles, and tho$e di$tances, by means of the In$trumental Ob$ervations.</P> <P>SALV. True; and of this there is no doubt to be made. Now it is nece$$ary that you ob$erve inten$ely, how in removing the Star from B to C, whereupon the angle alwayes grows more acute, the ray E B G goeth farther and farther off from the ray A B D in the part beneath the angle, as you may $ee in the line E C F, who$e inferiour part E C is more remote from the part A C, than is the part E B, but it can never happen, that by any what$oever immen$e rece$$ion, the lines A D and E F $hould totally $ever from each other, they being finally to go and conjoyn in the Star: and onely this may be $aid, that they would $eparate, and reduce them- $elves to parallels, if $o be the rece$$ion $hould be infinite, which <foot>Kk 2 ca$e</foot> <p n=>260</p> ca$e is not to be $uppo$ed. But becau$e (ob$erve well) the di$tance of the Firmament, in relation to the $mallne$$e of the Earth, as hath been $aid, is to be accounted, as if it were infinite; therefore the angle conteined betwixt the two rayes, that being drawn from the points A and E, go to determine in a fixed Star, is e$teemed nothing, and tho$e rayes held to be two parallel lines; and there- fore it is concluded, that then only may the New Star be affirmed to have been in the Firmament, when from the collating of the Ob$ervations made in divers places, the $aid angle is, by calcula- tion, gathered to be in$en$ible, and the lines, as it were, parallels. But if the angle be of a con$iderable quantity, the New Star mu$t of nece$$ity be lower than tho$e fixed; and al$o than the Moon, in ca$e the angle A B E $hould be greater than that which would be made in the Moons centre.</P> <P>SIMP. Then the remotene$$e of the Moon is not $o great, that a like angle $hould be ^{*}in$en$ible in her?</P> <marg>* Imperceptible.</marg> <P>SALV. No Sir; nay it is $en$ible, not onely in the Moon, but in the Sun al$o.</P> <P>SIMP. But if this be $o, it's po$$ible that the $aid angle may be ob$erved in the New Star, without nece$$itating it to be inferi- our to the Sun, a$well as to the Moon.</P> <P>SALV. This may very well be, yea, and is in the pre$ent ca$e, as you $hall $ee in due place; that is, when I $hall have made plain the way, in $uch manner that you al$o, though not very perfect in <I>A$tronomical</I> calculations, may clearly $ee, and, as it were, with your hands feel how that this Authour had it more in his eye to write in complacency of the <I>Peripateticks,</I> by palliating and di$- $embling $undry things, than to e$tabli$h the truth, by producing them with naked $incerity: therefore let us proceed forwards. By the things hitherto $poken, I $uppo$e that you comprehend very well how that the di$tance of the new Star can never be made $o immen$e, that the angle $o often named $hall wholly di$- appear, and that the two rayes of the Ob$ervators at the places A and E, $hall become altogether parallels: and you may con$e- quently comprehend to the full, that if the calculations $hould collect from the ob$ervations, that that angle was totally null, or that the lines were truly parallels, we $hould be certain that the ob$ervations were at lea$t in $ome $mall particular erroneous: But, if the calculations $hould give us the $aid lines to be $epara- ted not only to equidi$tance, that is, $o as to be parallel, but to have pa$t beyond that terme, and to be dilated more above than below, then mu$t it be re$olutely concluded, that the ob$ervations were made with le$$e accuratene$$e, and in a word, to be errone- ous; as leading us to a manife$t impo$$ibility. In the next place, you mu$t believe me, and $uppo$e it for true, that two right lines <foot>which</foot> <p n=>261</p> which depart from two points marked upon another right line, are then wider above than below, when the angles included between them upon that right line are greater than two right angles; and if the$e angles $hould be equal to two right angles, the lines would be parallels; but if they were le$s than two right angles, the lines would be concurrent, and being continued out would undoubted- ly inter$ect the triangle.</P> <P>SIMP. Without taking it upon tru$t from you, I know the $ame; and am not $o very naked of <I>Geometry,</I> as not to know a Propo$ition, which I have had occa$ion of reading very often in <I>Ari$totle,</I> that is, that the three angles of all triangles are equall to two right angles: $o that if I take in my Figure the triangle ABE, it being $uppo$ed that the line E A is right; I very well conceive, that its three angles A, E, B, are equal to two right angles; and that con$equently the two angles E and A are le$$e than two right angles, $o much as is the angle B. Whereupon widening the lines A B and E B ($till keeping them from moving out of the points A and E) untill that the angle conteined by them towards the parts B, di$appear, the two angles beneath $hall be equal to two right angles, and tho$e lines $hall be reduced to parallels: and if one $hould proceed to enlarge them yet more, the angles at the points E and A would become greater than two right angles.</P> <P>SALV. You are an <I>Archimedes,</I> and have freed me from the expence of more words in declaring to you, that when$oever the calculations make the two angles A and E to be greater than two right angles, the ob$ervations without more adoe will prove erro- neous. This is that which I had a de$ire that you $hould perfect- ly under$tand, and which I doubted that I was not able $o to make out, as that a meer <I>Peripatetick</I> Philo$opher might attain to the certain knowledg thereof. Now let us go on to what remains. And re-a$$uming that which even now you granted me, namely, that the new $tar could not po$$ibly be in many places, but in one alone, when ever the $upputations made upon the ob$ervations of the$e A$tronomers do not a$$ign it the $ame place, its nece$$ary that it be an errour in the ob$ervations, that is, either in taking the altitudes of the pole, or in taking the elevations of the $tar, or in the one or other working. Now for that in the many workings made with the combinations two by two, there are very few of the ob$ervations that do agree to place the $tar in the $ame $itua- tion; therefore the$e few onely may happily be the non-errone- ous, but the others are all ab$olutely fal$e.</P> <P>SAGR. It will be nece$$ary then to give more credit to the$e few alone, than to all the re$t together, and becau$e you $ay, that the$e which accord are very few, and I among$t the$e 12, do find two that $o accord, which both make the di$tance of the <foot>$tar</foot> <p n=>262</p> $tar from the centre of the Earth 4 $emidiameters, which are the$e, the fifth and $ixth, therefore it is more probable that the new $tar was elementary, than cele$tial.</P> <P>SALV. You mi$take the point; for if you note well it was not written, that the di$tance was exactly 4 $emidiameters, but about 4 $emidiameters; and yet you $hall $ee that tho$e two di$tances differed from each other many hundreds of miles. Here they are; you $ee that this fifth, which is 13389 <I>Italian</I> miles, exceeds the $ixth, which is 13100 miles, by almo$t 300 miles.</P> <P>SAGR. Which then are tho$e few that agree in placing the $tar in the $ame $ituation?</P> <P>SALV. They are, to the di$grace of this Author five workings, which all place it in the firmament, as you $hall $ee in this note, where I have $et down many other combinations. But I will grant the Author more than peradventure he would demand of me, which is in $um, that in each combination of the ob$ervations there is $ome error; which I believe to be ab$olutely nece$$ary; for the ob$ervations being four in number that $erve for one working, that is, two different altitudes of the Pole, and two different eleva- tions of the $tar, made by different ob$ervers, in different pla- ces, with different in$truments, who ever hath any $mall know- <marg><I>A$tronomical In- struments are very $ubject to errour.</I></marg> ledg of this art, will $ay, that among$t all the four, it is impo$$ible but there will be $ome error; and e$pecially $ince we $ee that in taking but one onely altitude of the Pole, with the $ame in$tru- ment, in the $ame place, by the $ame ob$erver, that hath re- peated the ob$ervation a thou$and times, there will $till be a titu- bation of one, or $ometimes of many minutes, as in this $ame book you may $ee in $everal places. The$e things pre$uppo$ed, I ask you <I>Simplicius</I> whether you believe that this Authour held the$e thirteen ob$ervators for wi$e, under$tanding and expert men in u$ing tho$e in$truments, or el$e for inexpert, and bunglers?</P> <P>SIMP. It mu$t needs be that he e$teemed them very acute and intelligent; for if he had thought them unskilful in the bu$ine$$e, he might have omitted his $ixth book as inconclu$ive, as being founded upon $uppo$itions very erroneous; and might take us for exce$$ively $imple, if he $hould think he could with their inex- pertne$$e per$wade us to believe a fal$e po$ition of his for truth.</P> <P>SALV. Therefore the$e ob$ervators being $uch, and that yet notwith$tanding they did erre, and $o con$equently needed cor- rection, that $o one might from their ob$ervations infer the be$t hints that may be; it is convenient that we apply unto them the lea$t and neere$t emendations and corrections that may be; $o that they do but $uffice to reduce the ob$ervations from impo$- $ibility to po$$ibility; $o as <I>v. gr.</I> if one may but correct a mani- fe$t errour, and an apparent impo$$ibility of one of their ob$er- <foot>vations</foot> <p n=>263</p> vations by the addition or $ub$traction of two or three minutes, and with that amendment to reduce it to po$$ibility, a man ought not to e$$ay to adju$t it by the addition or $ub$traction of fifteen, twenty, or fifty.</P> <P>SIMP. I think the Authour would not deny this: for granting that they are expert and judicious men, it ought to be thought that they did rather erre little than much.</P> <P>SALV. Ob$erve again; The places where the new Star is pla- ced, are $ome of them manife$tly impo$$ible, and others po$$ible. Ab$olutely impo$$ible it is, that it $hould be an infinite $pace $upe- riour to the fixed Stars, for there is no $uch place in the world; and if there were, the Star there $cituate would have been imper- ceptible to us: it is al$o impo$$ible that it $hould go creeping along the $uperficies of the Earth; and much le$$e that it $hould be within the $aid Terre$trial Globe. Places po$$ible are the$e that be in controver$ie, it not interferring with our under$tanding, that a vi$ible object in the likene$$e of a Star might be a$well above the Moon, as below it. Now whil$t one goeth about to compute by the way of Ob$ervations and Calculations made with the utmo$t certainty that humane diligence can attain unto what its place was, it is found that the greate$t part of tho$e Calculations make it more than infinitely $uperiour to the Firmament, others make it very neer to the $urface of the Earth, and $ome al$o under the $ame; and of the re$t, which place it in $ituations not impo$$ible, none of them agree with each other; in$omuch that it mu$t be confe$$ed, that all tho$e ob$ervations are nece$$arily fal$e; $o that if we would neverthele$s collect $ome fruit from $o many laborious calculations, we mu$t have recour$e to the corrections, amending all the ob$ervations.</P> <P>SIMP. But the Authour will $ay, that of the ob$ervations that a$$ign to the Star impo$$ible places, there ought no account to be made, as being extreamly erroneous and fal$e; and tho$e onely ought to be accepted, that con$titute it in places not impo$$ible: and among$t the$e a man ought to $eek, by help of the mo$t pro- bable, and mo$t numerous concurrences, not if the particular and exact $ituation, that is, its true di$tance from the centre of the Earth, at lea$t, whether it was among$t the Elements, or el$e a- mong$t the Cœle$tial bodies.</P> <P>SALV. The di$cour$e which you now make, is the $elf $ame that the Author made, in favour of his cau$e, but with too unrea- $onable a di$advantage to his adver$aries; and this is that princi- pal point that hath made me exce$$ively to wonder at the too great confidence that he expre$$ed to have, no le$s of his own authority, than of the blindne$s and inadvertency of the A$tronomers; in favour of whom I will $peak, and you $hall an$wer for the Author. <foot>And</foot> <p n=>264</p> And fir$t, I ask you, whether the A$tronomers, in ob$erving with their In$truments, and $eeking <I>v. gr.</I> how great the elevation of a Star is above the Horizon, may deviate from the truth, a$well in making it too great, as too little; that is, may erroneou$ly com- pute, that it is $ometime higher than the truth, and $ometimes low- er; or el$e whether the errour mu$t needs be alwayes of one kinde, to wit, that erring they alwayes make it too much, and ne- ver too little, or alwayes too little, and never too much?</P> <P>SIMP. I doubt not, but that it is as ea$ie to commit an errour the one way, as the other.</P> <P>SALV. I believe the Author would an$wer the $ame. Now of the$e two kinds of errours, which are contraries, and into which the ob$ervators of the new $tar may equally have fallen, applied to calculations, one $ort will make the $tar higher, and the other lower than really it is. And becau$e we have already agreed, that all the ob$ervations are fal$e; upon what ground would this Au- thor have us to accept tho$e for mo$t congruous with the truth, that $hew the $tar to have been near at hand, than the others that $hew it exce$$ively remote?</P> <P>SIMP. By what I have, as yet, collected of the Authors mind, I $ee not that he doth refu$e tho$e ob$ervations, and indagations that might make the $tar more remote than the Moon, and al$o than the Sun, but only tho$e that make it remote (as you your $elf have $aid) more than an infinite di$tance; the which di$tance, be- cau$e you al$o do refu$e it as impo$$ible, he al$o pa$$eth over, as being convicted of infinite fal$hood; as al$o tho$e ob$ervations are of impo$$ibility. Methinks, therefore, that if you would con- vince the Author, you ought to produce $upputations, more exact, or more in number, or of more diligent ob$ervers, which con$titute the $tar in $uch and $uch a di$tance above the Moon, or above the Sun, and to be brief, in a place po$$ible for it to be in, like as he produceth the$e twelve, which all place the $tar beneath the Moon in places that have a being in the world, and where it is po$$ible for it to be.</P> <P>SALV. But <I>Simplicius</I> yours and the Authors Equivocation lyeth in this, yours in one re$pect, and the Authors in another; I di$cover by your $peech that you have formed a conceit to your $elf, that the exorbitancies that are commited in the e$tabli$hing the di$tance of the Star do encrea$e $ucce$$ively, according to the proportion of the errors that are made by the In$trument, in tak- ing the ob$ervations, and that by conver$ion, from the greatne$s of the exorbitancies, may be argued the greatne$$e of the error; and that thereforefore hearing it to be infered from $uch an ob$er- vation, that the di$tance of the $tar is infinite, it is nece$$ary, that the errour in ob$erving was infinite, and therefore not to be amend- <foot>ed,</foot> <p n=>265</p> ed, and as $uch to be refu$ed; but the bu$ine$$e doth not $ucceed in that manner, my <I>Simplicius,</I> and I excu$e you for not having comprehended the matter as it is, in regard of your $mall experi- ence in $uch affairs; but yet cannot I under that cloak palliate the error of the Author, who di$$embling the knowledge of this which he did per$wade him$elf that we in good earne$t did not under- $tand, hath hoped to make u$e of our ignorance, to gain the bet- ter credit to his Doctrine, among the multitude of illiterate men. Therefore for an adverti$ement to tho$e who are more credulous then intelligent, and to recover you from error, know that its po$- $ible (and that for the mo$t part it will come to pa$$e) that an ob$ervation, that giveth you the $tar <I>v. gr.</I> at the di$tance of <I>Sa- turn,</I> by the adition or $ub$traction of but one $ole minute from the elevation taken with the in$trument, $hall make it to become infinitely di$tant; and therefore of po$$ible, impo$$ible, and by conver$ion, tho$e calculations which being grounded upon tho$e ob$ervations, make the $tar infinitely remote, may po$$ibly often- times with the addition or $ubduction of one $ole minute, reduce it to a po$$ible $cituation: and this which I $ay of a minute, may al- $o happen in the correction of half a minute, a $ixth part, and le$s. Now fix it well in your mind, that in the highe$t di$tances, that is <I>v. g.</I> the height of <I>Saturn,</I> or that of the fixed Stars, very $mall errors made by the Ob$ervator, with the in$trument, render the $cituation determinate and po$$ible, infinite & impo$$ible. This doth not $o evene in the $ublunary di$tances, and near the earth, where it may happen that the ob$ervation by which the Star is collected to be remote <I>v. g.</I> 4. Semidiameters terre$trial, may encrea$e or dimi- ni$h, not onely one minute but ten, and an hundred, and many more, without being rendred by the calculation either infinitely remote, or $o much as $uperior to the Moon. You may hence comprehend that the greatne$$e of the error (to $o $peak) in$tru- mental, are not to be valued by the event of the calculation, but by the quantity it $elf of degrees and minutes numbred upon the in$trument, and the$e ob$ervations are to be called more ju$t or le$s erroneous, which with the addition or $ub$traction of fewer minutes, re$tore the $tar to<*>a po$$ible $ituation; and among$t the po$$ible places, the true one may be believed to have been that, a- bout which a greater number of di$tances concurre upon calcula- ting the more exact ob$ervations.</P> <P>SIMP. I do not very well apprehend this which you $ay: nor can I of my $elf conceive how it can be, that in greater di$tances, greater exorbitancies can ari$e from the errour of one minute only, than in the $maller from ten or an hundred; and therefore would gladly under$tand the $ame.</P> <P>SALV. You $hall $ee it, if not Theorically, yet at lea$t Practi- <foot>Ll cally,</foot> <p n=>266</p> cally, by this $hort a$$umption, that I have made of all the combi- nations, and of part of the workings pretermitted by the Author, which I have calculated upon this $ame paper.</P> <P>SAGR. You mu$t then from ye$terday, till now, which yet is not above eighteen hours, have done nothing but compute, with- out taking either food or $leep.</P> <P>SALV. I have refre$hed my $elf both tho$e wayes; but truth is, make the$e $upputations with great brevity; and, if I may $peak the truth, I have much admired, that this Author goeth $o farre a- bout, and introduceth $o many computations no wi$e nece$sary to the que$tion in di$pute. And for a full knowledge of this, and al- $o to the end it may $oon be $een, how that from the ob$ervations of the A$tronomers, whereof this Author makes u$e, it is more pro- bably gathered, that the new $tar might have been above the Moon, and al$o above all the Planets, yea among$t the fixed $tars, and yet higher $till than they, I have tran$cribed upon this paper all the ob$ervations $et down by the $aid Authour, which were made by thirteen A$tronomers, wherein are noted the Polar alti- tude, and the altitudes of the $tar in the meridian, a$well the le$$er under the Pole, as the greater and higher, and they are the$e.</P> <table> <row><col></col><col><I>Tycho.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col></row> <row><col>Altitude of the Pole</col><col>55</col><col>58</col><col></col></row> <row><col>Altitude of the Star</col><col>84</col><col>00</col><col>the greate$t.</col></row> <row><col></col><col>27</col><col>57</col><col>the lea$t.</col></row> <row><col>And the$e are, according to the fir$t paper: but accor- ding to the $econd, the greate$t is ------------</col><col>27</col><col>45</col><col></col></row> </table> <table> <row><col></col><col><I>Hainzelius.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col></row> <row><col>Altitude of the Pole</col><col>48</col><col>22</col><col></col></row> <row><col>Altitude of the Star</col><col>76</col><col>34</col><col></col></row> <row><col></col><col>76</col><col>33</col><col>45</col></row> <row><col></col><col>76</col><col>35</col><col></col></row> <row><col></col><col>20</col><col>09</col><col>40</col></row> <row><col></col><col>20</col><col>09</col><col>30</col></row> <row><col></col><col>20</col><col>09</col><col>20</col></row> </table> <foot><I>Peucerus</I></foot> <p n=>267</p> <table> <row><col><I>Peucerus</I> and <I>Sculerus.</I></col><col></col><col></col><col><I>Landgravius.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>51</col><col>54</col><col>Altitude of the pole</col><col>51</col><col>18</col></row> <row><col>Altitude of the Star</col><col>79</col><col>56</col><col>Altitude of the Star</col><col>79</col><col>30</col></row> <row><col></col><col>23</col><col>33</col><col></col><col></col><col></col></row> </table> <table> <row><col></col><col><I>Camerarius.</I></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>52</col><col>24</col></row> <row><col>Altitude of the Star</col><col>80</col><col>30</col></row> <row><col></col><col>80</col><col>27</col></row> <row><col></col><col>80</col><col>26</col></row> <row><col></col><col>24</col><col>28</col></row> <row><col></col><col>24</col><col>20</col></row> <row><col></col><col>24</col><col>17</col></row> </table> <table> <row><col><I>Hagecius</I></col><col></col><col></col><col><I>Maurolycus.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>48</col><col>22</col><col>Altitude of the pole</col><col>38</col><col>30</col></row> <row><col>Altitude of the Star</col><col>20</col><col>15</col><col>Altitude of the Star</col><col>62</col><col>00</col></row> <row><col><I>Munocius.</I></col><col></col><col></col><col><I>Vr$inus.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>39</col><col>30</col><col>Altitude of the pole</col><col>49</col><col>24</col></row> <row><col>Altitude of the $tar</col><col>67</col><col>30</col><col>Altitude of the $tar</col><col>79</col><col>00</col></row> <row><col></col><col>11</col><col>30</col><col></col><col>22</col><col>00</col></row> <row><col><I>Reinholdus.</I></col><col></col><col></col><col><I>Buchius.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>51</col><col>18</col><col>Altitude of the pole</col><col>51</col><col>10</col></row> <row><col>Altitude of the $tar</col><col>79</col><col>30</col><col>Altitude of the $tar</col><col>79</col><col>20</col></row> <row><col></col><col>23</col><col>02</col><col></col><col>22</col><col>40</col></row> </table> <table> <row><col><I>Gemma.</I></col><col></col><col></col></row> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of the pole</col><col>50</col><col>50</col></row> <row><col>Altitude of the $tar</col><col>79</col><col>45</col></row> </table> <foot>Ll 2 Now</foot> <p n=>268</p> <P>Now to $ee my whole proceeding, we may begin from the$e calculations, which are four, omitted by the Author, perhaps be- cau$e they make again$t him, in regard they place the $tar above the moon by many $emidiameters of the Earth. The fir$t of which is this, computed upon the ob$ervations of the Landgrave of <I>Ha$$ia,</I> and <I>Tycho</I>; which are, even by the Authors conce$$ion, two of the mo$t exact ob$ervers: and in this fir$t, I will declare the order that I hold in the working; which $hall $erve for all the re$t, in that they are all made by the $ame rule, not varying in any thing, $ave in the quantity of the given $ummes, that is, in the number of the degrees of the Poles altitude, and of the new Stars elevation above the Horizon, the di$tance of which from the cen- tre of the Earth, in proportion to the $emidiameter of the terre- $trial Globe is $ought, touching which it nothing imports in this ca$e, to know how many miles that $emidiameter conteineth; whereupon the re$olving that, and the di$tance of places where the ob$ervations were made, as this Author doth, is but time and labour lo$t; nor do I know why he hath made the $ame, and e$pe- cially why at the la$t he goeth about to reduce the miles found, in- to $emidiameters of the Terre$trial Globe.</P> <P>SIMP. Perhaps he doth this to finde with $uch $mall mea$ures, and with their fractions the di$tance of the Star terminated to three or four inches; for we that do not under$tand your rules of Arith- metick, are $tupified in hearing your conclu$ions; as for in$tance, whil$t we read; Therefore the new Star or Comet was di$tant from the Earths centre three hundred $eventy and three thou$and eight hundred and $even miles; and moreover, two hundred and eleven, four chou$and ninety $evenths 373807 211/4097, and upon the$e preci$e punctualities, wherein you take notice of $uch $mall mat- ters, we do conceive it to be impo$$ible, that you, who in our cal- culations keep an account of an inch, can at the clo$e deceive us $o much as an hundred miles.</P> <P>SALV. This your rea$on and excu$e would pa$$e for currant, if in a di$tance of thou$ands of miles, a yard over or under were of any great moment, and if the $uppo$itions that we take for true, were $o certain, as that they could a$$ure us of producing an indubitable truth in the conclu$ion; but here you $ee in the twelve workings of the Author, the di$tances of the Star, which from them one may conclude to have been different from each other, (and therefore wide of the truth) for many hundreds and thou- $ands of miles: now whil$t that I am more than certain, that that which I $eek mu$t needs differ from the truth by hundreds of miles, to what purpp$e is it to be $o curious in our calculations, for fear of mi$$ing the quantity of an inch? But let us proceed, at la$t, to the working, which I re$olve in this manner. <I>Tycho,</I> as may be <foot>$een</foot> <p n=>269</p> $een in that $ame note ob$erved the $tar in the polar altitude of 55 degrees and 58 <I>mi. pri.</I> And the polar altitude of the <I>Landgrave</I> was 51 degrees and 18 <I>mi. pri.</I> The altitude of the $tar in the Me- ridian taken by <I>Tycho</I> was 27 degrees 45 <I>mi. pri.</I> The <I>Land- grave</I> found its altitude 23 degrees 3 <I>mi. pri.</I> The which altitudes are the$e noted here, as you $ee.</P> <table> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Tycho</I> Pole</col><col>55</col><col>58</col><col>* 27</col><col>45</col></row> <row><col><I>Landgr.</I> Pole</col><col>51</col><col>18</col><col>* 23</col><col>3</col></row> </table> <P>This done, $ub$tract the le$$e from the greater, and there remains the$e differences here underneath.</P> <table> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col></col><col>4</col><col>40</col></row> <row><col></col><col>4</col><col>42</col></row> <row><col>Parall.</col><col></col><col>2</col></row> </table> <P>Where the difference of the poles altitudes 4 <I>gr. 4 mi. pr.</I> is le$$e than the difference of the altitudes of the Star 4 <I>gr. 42 mi. pr.</I> and therefore we have the difference of parallaxes, 0 <I>gr. 2 mi. pri.</I> The$e things being found, take the Authours own figure [<I>Fig. 2.</I>] in which the point B is the $tation of the <I>Landgrave,</I> D the $tation of <I>Tycho,</I> C the place of the $tar, A the centre of the Earth, A B E the vertical line of the <I>Landgrave,</I> A D F <table> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col>Its chord 8142 of tho$e</col></row> <row><col>Ang. B A D</col><col>4</col><col>40</col><col>parts, whereof the $emid.</col></row> <row><col>B D F</col><col>92</col><col>20</col><col>A B is an 100000.</col></row> </table> <table> <row><col>B D C</col><col>154</col><col>45</col><col>Sines</col><col>42657</col></row> <row><col>B C D</col><col>0</col><col>2</col><col></col><col>58</col></row> </table> <table> <row><col>58</col><col>42657</col><col>8142</col></row> <row><col></col><col>8142</col><col></col></row> <row><col></col><col>85314</col><col></col></row> <row><col></col><col>170628</col><col></col></row> <row><col></col><col>42657</col><col></col></row> <row><col></col><col>341256</col><col></col></row> <row><col></col><col>59</col><col></col></row> <row><col>58</col><col>3473</col><col>13294</col></row> <row><col></col><col>571</col><col></col></row> <row><col></col><col>5</col><col></col></row> </table> of <I>Tycho,</I> and the angle B C D the difference of Parallaxes. And <foot>be-</foot> <p n=>270</p> becau$e the angle B A D, conteined between the vertical lines, is equal to the difference of the Polar altitudes, it $hall be 4<I>gr. 40m.</I> which I note here apart; and I finde the chord of it by the Table of Arches and Chords, and $et it down neer unto it, which is 8142 parts, of which the $emidiameter A B is 100000. Next, I finde the angle B D C with ea$e, for the half of the angle B A D, which is 2 <I>gr. 20 m.</I> added to a right angle, giveth the angle B D F 92 <I>gr.</I> 20 <I>m.</I> to which adding the angle C D F, which is the di$tance from the vertical point of the greate$t altitude of the Star, which here is 62 <I>gr. 15 m.</I> it giveth us the quantity of the angle B D C, 154 <I>grad. 45 min.</I> the which I $et down together with its Sine, taken out of the Table, which is 42657, and under this I note the angle of the Parallax B C D 0 <I>gr. 2 m.</I> with its Sine 58. And becau$e in the Triangle B C D, the $ide D B is to the $ide B C; as the $ine of the oppo$ite angle B C D, to the $ine of the oppo$ite angle B D C: therefore, if the line B D were 58. B C would be 42657. And becau$e the Chord D B is 8142. of tho$e parts whereof the $emidiameter B A is 100000. and we $eek to know how many of tho$e parts is B C; therefore we will $ay, by the Golden Rule, if when B D is 58. B G is 42657. in ca$e the $aid D B were 8142. how much would B C be? I multiply the $econd term by the third, and the product is 347313294. which ought to be divided by the fir$t, namely, by 58. and the quotient $hall be the number of the parts of the line B C, whereof the $e- midiameter A B is 100000. And to know how many $emidiame- ters B A, the $aid line B C doth contein, it will be nece$$ary anew to divide the $aid quotient $o found by 100000. and we $hall have the number o$ $emidiameters conteined in B G. Now the num- ber 347313294. divided by 58. giveth 5988160 1/4. as here you may $ee.</P> <table> <row><col></col><col>5988160 1/4</col></row> <row><col>58</col><col>347313294</col></row> <row><col></col><col>5717941</col></row> <row><col></col><col>543</col></row> </table> <P>And this divided by 100000. the product is 59 88160/100000</P> <table> <row><col>1 |00000</col><col>| 59 |</col><col>88160.</col></row> </table> <P>But we may much abbreviate the operation, dividing the fir$t quotient found, that is, 347313294. by the product of the multi- plication of the two numbers 58. and 100000. that is,</P> <foot>59</foot> <p n=>271</p> <table> <row><col></col><col>59</col><col></col></row> <row><col>58 00000</col><col>3473</col><col>13294</col></row> <row><col></col><col>571</col><col></col></row> <row><col></col><col>5</col><col></col></row> </table> <P>And this way al$o there will come forth 59 5113294/5800000</P> <P>And $o many $emidiameters are contained in the line B C, to which one being added for the line A B, we $hall have little le$$e than 61. $emidiameters for the two lines A B C; and therefore the right di$tance from the centre A, to the Star C, $hall be more than 60. $emidiameters, and therefore it is $uperiour to the Moon, according to <I>Ptolomy,</I> more than 27. $emidiameters, and according to <I>Copernicus,</I> more than 8. $uppo$ing that the di$tance of the Moon from the centre of the Earth by <I>Copernicus</I> his account is what the Author maketh it, 52 $emidiameters. With this $ame working, I find by the ob$ervations of <I>Camerarius,</I> and of <I>Muno- $ius,</I> that the Star was $ituate in that $ame di$tance, to wit, $ome- what more than 60. $emidiameters. The$e are the ob$ervations, and the$e following next after them the calculations.</P> <table> <row><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitude of <I>Camerar.</I></col><col>52</col><col>24</col><col>Altitude of</col><col>24</col><col>28</col></row> <row><col>the Pole <I>Muno$.</I></col><col>39</col><col>30</col><col>the Star</col><col>11</col><col>30</col></row> <row><col>Differences of the</col><col>12</col><col>54</col><col>Differences</col><col>12</col><col>58</col></row> <row><col>Polar Altitudes</col><col></col><col></col><col>of the alt. of *</col><col>12</col><col>54</col></row> <row><col></col><col>Difference of Parallaxes</col><col></col><col></col><col>00</col><col>04. ang. BCD.</col></row> </table> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col></col><col>B A D</col><col>12</col><col>54</col><col>and its chord or $ubten$e 22466.</col><col></col></row> <row><col>Angles</col><col>B D C</col><col>161</col><col>59</col><col>Sines</col><col>30930</col></row> <row><col></col><col>B C D</col><col>00</col><col>04</col><col></col><col>116</col></row> </table> <P><I>The Golden Rule.</I></P> <table> <row><col></col><col>22466</col><col></col></row> <row><col>116</col><col>30930</col><col>22466</col></row> <row><col></col><col>673980</col><col></col></row> <row><col></col><col>202194</col><col></col></row> <row><col></col><col>67398</col><col></col></row> </table> <table> <row><col></col><col>59</col><col>_______</col><col>Di$tance B C 59. and</col></row> <row><col>116</col><col>6948</col><col>73380</col><col>almo$t 60. $emidiameters.</col></row> <row><col></col><col>1144</col><col></col><col></col></row> <row><col></col><col>10</col><col></col><col></col></row> </table> <foot>The</foot> <p n=>272</p> <P>The next working is made upon two ob$ervations of <I>Tycho,</I> and of <I>Muno$ius,</I> from which the Star is calculated to be di$tant from the Centre of the Earth 478 Semidiameters and more.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitudes</col><col><I>Tycho.</I></col><col>55</col><col>58</col><col>Altitude</col><col>84</col><col>00</col></row> <row><col>of the Pole.</col><col><I>Muno$.</I></col><col>39</col><col>30</col><col>of the Star.</col><col>67</col><col>30</col></row> </table> <table> <row><col>Differences of the</col><col>16</col><col>28</col><col>Differ. of the</col><col>16 30</col></row> <row><col>Polar Altitudes.</col><col></col><col></col><col>Alt. of the *</col><col>16 28</col></row> <row><col></col><col>Difference of Parallax.</col><col></col><col></col><col>0 2 and ang. BCD</col></row> </table> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col></col><col><I>B A D.</I></col><col>16</col><col>28</col><col>its chord</col><col>28640</col></row> <row><col>Angles</col><col><I>B D C.</I></col><col>104</col><col>14</col><col>Sines</col><col>96930</col></row> <row><col></col><col><I>B C D.</I></col><col>0</col><col>2</col><col></col><col>58</col></row> </table> <P><I>The Golden Rule.</I></P> <table> <row><col>58</col><col>96930</col><col>28640</col></row> <row><col></col><col>28640</col><col></col></row> <row><col></col><col>3877200</col><col></col></row> <row><col></col><col>58158</col><col></col></row> <row><col></col><col>77544</col><col></col></row> <row><col></col><col>19386</col><col></col></row> <row><col></col><col>478</col><col></col></row> <row><col>58</col><col>27760</col><col>75200</col></row> <row><col></col><col>4506</col><col></col></row> <row><col></col><col>53</col><col></col></row> </table> <P>The$e workings following make the Star remote from the Cen- tre, more than 358 Semidiameters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col>Altitudes</col><col><I>Peucerus</I></col><col>51</col><col>54</col><col>Altitude</col><col>79</col><col>56</col></row> <row><col>of the Pole.</col><col><I>Muno$ius</I></col><col>39</col><col>30</col><col>of the *</col><col>47</col><col>30</col></row> <row><col></col><col></col><col>12</col><col>24</col><col></col><col>12</col><col>26</col></row> <row><col></col><col></col><col></col><col></col><col></col><col>12</col><col>24</col></row> <row><col></col><col></col><col></col><col></col><col></col><col>0</col><col>2</col></row> </table> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col></col><col><I>B A D.</I></col><col>12</col><col>24</col><col>its chord</col><col>21600</col></row> <row><col>Angles</col><col><I>B D C.</I></col><col>106</col><col>16</col><col>Sines</col><col>95996</col></row> <row><col></col><col><I>B C D.</I></col><col>0</col><col>2</col><col></col><col>58</col></row> </table> <foot><I>The</I></foot> <p n=>273</p> <P>The Golden Rule.</P> <table> <row><col>58</col><col>---- 95996</col><col>---- 21600</col></row> <row><col></col><col>21600</col><col></col></row> <row><col></col><col>57597600</col><col></col></row> <row><col></col><col>95996</col><col></col></row> <row><col></col><col>191992</col><col></col></row> <row><col></col><col>357</col><col></col></row> <row><col>58</col><col>20735</col><col>13600</col></row> <row><col></col><col>3339</col><col></col></row> <row><col></col><col>42</col><col></col></row> </table> <P>From this other working the $tar is found to be di$tant from the centre more than 716. $emidiameters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col></row> <row><col>Altitudes</col><col><I>Landgr.</I></col><col>51</col><col>18</col><col>Altitude</col><col>79</col><col>30</col><col>00</col></row> <row><col>of the Pole</col><col><I>Hainzel.</I></col><col>48</col><col>22</col><col>of the Star</col><col>76</col><col>33</col><col>45</col></row> <row><col></col><col></col><col>2</col><col>56</col><col></col><col>2</col><col>56</col><col>15</col></row> <row><col></col><col></col><col></col><col></col><col></col><col>2</col><col>56</col><col></col></row> <row><col></col><col></col><col></col><col></col><col></col><col>0</col><col>00</col><col>15</col></row> </table> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col><col></col><col></col></row> <row><col></col><col>B A D</col><col>2</col><col>56</col><col>00</col><col>its Chord</col><col>5120</col></row> <row><col>Angles</col><col>B D C</col><col>101</col><col>58</col><col>00</col><col>Sines</col><col>97845</col></row> <row><col></col><col>B C D</col><col>0</col><col>00</col><col>15</col><col></col><col>7</col></row> </table> <P>The Golden Rule.</P> <table> <row><col>7</col><col>---- 97845</col><col>---- 5120</col></row> <row><col></col><col>5120</col><col></col></row> <row><col></col><col>1956900</col><col></col></row> <row><col></col><col>57845</col><col></col></row> <row><col></col><col>489225</col><col></col></row> <row><col></col><col>715</col><col></col></row> <row><col>7</col><col>5009</col><col>66400</col></row> <row><col></col><col>4</col><col></col></row> </table> <P>The$e as you $ee are five workings which place the $tar very much above the Moon. And here I de$ire you to con$ider upon that particular, which even now I told you, namely, that in great <foot>Mm di-</foot> <p n=>274</p> di$tances, the mutations, or if you plea$e corrections, of a ve- ry few minutes, removeth the $tar a very great way farther off. As for example, in the fir$t of the$e workings, where the calcu- lation made the $tar 60. $emidiameters remote from the centre, with the Parallax of 2. minutes; he that would maintain that it was in the Firmament, is to correct in the ob$ervations but onely two minutes, nay le$$e, for then the Parallax cea$eth, or be- commeth $o $mall, that it removeth the $tar to an immen$e di- $tance, which by all is received to be the Firmament. In the $e- cond indagation, or working, the correction of le$$e than 4 <I>m. prim.</I> doth the $ame. In the third, and fourth, like as in the fir$t, two minutes onely mount the $tar even above the Firmament. In the la$t preceding, a quarter of a minute, that is 15. $econds, gives us the $ame. But it doth not $o occur in the $ublunary alti- tudes; for if you fancy to your $elf what di$tance you mo$t like, and go about to correct the workings made by the Au- thour, and adju$t them $o as that they all an$wer in the $ame determinate di$tance, you will find how much greater correcti- ons they do require.</P> <P>SAGR. It cannot but help us in our fuller under$tanding of things, to $ee $ome examples of this which you $peak of.</P> <P>SALV. Do you a$$ign any what$oever determinate $ublunary di$tance at plea$ure in which to con$titute the $tar, for with $mall ado we may a$$ertain our $elves whether corrections like to the$e, which we $ee do $uffice to reduce it among$t the fixed $tars, will reduce it to the place by you a$$igned.</P> <P>SAGR. To take a di$tance that may favour the Authour, we will $uppo$e it to be that which is the greate$t of all tho$e found by him in his 12 workings; for whil$t it is in controver$ie be- tween him and A$tronomers, and that they affirm the $tar to have been $uperiour to the Moon, and he that it was inferiour, very $mall $pace that he proveth it to have been lower, giveth him the victory.</P> <P>SALV. Let us therefore take the $eventh working wrought upon the ob$ervations of <I>Tycho</I> and <I>Thaddæus Hagecius,</I> by which the Authour found the $tar to have been di$tant from the centre 32. $emidiameters, which $ituation is mo$t favourable to his purpo$e; and to give him all advantages, let us moreover place it in the di$tance mo$t disfavouring the <I>A$tronomers,</I> which is to $ituate it above the Firmament. That therefore being $up- po$ed, let us $eek in the next place what corrections it would be ne- ce$$ary to apply to his other 11 workings. And let us begin at the fir$t calculated upon the ob$ervations of <I>Hainzelius</I> and <I>Mauroice</I>; in which the Authour findeth the di$tance from the centre about 3. $emidiameters with the Parallax of 4 <I>gr. 42 m. 30. $ec.</I> Let <foot>us</foot> <p n=>275</p> us $ee whether by withdrawing it 20. minutes onely, it will ri$e to the height of 32. $emidiameters: See the $hort and true opera- tion. Multiply the $ine of the angle B D C, by the $ine of the <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col></row> <row><col><I>Hainzelius</I></col><col>Pole</col><col>48</col><col>32</col><col>---- *</col><col>76</col><col>34</col><col>30</col></row> <row><col><I>Maurolicus</I></col><col>Pole</col><col>38</col><col>30</col><col>---- *</col><col>62</col><col>00</col><col>00</col></row> <row><col></col><col></col><col>9</col><col>52</col><col></col><col>14</col><col>34</col><col>30</col></row> <row><col></col><col></col><col></col><col></col><col></col><col>9</col><col>52</col><col>00</col></row> <row><col></col><col></col><col></col><col></col><col>Parallax</col><col>4</col><col>42</col><col>30</col></row> </table> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col><col></col><col></col></row> <row><col></col><col>B A D</col><col>9</col><col>52</col><col>00</col><col>Chord</col><col>17200</col></row> <row><col>Angles</col><col>B D C</col><col>108</col><col>21</col><col>30</col><col>Sine</col><col>94910</col></row> <row><col></col><col>B C D</col><col>0</col><col>20</col><col>00</col><col>Sine</col><col>582</col></row> </table> <table> <row><col></col><col>94910</col><col></col></row> <row><col></col><col>17200</col><col></col></row> <row><col></col><col>18982000</col><col></col></row> <row><col></col><col>66437</col><col></col></row> <row><col></col><col>9491</col><col></col></row> <row><col></col><col>28</col><col></col></row> <row><col>582</col><col>16324</col><col>52000</col></row> <row><col></col><col>4688</col><col></col></row> <row><col></col><col>2</col><col></col></row> </table> chord B D, and divide the product, the five la$t figures being cut off by the $ine of the Parallax, and the quotient will be 28. $e- midiameters, and an half, $o that though you make a correction of 4 <I>gr. 22 min. 30 $ec.</I> taken from 4 <I>gr. 42 min. 30 $ec.</I> it $hall not elevate the $tar to the altitude of 32. $emidiameters, which correction for <I>Simplicius</I> his under$tanding it, is of 262. minutes, and an half.</P> <P>In the $econd operation made upon the ob$ervations of <I>Hain- zelius,</I> and <I>Sculerus,</I> with the Parallax of 0 <I>gr. 8 min. 30 $ec.</I> the $tar is found in the height of 25. $emidiameters or therea- bouts, as may be $een in the $ub$equent working.</P> <table> <row><col>B D</col><col>Chord</col><col>6166</col></row> <row><col>B D C</col><col>Sines</col><col>97987</col></row> <row><col>B C D</col><col></col><col>247</col></row> </table> <foot>Mm 2 97987</foot> <p n=>276</p> <table> <row><col></col><col>97987</col><col></col></row> <row><col></col><col>6166</col><col></col></row> <row><col></col><col>587922</col><col></col></row> <row><col></col><col>587922</col><col></col></row> <row><col></col><col>97987</col><col></col></row> <row><col></col><col>587922</col><col></col></row> <row><col></col><col>24</col><col></col></row> <row><col>247</col><col>6041</col><col>87842</col></row> <row><col></col><col>1103</col><col></col></row> <row><col></col><col>11</col><col></col></row> </table> <P>And bringing back the Parallax 0 <I>gr. 8 m. 30 $ec.</I> to 7 <I>gr.</I> 7 <I>m.</I> who$e $ine is 204, the $tar elevateth to 30 $emidiameters or thereabouts; therefore the correction of 0 <I>gr. 1 mi. 30 $ec.</I> doth not $uffice.</P> <table> <row><col></col><col>20</col><col></col></row> <row><col>204</col><col>6041</col><col>87342</col></row> <row><col></col><col>1965</col><col></col></row> <row><col></col><col>12</col><col></col></row> </table> <P>Now let us $ee what correction is requi$ite for the third work- ing made upon the ob$ervations of <I>Hainzelius</I> and <I>Tycho,</I> which rendereth the $tar about 19 $emidiameters high, with the Pa- rallax of 10 <I>m. pri.</I> The u$ual angles and their $ines, and chord found by the Authour, are the$e next following; and they re- move the $tar (as al$o in the Authours working) 19 $emidia- meters from the centre of the Earth. It is nece$$ary therefore for the rai$ing of it, to dimini$h the Parallax according to the Rule which he likewi$e ob$erveth in the ninth working. Let us there- fore $uppo$e the Parallax to be 6 <I>m. prim.</I> who$e $ine is 175, and the divi$ion being made, there is found likewi$e le$$e than 31 $emidiameters for the $tars di$tance. And therefore the correcti- on of 4 <I>min. prim.</I> is too little to $erve the Authours purpo$e.</P> <table> <row><col></col><col>B A D</col><col>7</col><col>36</col><col>Chord</col><col>13254</col></row> <row><col>Angles</col><col>B D C</col><col>155</col><col>52</col><col>Sine</col><col>40886</col></row> <row><col></col><col>B C D</col><col>0</col><col>10</col><col>Sine</col><col>291</col></row> </table> <foot>13254</foot> <p n=>277</p> <table> <row><col></col><col>13254</col><col></col><col></col><col></col></row> <row><col></col><col>40886</col><col></col><col></col><col></col></row> <row><col></col><col>79524</col><col></col><col></col><col></col></row> <row><col></col><col>106032</col><col></col><col></col><col></col></row> <row><col></col><col>106032</col><col></col><col></col><col></col></row> <row><col></col><col>53016</col><col></col><col></col><col></col></row> <row><col></col><col>18</col><col></col><col></col><col>30</col></row> <row><col>291</col><col>5419</col><col>03044</col><col>175</col><col>5419</col></row> <row><col></col><col>250</col><col></col><col></col><col>16</col></row> <row><col></col><col>181</col><col></col><col></col><col></col></row> </table> <P>Let us come to the fourth working, and the re$t with the $ame rule, and with the chords and $ines found out by the Authour him$elf; in this the Parallax is 14 <I>m. prim.</I> and the height found le$$e than 10 $emidiameters, and dimini$hing the Parallax from 14 <I>min.</I> to 4 <I>min.</I> yet neverthele$$e you $ee that the $tar doth not elevate full 31 $emidiameters. Therefore 10 <I>min.</I> in 14 <I>min.</I> doth not $uffice.</P> <table> <row><col></col><col>B A D</col><col>Chord</col><col>8142</col></row> <row><col>Angles</col><col>B D C</col><col>Sine</col><col>43235</col></row> <row><col></col><col>B C D</col><col>Sine</col><col>407</col></row> </table> <table> <row><col></col><col>43235</col><col></col></row> <row><col></col><col>8142</col><col></col></row> <row><col></col><col>86470</col><col></col></row> <row><col></col><col>172940</col><col></col></row> <row><col></col><col>43235</col><col></col></row> <row><col></col><col>345880</col><col></col></row> <row><col></col><col>30</col><col></col></row> <row><col>116</col><col>3520</col><col>19370</col></row> <row><col></col><col>4</col><col></col></row> </table> <P>In the fifth operation of the Authour we have the $ines and the chord as you $ee, and the Parallax is 0 <I>gr. 42 m. 30 $ec.</I> which rendereth the height of the $tar about 4 $emidiameters, and cor- recting the Parallax, with reducing it from 0 <I>gr. 42 m. 30 $ec.</I> to 0 <I>gr. 5 m.</I> onely, doth not $uffice to rai$e it to $o much as 28 $e- midiameters, the correction therefore of 0 <I>gr. 37 m. 30 $ec.</I> is too little.</P> <table> <row><col></col><col>B A D</col><col>Chord</col><col>4034</col></row> <row><col>Angles</col><col>B D C</col><col>Sine</col><col>97998</col></row> <row><col></col><col>B C D</col><col></col><col>1236</col></row> </table> <foot>97998</foot> <p n=>278</p> <table> <row><col></col><col>97998</col><col></col></row> <row><col></col><col>4034</col><col></col></row> <row><col></col><col>391992</col><col></col></row> <row><col></col><col>293994</col><col></col></row> <row><col></col><col>391992</col><col></col></row> <row><col></col><col>27</col><col></col></row> <row><col>145</col><col>3953</col><col>23932</col></row> <row><col></col><col>1058</col><col></col></row> <row><col></col><col>3</col><col></col></row> </table> <P>In the $ixth operation the chord, the $ines and Parallax are as followeth, and the $tar is found to be about 4 $emidiameters; let us $ee whether it will be reduced, abating the Parallax from 8 <I>m.</I> to 1 <I>m.</I> onely; Here is the operation, and the $tar rai$ed but to 27. $emidiameters or thereabout; therefore the correction of 7 <I>m.</I> in 8 <I>m.</I> doth not $uffice.</P> <table> <row><col>B D</col><col>Chord</col><col>1920</col></row> <row><col>B D C</col><col>Sine</col><col>40248</col></row> <row><col>B C D 8 <I>gr.</I></col><col>Sine</col><col>233</col></row> </table> <table> <row><col></col><col>40248</col><col></col></row> <row><col></col><col>1920</col><col></col></row> <row><col></col><col>804960</col><col></col></row> <row><col></col><col>362232</col><col></col></row> <row><col></col><col>40248</col><col></col></row> <row><col></col><col>26</col><col></col></row> <row><col>29</col><col>772</col><col>76160</col></row> <row><col></col><col>198</col><col></col></row> <row><col></col><col>1</col><col></col></row> </table> <P>In the eighth operation the chord, the $ines, and the Parallax, as you $ee, are the$e en$uing, and hence the Authour calculates the height of the $tar to be 1. $emidiameter and an half, with the Parallax of 43. <I>min.</I> which reduced to 1 <I>min.</I> yet notwith$tand- ing giveth the $tar le$$e remote than 24. $emidiameters, the corre- ction therefore of 42. <I>min.</I> is not enough.</P> <table> <row><col>B D</col><col>Chord</col><col>1804</col></row> <row><col>B D C</col><col>Sine</col><col>36643</col></row> <row><col>B C D</col><col>Sine</col><col>29</col></row> </table> <foot>36643</foot> <p n=>279</p> <table> <row><col></col><col>36643</col><col></col></row> <row><col></col><col>1804</col><col></col></row> <row><col></col><col>146572</col><col></col></row> <row><col></col><col>293144</col><col></col></row> <row><col></col><col>36643</col><col></col></row> <row><col></col><col>22</col><col></col></row> <row><col>29</col><col>661</col><col>03972</col></row> <row><col></col><col>83</col><col></col></row> <row><col></col><col>2</col><col></col></row> </table> <P>Let us now $ee the ninth. Here is the chord, the $ines and the Parallax which is 15 <I>m.</I> From whence the Authour calcu- lates the di$tance of the $tar from the $uperficies of the Earth to be le$$e than a ^{*} $even and fortieth part of a $emidiameter, <marg>* Here the La- tine ver$ion is erro- neous, making it a fortieth part of, <I>&c.</I></marg> but this is an errour in the calcultaion, for it cometh forth truly, as we $hall $ee here below, more than a $ifth: See here the quo- tienr is 90/436, which is more than one fifth.</P> <table> <row><col>B D</col><col>Chord</col><col>232</col></row> <row><col>B D C</col><col>Sine</col><col>39046</col></row> <row><col>B C D</col><col>Sine</col><col>436</col></row> </table> <table> <row><col></col><col>39046</col><col></col></row> <row><col></col><col>232</col><col></col></row> <row><col></col><col>78092</col><col></col></row> <row><col></col><col>117138</col><col></col></row> <row><col></col><col>78092</col><col></col></row> <row><col>436</col><col>90</col><col>58672</col></row> </table> <P>That which the Authour pre$ently after $ubjoyns in way of amending the ob$ervations, that is, that it $u$$iceth not to re- duce the difference of Parallax, neither to a minute, nor yet to the eighth part of a minute is true. But I $ay, that neither will the tenth part of a minute reduce the height of the $tar to 32. $emidiameters; for the $ine of the tenth part of a minute, that is of $ix $econds, is 3; by which if we according to our Rule $hould divide 90. or we may $ay, if we $hould divide 9058672. by 300000. the quotient will be 30 58672/100000, that is little more than 30. $emidiameters and an half.</P> <P>The tenth giveth the altitude of the $tar one fifth of a $emi- diameter, with the$e angles, $ines, and Parallax, that is, 4 <I>gr.</I> <foot>30</foot> <p n=>280</p> 30 <I>m.</I> which I $ee that being reduced from 4 <I>gr. 30 min.</I> to 2 <I>min.</I> yet neverthele$$e it elevates not the $tar to 29. $emidiameters.</P> <table> <row><col>B D</col><col></col><col>Chord</col><col>1746</col></row> <row><col>B D C</col><col></col><col>Sine</col><col>92050</col></row> <row><col>B C D</col><col>4 <I>gr. 30 m.</I></col><col>Sine</col><col>7846</col></row> </table> <table> <row><col></col><col>92050</col><col></col></row> <row><col></col><col>17460</col><col></col></row> <row><col></col><col>552300</col><col></col></row> <row><col></col><col>36820</col><col></col></row> <row><col></col><col>64435</col><col></col></row> <row><col></col><col>9205</col><col></col></row> <row><col></col><col>27</col><col></col></row> <row><col>58</col><col>1607</col><col>19300</col></row> <row><col></col><col>441</col><col></col></row> <row><col></col><col>4</col><col></col></row> </table> <P>The eleventh rendereth the $tar to the Authour remote about 13. $emidiameters, with the Parallax of 55. <I>min.</I> let us $ee, re- ducing it to 20 <I>min.</I> whether it will exalt the $tar: See here the calculation elevates it to little le$$e than 33. $emidiameters, the correction therefore is little le$$e than 35. <I>min.</I> in 55. <I>min.</I></P> <table> <row><col>B D</col><col></col><col>Chord</col><col>19748</col></row> <row><col>B D C</col><col></col><col>Sine</col><col>96166</col></row> <row><col>B C D</col><col>o <I>gr. 55 m.</I></col><col>Sine</col><col>1600</col></row> </table> <table> <row><col></col><col>96166</col><col></col></row> <row><col></col><col>19748</col><col></col></row> <row><col></col><col>639328</col><col></col></row> <row><col></col><col>384664</col><col></col></row> <row><col></col><col>673162</col><col></col></row> <row><col></col><col>865494</col><col></col></row> <row><col></col><col>96166</col><col></col></row> <row><col></col><col>32</col><col></col></row> <row><col>582</col><col>18990</col><col>56168</col></row> <row><col></col><col>1536</col><col></col></row> <row><col></col><col>56</col><col></col></row> </table> <P>The twelfth with the Parallax of 1. <I>gr. 36. min.</I> maketh the $tar le$$e high than 6. $emidiameters, reducing the Parallax to 20 <I>min.</I> it carrieth the $tar to le$$e than 30. $emidiameters di- $tance, therefore the correction of 1 <I>gr. 16. min.</I> $ufficeth not.</P> <foot>B D</foot> <p n=>281</p> <table> <row><col>B D</col><col></col><col>Chord</col><col>17258</col></row> <row><col>B D C</col><col></col><col>Sine</col><col>96150</col></row> <row><col>B C D</col><col>1 <I>gr. 36 m.</I></col><col>Sine</col><col>2792</col></row> </table> <table> <row><col></col><col>17258</col><col></col></row> <row><col></col><col>96150</col><col></col></row> <row><col></col><col>862900</col><col></col></row> <row><col></col><col>17258</col><col></col></row> <row><col></col><col>103548</col><col></col></row> <row><col></col><col>155322</col><col></col></row> <row><col></col><col>28</col><col></col></row> <row><col>582</col><col>16593</col><col>56700</col></row> <row><col></col><col>4957</col><col></col></row> <row><col></col><col>29</col><col></col></row> </table> <head><I>The$e are the Corrections of the Parallaxes of the ten workings of the Author, to reduce the Star to the altitude of</I> 32 <I>Semidiameters.</I></head> <table> <row><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col><I>$ec.</I></col></row> <row><col>04</col><col>22</col><col>30</col><col>in</col><col>04</col><col>42</col><col>30</col></row> <row><col>00</col><col>04</col><col>00</col><col>in</col><col>00</col><col>10</col><col>00</col></row> <row><col>00</col><col>10</col><col>00</col><col>in</col><col>00</col><col>14</col><col>00</col></row> <row><col>00</col><col>37</col><col>00</col><col>in</col><col>00</col><col>42</col><col>30</col></row> <row><col>00</col><col>07</col><col>00</col><col>in</col><col>00</col><col>18</col><col>00</col></row> <row><col>00</col><col>42</col><col>00</col><col>in</col><col>00</col><col>43</col><col>00</col></row> <row><col>00</col><col>14</col><col>50</col><col>in</col><col>00</col><col>15</col><col>00</col></row> <row><col>04</col><col>28</col><col>00</col><col>in</col><col>04</col><col>30</col><col>00</col></row> <row><col>00</col><col>35</col><col>00</col><col>in</col><col>00</col><col>55</col><col>00</col></row> <row><col>01</col><col>16</col><col>00</col><col>in</col><col>01</col><col>36</col><col>00</col></row> </table> <table> <row><col>216</col><col>296.60</col></row> <row><col>540</col><col>240.9</col></row> <row><col>765</col><col>836.540</col></row> </table> <P>From hence we $ee, that to reduce the Star to 32. Semidiame- ters in altitude, it is requi$ite from the $um of the Parallaxes 836. to $ubtract 756. and to reduce them to 80. nor yet doth that correction $uffice.</P> <foot>Nn Here</foot> <p n=>282</p> <P>Here we $ee al$o, (as I have noted even now) that $hould the Authour con$ent to a$$ign the di$tance of 32. Semidiameters for the true height of the Star, the correction of tho$e his 10. workings, (I $ay 10. becau$e the $econd being very high, is reduced to the height of 32. Semidiameters, with 2. minutes correction) <*>o make them all to re$tore the $aid Star to that di$tance, would require $uch a reduction of Parallaxes, that among$t the whole number of $ub $tractions they $hould make more than 756 <I>m. pr.</I> whereas in the 5. calculated by me, which do place the Star above the Moon, to correct them in $uch $ort, as to con$titute it in the Firmament, the correction onely of 10. minutes, and one fourth $ufficeth.</P> <P>Now adde to the$e, other 5. workings, that place the Star pre- ci$ely in the Firmament, without need of any correction at all, and we $hall have ten workings or indagations that agree to place it in the Firmament, with the correction onely of 5. of them (as hath been $een) but 10. <I>m.</I> and 15 <I>$ec.</I> Whereas for the correcti- on of tho$e 10. of the Authour, to reduce them to the altitude of 32. $emidiameters, there will need the emendations of 756 mi- nutes in 836. that is, there mu$t from the $umme 836 be $ub$tra- cted 756. if you would have the Star elevated to the altitude of 32. $emidiameters, and yet that correction doth not fully $erve.</P> <P>The workings that immediately without any correction free the Star from Parallaxes, and therefore place it in the Firmament, and that al$o in the remote$t parts of it, and in a word, as high as the Pole it $elf, are the$e 5. noted here.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Camerar.</I></col><col>Polar altit.</col><col>52</col><col>24</col><col>Altit. of the Star</col><col>80</col><col>26</col></row> <row><col><I>Peucerus</I></col><col></col><col>51</col><col>54</col><col></col><col>79</col><col>56</col></row> <row><col></col><col></col><col>0</col><col>30</col><col></col><col>0</col><col>30</col></row> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Landgrav.</I></col><col>Polar altit.</col><col>51</col><col>18</col><col>Altit. of the Star</col><col>79</col><col>30</col></row> <row><col><I>Hainzel.</I></col><col></col><col>48</col><col>22</col><col></col><col>76</col><col>34</col></row> <row><col></col><col></col><col>2</col><col>56</col><col></col><col>2</col><col>56</col></row> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Tycho</I></col><col>Polar altit.</col><col>55</col><col>58</col><col>Altit. of the Star</col><col>84</col><col>00</col></row> <row><col><I>Peucerus</I></col><col></col><col>51</col><col>54</col><col></col><col>79</col><col>56</col></row> <row><col></col><col></col><col>4</col><col>4</col><col></col><col>4</col><col>4</col></row> <foot><I>Reinold</I></foot> <p n=>283</p> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Reinhold.</I></col><col>Polar altit.</col><col>51</col><col>18</col><col>Altit. of the Star</col><col>79</col><col>30</col></row> <row><col><I>Hainzel.</I></col><col></col><col>48</col><col>22</col><col></col><col>36</col><col>34</col></row> <row><col></col><col></col><col>2</col><col>56</col><col></col><col>2</col><col>56</col></row> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Camerar.</I></col><col>Polar altit.</col><col>52</col><col>24</col><col>Altit. of the Star</col><col>24</col><col>17</col></row> <row><col><I>Hagecius</I></col><col></col><col>48</col><col>22</col><col></col><col>20</col><col>15</col></row> <row><col></col><col></col><col>4</col><col>2</col><col></col><col>4</col><col>2</col></row> </table> <P>Of the remaining combinations that might be made of the Ob- $ervations of all the$e A$tronomers, tho$e that make the Stars $ub- lime to an infinite di$tance, are many in number, namely, about 30. more than tho$e who give the Star, by calculation, to be be- low the Moon; and becau$e (as it was agreed npon between us) it is to be believed that the Ob$ervators have erred rather little than much, it is a manife$t thing that the corrections to be applied to the Ob$ervaations, which make the $tar of an infinite altitude, to reduce it lower, do $ooner, and with le$$er amendment place it in the Firmament, than beneath the Moon; $o that all the$e applaud the opinion of tho$e who put it among$t the fixed Stars. You may adde, that the corrections required for tho$e emendations, are much le$$er than tho$e, by which the Star from an unlikely proxi- mity may be removed to the height more favourable for this Au- thour, as by the foregoing examples hath been $een; among$t which impo$$ible proximities, there are three that $eem to remove the Star from the Earths centre, a le$$e di$tance than one Semidi- ameter, making it, as it were, to turn round under ground, and the$e are tho$e combinations, wherein the Polar altitude of one of the Ob$ervators being greater than the Polar altitude of the other, the elevation of the Star taken by the fir$t, is le$$er than the elation of the Star taken by the latter.</P> <P>The fir$t of the$e is this of the <I>Landgrave</I> with <I>Gemma,</I> where the Polar altitude of the <I>Landgrave 51 gr. 18 min.</I> is greater than the Polar altitude of <I>Gemma,</I> which is 50 <I>gr. 50 m.</I> But the altitude of the Star of the <I>Landgrave 79 gr. 30 min.</I> is le$$er than that of the Star, of <I>Gemma 79 gr. 45 min.</I></P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Landgrave</I></col><col>Polar altit.</col><col>51</col><col>18</col><col>Altit. of the Star</col><col>79</col><col>30</col></row> <row><col><I>Gemma</I></col><col></col><col>50</col><col>50</col><col></col><col>79</col><col>45</col></row> </table> <foot>Nn 2 The</foot> <p n=>284</p> <P>The other two are the$e below.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col><I>gr.</I></col><col><I>m.</I></col></row> <row><col><I>Bu$chius.</I></col><col>Polar Altitude</col><col>51</col><col>10</col><col>Altit. of the Star</col><col>79</col><col>20</col></row> <row><col><I>Gemma.</I></col><col></col><col>50</col><col>50</col><col></col><col>79</col><col>45</col></row> <row><col><I>Reinholdus.</I></col><col>Polar Altitude</col><col>51</col><col>18</col><col>Altit. of the Star</col><col>79</col><col>30</col></row> <row><col><I>Gemma.</I></col><col></col><col>50</col><col>50</col><col></col><col>79</col><col>45</col></row> </table> <P>From what I have hitherto demon$trated, you may gue$$e how much this fir$t way of finding out the di$tance of the Star, and proving it $ublunary introduced by the Authour, maketh again$t him$elf, and how much more probably and clearly the di$tance thereof is collected to have been among$t the more remote fixed Stars.</P> <P>SIMP. As to this particular, I think that the inefficacy of the Authors demonftrations is very plainly di$covered; But I $ee that all this was compri$ed in but a few leaves of his Book, and it may be, that $ome other of his Arguments are more conclu$ive then the$e fir$t.</P> <P>SALV. Rather they mu$t needs be le$$e valid, if we will take tho$e that lead the way for a proof of the re$t: For (as it is clear) the uncertainty and inconclu$ivene$$e of tho$e, is manife$tly ob- $erved to derive it $elf from the errours committed in the in$tru- mental ob$ervations, upon which the Polar Altitude, and height of the Star was thought to have been ju$tly taken, all in effect having ea$ily erred; And yet to find the Altitude of the Pole, A- $tronomers have had Ages of time to apply them$elves to it, at their lea$ure: and the Meridian Altitudes of the Star are ea$ier to be ob$erved, as being mo$t terminate, and yielding the Ob$ervator $ome time to continue the $ame, in regard they change not $en$ibly, in a $hort time, as tho$e do that are remote from the Meridian. And if this be $o, as it is mo$t certain, what credit $hall we give to Calcu- lations founded upon Ob$ervations more numerous, more difficult to be wrought, more momentary in variation, and we may add, with In$truments more incommodious and erroneous? Upon a $light peru$al of the en$uing demon$trations, I $ee that the Com- putations are made upon Altitudes of the Star taken in different Vertical Circles, which are called by the Arabick name, <I>Azimuths</I>; in which ob$ervations moveable in$truments are made u$e of, not on- ly in the Vertical Circles, but in the Horizon al$o, at the $ame time; in$omuch that it is requi$ite in the $ame moment that the altitude is taken, to have ob$erved, in the Horizon, the di$tance of the Vir- <foot>tical</foot> <p n=>285</p> tical point in which the Star is, from the Meridian; Moreover, after a con$iderable interval of time, the operation mu$t be re- peated, and exact account kept of the time that pa$$ed, tru$ting either to Dials, or to other ob$ervations of the Stars. Such an <I>Olio</I> of Ob$ervations doth he $et before you, comparing them with $uch another made by another ob$erver in another place with a- nother different in$trument, and at another time; and from this the Authour $eeks to collect what would have been, the Elevations of the Star, and Horizontal Latitudes happened in the time and hour of the other fir$t ob$ervations, and upon $uch a coæquation he in the end grounds his account. Now I refer it to you, what credit is to be given to that which is deduced from $uch like workings. Moreover, I doubt not in the lea$t, but that if any one would tor- ture him$elf with $uch tedious computations, he would find, as in tho$e aforegoing, that there were more that would favour the ad- ver$e party, than the Authour: But I think it not worth the while to take $o much pains in a thing, which is not, among$t tho$e prima- ry ones, by us under$tood.</P> <P>SAGR. I am of your Opinion in this particular: But this bu$i- ne$$e being environed with $o many intricacies, uncertainties, and errours, upon what confidence have $o many A$tronomers po$itive- ly pronounced the new Star to have been $o high?</P> <P>SALV. Upon two $orts of ob$ervations mo$t plain, mo$t ea$ie, and mo$t certain; one only of which is more than $ufficient to a$$ure us, that it was $cituate in the Firmament, or at lea$t by a great di$tance $uperiour to the Moon. One of which is taken from the equality, or little differing inequality of its di$tances from the Pole, a$well whil$t it was in the lowe$t part of the Meridian, as when it was in the uppermo$t: The other is its having perpetual- ly kept the $ame di$tances from certain of the fixed Stars, adjacent to it, and particularly from the eleventh of <I>Ca$$iopea,</I> no more remote from it than one degree and an half; from which two par- ticulars is undoubtedly inferred, either the ab$olute want of Paral- lax, or $uch a $malne$$e thereof, that it doth a$$ure us with very expeditious Calculations of its great di$tance from the Earth.</P> <P>SAGR. But the$e things, were they not known to this Author? and if he $aw them, what doth he $ay unto them?</P> <P>SALV. We are wont to $ay, of one that having no reply that is able to cover his fault, produceth frivolous excu$es, <I>cerca di at- taccar$i alle funi del cielo,</I> [He $trives to take hold of the Cords of Heaven;] but this Authour runs, not to the Cords, but to the Spi- ders Web of Heaven; as you $hall plainly $ee in our examination of the$e two particulars even now hinted. And fir$t, that which $heweth us the Polar di$tances of the Ob$ervators one by one, I have noted down in the$e brief Calculations; For a full under- <foot>$tand-</foot> <p n=>286</p> $tanding of which, I ought fir$t to adverti$e you, that when ever the new Star, or other Phænomenon is near to the earth, turning with a Diurnal motion about the Pole, it will $eem to be farther off from the $aid Pole, whil$t it is in the lower part of the Meridi- an, then whil$t it is above, as in this Figure [<I>being fig. third of this Dial.</I>] may be $een. In which the point T. denotes the cen- tre of the Earth; O th<*> place of the Ob$ervator; the Arch VPC the Firmament; P. the Pole. The <I>Phænomenon,</I> [<I>or appearance</I>] moving along the Circle F S. is $een one while under the Pole by the Ray O F C. and another while above, according to the Ray O S D. $o that the places $een in the Firmament are D. and C. but the true places in re$pect of the Centre T, are B, and A, equidi- $tant from the Pole. Where it is manife$t that the apparent place of the <I>Phænomenon</I> S, that is the point D, is nearer to the Pole than the other apparent place C, $een along the Line or Ray O F C, which is the fir$t thing to be noted. In the $econd place you mu$t note that the exces of the apparent inferiour di$tance from the Pole, over and above the apparent $uperiour di$tance from the $aid Pole, is greater than the Inferiour Parallax of the <I>Phænomenon,</I> that is, I $ay, that the exce$$e of the Arch C P, (the apparent inferior di- $tance) over and above the Arch P D, (the apparent $uperior di- $tance) is greater then the Arch C A, (that is the inferiour Para- lax.) Which is ea$ily proved; for the Arch C P. more exceedeth P D, then P B; P B, being bigger than P D, but P B. is equal to P A, and the exce$$e of C P, above P A, is the arch, C A, there- fore the exce$$e of the arch C P above the arch P D, is great- er than the arch C A, which is the parallax of the Phænomenon placed in F, which was to be demon$trated. And to give all ad- vantages to the Author, let us $uppo$e that the parallax of the $tar in F, is the whole exce$$e of the arch C P (that is of the inferiour di$tance from the pole) above the arch P D (the inferiour di- $tance.) I proceed in the next place to examine that which the ob$ervations of all A$tronomers cited by the Authour giveth us, among$t which, there is not one that maketh not again$t him$elf and his purpo$e. And let us begin with the$e of <I>Bu$chius,</I> who findeth the $tars di$tance from the pole, when it was $uperiour, to be 28 <I>gr. 10 m.</I> and the inferiour to be 28 <I>gr. 30 m.</I> $o that the ex- ce$$e is 0 <I>gr. 20 m.</I> which let us take (in favour of the Author) as if it all were the parallax of the $tar in F, that is the angle T F O. Then the di$tance from the <I>Vertex</I> [or Zenith] that is the arch C V, is 67 <I>gr. 20 m.</I> The$e two things being found, prolong the line C O, and from it let fall the perpendicular T I, and let us con$ider the triangle T O I, of which the angle I is right angle, and the angle I O T known, as being vertical to the angle V O C, the di$tance of the $tar from the <I>Vertex,</I> Moreover in the triangle <foot>T I F,</foot> <p n=>287</p> T I F, which is al$o rectangular, there is known the angle F, ta- ken by the parallax. Then note in $ome place apart the two an- gles I O T and I F T, and of them take the $ines, which are here $et down to them, as you $een. And becau$e in the triangle I O T, the $ine T I is 92276. of tho$e parts, whereof the whole $ine TO is 100000; and moreover in the triangle I F T, the $ine T I is 582. of tho$e parts, whereof the whole $ine T F is 100000, to find how many T F is of tho$e parts, whereof T O is 100000; we will $ay by the Rule of three: If T I be 582. T F is an 100000. but if T I were 92276. how much would T F be. Let us multiply 92276. by 100000. and the product will be 9227600000. and this mu$t be divided by 582. and the quotient will be 15854982. and $o many $hall there be in T F of tho$e parts, of which there are in T O an 100000. So that if it were required to know how many lines T O, are in T F, we would divide 15854982 by 100000. and there will come forth 158. and very near an half; and $o many diameters $hall be the di$tance of the $tar F, from the centre T, and to abreviate the opera- tion, we $eeing, that the product of the multiplication of 92276. by 100000, ought to be divided fir$t by 582, and then the quo- tient of that divi$ion by 100000. we may without multiplying 92276. by 100000. and with one onely divi$ion of the $ine 92276. by the $ine 582. $oon obtain the $ame $olution, as may be $een there below; where 92276. divided by 582. giveth us the $aid 158 1/2, or thereabouts. Let us bear in mind therefore, that the onely divi$ion of the $ine T I, as the $ine of the angle T O I by the $ine T I, as the $ine of the angle I F T, giveth us the di- $tance $ought T F, in $o many diameters T O.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I O T</col><col>67</col><col>20</col><col>Sines</col><col>92276</col></row> <row><col></col><col>I F T</col><col>0</col><col>20</col><col></col><col>582</col></row> </table> <table> <row><col>T I</col><col>T F</col><col>T I</col><col>T F</col></row> <row><col>582</col><col>10000</col><col>92276</col><col>0</col></row> </table> <table> <row><col></col><col>15854982</col><col></col></row> <row><col>582</col><col>9227600000</col><col></col></row> <row><col></col><col>3407002746</col><col></col></row> <row><col></col><col>49297867</col><col></col></row> <row><col></col><col>325414</col><col></col></row> <row><col>100000</col><col>158</col><col>54982</col></row> <row><col></col><col>158</col><col></col></row> <row><col>582</col><col>92276</col><col></col></row> <row><col></col><col>34070</col><col></col></row> <row><col></col><col>492</col><col></col></row> <row><col></col><col>3</col><col></col></row> </table> <foot>See</foot> <p n=>288</p> <P>See next that which the ob$ervations of <I>Peucerus</I> giveth us, in which the inferiour di$tance from the Pole is 28 <I>gr. 21 m.</I> and the $uperiour 28 <I>gr. 2 m.</I> the difference 0 <I>gr. 19 m.</I> and the di$tance from the vertical point 66 <I>gr. 27 m.</I> from which particulars is ga- thered the $tars di$tance from the centre almo$t 166 $emedia- meters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I A C</col><col>66</col><col>27</col><col>Sines</col><col>91672</col></row> <row><col></col><col>I E C</col><col>0</col><col>19</col><col></col><col>553</col></row> </table> <table> <row><col></col><col>165 427/553</col></row> <row><col>553</col><col>91672</col></row> <row><col></col><col>36397</col></row> <row><col></col><col>312</col></row> <row><col></col><col>4</col></row> </table> <P>Here take what <I>Tycho</I> his ob$ervation holdeth forth to us, in- terpreted with greate$t favour to the adver$ary; to wit, the inferi- our di$tance from the pole is 28 <I>gr. 13 m.</I> and the $uperiour 28 <I>gr. 2 m.</I> omitting the difference which is 0 <I>gr. 11 m.</I> as if all were one Parallax; the di$tance from the vertical point 62 <I>gr. 15 m.</I> Behold here below the operation, and the di$tance of the $tar from the centre found to be 976 9/16 $emidiameters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I A C</col><col>62</col><col>15</col><col>Sines</col><col>88500</col></row> <row><col></col><col>I E C</col><col>0</col><col>11</col><col></col><col>320</col></row> </table> <table> <row><col></col><col>276 9/16</col></row> <row><col>320</col><col>88500</col></row> <row><col></col><col>2418</col></row> <row><col></col><col>1</col></row> </table> <P>The ob$ervation of <I>Reinholdus,</I> which is the next en$uing, giv- eth us the di$tance of the Star from the Centre 793. Semidia- meters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I A C</col><col>66</col><col>58</col><col>Sines</col><col>92026</col></row> <row><col></col><col>I E C</col><col>0</col><col>4</col><col></col><col>116</col></row> </table> <table> <row><col></col><col>793 3<*>/116</col></row> <row><col>116</col><col>92026</col></row> <row><col></col><col>10888</col></row> <row><col></col><col>33</col></row> </table> <foot>From</foot> <p n=>289</p> <P>From the following ob$ervation of the <I>Landgrave,</I> the di$tance of the Star from the Centre is made to be 1057, Semidiameters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I A C</col><col>66</col><col>57</col><col>Sines</col><col>92012</col></row> <row><col></col><col>I E C</col><col>0</col><col>3</col><col></col><col>87</col></row> </table> <table> <row><col></col><col>1057 <*>/87</col></row> <row><col>87</col><col>92012</col></row> <row><col></col><col>5663</col></row> <row><col></col><col>5</col></row> </table> <P>Two of the mo$t favourable ob$ervations for the Authour be- ing taken from <I>Camerarius,</I> the di$tance of the Star from the Cen- tre is found to be 3143 Semidiameters.</P> <table> <row><col></col><col></col><col><I>gr.</I></col><col><I>m.</I></col><col></col><col></col></row> <row><col>Angles</col><col>I A C</col><col>65</col><col>43</col><col>Sines</col><col>91152</col></row> <row><col></col><col>I E C</col><col>0</col><col>1</col><col></col><col>29</col></row> </table> <table> <row><col></col><col>3143</col></row> <row><col>29</col><col>91152</col></row> <row><col></col><col>4295</col></row> </table> <P>The Ob$ervation of <I>Muno$ius</I> giveth no <I>Parallax,</I> and there- fore rendreth the new Star among$t the highe$t of the fixed. That of <I>Hainzelius</I> makes it infinitely remote, but with the correction of an half <I>min. prim.</I> placeth it among$t the fixed Stars. And the $ame is collected from <I>Vr$inus,</I> with the correction of 12. <I>min. prim.</I> The other A$tronomers have not given us the di$tance above and below the Pole, $o that nothing can be concluded from them. By this time you $ee, that all the ob$ervations of all the$e men con$pire again$t the Author, in placing the Star in the Heavenly and high- e$t Regions.</P> <P>SAGR. But what defence hath he for him$elf again$t $o manife$t contradictions?</P> <P>SALV. He betakes him$elf to one of tho$e weak threads which I $peak of; $aying that the <I>Parallaxes</I> come to be le$$ened by means of the refractions, which opperating contrarily $ublimate the <I>Phæ- nomenon,</I> whereas the <I>Parallaxes</I> aba$e it. Now of what little $tead this lamentable refuge is, judge by this, that in ca$e that effectof the refractions were of $uch an efficacy, as that which not long time $ince $ome A$tronomers have introduced, the mo$t that they could work touching the elevating a <I>Phæuomenon</I> above the Horizon <foot>Oo more</foot> <p n=>290</p> more than truth, when it is before hand 23. or 24. Degrees high, would be the le$$ening its <I>Parallax</I> about 3. minutes, the which abatement is too $mall to pull down the Star below the Moon, and in $ome ca$es is le$$e than the advantage given him by us in admit- ting that the exce$$e of the inferiour di$tance from the Pole above the Superiour, is all <I>Parallax,</I> the which advantage is far more clear and palpable than the effect of Refracton, of the greatne$$e of which I $tand in doubt, and not without rea$on. But be$ides, I demand of the Author, whether he thinks that tho$e A$tronomers, of who$e ob$ervations he maketh u$e, had knowledge of the$e ef- fects of Refractions, and con$idered the $ame, or no; if they did know and con$ider them, it is rea$onable to think that the, kept ac- count of them in a$$igning the true Elevation of the Star, making in tho$e degrees of Altitude di$covered with the In$truments, $uch abatements as were convenient on the account of the alterations made by the Refractions; in$omuch that the di$tances by them de- livered, were in the end tho$e corrected and exact, and not the ap- parent and fal$e ones. But if he think that tho$e Authors made no reflection upon the $aid Refractions, it mu$t be confe$$ed, that they had in like manner erred in determining all tho$e things which cannot be perfectly adju$ted without allowance for the Refracti- ons; among$t which things one is the preci$e inve$tigation of the Polar Altitudes, which are commonly taken from the two Meridi- an Altitudes of $ome of the fixed Stars that are con$tantly vi$ible, which Altitudes will come to be altered by Refraction in the $ame manner, ju$t as tho$e of the new Star; $o that the Polar Altitude that is deduced from them, will prove to be defective, and to par- take of the $elf $ame want which this Author a$$igns to the Alti- tudes a$cribed to the new Star, to wit, both that and the$e will be with equal fal$hood placed higher than really they are. But any $uch errour, as far as concerns our pre$ent bu$ine$$e, doth no pre- judce at all: For we not needing to know any more, but onely the difference between the two di$tances of the new Star from the Pole at $uch time as it was inferiour and $uperiour, it is evident that $uch di$tances would be the $ame, taking the alteration of Refra- ction commonly for the Star and for the Pole, or for them when commonly amended. The Authors Argument would indeed have had $ome $trength, though very $mall, if he had a$$ured us that the Altitude of the Pole had been once preci$ely a$$igned, and cor- rected from the errour depending on refraction, from which a- gain the A$tronomers had not kept them$elves in a$$igning the al- titudes of the new Star; but he hath not a$certained us of that, nor perhaps could he have done, nor haply, (and this is more pro- bable) was that caution wanting in the Ob$ervators.</P> <P>SAGR. This argument is in my judgment $ufficiently an$wer- <foot>ed;</foot> <p n=>291</p> ed; therefore tell me how he di$-ingageth him$elf in the next place from that particular of the Stars having con$tantly kept the $ame di$tance from the fixed Stars circumjacent to it.</P> <P>SALV. He betakes him$elf, in like manner, to two threads, yet more unable to uphold him than the former: one of which is like- wi$e fa$tened to refraction, but $o much le$s firmly, in that he $aith, that refraction operating upon the new Star, and $ublimating it higher than its true $ituation, maketh the $eeming di$tances un- tain to be di$tingui$hed from the true, when compared to the cir- cumpo$ed fixed Stars that environ it. Nor can I $ufficiently ad- mire how he can di$$emble his knowing how that the $ame refra- ction will work alike upon the new Star, as upon the antient one its neighbour, elevating both equally, $o as that $uch a like acci- dent altereth not the $pace betwixt them. His other $ubterfuge is yet more unhappy, and carryeth with it much of ridiculous, it be- ing founded upon the errour that may ari$e in the in$trumen talo- peration it $elf; whil$t that the Ob$ervator not being able to con$titute the centre of the eyes pupil in the centre of the Sex- tant (an In$trument imployed in ob$erving the di$tance between two Stars) but holding it elevated above that centre, as much as the $aid pupil is di$tant from I know not what bone of the cheek, again$t which the end of the In$trument re$teth, there is formed in the eye an angle more acute than that which is made by the $ides of the In$trument; which angle of rayes differeth al$o from it $elf, at $uch time as a man looketh upon Stars, not much elevated above the Horizon, and the $ame being afterwards placed at a great height; that angle, $aith he, is made different, while the In- $trument goeth a$cending, the head $tanding $till: but if in moun- ting the In$trument, the neck $hould bend backwards, and the head go ri$ing, together with the In$trument, the angle would then continue the $ame. So that the Authours an$wer $uppo$eth that the Ob$ervators in u$ing the In$trument have not rai$ed the head, as they ought to have done; a thing which hath nothing of likeli- hood in it. But granting that $o it had been, I leave you to judge what difference can be between two acute angles of two equicru- ral triangles, the $ides of one of which triangles are each four [<I>Italian] Braces</I> [<I>i.e.</I> about three Engli$h yards] and tho$e of the other, four braces within the quantity of the diameter of a Pea; for the differences cannot be ab$olutely greater between the length of the two vi$ive rayes, whil$t the line is drawn perpendicularly from the centre of the pupil, upon the plain of the Rule of the Sextant (which line is no bigger than the breath of the thumb) and the length of the $ame rayes, whil$t elevating the Sextant, without rai$ing the head together with it, that $ame line no longer falleth perpendicularly upon the $aid plane, but inclineth, making <foot>Oo 2 the</foot> <p n=>292</p> the angle towards the circumference $omething acute. But wholly to free this Authour from the$e unhappy lies, let him know, (in re- gard it appears that he is not very skilful in the u$e of A$tronomi- call In$truments) that in the $ides of the Sextant or Quadrant <marg>* Traguardi.</marg> there are placed two ^{*} Sights, one in the centre, and the other at the other at the oppo$ite end, which are rai$ed an inch or more a- bove the plane of the Rule; and through the tops of tho$e $ights the ray of the eye is made to pa$$e, which eye likewi$e is held an hands breadth or two, or it may be more, from the In$trument; $o that neither the pupil, nor any bone of the cheek, nor of the whole body toucheth or $tayeth it $elf upon the In$trument, nor much le$$e is the In$trument upheld or mounted in the armes, e$pecially if it be one of tho$e great ones, as is u$ual, which weighing tens, hundreds, and al$o thou$ands of pounds, are placed upon very $trong feet or frames: $o that the whole objection vani$heth. The$e are the $ubterfuges of this Authour, which, though they were all of $teel, would not $ecure him the hundredth part of a minute; and with the$e he conceits to make us believe, that he hath com- pen$ated that difference, which importeth more than an hundred minutes; I mean, that of the not ob$erving a notable difference in the di$tances between one of the fixed $tars, and the new $tar in in any of their circulations; which, had it been neer to the Moon, it ought to have been very con$picuous to the meer $ight, without any In$trument, e$pecially comparing it with the eleventh of <I>Ca$- $iopeia,</I> its neighbour, within 1 <I>gr. 30 m.</I> which ought to have va- ried from it more than two diameters of the moon, as the more intelligent A$tronomers of t' o$e times do well note.</P> <P>SAGR. Methinks I $ee th<*>t unfortunate Husbandman, who af- ter all his expected crops, have been beaten down and de$troyed by a $torm, goeth up and down with a langui$hing and down-ca$t look, gleaning up every $mall ear that would not $uffice to keep a chicken alive one $ole day.</P> <P>SALV. Truly, this Authour came out too $lenderly provided with armes again$t the a$$ailants of the Heavens inalterability, and with too brittle a chain attempted to pull down the new $tar of <I>Ca$$iopeia</I> from the highe$t Regions, to the$e $o low and elementa- ry. And for that I think that we have $ufficiently demon$trated the va$t difference that is between the arguments of tho$e A$tro- nomers, and of this their Antagoni$t, it will be convenient that we leave this particular, and return to our principal matter; in which there pre$ents it $elf to our con$ideration the annual motion com- monly a$cribed to the Sun, but by <I>Aristarchus Samius</I> fir$t of all, and after by <I>Copernicus</I> taken from the Sun, and transferred upon the Earth; again$t which Hypothe$is, methinks I $ee <I>Simplicius</I> to come $trongly provided, and particularly with the $word and <foot>buckler</foot> <p n=>293</p> buckler of the little Treati$e of <I>Conclu$ions,</I> or Di$qui$itions Ma- thematical, the oppugnations of which it would be good to be- gin to produce.</P> <P>SIMP. I will, if you $o plea$e, re$erve them to the la$t, as tho$e that are of late$t invention.</P> <P>SALV. It will therefore be nece$$ary, that in conformity to the method hitherto ob$erved, you do orderly, one by one, propound the arguments, on the contrary, a$well of <I>Ari$totle,</I> as of the o- ther ancients, which $hall be my task al$o, that $o nothing may e- $cape our $trict con$ideration and examination; and likewi$e <I>Sa- gredus,</I> with the vivacity of his wit, $hall interpo$e his thoughts, as he $hall finde him$elf inclined.</P> <P>SAGR. I will do it with my wonted freedome; and your com- mands $hall oblige you to excu$e me in $o doing.</P> <P>SALV. The favour will challenge thanks, and not an excu$e. But now let <I>Simplicius</I> begin to propo$e tho$e doubts which di$- $wade him from believing that the Earth, in like manner, as the other pianets, may move round about a fixed centre.</P> <P>SIMP. The fir$t and greate$t difficulty is the repugnance and incompatibility that is between being in the centre, and being far from it; for if the Terre$trial Globe were to move in a year by the circumference of a circle, that is, under the Zodiack, it is im- po$$ible that it $hould, at the $ame time, be in the centre of the Zo- diack; but that the Earth is in the $aid centre <I>Aristotle, Ptolomy,</I> and others have many wayes proved.</P> <P>SALV. You very well argue, aud there is no que$tion but that one that would make the Earth to move in the circumference of a circle, mu$t fir$t of nece$$ity prove, that it is not in the centre of that $ame circle; it now followeth, that we enquire, whether the Earth be, or be not in that centre, about which, I $ay, that it tur- neth, and you $ay that it is fixed; and before we $peak of this, it is likewi$e nece$$ary that we declare our $elves, whether you and I have both the $ame conceit of this centre, or no. Therefore tell me, what and where is this your intended centre?</P> <P>SIMP. When I $peak of the centre, I mean that of the Uni- ver$e, that of the World, that of the Starry Sphere.</P> <P>SALV. Although I might very rationally put it in di$pute, whe- ther there be any $uch centre in nature, or no; being that neither <marg><I>It hath not been hitherto proved by any, whether the World be finite or infinite.</I></marg> you nor any one el$e hath ever proved, whether the World be fi- nite and figurate, or el$e infinite and interminate; yet neverthele$s granting you, for the pre$ent, that it is finite, and of a terminate Spherical Figure, and that thereupon it hath its centre; it will be requi$ite to $ee how credible it is that the Earth, and not rather $ome other body, doth po$$e$$e the $aid centre.</P> <P>SIMP. That the world is finite, terminato, and $pherical, <I>Ari-</I> <foot><I>$totle</I></foot> <p n=>294</p> <I>$totle</I> proveth with an hundred demon$trations.</P> <marg><I>The Demon$tra- tions of</I> Ari$totle <I>to Prove that the Vniver$e is finite, are all nullified by denying it to be moveable.</I></marg> <P>SALV. All which in the end are reduced to one alone, and that one to none at all; for if I deny his a$$umption, to wit, that the Univer$e is moveable, all his demon$trations come to nothing, for he onely proveth the Univer$e to be finite and terminate, for that it is moveable. But that we may not multiply di$putes, let it be granted for once, that the World is finite, $pherical, and hath its centre. And $eeing that that centre and figure is argued from its mobility, it will, without doubt, be very rea$onable, if from the circular motions of mundane bodies we proceed to the particular inve$tigation of that centres proper place: Nay <I>Ari$totle</I> him$elf <marg><I>Ari$totle makes that point to be the centre of the Uni- ver$e about which all the Cele$tial Spheres do revolve.</I></marg> hath argued and determined in the $ame manner, making that $ame to be the centre of the Univer$e about which all the Cœle- le$tial Spheres revolve, and in which he beleived the Terre$trial Globe to have been placed. Now tell me <I>Simplicius,</I> if <I>Ari$totle</I> <marg><I>A que$tion is put, in ca$e that if</I> Ari$totle <I>were forced to receive one of two propo$i- tions that make a- gain$t his doctrine, which he would admit.</I></marg> $hould be con$trained by evident experience to alter in part this his di$po$ure and order of the Univer$e, and confe$$e him$elf to have been deceived in one of the$e two propo$itions, namely, ei- ther in placing the Earth in the centre, or in $aying, that the Cœle$tial Spheres do move about that centre, which of the two confe$$ions think you would he choo$e?</P> <P>SIMP. I believe, that if it $hould $o fall out, the <I>Peripate- ticks.</I></P> <P>SALV. I do not ask the <I>Peripateticks,</I> I demand of <I>Ari$totle,</I> for as to tho$e, I know very well what they would reply; they, as ob$ervant and humble va$$als of <I>Ari$totle,</I> would deny all the ex- periments and all the ob$ervations in the World, nay, would al$o refu$e to $ee them, that they might not be forced to acknowledg them, and would $ay that the World $tands as <I>Ari$totle</I> writeth, and not as nature will have it, for depriving them of the $hield of his Authority, with what do you think they would appear in the field? Tell me therefore what you are per$waded <I>Ari$totle</I> him- $elf would do in the ca$e.</P> <P>SIMP. To tell you the truth, I know not how to re$olve which of the two inconveniences is to be e$teemed the le$$er.</P> <P>SALV. Apply not I pray you this term of inconvenience to a thing which po$$ibly may of nece$$ity be $o. It was an inconveni- ence to place the Earth in the centre of the Cœle$tial revolutions; but $eeing you know not to which part he would incline, I e- $teeming him to be a man of great judgment, let us examine which of the two choices is the more rational, and that we will hold that <I>Ari$totle</I> would have received. Rea$$uming therefore our di$cour$e from the beginning, we $uppo$e with the good liking of <I>Ari$totle,</I> that the World (of the magnitude of which we have no $en$ible notice beyond the fixed $tars) as being of a $pherical <foot>figure;</foot> <p n=>295</p> figure; and moveth circularly, hath nece$$arily, and in re$pect of its figure a centre; and we being moreover certain, that within the $tarry Sphere there are many Orbs, the one within another, with their $tars, which likewi$e do move circulary, it is in di$pute whether it is mo$t rea$onable to believe and to $ay that the$e con- teined Orbs do move round the $aid centre of the World, or el$e about $ome other centre far remote from that? Tell me now <I>Sim- plicius</I> what you think concerning this particular.</P> <P>SIMP. If we could $tay upon this onely $uppo$ition, and that <marg><I>Its more ratio- nal that the Orb conteining, and the parts conteined, do move all about one centre, than uoon divers.</I></marg> we were $ure that we might encounter nothing el$e that might di- $turb us, I would $ay that it were much more rea$onable to af- firm that the Orb containing, and the parts contained, do all move about one common centre, than about divers.</P> <P>SALV. Now if it were true that the centre of the World is the <marg><I>If the centre of the World be the $ame with that a- bout which the via- nees move the Sun and not the Earth is placed in it.</I></marg> $ame about which the Orbs of mundane bodies, that is to $ay, of the Planets, move, it is mo$t certain that it is not the Earth, but the Sun rather that is fixed in the centre of the World. So that as to this fir$t $imple and general apprehen$ion, the middle place belongeth to the Sun, and the Earth is as far remote from the centre, as it is from that $ame Sun.</P> <P>SIMP. But from whence do you argue that not the Earth, but the Sun is in the centre of the Planetary revolutions?</P> <P>SALV. I infer the $ame from mo$t evident, and therefore ne- ce$$arily concludent ob$ervations, of which the mo$t palpable to <marg><I>Ob$ervations from whence it is col- lected that the Sun and not the Earth is in the centre of the Cele$tial revo- lutions.</I></marg> exclude the Earth from the $aid centre, and to place the Sun therein, are, the $eeing all the Planets one while neerer and ano- ther while farther off from the Earth with $o great differences, that for example, <I>Venus</I> when it is at the farthe$t, is $ix times more remote from us, than when it is neere$t, and <I>Mars</I> ri$eth almo$t eight times as high at one time as at another. See therefore whe- ther <I>Ari$totle</I> was not $omewhat mi$taken in thinking that it was at all times couidi$tant from us.</P> <P>SIMP. What in the next place are the tokens that their moti- ons are about the Sun?</P> <P>SALV. It is argued in the three $uperiour planets <I>Mars, Jupi- ter,</I> and <I>Saturn,</I> in that we find them alwayes neere$t to the Earth when they are in oppo$ition to the Sun, and farthe$t off when they are towards the conjunction, and this approximatian and rece$$ion importeth thus much that <I>Mars</I> neer at hand, ap- peareth very neer 60 times greater than when it is remote. As to <marg><I>The mutation of figure in</I> Venus <I>argueth its motion to be about the Sun.</I></marg> <I>Venus</I> in the next place, and to <I>Mercury,</I> we are certain that they revolve about the Sun, in that they never move far from him, and in that we $ee them one while above and another while <marg><I>The Moon can- not $eperate from the Earth.</I></marg> below it, as the mutations of figure in <I>Venus</I> nece$$arily argueth. Tonchiug the Moon it is certain, that $he cannot in any way <foot>$e-</foot> <p n=>296</p> $eperate from the Earth, for the rea$ons that $hall be more di$tinct- ly alledged hereafter.</P> <P>SAGR. I expect that I $hall hear more admirable things that depend upon this annual motion of the Earth, than were tho$e dependant upon the diurnal revolution.</P> <marg><I>The annual mo- tion of the Earth mixing with the motions of the o- ther Planets pro- duce extravagant appearances.</I></marg> <P>SALV. You do not therein erre: For as to the operation of the diurnal motion upon the Cele$tial bodies, it neither was, nor can be other, than to make the Univer$e $eem to run precipitately the contrary way; but this annual motion intermixing with the particular motions of all the planets, produceth very many ex- travagancies, which have di$armed and non-plu$t all the greate$t Scholars in the World. But returning to our fir$t general appre- hen$ions, I reply that the centre of the Cele$tial conver$ions of the five planets <I>Saturn, Jupiter, Mars, Venus</I> and <I>Mercury,</I> is the Sun; and $hall be likewi$e the centre of the motion of the Earth, if we do but $ucceed in our attempt of placing it in Hea- ven. And as for the Moon, this hath a circular motion about the Earth, from which (as I $aid before) it can by no means alienate it $elf, but yet doth it not cea$e to go about the Sun together with the Earth in an annual motion.</P> <P>SIMP. I do not as yet very well apprehend this $tructure, but it may be, that with making a few draughts thereof, one may bet- ter and more ea$ily di$cour$e concerning the $ame.</P> <P>SALV. Tis very true: yea for your greater $atisfaction and ad- mi<*>ation together, I de$ire you, that you would take the pains to draw the $ame; and to $ee that although you think you do not apprehend it, yet you very perfectly under$tand it; And onely by an$wering to my interrogations you $hall de$igne it punctually. <marg><I>The Sy$teme of the Univer$e de- $igned from the ap- pearances.</I></marg> Take therefore a $heet of paper and Compa$les; And let this white paper be the immen$e expan$ion of the Univer$e; in which you are to di$tribute and di$po$e its parts in order, according as rea$on $hall direct you. And fir$t, in regard that without my in- $truction you verily believe that the Earth is placed in this Uni- ver$e, therefore note a point at plea$ure, about which you in- tend it to to be placed, and mark it with $ome characters.</P> <P>SIMP. Let this mark A be the place of the Terre$trial Globe.</P> <P>SALV. Very well. I know $econdly, that you under$tand per- fectly that the $aid Earth is not within the body of the Sun, nor $o much as contiguous to it, but di$tant for $ome $pace from the $ame, and therefore a$$ign to the Sun what other place you be$t like, as remote from the Earth as you plea$e, and mark this in like manner.</P> <P>SIMP. Here it is done: Let the place of the Solar body be O.</P> <P>SALV. The$e two being con$tituted, I de$ire that we may <foot>think</foot> <p n=>297</p> think of accomodating the body of <I>Venus</I> in $uch a manner that its $tate and motion may agree with what $en$ible experiments do $hew us; and therefore recall to mind that. which either by the pa$t di$cour$es, or your own ob$ervations you have learnt to be- fal that $tar, and afterwards a$$ign unto it that $tate which you think agreeth with the $ame.</P> <P>SIMP. Suppo$ing tho$e <I>Phænomena</I> expre$$ed by you, and which I have likewi$e read in the little treati$e of Conclu$ions, to <fig> be true, namely, that that $tar never recedes from the Sun beyond $uch a determinate $pace of 40 degrees or thereabouts, $o as that it never cometh either to appo$ition with the Sun, or $o much as to quadrature, or yet to the $extile a$pect; and more than that, <marg>Venus <I>very greas towards the re$pe- ctive conjunction and very $mall to- wards the matu- tine.</I></marg> $uppo$ing that it $heweth at one time almo$t 40 times greater than at another; namely, very great, when being retrograde, it goeth to the ve$pertine conjnnction of the Sun, and very $mall when with a <foot>Pp motion</foot> <p n=>298</p> motion $traight forwards, it goeth to the matutine conjunction; and moreover it being true, that when it appeareth bigge it $hews with a corniculate figure, and when it appeareth little, it $eems perfectly round, the$e appearances, I $ay, being true, I do not $ee how one can choo$e but affirm the $aid $tar to revolve in a circle a- <marg>Venus <I>nece$$a- rily proved to move about the Sun.</I></marg> bout the Sun, for that the $aid circle cannot in any wi$e be $aid to encompa$$e or to contain the Earth within it, nor to be inferi- our to the Sun, that is between it and the Earth, nor yet $upe- riour to the Sun. That circle cannot incompa$$e the Earth, be- cau$e <I>Venus</I> would then $ometimes come to oppofition with the Sun; it cannot be inferiour, for then <I>Venus</I> in both its conjuncti- ons with the Sun would $eem horned; nor can it be $uperiour, for then it would alwayes appear round, and never cornicular; and therefore for receit of it I will draw the circle CH, about the Sun, without encompa$$ing the Earth.</P> <P>SALV. Having placed <I>Venus,</I> it is requi$ite that you think of <I>Mercury,</I> which, as you know, alwayes keeping about the Sun, doth recede le$$e di$tance from it than <I>Venus</I>; therefore con$ider with your $elf, what place is mo$t convenient to a$$ign it.</P> <marg><I>The revolution of</I> Mercury <I>concluded to be about the Sun, within the Orb of</I> Venus.</marg> <P>SIMP. It is not to be que$tioned, but that this Planet imitat- ing <I>Venus,</I> the mo$t commodious place for it will be, a le$$er cir- cle within this of <I>Venus,</I> in like manner about the Sun, being that of its greate$t vicinity to the Sun, an argument, an evidence $ufficiently proving the vigour of its illumination, above that of <I>Venus,</I> and of the other Planets, we may therefore upon the$e con$iderations draw its Circle, marking it with the Characters BG.</P> <marg>Mars <I>nece$$arily includeth within its Orb the Earth, and al$o the Sun.</I></marg> <P>SALV. But <I>Mars,</I> Where $hall we place it?</P> <P>SIMP. <I>Mars,</I> Becau$e it comes to an oppo$ition with the Sun, its Circle mu$t of nece$$ity encompa$s the Earth; But I $ee that it mu$t nece$$arily encompa$s the Sun al$o, for coming to conjuncti- on with the Sun, if it did not move over it, but were below it, it would appear horned, as <I>Venus</I> and the Moon; but it $hews al- wayes round, and therefore it is nece$$ary, that it no le$s includ- <marg>Mars <I>at its oppo- $ition to the Sun $hews to be $ixty times bigger than towards the con- junction.</I></marg> eth the Sun within its circle than the Earth. And becau$e I re- member that you did $ay, that when it is in oppo$ition with the Sun, it $eems 60 times bigger than when it is in the conjunction, me thinks that a Circle about the Centre of the Sun, and that tak- eth in the earth, will very well agree with the$e <I>Phænomena,</I> which I do note and mark D I, where <I>Mars</I> in the point D, is near to the earth, and oppo$ite to the Sun; but when it is in the point I, it is at Conjuction with the Sun, but very far from the Earth. <marg>J<*>piter <I>and</I> Sa- turn <I>do likewi$e en- compa$$e the Earth, and the Sun.</I></marg> And becau$e the $ame appearances are ob$erved in <I>Jupiter</I> and <I>Saturn,</I> although with much le$$er difference in <I>Jupiter</I> than in <I>Mars,</I> and with yet le$$e in <I>Saturn</I> than in <I>Jupiter</I>; me thinks I <foot>un-</foot> <p n=>299</p> under$tand that we $hould very commodiou$ly $alve all the <I>Phæ- nomena</I> of the$e two Planets, with two Circles, in like manner, drawn about the Sun, and this fir$t for <I>Jupiter,</I> marking it E L, and another above that for <I>Saturn</I> marked F M.</P> <marg><I>The approxima- tion and rece$$ion of the three $uperiour Planets, importeth double the Suns di- $tance.</I></marg> <P>SALV. You have behaved your $elf bravely hitherto. And becau$e (as you $ee) the approach and rece$$ion of the three Su- periour Planets is mea$ured with double the di$tance between the Earth and Sun, this maketh greater difference in <I>Mars</I> than in <I>Ju-</I> <marg><I>The difference of the apparent mag- nitude le$$e in</I> Sa- turn, <I>than in</I> Jupi- ter, <I>an dn</I> Jupiter <I>than in</I> Mars, <I>and why.</I></marg> <I>piter,</I> the Circle D I, of <I>Mars,</I> being le$$er than the Circle E L, of <I>Jupiter,</I> and likewi$e becau$e this E L, is le$$e than this Circle F M, of <I>Saturn,</I> the $aid difference is al$o yet le$$er in <I>Saturn</I> than in <I>Jupiter,</I> and that punctually an$wereth the <I>Phænomena.</I> It remains now that you a$$ign a place to the Moon.</P> <P>SIMP. Following the $ame Method (which $eems to me very <marg><I>The Moons Orb invironeth the Earth, but not the Sun.</I></marg> conclu$ive) in regard we $ee that the Moon cometh to conjunction and oppo$ition with the Sun, it is nece$$ary to $ay, that its circle encompa$$eth the Earth, but yet doth it not follow, that it mu$t environ the Sun, for then at that time towards its conjunction, it would not $eem horned, but alwayes round and full of Light. Moreover it could never make, as it often doth, the Eclip$e of the Sun, by interpo$ing betwixt it and us; It is nece$$ary therefore to a$$ign it a circle about the Earth, which $hould be this N P, $o that being con$tituted in P, it will appear from the Earth A, to be in conjunction with the Sun, and placed in N, it appeareth oppo$ite to the Sun, and in that po$ition it may fall under the Earths $ha- dow, and be ob$cured.</P> <P>SALV. Now, <I>Simplicius,</I> what $hall we do with the fixed $tars? Shall we $uppo$e them $cattered through the immen$e abi$- $es of the Univer$e, at different di$tances, from any one determi- nate point; or el$e placed in a $uperficies $pherically di$tended a- bout a centre of its own, $o that each of them may be equi- di$tant from the $aid centre?</P> <marg><I>The probable $ituation of the fixed $tars.</I></marg> <P>SIMP. I would rather take a middle way; and would a$$ign them an Orb de$cribed about a determinate centre and comprized within two $pherical $uperficies, to wit, one very high, and con- cave, and the other lower, and convex, betwixt which I would <marg><I>Which ought to be accounted the $phere of the Vm- ver$e.</I></marg> con$titute the innumerable multitude of $tars, but yet at divers al- titudes, and this might be called the Sphere of the Univer$e, contein- ing within it the Orbs of the planets already by us de$cribed.</P> <P>SALV. But now we have all this while, <I>Simplicius,</I> di$po$ed the mundane bodies exactly, according to the order of <I>Copernicus,</I> and we have done it with your hand; and moreover to each of them you have a$$igned peculiar motions of their own, except to the Sun, the Earth, and $tarry Sphere; and to <I>Mercury</I> with <I>Venus,</I> you have a$cribed the circular motion about the Sun, <foot>Pp 2 with-</foot> <p n=>300</p> without encompa$$ing the Earth; about the $ame Sun you make the three $uperiour Planets <I>Mars, Jupiter,</I> and <I>Saturn,</I> to move, comprehending the Earth within their circles. The Moon in the next place can move in no other manner than about the Earth, without taking in the Sun, and in all the$e motions you agree al$o with the $ame <I>Copernicus.</I> There remains now three things to be decided between the Sun, the Earth, and fixed $tars, namely, <marg><I>Re$t, the annual motion and the di- urnal ought to be di$tributed be- twixt the Sun, Earth, and Fir- mament.</I></marg> Re$t, which $eemeth to belong to the Earth; the annual motion under the Zodiack, which appeareth to pertain to the Sun; and the diurnal motion, which $eems to belong to the Starry Sphere, and to be by that imparted to all the re$t of the Univer$e, the Earth excepted, And it being true that all the Orbs of the Planets, I <marg><I>In a moveable $phere, it $eemeth more vea$onable that its centre be $table, than any o- ther of its parts.</I></marg> mean of <I>Mercury, Venus, Mars, Jupiter,</I> and <I>Saturn,</I> do move about the Sun as their centre; re$t $eemeth with $o much more rea$on to belong to the $aid Sun, than to the Earth, in as much as in a moveable Sphere, it is more rea$onable that the centre $tand $till, than any other place remote from the $aid centre; to the Earth therefore, which is con$tituted in the mid$t of move- able parts of the Univer$e, I mean between <I>Venus</I> and <I>Mars,</I> one of which maketh its revolution in nine moneths, and the other in two years, may the motion of a year very commodiou$ly be a$- <marg><I>Granting to the Earth the annual, it mu$t of nece$$ity al$o have the diur- nal motion a$$ign- ed to it.</I></marg> $igned, leaving re$t to the Sun. And if that be $o, it followeth of nece$$ary con$equence, that likewi$e the diurnal motion be- longeth to the Earth; for, if the Sun $tanding $till, the Earth $hould not revolve about its $elf, but have onely the annual mo- tion about the Sun, our year would be no other than one day and one night, that is $ix moneths of day, and $ix moneths of night, as hath already been $aid. You may con$ider withal how commo- diou$ly the precipitate motion of 24 hours is taken away from the Univer$e, and the fixed $tars that are $o many Suns, are made in conformity to our Sun to enjoy a perpetual re$t. You $ee more- over what facility one meets with in this rough draught to render the rea$on of $o great appearances in the Cele$tial bodies.</P> <P>SAGR. I very well perceive that facility, but as you from this $implicity collect great probabilities for the truth of that Sy$tem, others haply could make thence contrary deductions; doubting, not without rea$on, why that $ame being the ancient Sy$teme of <I>Pythagoreans,</I> and $o well accommodated to the <I>Phænomena,</I> hath in the $ucce$$ion of $o many thou$and years had $o few fol- lowers, and hath been even by <I>Ari$totle</I> him$elf refuted, and $ince that <I>Copernicus</I> him$elf hath had no better fortune.</P> <P>SALV. If you had at any time been a$$aulted, as I have been, many and many a time, with the relation of $uch kind of frivolous rea$ons, as $erve to make the vulgar contumacious, and difficult to be per$waded to hearken, (I will not $ay to con$ent) to this novel- <foot>ty,</foot> <p n=>301</p> ty, I believe that you wonder at the paucity of tho$e who are fol- lowers of that opinion would be much dimini$hed. But $mall re- <marg><I>Di$cour$es more than childi$h, $erve to keep fools in the opinion of the Earths $tability.</I></marg> gard in my judgement, ought to be had of $uch thick $culs, as think it a mo$t convincing proof to confirm, and $teadfa$tly $ettle them in the belief of the earths immobility, to $ee that if this day they cannot Dine at <I>Con$tantinople,</I> nor Sup in <I>Jappan,</I> that then the Earth as being a mo$t grave body cannot clamber above the Sun, and then $lide headlong down again; Of $uch as the$e I $ay, who$e number is infinite, we need not make any reckoning, nor need we to record their foolieries, or to $trive to gain to our $ide as our partakers in $ubtil and $ublime opinions, men in who$e de- finition the kind onely is concerned, and the difference is wanting. Moreover, what ground do you think you could be able to gain, with all the demon$trations of the World upon brains $o $tupid, as are not able of them$elves to know their down right follies? But my admiration, <I>Sagredus,</I> is very different from yours, you won- der that $o few are followers of the <I>Pythagorean</I> Opinion; and I am amazed how there could be any yet left till now that do em- brace and follow it: Nor can I $ufficiently admire the eminencie of <marg><I>A declaration of the improbabi- lity of</I> Copernicus <I>his opinion.</I></marg> tho$e mens wits that have received and held it to be true, and with the $prightline$$e of their judgements offered $uch violence to their own $ences, as that they have been able to prefer that which their rea$on dictated to them, to that which $en$ible experiments re- pre$ented mo$t manife$tly on the contrary. That the rea$ons again$t the Diurnal virtiginous revolution of the Earth by you already ex- amined, do carry great probability with them, we have already $een; as al$o that the <I>Ptolomaicks,</I> and <I>Ari$totelicks,</I> with all their Sectators did receive them for true, is indeed a very great argument of their efficacie; but tho$e experiments which apertly contradict the annual motion, are of yet $o much more manife$tly repugnant, <marg><I>Rea$ons and di$- cour$e in</I> Ari$tar- cus <I>and</I> Coperni- cus <I>prevailed over manife$t $ence.</I></marg> that (I $ay it again) I cannot find any bounds for my admiration, how that rea$on was able in <I>Ari$tarchus</I> and <I>Copernicus,</I> to com- mìt $uch a rape upon their Sences, as in de$pight thereof, to make her $elf mi$tre$s of their credulity.</P> <P>SAGR. Are we then to have $till more of the$e $trong oppo$iti- ons again$t this annual motion?</P> <P>SALV. We are, and they be $o evident and $en$ible, that if a $ence more $ublime and excellent than tho$e common and vulgar, did not take part with rea$on, I much fear, that I al$o $hould have been much more aver$e to the <I>Copernican</I> Sy$teem than I have been $ince the time that a clearer lamp than ordinary hath enlightned me.</P> <P>SAGR. Now therefore <I>Salviatus,</I> let us come to joyn battail for every word that is $pent on any thing el$e, I take to be ca$t a- way.</P> <foot>SALV.</foot> <p n=>302</p> <P>SALV. I am ready to $erve you. You have already $een me draw the form of the <I>Copernican</I> Sy$teme; again$t the truth of <marg>Mars <I>makes an hot a$$ault upon the</I> Copernican <I>Sy- $teme.</I></marg> which <I>Mars</I> him$elf, in the fir$t place, makes an hot charge; who, in ca$e it were true, that its di$tances from the earth $hould $o much vary, as that from the lea$t di$tance to the greate$t, there were twice as much difference, as from the earth to the Sun; it would be nece$$ary, that when it is neare$t unto us, its <I>di$cus</I> would $hew more than 60. times bigger than it $eems, when it is farthe$t from us; neverthele$s that diver$ity of apparent magnitude is not to be $een, nay in its oppo$ition with the Sun, when its neare$t to the Earth, it doth not $hew $o much as quadruple and quintuple in bigne$s, to what it is, when towards the conjunction it cometh to be occulted under the Suns rayes. Another and greater difficulty doth <I>Venus</I> exhibit; For if revolving about the Sun, as <I>Copernicus</I> <marg><I>The</I> Phænome- na <I>of</I> Venus <I>appear contrary to the Sy- $teme of</I> Coperni- cus.</marg> affirmeth, it were one while above, & another while below the $ame, receding and approaching to us $o much as the Diameter of the cir- cle de$cribed would be, at $uch time as it $hould be below the Sun, and neare$t to us, its <I>di$cus</I> would $hew little le$s than 40 times big- ger than when it is above the Sun, near to its other conjunction; yet neverthele$$e, the difference is almo$t imperceptible Let us add an- <marg><I>Another diffi- culty rai$ed by</I> Ve- nus <I>again$t</I> Coper- nicus.</marg> other difficulty, that in ca$e the body of <I>Venus</I> be of it $elf dark, and onely $hineth as the Moon, by the illumination of the Sun, which $eemeth mo$t rea$onable; it would $hew forked or horned at $uch time as it is under the Sun, as the Moon doth when $he is in like manner near the Sun; an accident that is not to be di$covered in her. Whereupon <I>Copernicus</I> affirmeth, that either $he is light of <marg>Venus, <I>according to</I> Copernicus, <I>ei- ther lucid in it $elf, or el$e of a tran$parent $ub- $tance.</I></marg> her $elf, or el$e that her $ub$tance is of $uch a nature, that it can imbue the Solar light, and tran$mit the $ame through all its whole depth, $o as to be able to appear to us alwayes $hining; and in this manner <I>Copernicus</I> excu$eth the not changing figure in <I>Venus</I>: but of her $mall variation of Magnitude, he maketh no mention at all; <marg>Copernicus <I>$peak- eth nothing of the $mall variation of bigne$s in</I> Venus <I>and in</I> Mars.</marg> and much le$s of <I>Mars</I> than was needful; I believe as being una- ble $o well as he de$ired to $alve a <I>Phænomenon</I> $o contrary to his Hypothe$is, and yet being convinced by $o many other occurrences and rea$ons he maintained, and held the $ame Hypothe$is to be true. Be$ides the$e things, to make the Planets, together with the Earth, to move above the Sun as the Centre of their conver$ions, and the <marg><I>The moon much di$turbeth the or- der of the other Planets.</I></marg> Moon onely to break that order, and to have a motion by it $elf about the earth; and to make both her, the Earth, and the whole Elementary <I>Sphere,</I> to move all together about the Sun in a year, this $eemeth to pervert the order of this Sy$teme, which rendreth it unlikely and fal$e. The$e are tho$e difficulties that make me wonder how <I>Aristarchus</I> and <I>Copernicus,</I> who mu$t needs have ob- $erved them, not having been able for all that to $alve them, have yet notwith$tanding by other admirable occurrences been induced <foot>to</foot> <p n=>303</p> to con$ide $o much in that which rea$on dictated to them, as that they have con$idently affirmed that the $tructure of the Univer$e could have no other figure than that which they de$igned to them- $elves. There are al$o $everal other very $erious and curious doubts, not $o ea$ie to be re$olved by the middle $ort of wits, but yet pe- netrated and declared by <I>Coperninus,</I> which we $hall defer till by and by, after we have an$wered to other objections that $eem to make again$t this opinion. Now coming to the declarations and an$wers to tho$e three before named grand Objections, I $ay, that the two fir$t not onely contradict not the <I>Copernican</I> Sy$teme, but <marg><I>An$wers to the three first objecti- ons again$t the</I> Co- pernican <I>Sy$teme.</I></marg> greatly and ab$olutely favour it; For both <I>Mars</I> and <I>Venus</I> $eems unequal to them$elves, according to the proportions a$$igned; and <I>Venus</I> under the Sun $eemeth horned, and goeth changing figures in it $elf exactly like the Moon.</P> <P>SAGR. But how came this to be concealed from <I>Copernicus,</I> and revealed to you?</P> <P>SALV. The$e things cannot be comprehended, $ave onely by the $en$e of $eeing, the which by nature was not granted to man $o perfect, as that it was able to attain to the di$covery of $uch dif- ferences; nay even the very in$trument of $ight is an impediment to it $elf: But $ince that it hath plea$ed God in our age to vouch- $afe to humane ingenuity, $o admirable an invention of perfecting our $ight, by multiplying it four, $ix, ten, twenty, thirty, and four- ty times, infinite objects, that either by rea$on of their di$tance, or for their extream $mallne$$e were invi$ible unto us, have by help of the Tele$cope been rendered vi$ible.</P> <P>SAGR. But <I>Venus</I> and <I>Mars</I> are none of the objects invi$ible for their di$tance or $mallne$$e, yea, we do di$cern them with our bare natural $ight; why then do we not di$tingui$h the differences of their magnitudes and figures?</P> <P>SALV. In this, the impediment of our very eye it $elf hath a <marg><I>Therea$on whence it happens that</I> Ve- nus <I>and</I> Mars <I>do not appear to vary magnitude $o much as is requi$ite.</I></marg> great $hare, as but even now I hinted, by which the re$plendent and remote objects are not repre$ented to us $imple and pure; but gives them us fringed with $trange and adventitious rayes, $o long and den$e, that their naked body $heweth to us agrandized ten, twen- ty, an hundred, yea a thou$and times more than it would appear, if the capillitious rayes were taken away.</P> <P>SAGR. Now I remember that I have read $omething on this $ubject, I know not whether in the Solar Letters, or in the <I>Sag- giatore</I> of our common Friend, but it would be very good, a$well for recalling it into my memory, as for the information of <I>Simpli- cius,</I> who it may be never $aw tho$e writings, that you would de- clare unto us more di$tinctly how this bu$ine$$e $tands, the know- ledge whereof I think to be very nece$$ary for the a$$i$ting of us to under$tand that of which we now $peak.</P> <foot>SIMP.</foot> <p n=>304</p> <P>SIMP. I mu$t confe$$e that all that which <I>Salviatus</I> hath $po- ken is new unto me, for truth is, I never have had the curio$ity to read tho$e Books, nor have I hitherto given any great credit to the Tele$cope newly introduced; rather treading in the $teps of o- <marg><I>The operations of the Tele$cope ac- counted fallacies by the</I> Peripateticks.</marg> ther <I>Peripatetick</I> Philo$ophers my companions, I have thought tho$e things to be fallacies and delu$ions of the Chry$tals, which others have $o much admired for $tupendious operations: and therefore if I have hitherto been in an errour, I $hall be glad to be freed from it, and allured by the$e novelties already heard from you, I $hall the more attentively hearken to the re$t.</P> <P>SALV. The confidence that the$e men have in their own ap- prehen$ivene$$e, is no le$s unrea$onable than the $mall e$teem they have of the judgment of others: yet its much that they $hould e- $teem them$elves able to judge better of $uch an in$trument, with- out ever having made trial of it, than tho$e who have made, and daily do make a thou$and experiments of the $ame: But I pray you, let us leave this kind of pertinacious men, whom we can- not $o much as tax without doing them too great honour. And re- <marg><I>Shining objects $eem environed with adventitious rayes.</I></marg> turning to our purpo$e, I $ay, that re$plendent objects, whether it is that their light doth refract on the humidity that is upon the pupils, or that it doth reflect on the edges of the eye-browes, dif- fu$ing its reflex rayes upon the $aid pupils, or whether it is for $ome other rea$on, they do appear to our eye, as if they were environ'd with new rayes, and therefore much bigger than their bodies would repre$ent them$elves to us, were they dive$ted of tho$e ir- <marg><I>The rea$on why luminous bodies ap- pear enlarged much the more, by how much they are le$$er.</I></marg> radiations. And this aggrandizement is made with a greater and greater proportion, by how much tho$e lucid objects are le$$er and le$$er; in the $ame manner for all the world, as if we $hould $up- po$e that the augmentation of $hining locks were <I>v.g.</I> four inches, which addition being made about a circle that hath four inches di- ameter would increa$e its appearance to nine times its former big- ne$$e: but---------</P> <P>SIMP. I believe you would have $aid three times; for adding four inches to this $ide, and four inches to that $ide of the diame- ter of a circle, which is like wi$e four inches, its quantity is there- by tripled, and not made nine times bigger.</P> <P>SALV. A little more <I>Geometry</I> would do well, <I>Simplicius.</I> <marg><I>Superficial fi- gures encrea$<*>ing proportion àouble to their lines.</I></marg> True it is, that the diameter is tripled, but the $uperficies, which is that of which we $peak, increa$eth nine times: for you mu$t know, <I>Simplicius,</I> that the $uperficies of circles are to one another, as the $quares of their diameters; and a circle that hath four inches diameter is to another that hath twelve, as the $quare of four to the $quare of twelve; that is, as 16. is to 144 and therefore it $hall be increa$ed nine times, and not three; this, by way of adverti$e- ment to <I>Simplicius.</I> And proceeding forwards, if we $hould add <foot>the</foot> <p n=>305</p> the $aid irradiation of four inches to a circle that hath but two in- ches of diameter onely, the diameter of the irradiation or Gar- land would be ten inches, and the $uperficial content of the circle would be to the <I>area</I> of the naked body, as 100. to 4. for tho$e are the $quares of 10. and of 2. the agrandizement would there- fore be 25. times $o much; and la$tly, the four inches of hair or fringe, added to a $mall circle of an inch in diameter, the $ame would be increa$ed 81. times; and $o continually the augmenta- tions are made with a proportion greater and greater, according as the real objects that increa$e, are le$$er and le$$er.</P> <P>SAGR. The doubt which puzzled <I>Simplicius</I> never troubled me, but certain other things indeed there are, of which I de$ire a more di$tinct under$tanding; and in particular, I would know up- on what ground you affirm that the $aid agrandizement is alwayes equal in all vi$ible objects.</P> <marg><I>Objects the more vigorous they are in light, the more they do $eem to in- crea$e.</I></marg> <P>SALV. I have already declared the $ame in part, when I $aid, that onely lucid objects $o increa$ed, and not the ob$cure; now I adde what remaines, that of the re$plendent objects tho$e that are of a more bright light, make the reflection greater and more re- $plendent upon our pupil; whereupon they $eem to augment much more than the le$$e lucid: and that I may no more inlarge my $elf upon this particular, come we to that which the true Mi- $tris of <I>Astronomy,</I> Experience, teacheth us. Let us this evening, when the air is very ob$cure, ob$erve the $tar of <I>Jupiter</I>; we $hall $ee it very glittering, and very great; let us afterwards look <marg><I>An ea$ie expe- riment that $hew- eth the increa$e in the $tars, by means of the adventitious rays.</I></marg> through a tube, or el$e through a $mall trunk, which clutching the hand clo$e, and acco$ting it to the eye, we lean between the palm of the hands and the fingers, or el$e by an hole made with a $mall needle in a paper; and we $hall $ee the $aid $tar dive$ted of its beams, but $o $mall, that we $hall judge it le$$e, even than a $ixti- eth part of its great glittering light $een with the eye at liberty: we may afterwards behold the <I>Dog-$tars</I> beautiful and bigger than <marg>Jupiter <I>augments le$$e than the</I> Dog- $tar.</marg> any of the other fixed $tars, which $eemeth to the bare eye no great matter le$$e than <I>Jupiter</I>; but taking from it, as before, the irradiation, its <I>Di$cus</I> will $hew $o little, that it will not be thought the twentieth part of that of <I>Jupiter,</I> nay, he that hath not very good eyes, will very hardly di$cern it; from whence it may be rationally inferred, that the $aid $tar, as having a much more lively light than <I>Jupiter,</I> maketh its irradiation greater than <I>Jupi- ter</I> doth his. In the next place, as to the irradiation of the Sun and Moon, it is as nothing, by means of their magnitude, which <marg><I>The</I> Sun <I>and</I> Moon <I>increa$e lis- tle.</I></marg> po$$e$$eth of it $elf alone $o great a $pace in our eye, that it lea- veth no place for the adventitious rayes; $o that their faces $eem clo$e clipt, and terminate. We may a$$ure our $elves of the $ame truth by another experiment which I have often made triall of; <foot>Qq we</foot> <p n=>306</p> <marg><I>It is $een by ma- nife$t experience, that the more $plendid bodies do much more irradi- ate than the le$$e lucid.</I></marg> we may a$$ure our $elves, I $ay, that bodies $hining with mo$t| live- ly light do irradiate, or beam forth rayes more by far than tho$e that are of a more langui$hing light. I have many times $een <I>Ju- piter</I> and <I>Venus</I> together twenty or thirty degrees di$tant from the Sun, and the air being very dark, <I>Venus</I> appeared eight or ten times bigger than <I>Jupiter,</I> being both beheld by the eye at liber- ty; but being beheld afterwards with the Tele$cope, the <I>Di$cus</I> of <I>Jupiter</I> di$covered it $elf to be four or more times greater than that of <I>Venus,</I> but the vivacity of the $plendour of <I>Venus</I> was in- comparably bigger than the langui$hing light of <I>Jupiter</I>; which was only becau$e of <I>Jupiters</I> being far from the Sun, and from us; and <I>Venus</I> neer to us, and to the Sun. The$e things premi$ed, it will not be difficult to comprehend, how Mars, when it is in oppo- $ition to the Sun, and therefore neerer to the Earth by $even times, and more, than it is towards the conjunction, cometh to appear $carce four or five times bigger in that $tate than in this, when as it $hould appear more than fifty times $o much; of which the only irradiation is the cau$e; for if we dive$t it of the adventitious rayes, we $hall find it exactly augmented with the due proportion: but to take away the capillitious border, the Tele$cope is the be$t <marg><I>The</I> Tele$cope <I>is the be$t means to take away the ir- radiations of the Stars.</I></marg> and only means, which inlarging its <I>Di$cus</I> nine hundred or a thou$and times, makes it to be $een naked and terminate, as that of the Moon, and different from it $elf in the two po$itions, ac- <marg><I>Another $econd rea$on of the $mall apparent increa$e of</I> Venus.</marg> cording to its due proportions to an hair. Again, as to <I>Venus,</I> that in its ve$pertine conjunction, when i<*> is below the Sun, ought to $hew almo$t fourty times bigger than in the other matutine con- junction, and yet doth not appear $o much as doubled; it happen- eth, be$ides the effect of the irradiation, that it is horned; and its cre$cents, be$ides that they are $harp, they do receive the Suns light obliquely, and therefore emit but a faint $plendour; $o that as being little and weak, its irradiation becometh the le$$e ample and vivacious, than when it appeareth to us with its Hemi$phere all $hining: but now the Tele$cope manife$tly $hews its hornes to have been as terminate and di$tinct as tho$e of the Moon, and appear, as it were, with a great circle, and in a proportion tho$e well neer fourty times greater than its $ame <I>Di$cus,</I> at $uch time as it is $uperiour to the Sun in its ultimate matutine apparition.</P> <P>SAGR. Oh, <I>Nicholas Copernicus,</I> how great would have been thy joy to have $een this part of thy Sy$teme, confirmed with $o manife$t experiments!</P> <marg>Copernicus <I>per- $waded by rea$ons contrary to $en$ible experiments.</I></marg> <P>SALV. Tis true. But how much le$$e the fame of his $ublime wit among$t the intelligent? when as it is $een, as I al$o $aid before, that he did con$tantly continue to affirm (being per$waded thereto by rea$on) that which $en$ible experiments $eemed to contradict; for I cannot cea$e to wonder that he $hould con$tantly per$i$t in $aying, that <I>Venus</I> revolveth about the Sun, and is more than $ix <p n=>307</p> times farther from us at one time, than at another; and al$o $eem- eth to be alwayes of an equal bigne$s, although it ought to $hew forty times bigger when neare$t to us, than when farthe$t off.</P> <P>SAGR. But in <I>Jupiter, Saturn</I> and <I>Mercury,</I> I believe that the differences of their apparent magnitudes, $hould $eem punctu- ally to an$wer to their different di$tances.</P> <P>SALV. In the two Superiour ones, I have made preci$e ob- $ervations yearly for this twenty two years la$t pa$t: In <I>Mercury</I> <marg>Mercury <I>admit- teth not of clear ob$ervations.</I></marg> there can be no ob$ervation of moment made, by rea$on it $uf- fers not it $elf to be $een, $ave onely in its greate$t digr$$ieons from the Sun, in which its di$tances from the earth are in$en$ibly unequal, and tho$e differences con$equently not to be ob$erved; as al$o its mutations of figures which mu$t ab$olutely happen in it, as in <I>Venus.</I> And if we do $ee it, it mu$t of nece$$ity appear in form of a Semicircle, as <I>Venus</I> likewi$e doth in her greate$t digre$$ions; but its <I>di$cus</I> is $o very $mall, and its $plendor $o very great, by rea$on of its vicinity to the Sun, that the virtue of the Tele$cope doth not $uffice to clip its tre$$es or adventitious rayes, $o as to make them appear $haved round about. It re- <marg><I>The difficulties removed that ari$e from the Earths moving about the Sun, not $olitarily, but in con$ort with the Moon.</I></marg> mains, that we remove that which $eemed a great inconvenience in the motion of the Earth, namely that all the Planets moving about the Sun, it alone, not $olitary as the re$t, but in company with the Moon, and the whole Elementary Sphear, $hould move round about the Sun in a year; and that the $aid Moon withal $hould move every moneth about the earth. Here it is nece$$ary once again to exclaim and extol the admirable per$picacity of <I>Co- pernicus,</I> and withal to condole his misfortune, in that he is not now alive in our dayes, when for removing of the $eeming ab- $urdity of the Earth and Moons motion in con$ort we $ee <I>Jupi- ter,</I> as if it were another Earth, not in con$ort with the Moon, but accompanied by four Moons to rovolve about the Sun in 12. years together, with what ever things the Orbs of the four Medi- cæan Stars can contain within them.</P> <P>SALV. Why do you call the four jovial Planets, Moons?</P> <P>SAGR. Such they would $eem to be to one that $tanding in <marg><I>The</I> Medicean <I>Stars areas it were four Moons about</I> Jupiter.</marg> <I>Jupiter</I> $hould behold them; for they are of them$elves dark, and receive their light from the Sun, which is manife$t from their be- ing eclip$ed, when they enter into the cone of <I>Jupiters</I> $hadow: and becau$e onely tho$e their Hemi$pheres, that look towards the Sun are illuminated, to us that are without their Orbs, and near- er to the Sun, they $eem alwayes <I>lucid,</I> but to one that $hould be in <I>Jupiter,</I> they would $hew all illuminated, at $uch time as they were in the upper parts of their circles; but in the parts inferi- our, that is between <I>Jupiter</I> and the Sun, they would from <I>Ju- piter</I> be ob$erved to be horned; and in a word they would, to <foot>Qq 2 the</foot> <p n=>308</p> the ob$ervators $tanding in <I>Jupiter,</I> make the $elf $ame changes of Figure, that to us upon the Earth, the Moon doth make. You $ee now how the$e three things, which at $ir$t $eémed di$$onant, do admirably accord with the <I>Copernican</I> Sy$teme. Here al$o by the way may <I>Simplicius</I> $ee, with what probability one may con- clude, that the Sun and not the Earth, is in the Centre of the <I>Planetary</I> conver$ions. And $ince the Earth is now placed a- mong$t mundane Bodies, that undoubtedly move about the Sun, to wit, above <I>Mercury</I> and <I>Venus,</I> and below <I>Saturn, Jupiter,</I> and <I>Mars</I>; $hall it not be in like manner probable, and perhaps nece$$ary to grant, that it al$o moveth round?</P> <P>SIMP. The$e accidents are $o notable and con$picuous, that it is not po$$ible, but that <I>Ptolomy</I> and others his Sectators, $hould have had knowledge of them, and having $o, it is likewi$e nece$- $ary, that they have found a way to render rea$ons of $uch, and $o $en$ible appearances that were $ufficient, and al$o congruous and probable, $eeing that they have for $o long a time been re- ceived by $uch numbers of learned men.</P> <marg><I>The Principal $cope of A$trono- mers, is to give a rea$on of appear- ances.</I></marg> <P>SALV. You argue very well; but you know that the principal $cope of <I>A$tronomers,</I> is to render only rea$on for the appearances in the Cæle$tial Bodies, and to them, and to the motions of the Stars, to accomodate $uch $tructures and compo$itions of Circles, that the motions following tho$e calculations, an$wer to the $aid appearances, little $crupling to admit of $ome exorbitances, that indeed upon other accounts they would much $tick at. And <I>Co-</I> <marg>Copernicus <I>re- $tored A$tronomy upon the $uppo$iti- ous of</I> Ptolomy:</marg> <I>pernic us</I> him$elf writes, that he had in his fir$t $tudies re$tored the Science of <I>A$tronomy</I> upon the very $uppo$itions of <I>Ptolomy,</I> and in $uch manner corrected the motions of the Planets, that the computations did very exactly agree with the <I>Phænomena,</I> and the <I>Phænomena</I> with the $upputations, in ca$e that he took the Planets $everally one by one. But he addeth, that in going a- bout to put together all the $tructures of the particular Fabricks, there re$ulted thence a Mon$ter and <I>Chimæra,</I> compo$ed of mem- bers mo$t di$proportionate to one another, and altogether incom- patible; So that although it $atisfied an <I>A$tronomer</I> meerly <I>A- rithmetical,</I> yet did it not afford $atisfaction or content to the <marg><I>What moved</I> Co- pernicus <I>to e$ta- bli$h his Sy$teme.</I></marg> <I>A$tronomer Phylo$ophical.</I> And becau$e he very well under- $tood, that if one might $alve the Cæle$tial appearances with fal$e a$$umptions in nature, it might with much more ea$e be done by true $uppo$itions, he $et him$elf diligently to $earch whether a- ny among$t the antient men of fame, had a$cribed to the World any other $tructure, than that commonly received by <I>Ptolomy</I>; and finding that $ome <I>Pythagoreans</I> had in particular a$$igned the Diurnal conver$ion to the Earth, and others the annual mo- tion al$o, he began to compare the appearances, and particulari- <foot>ties</foot> <p n=>309</p> ties of the Planets motions, with the$e two new $uppo$itions, all which things jumpt exactly with his purpo$e; and $eeing the whole corre$pond, with admirable facility to its parts, he imbraced this new Sy$teme, and it took up his re$t.</P> <P>SIMP. But what great exorbitancies are there in the <I>Ptolo- maick</I> Sy$teme, for which there are not greater to be found in this of <I>Copernicus</I>?</P> <P>SALV. In the <I>Ptolomaick Hypothe$is</I> there are di$ea$es, and in <marg><I>Inconveniencies that are in the Sy- $teme of</I> Ptolomy.</marg> the <I>Copernican</I> their cures. And fir$t will not all the Sects of <I>Phylo$ophers,</I> account it a great inconvenience, that a body na- turally moveable in circumgyration, $hould move irregularly upon its own Centre, and regularly upon another point? And yet there are $uch deformed motions as the$e in the <I>Ptolomæan</I> Hypo- the$is, but in the <I>Copernican</I> all move evenly about their own Centres. In the <I>Ptolomaick,</I> it is nece$$ary to a$$ign to the Cæ- le$tial bodies, contrary motions, and to make them all to move, from Ea$t to We$t, and at the $ame time, from We$t to Ea$t; But in the <I>Copernican,</I> all the Cæle$tial revolutions are towards one onely way, from We$t to Ea$t. But what $hall we $ay of the apparent motion of the Planets, $o irregular, that they not on- ly go one while $wift, and another while $low, but $ometimes wholly $eace to move; and then after a long time return back a- gain? To $alve which appearances <I>Ptolomie</I> introduceth very great <I>Epicicles,</I> accommodating them one by one to each Planet, with $ome rules of incongruous motions, which are all with one $in- gle motion of the Earth taken away. And would not you, <I>Sim- plicius,</I> call it a great ab$urditie, if in the <I>Ptolomaick</I> Hypothe- $is, in which the particular Planets, have their peculiar Orbs a$- $igned them one above another, one mu$t be frequently forced to $ay, that <I>Mars,</I> con$tituted above the Sphære of the Sun, doth $o de$cend, that breaking the Solar Orb, it goeth under it, and approacheth neaer to the Earth, than to the Body of the Sun, and by and by immea$urably a$cendeth above the $ame? And yet this, and other exorbitancies are remedied by the Soul and fingle annual motion of the Earth.</P> <P>SAGR. I would gladly be bettter informed how the$e $tations, and retrograde and direct motions, which did ever $eem to me great improbalities, do accord in this <I>Copernican</I> Sy$teme.</P> <marg><I>Its a great Ar- gument in favour of</I> Copernicus, <I>that he obviates the $ta- tions & retrograda- tions of the motions of the Planets.</I></marg> <P>SALV. You $hall $ee them $o to accord, <I>Sagredus,</I> that this onely conjecture ought to be $ufficient to make one that is not more than pertinacious or $tupid, yield, a$$ent to all the re$t of this Doctrine. I tell you therefore, that nothing being altered in the motion of <I>Saturn,</I> which is 30 years, in that of <I>Jupiter,</I> which is 12, in that of <I>Mars,</I> which is 2, in that of <I>Venus,</I> which is 9. moneths, in that of <I>Mercury,</I> which is 80. <foot>dayes,</foot> <p n=>310</p> dayes, or thereabouts, the $ole annual motion of the Earth be- tween <I>Mars</I> and <I>Venus,</I> cau$eth the apparent inequalities in all <marg><I>The $ole annual motion of the Earth cau$eth great inequality of motions in the five Planets.</I></marg> the five $tars before named. And for a facile and full under- $tanding of the whole, I will de$cribe this figure of it. There- fore $uppo$e the Sun to be placed in the centre O, about which we will draw the Orb de$cribed by the Earth, with the an- nual motion B G M, and let the circle de$cribed, <I>v. gr.</I> by <I>Jupiter</I> about the Sun in 12. years, be this BGM, and in the <fig> <marg><I>A demon$tration of the inequalities of the three $uperiour Planets dependent on the annual mo- tion of the Earth.</I></marg> $tarry $phere let us imagine the Zodiack Y V S. Again, in the annual Orb of the Earth let us take certain equal arches, B C, C D, E F, F G, G H, H I, I K, K L, L M, and in the Sphere of <I>Jupiter</I> let us make certain other arches, pa$$ed in the $ame times in which the Earth pa$$eth hers, which let be B C, C D, D E, E F, F G, G H, H I, I K, K L, L M, which $hall each be proportionally le$$e than the$e marked in the Earths Orb, like as the motion of <I>Jupiter</I> under the Zodiack is $lower than the annual. Suppo$ing now, that when the Earth is in B, <I>Jupiter</I> is in B, it $hall appear to us in the Zodiack to be in P, de$cribing <foot>the</foot> <p n=>311</p> the right line B B P. Next $uppo$e the Earth to be moved from B to C, and <I>Jupiter</I> from B to C, in the $ame time; <I>Iupiter</I> $hall appear to have pa$$ed in the Zodiack to Q, and to have moved $traight forwards, according to the order of the $ignes P Q. In the next place, the Earth pa$$ing to D, and <I>Iupiter</I> to D, it $hall be $een in the Zodiack in R, and from E, <I>Iupi- ter</I> being come to E; will appear in the Zodiack in S, having all this while moved right forwards. But the Earth afterwards beginning to interpo$e more directly between <I>Iupiter</I> and the Sun, $he being come to F, and <I>Iupiter</I> to F, he will appear in T, to have already begun to return apparently back again un- der the Zodiack, and in that time that the Earth $hall have pa$- ed the arch E F, <I>Iupiter</I> $hall have entertained him$elf between the points S T, and $hall have appeared to us almo$t motion- le$$e and $tationary. The Earth being afterwards come to G, and <I>Iupiter</I> to G, in oppo$ition to the Sun, it $hall be vi$ible in the Zodiack at V, and much returned backwards by all the arch of the Zodiack T V; howbeit that all the way pur$uing its even cour$e it hath really gone forwards not onely in its own circle, but in the Zodiack al$o in re$pect to the centre of the $aid Zodi- ack, and to the Sun placed in the $ame. The Earth and <I>Iupiter</I> again continuing their motions, when the Earth is come to H, and <I>Iupiter</I> to H, it $hall $eem very much gone backward in the Zodiack by all the arch V X. The Earth being come to I, and <I>Iupiter</I> to I, it $hall be apparently moved in the Zodiack by the lit- tle $pace X Y, and there it will $eem $tationary. When after- wards the Earth $hall be come to K, and <I>Iupiter</I> to K; in the Zodiack he $hall have pa$$ed the arch Y N in a direct motion; and the Earth pur$uing its cour$e to L, $hall $ee <I>Iupiter</I> in L, in the point Z. And la$tly <I>Iupiter</I> in M $hall be $een from the Earth M, to have pa$$ed to A, with a motion $till right forwards; and its whole apparent retrogadation in the Zodiack $hall an$wer to the arch S Y, made by <I>Iupiter,</I> whil$t that he in his own circle pa$$eth the arch E I, and the Earth in hers the arch E I. And <marg><I>Retrogradations more frequent in</I> Saturn, <I>le$$e in</I> Ju- piter, <I>and yet le$$e in</I> Mars, <I>and why.</I></marg> this which hath been $aid, is intended of <I>Saturn</I> and of <I>Mars</I> al$o; and in <I>Saturn</I> tho$e retrogradations are $omewhat more frequent than in <I>Jupiter,</I> by rea$on that its motion is a little $lower than that of <I>Jupiter,</I> $o that the Earth overtaketh it it in a $horter $pace of time; in <I>Mars</I> again they are more rare, for that its motion is more $wift than that of <I>Jupiter.</I> Whereupon the Earth con$umeth more time in recovering it. Next as to <I>Venus</I> and <I>Mercury,</I> who$e Circles are comprehended by that <marg><I>The Retrograda- tion of</I> Venus <I>and</I> Mercury <I>demon- $trated by</I> Apollo- nius <I>and</I> Coperni- cus.</marg> of the Earth, their $tations and regre$$ions appear to be occa$i- oned, not by their motions that really are $uch, but by the anual motion of the $aid Earth, as <I>Copernicus</I> exellently demon$trateth, <foot>to-</foot> <p n=>312</p> together with <I>Appollonius Pergæus</I> in <I>lib.</I> 5. of his Revolutions, <I>Chap.</I> 35.</P> <P>You $ee, Gentlemen, with what facility and $implicity the annu- <marg><I>The annual mo- tion of the Earth mo$t apt to render a rea$on of the ex- orbttances of the five Planets.</I></marg> al motion, were it appertaining to the Earth, is accommodated to render a rea$on of the apparent exorbitances, that are ob$erved in the motions of the five Planets, <I>Saturn, Jupiter, Mars, Ve- nus</I> and <I>Mercury,</I> taking them all away, and reducing them to <marg><I>The Sun it $elf te$tifieth the annu- al motion to belong to the Earth.</I></marg> equal and regular motions. And of this admirable effect, <I>Ni- cholas Copernicus,</I> hath been the fir$t that hath made the rea$on plain unto us. But of another effect, no le$$e admirable than this, and that with a knot, perhaps more difficult to unknit, bindeth the wit of man, to admit this annual conver$ion, and to leave it to our Terre$trial Globe; a new and unthought of con- jecture ari$eth from the Sun it $elf, which $heweth that it is unwil- ling to be $ingular in $hifting, of this atte$tation of $o eminent a conclu$ion, rather as a te$timony beyond all exception, it hath de$ired to be heard apart. Hearken then to this great and new wonder.</P> <marg><I>The Lyncæan Academick the fir$t di$coverer of the Solar $pots, and all the other cele- $tial novelties.</I></marg> <P>The fir$t di$coverer and ob$erver of the <I>Solar</I> $pots, as al$o of all the other Cœle$tial novelties, was our <I>Academick Lincæus</I>; and he di$covered them <I>anno</I> 1610. being at that time Reader of the <I>Mathematicks,</I> in the Colledge of <I>Padua,</I> and there, and in <I>Ve- nice,</I> he di$cour$ed thereof with $everal per$ons, of which $ome <marg><I>The hi$tory of the proceedings of the Academian for a long time a- bout the ob$ervati- on of the Solar $pots.</I></marg> are yet living: And the year following, he $hewed them in <I>Rome</I> to many great per$onages, as he relates in the fir$t of his Letters to <I>Marcus Vel$erus,</I> ^{*} Sheriffe of <I>Augu$ta.</I> He was the fir$t that again$t the opinions of the too timorous and too jealous <marg>* Duumviro.</marg> a$$ertors of the Heavens inalterability, affirmed tho$e $pots to be matters, that in $hort times were produced and di$$olved: for as to place, they were contiguous to the body of the Sun, and re- volved about the $ame; or el$e being carried about by the $aid Solar body, which revolveth in it $elfe about its own Centre, in the $pace almo$t of a moneth, do fini$h their cour$e in that time; which motion he judged at fir$t to have been made by the Sun a- bout an Axis erected upon the plane of the Ecliptick; in regard that the arches de$cribed by the $aid $pots upon the <I>Di$cus</I> of the Sun appear unto our eye right lines, and parallels to the plane of the Ecliptick: which therefore come to be altered, in part, with $ome accidental, wandring, and irregular motions, to which they are $ubject, and whereby tumultuarily, and without any order they $ucce$$ively change $ituations among$t them$elves, one while crouding clo$e together, another while di$$evering, and $ome dividing them$elves into many and very much changing fi- gures, which, for the mo$t part, are very unu$ual. And albeit tho$e $o incon$tant mutations did $omewhat alter the primary pe- <foot>riodick</foot> <p n=>313</p> riodick cour$e of the $aid $pots, yet did they not alter the opini- on of our friend, $o as to make him believe, that they were any e$$ential and fixed cau$e of tho$e deviations, but he continued to hold, that all the apparent alterations derived them$elves from tho$e accidental mutations: in like manner, ju$t as it would hap- pen to one that $hould from far di$tant Regions ob$erve the mo- tion of our Clouds; which would be di$covered to move with a mo$t $wift, great, and con$tant motion, carried round by the di- urnal <I>Vertigo</I> of the Earth (if haply that motion belong to the $ame) in twenty four hours, by circles parallel to the Equinocti- al, but yet altered, in part, by the accidental motions cau$ed by the winds, which drive them, at all adventures, towards different quarters of the World. While this was in agitation, it came to pa$s that <I>Vel$erus</I> $ent him two Letters, written by a certain per- <marg>* This Authors true name is <I>Chri- $topher Scheiner us</I> a Je$uit, and his Book here meant is intituled, <I>Apel- les po$t tabulam.</I></marg> $on, under the feigned name of ^{*} <I>Apelles,</I> upon the $ubject of the$e Spots, reque$ting him, with importunity, to declare his thoughts freely upon tho$e Letters, and withall to let him know what his opinion was touching the e$$ence of tho$e $pots; which his reque$t he $atisfied in 3 Letters, $hewing fir$t of all howvain the conjectures of <I>Apelles</I> were; & di$covering, $econdly, his own opi- nions; withal foretelling to him, that <I>Apelles</I> would undoubtedly be better advi$ed in time, and turn to his opinion, as it afterwards came to pa$s. And becau$e that our Academian (as it was al$o the judgment of many others that were intelligent in Natures $e- crets) thought he had in tho$e three Letters inve$tigated and de- mon$trated, if not all that could be de$ired, or required by hu- mane curio$ity, at lea$t all that could be attained by humane rea$on in $uch a matter, he, for $ome time (being bu$ied in other $tudies) intermitted his continual ob$ervations, and onely in com- placency to $ome friend, joyned with him, in making now and then an abrupt ob$ervation: till that he, and after $ome years, <marg>* La mia villa delle Selue.</marg> we, being then at my ^{*} Country-$eat, met with one of the $olita- ry Solar $pots very big, and thick, invited withal by a clear and con$tant $erenity of the Heavens, he, at my reque$t, made ob$er- vations of the whole progre$$e of the $aid $pot, carefully marking upon a $heet of paper the places that it was in every day at the time of the Suns coming into the Meridian; and we having found that its cour$e was not in a right line, but $omewhat incurvated, we came to re$olve, at la$t, to make other ob$ervations from time to time; to which undertaking we were $trongly induced by a conceit, that accidentally came into the minde of my Gue$t, which he imparted to me in the$e or the like words.</P> <P>In my opinion, <I>Philip,</I> there is a way opened to a bu$ine$s of very great con$equence. For if the Axis about which the Sun turneth be not erect perpendicularly to the plane of the Eclip- <foot>Rr tick,</foot> <p n=>314</p> <marg><I>A concipt that came $uddenly in- to the minde of the Academian</I> Lyncæus <I>concern- ing the great con- $equence that fol- lowed upon the mo- tion of the Solar $pots.</I></marg> tick, but is inclined upon the $ame, as its crooked cour$e, but e- ven now ob$erved, makes me believe, we $hall be able to make $uch conjectures of the $tates of the Sun and Earth, as neither $o $olid or $o rational have been hitherto deduced from any other ac- cident what$oever. I being awakened at $o great a promi$e, im- portun'd him to make a free di$covery of his conceit unto me. And he continued his di$cour$e to this purpo$e. If the Earths <marg><I>Extravagant mu- tations to be ob$er- ved in the motions of the $pots, fore- $een by the Aca- demick, in ca$e the Earth had the annual motion.</I></marg> motion were along the Ecliptique about the Sun; and the Sun were con$tituted in the centre of the $aid Ecliptick, and therein revolved in its $elf, not about the Axis of the $aid Ecliptique (which would be the Axis of the Earths annual motion) but up- on one inclined, it mu$t needs follow, that $trange changes will repre$ent them$elves to us in the apparent motions of the Solar $pots, although the $aid Axis of the Sun $hould be $uppo$ed to per$i$t perpetually and immutably in the $ame inclination, and in one and the $ame direction towards the $elf-$ame point of the Univer$e. Therefore the Terre$trial Globe in the annual motion moving round it, it will fir$t follow, that to us, carried about by the $ame, the cour$es of the $pots $hall $ometimes $eem to be made in right lines, but this only twice a year, and at all other times $hall appear to be made by arches in$en$ibly incurvated. Secondly, the curvity of tho$e arches for one half of the year, will $hew inclined the contrary way to what they will appear in the other half; that is, for $ix moneths the convexity of the ar- ches $hall be towards the upper part of the Solar <I>Di$cus,</I> and for the other $ix moneths towards the inferiour. Thirdly, the $pots be- ginning to appear, and (if I may $o $peak) to ri$e to our eye from the left $ide of the Solar <I>Di$cus,</I> and going to hide them$elves and to $et in the right $ide, the Oriental termes, that is, of their fir$t appearings for $ix moneths, $hall be lower than the oppo$ite termes of their occultations; and for other $ix moneths it $hall happen contrarily, to wit, that the $aid $pots ri$ing from more e- levated points, and from them de$cending, they $hall, in their cour$es, go and hide them$elves in lower points; and onely for two dayes in all the year $hall tho$e termes of ri$ings and $et- tings be equilibrated: after which freely beginning by $mall de- grees the inclination of the cour$es of the $pots, and day by day growing bigger, in three moneths, it $hall arrive at its greate$t obliquity, and from thence beginning to dimini$h, in $uch another time it $hall reduce it $elf to the other <I>Æquilibrium.</I> It $hall hap- pen, for a fourth wonder, that the cour$e of the greate$t obli- quity $hall be the $ame with the cour$e made by the right line, and in the day of the Libration the arch of the cour$e $hall $eem more than ever incurvated. Again, in the other times, accord- ing as the pendency $hall $ucce$$ively dimini$h, and make its ap- <foot>proach</foot> <p n=>315</p> proach towards the <I>Æquilibrium,</I> the incurvation of the arches of the cour$es on the contrary $hall, by degrees, increa$e.</P> <P>SAGR. I confe$$e, <I>Salviatus,</I> that to interrupt you in your Di$cour$e is ill manners, but I e$teem it no le$$e rudene$s to per- mit you to run on any farther in words, whil$t they are, as the $aying is, ca$t into the air: for, to $peak freely, I know not how to form any di$tinct conceit of $o much as one of the$e conclu$i- ons, that you have pronounced; but becau$e, as I thus general- ly and confu$edly apprehend them, they hold forth things of ad- mirable con$equence, I would gladly, $ome way or other, be made to under$tand the $ame.</P> <P>SALV. The $ame that befalls you, befell me al$o, whil$t my Gue$t tran$ported me with bare words; who afterwards a$$i$ted my capacity, by de$cribing the bu$ine$$e upon a material In$tru- <marg><I>The fir$t Ac- cident to be ob$er- ved in the motion of the Solar $pots; and con$equently all the re$t explai- ned.</I></marg> ment, which was no other than a $imple Sphere, making u$e of $ome of its circles, but to a different purpo$e from that, to which they are commonly applied. Now I will $upply the defect of the Sphere, by drawing the $ame upon a piece of paper, as need $hall require. And to repre$ent the fir$t accident by me propoun- ded, which was, that the cour$es or journeys of the $pots, twice a year, and no more, might be $een to be made in right lines, let us $uppo$e this point O [<I>in Fig.</I> 4.] to be the centre of the grand Orb, or, if you will, of the Ecliptick, and likewi$e al$o of the Globe of the Sun it $elf; of which, by rea$on of the great di- $tance that is between it and the Earth, we that live upon the Earth, may $uppo$e that we $ee the one half: we will therefore de$cribe this circle A B C D about the $aid centre O, which repre- $enteth unto us the extream term that divideth and $eparates the Hemi$phere of the Sun that is apparent to us, from the other that is occult. And becau$e that our eye, no le$$e than the centre of the Earth, is under$tood to be in the plane of the Ecliptick, in which is likewi$e the centre of the Sun, therefore, if we $hould fancy to our $elves the body of the Sun to be cut thorow by the $aid plane, the $ection will appear to our eye a right line, which let be B O D, and upon that a perpendicular being let fall AOC, it $hall be the Axis of the $aid Ecliptick, and of the annual mo- tion of the Terre$trial Globe. Let us next $uppo$e the Solar body (without changing centre) to revolve in it $elf, not about the Axis A O C (which is the erect Axis upon the plane of the E- cliptick) but about one $omewhat inclined, which let be this E O I, the which fixed and unchangeable Axis maintaineth it $elf perpetually in the $ame inclination and direction towards the $ame points of the Firmament, and of the Univer$e. And be- cau$e, in the revolutions of the Solar Globe, each point of its $u- perficies (the Poles excepted) de$cribeth the circumference of a <foot>Rr 2 circle</foot> <p n=>316</p> circle, either bigger or le$$er, according as it is more or le$$e re- mote from the $aid Poles, let us take the point F, equally di$tant from them, and draw the diameter F O G, which $hall be perpen- dicular to the Axis E I, and $hall be the diameter of the grand circle de$cribed about the Poles E I. Suppo$ing not that the Earth, and we with her be in $uch a place of the Ecliptick, that the Hemi$phere of the Sun to us apparent is determin'd or bound- ed by the circle A B C D, which pa$$ing (as it alwayes doth) by the Poles A C, pa$$eth al$o by E I. It is manife$t, that the grand circle, who$e diameter is FG, $hall be erect to the circle A B C D, to which the ray that from our eye falleth upon the centre O, is perpendicular; $o that the $aid ray falleth upon the plane of the circle, who$e diameter is F G, and therefore its circumference will appear to us a right line, and the $elf $ame with F G, where- upon if there $hould be in the point F, a $pot, it comming after- wards to be carried about by the Solar conver$ion, would, upon the $urface of the Sun, trace out the circumference of that cir- cle, which $eems to us a right line. Its cour$e or pa$$age will therefore $eem $traight. And $traight al$o will the motion of the other $pots appear, which in the $aid revolution $hall de$cribe le$- $er circles, as being all parallel to the greater, and to our eye placed at an immen$e di$tance from them. Now, if you do but con$ider, how that after the Earth $hall in $ix moneths have run thorow half the grand Orb, and $hall be $ituate oppo$ite to that Hemi$phere of the Sun, which is now occult unto us, $o as that the boundary of the part that then $hall be $een, may be the $elf $ame A B C D, which al$o $hall pa$$e by the Poles E I; you $hall under$tand that the $ame will evene in the cour$es of the $pots, as before, to wit, that all will appear to be made by right lines. But becau$e that that accident takes not place, $ave one- ly when the terminator or boundary pa$$eth by the Poles E I, and the $aid terminator from moment to moment, by meanes of the Earths annual motion, continually altereth, therefore its pa$- $age by the fixed Poles E I, $hall be momentary, and con$equent- ly momentary $hall be the time, in which the motions of tho$e $pots $hall appear $traight. From what hath been hitherto $poken one may comprehend al$o how that the apparition and beginning of the motion of the $pots from the part F, proceeding towards G, their pa$$ages or cour$es are from the left hand, a$cending to- wards the right; but the Earth being placed in the part diame- trically oppo$ite the appearance of the $pots about G, $hall $till be to the left hand of the beholder, but the pa$$age $hall be de$- cending towards the right hand F. Let us now de$cribe the Earth te be $ituate one fourth part farther di$tant from its pre$ent $tate, and let us draw, as in the other figure, the terminator A B C D, <foot>[<I>as</I></foot> <p n=>317</p> [<I>as in Fig.</I> 5.] and the Axis, as before A C, by which the plane of our Meridian would pa$$e, in which plane $hould al$o be the Axis of the Suns revolution, with its Poles, one towards us, that is, in the apparent Hemi$phere, which Pole we will repre$ent by the point E, and the other $hall fall in the occult Hemi$phere, and I mark it I. Inclining therefore the Axis E I, with the $upe- riour part E, towards us, the great circle de$cribed by the Suns conver$ion, $hall be this B F D G, who$e half by us $een, name- ly B F D, $hall no longer $eem unto us a right line, by rea$on the Poles E I are not in the circumference A B C D, but $hall appear incurvated, and with its convexity towards the inferiour part C. And it is manife$t, that the $ame will appear in all the le$$er cir- cles parallel to the $ame B F D. It is to be under$tood al$o, that when the Earth $hall be diametrically oppo$ite to this $tate, $o that it $eeth the other Hemi$phere of the Sun, which now is hid, it $hall of the $aid great circle behold the part D G B incurved, with its convexity towards the $uperiour part A; and the cour- $es of the $pots in the$e con$titutions $hall be fir$t, by the arch B F D, and afterwards by the other D G B, and the fir$t appari- tions and ultimate occultations made about the points B and D, $hall be equilibrated, and not tho$e that are more or le$$e eleva- ted than the$e. But if we con$titute the Earth in $uch a place of the Ecliptick, that neither the boundary A B C D, nor the Meridian A C, pa$$eth by the Poles of the Axis E I, as I will $hew you anon, drawing this other Figure [<I>viz. Fig.</I> 6.] wherein the apparent or vi$ible Pole E falleth between the arch of the termi- nator A B, and the $ection of the Meridian A C; the diameter of the great circle $hall be F O G, and the apparent $emicircle F N G, and the occult $emicircle G S F, the one incurvated with its convexity N towards the inferiour part, and the other al$o bending with its convexity S towards the upper part of the Sun. The ingre$$ions and exitions of the $pots, that is, the termes F and G $hall not be librated, as the two others B and D; but F $hall be lower, and G higher: but yet with le$$er difference than in the fir$t Figure. The arch al$o F N G $hall be incurva- ted, but not $o much as the precedent B F D; $o that in this po- $ition the pa$$ages or motions of the $pots $hall be a$cendent from the left $ide F, towards the right G, and $hall be made by curved lines. And imagining the Earth to be con$tituted in the po$ition diametrically oppo$ite; $o that the Hemi$phere of the Sun, which was before the occult, may be the apparent, and ter- minated by the $ame boundary A B C D, it will be manife$tly di$cerned, that the cour$e of the $pots $hall be by the arch G S F, beginning from the upper point G, which $hall then be likewi$e from the left hand of the beholder, and going to determine, de$- <foot>cending</foot> <p n=>318</p> $cending towards the right, in the point F. What I have hi- therto $aid, being under$tood, I believe that there remains no difficulty in conceiving how $rom the pa$$ing of the terminator of the Solar Hemi$pheres by the Poles of the Suns conver$ion, or neer or far from the $ame, do ari$e all the differences in the appa- rent cour$es of the $pots; $o that by how much the more tho$e Poles $hall be remote from the $aid terminator, by $o much the more $hall tho$e cour$es be incurvated, and le$$e oblique; whereupon at the $ame di$tance, that is, when tho$e Poles are in the $ection of the Meridian, the incurvation is reduced to the greate$t, but the obliquity to the lea$t, that is to <I>Æquilibrium,</I> as the $econd of the$e three la$t figures [<I>viz. Fig.</I> 5.] demon$trateth. On the contrary, when the Poles are in the terminator, as the fir$t of the$e three figures [<I>viz. Fig.</I> 4.] $heweth the inclination is at the greate$t, but the incurvation at the lea$t, and reduced to rectitude. The terminator departing from the Poles, the curvity begins to grow $en$ible, the obliquity all the way encrea$ing, and the inclination growing le$$er.</P> <P>The$e are tho$e admirable and extravagant mutations, that my Gue$t told me would from time to time appear in the progre$$es of the Solar $pots, if $o be it $hould be true that the annual mo- tion belonged to the Earth, and that the Sun being con$tituted in the centre of the Ecliptick, were revolved in it $elf upon an Axis, not erect, but inclined to the Plane of the $aid Eclip- tick.</P> <P>SAGR. I do now very well apprehend the$e con$equences, and believe that they will be better imprinted in my fancy, when I $hall come to reflect upon them, accommodating a Globe to tho$e inclinations, and then beholding them from $everal pla- ces. It now remains that you tell us what followed afterwards touching the event of the$e imaginary con$equences.</P> <marg><I>The events be- ing ob$erved, were an$werable to the predictions.</I></marg> <P>SALV. It came to pa$$e thereupon, that continuing many $e- veral moneths to make mo$t accurate ob$ervations, noting down with great exactne$$e the cour$es or tran$itions of $undry $pots at divers times of the year, we found the events punctually to cor- re$pond to the predictions.</P> <P>SAGR. <I>Simplicius,</I> if this which <I>Salviatus</I> $aith be true; (nor can we di$tru$t him upon his word) the <I>Ptolomeans</I> and <I>Aristo- teleans</I> hadneed of $olid arguments, $trong conjectures, and well grounded experiments to counterpoi$e an objection of $o much weight, and to $upport their opinion from its final over- throw.</P> <P>SIMP. Fair and $oftly good Sir, for haply you may not yet be got $o far as you per$wade your $elf you are gone. And though I am not an ab$olute ma$ter of the $ubject of that narra- <foot>tion</foot> <p n=>319</p> tion given us by <I>Salviatus</I>; yet do I not find that my Logick, <marg><I>Though the an- nual motion a$$ign- ed to the Earth an- $werth to the</I> Phæ- nomena <I>of the $o- lar $pots, yet doth it not follow by con- ver$ion that from the</I> Phænomena <I>of the $pots one may infor the annual motion to belong to the Earth.</I></marg> whil$t I have a regard to form, teacheth me, that that kind of ar- gumentation affords me any nece$$ary rea$on to conclude in fa- vour of the <I>Copernican</I> Hypothe$is, that is, of the $tability of the Sun in the centre of the Zodiack, and of the mobility of the Earth under its circumference. For although it be true, that the $aid conver$ion of the Sun, and cirnition of the Earth being granted, there be a nece$$ity of di$cerning $uch and $uch $trange extravagancies as the$e in the $pots of the Sun, yet doth it not follow that arguing <I>per conver$um,</I> from finding $uch like un- u$ual accidents in the Sun, one mu$t of nec$$ity conclude the Earth to move by the circumference, and the Sun to be placed in the centre of the Zodiack. For who $hall a$$ertain me that the like irregularities may not as well be vi$ible in the Sun, it being moveable by the Ecliptick, to the inhabitants of the Earth, it being al$o immoveable in the centre of the $ame? Unle$$e you demon$trate to me, that there can be no rea$on given for that ap- pearance, when the Sun is made moveable, and the Earth $table, I will not alter my opinion and belief that the Sun moveth, and the Earth $tandeth $till.</P> <P>SAGR. <I>Simplicius</I> behaveth him$elf very bravely, and argueth very $ubtilly in defence of the cau$e of <I>Ari$totle</I> and <I>Ptolomy</I>; and if I may $peak the truth, mythinks that the conver$ation of <I>Salviatus,</I> though it have been but of $mall continuance, hath much farthered him in di$cour$ing $ilogi$tically. An effect which I know to be wrought in others as well as him. But as to finding and judging whether competent rea$on may be rendered of the apparent exorbitancies and irregularities in the motions of the $pots, $uppo$ing the Earth to be immoveable, and the Sun moveable, I $hall expect that <I>Salviatus</I> manife$t his opinion to us, for it is very probable that he he hath con$idered of the $ame, and collected together whatever may be $aid upon the point.</P> <P>SALV. I have often thought thereon, and al$o di$cour$ed thereof with my Friend and Gue$t afore-named; and touching what is to be produced by Philo$ophers and A$tronomers, in de- fence of the ancient Sy$teme, we are on one hand certain, cer- tain I $ay, that the true and pure <I>Peripateticks</I> laughing at $uch <marg><I>The Pure</I> Peri- patetick <I>Philo$o- phers will laugh at the $pots and their</I> Phænomena, <I>as illu$ions of the Chry$tals in the Tele$cope.</I></marg> as employ them$elves in $uch, to their thinking, in$ipid foole- ries, will cen$ure all the$e <I>Phænomena</I> to be vain illu$ions of the Chri$tals; and in this manner will with little trouble free them- $elves from the obligation of $tudying any more upon the $ame. Again, as to the A$tronomical Philo$ophers, after we have with $ome diligence weighed that which may be alledged as a mean between tho$e two others, we have not been able to find out an <foot>an$wer</foot> <p n=>320</p> an$wer that $ufficeth to $atis$ie at once the cour$e of the $pots, and the di$cour$e of the Mind. I will explain unto you $o much as I remember thereof, that $o you may judge thereon as $eems be$t unto you.</P> <P>Suppo$ing that the apparent motions of the Solar $pots are the $ame with tho$e that have been above declared, and $uppo$ing the Earth to be immoveable in the centre of the Ecliptick, in who$e circumference let the center of the Sun be placed; it is nece$$ary that of all the differences that are $een in tho$e motions, the cau- $es do re$ide in the motions that are in the body of the Sun: Which in the fir$t place mu$t nece$$arily revolve in it $elf (<I>i. e.</I> <marg><I>If the Earth be immoveable in the centre of the Zodi- ack, there mu$t be a$cribed to the Sun four $everal moti- ons, as is declared at length.</I></marg> about its own axis) carrying the $pots along therewith; which $pots have been $uppo$ed, yea and proved to adhere to the So- lar $uperficies. It mu$t $econdly be confe$t, that the Axis of the Solar conver$ion is not parallel to the Axis of the Ecliptick, that is as much as to $ay, that it is not perpendicularly erected upon the Plane of the Ecliptick, becau$e if it were $o, the cour$es and exitions of tho$e $pots would $eem to be made by right lines pa- rallel to the Ecliptick. The $aid Axis therefore is inclining, in regard the $aid cour$es are for the mo$t part made by curve lines. It will be nece$$ary in the third place to grant that the inclinati- on of this Axis is not fixed, and continually extended towards one and the $ame point of the Univer$e, but rather that it doth alwayes from moment to moment go changing its direction; for if the pendency $hould always look towards the $elf $ame point, the cour$es of the $pots would never change appearance; but appearing at one time either right or curved, bending upwards or downwards, a$cending or de$cending, they would appear the $ame at all times. It is therefore nece$$ary to $ay, that the $aid Axis is convertible; and is $ometimes found to be in the Plane of the circle that is extreme, terminate, or of the vi$ible Hemi$phere, I mean at $uch time as the cour$es of the $pots $eem to be made in right lines, and more than ever pendent, which happeneth twice a year; and at other times found to be in the Plane of the Meridian of the Ob$ervator, in $uch $ort that one of its Poles falleth in the vi$ible Hemi$phere of the Sun, and the other in the occult; and both of them remote from the ex- treme points, or we may $ay, from the poles of another Axis of the Sun, which is parallel to the Axis of the Ecliptick; (which $econd Axis mu$t nece$$arily be a$$igned to the Solar Globe) re- mote, I $ay, as far as the inclination of the Axis of the revolution of the $pots doth import; and moreover that the Pole falling in the apparent Hemi$phere, is one while in the $uperiour, another while in the inferiour part thereof; for that it mu$t be $o, the cour$es them$elves do manife$tly evince at $uch time as they are <foot>equi-</foot> <p n=>321</p> equilibrated, and in their greate$t curvity, one while with their convexity towards the upper part, and another while towards the lower part of the Solar <I>Di$cus.</I> And becau$e tho$e po$itions are in continuall alteration, making the in- clinations and incurvations now greater, now le$$er, and $ome- times reduce them$elves, the fir$t $ort to perfect libration, and the $econd to perfect perpendicularity, it is nece$$ary to a$$ert that the $elf $ame Axis of the monethly revolution of the $pots hath a particular revolution of its own, whereby its Poles de$cribe two circles about the Poles of another Axis, which for that rea- $on ought (as I have $aid) to be a$$igned to the Sun, the $emidi- ameter of which circles an$wereth to the quantity of the incli- nation of the $aid Axis. And it is nece$$ary, that the time of its Period be a year; for that $uch is the time in which all the ap- pearances and differences in the cour$es of the $pots do return. And that the revolution of this Axis, is made about the Poles of the other Axis parallel to that of the Ecliptick, & not about other points, the greate$t inclinations and greate$t incurvations, which are always of the $ame bigne$s, do clearly prove. So that finally, to maintain the Earth fixed in the centre, it will be nece$$ary to a$- $ign to the Sun, two motions about its own centre, upon two $eve- ral Axes, one of which fini$heth its conver$ion in a year, and the other in le$$e than a moneth; which a$$umption $eemeth, to my under$tanding, very hard, and almo$t impo$$ible; and this de- pendeth on the nece$$ity of a$cribing to the $aid Solar body two other motions about the Earth upon different Axes, de$cribing with one the Ecliptick in a year, and with the other forming $pi- rals, or circles parallel to the Equinoctial one every day: whereupon that third motion which ought to be a$$igned to the Solar Clobe about its own centre (I mean not that almo$t monethly, which carrieth the $pots about, but I $peak of that o- ther which ought to pa$$e thorow the Axis and Poles of this monethly one) ought not, for any rea$on that I $ee, to fini$h its Period rather in a year, as depending on the annual motion by the Ecliptick, than in twenty four hours, as depending on the diurnal motion upon the Poles of the Equinoctial. I know, that what I now $peak is very ob$cure, but I $hall make it plain unto you, when we come to $peak of the third motion annual, a$$ign- ed by <I>Copernicus,</I> to the Earth. Now if the$e four motions, $o incongruous with each other, (all which it would be nece$$ary to a$$ign to the $elf $ame body of the Sun) may be reduced to one $ole and $imple motion, a$$igned the Sun upon an Axis that never changeth po$ition, and that without innovating any thing in the motions for $o many other cau$es a$$igned to the Terre$trial Globe, may $o ea$ily $alve $o many extravagant appearances in <foot>S$ the</foot> <p n=>322</p> the motions of the Solar $pots, it $eemeth really that $uch an Hypothe$is ought not to be rejected.</P> <P>This, <I>Simplicius,</I> is all that came into the minds of our friend, and my $elf, that could be alledged in explanation of this <I>Phæno- menon</I> by the <I>Copernicans,</I> and by the <I>Ptolomæans,</I> in defence of their opinions. Do you inferre from thence what your judg- ment per$wades you.</P> <P>SIMP. I acknowledge my $elf unable to interpo$e in $o im- portant a deci$ion: And, as to my particular thoughts, I will $tand neutral; and yet neverthele$$e I hope that a time will come, when our minds being illumin'd by more lofty contempla- tions than the$e our humane rea$onings, we $hall be awakened and freed from that mi$t which now is $o great an hinderance to our $ight.</P> <P>SAGR. Excellent and pious is the coun$el taken by <I>Simpli- cius,</I> and worthy to be entertained and followed by all, as that which being derived from the highe$t wi$dome and $upreame$t authority, may onely, with $ecurity be received. But yet $o far as humane rea$on is permitted to penetrate, confining my $elf within the bounds of conjectures, and probable rea$ons, I will $ay a little more re$olutely than <I>Simplicius</I> doth, that among$t all the ingenuous $ubtilties I ever heard, I have never met with any thing of greater admiration to my intellect, nor that hath more ab$olutely captivated my judgment, (alwayes excepting pure Geometrical and Arithmetical Demon$trations) than the$e two conjectures taken, the one from the $tations and retrograda- tions of the five Planets, and the other from the$e irregularities of the motions of the Solar $pots: and becau$e they $eem to me $o ea$ily and clearly to a$$ign the true rea$on of $o extravagant ap- pearances, $hewing as if they were but one $ole $imple motion, mixed with $o many others, $imple likewi$e, but different from each other, without introducing any difficulty, rather with obvi- ating tho$e that accompany the other Hypothe$is; I am think- ing that I may rationally conclude, that tho$e who contumaci- ou$ly with$tand this Doctrine, either never heard, or never un- der$tood, the$e $o convincing arguments.</P> <P>SALV. I will not a$cribe unto them the title either of con- vincing, or non-convincing; in regard my intention is not, as I have $everal times told you, to re$olve any thing upon $o high a que$tion, but onely to propo$e tho$e natural and A$tronomicall rea$ons, which, for the one and other Sy$teme, may be produced by me, leaving the determination to others; which determinati- on cannot at la$t, but be very manife$t: for one of the two po$i- tions being of nece$$ity to be true, and the other of nece$$ity to be fal$e, it is a thing impo$$ible that (alwayes confining our $elves <foot>within</foot> <p n=>323</p> within the limits of humane doctrine) the rea$ons alledged for the true Hypothe$is $hould not manife$t them$elves as concludent as tho$e for the contrary vain and ineffectual.</P> <P>SAGR. It will be time therefore, that we hear the objections of the little Book of^{*} <I>Conclu$ions,</I> or Di$qui$itions which <I>Simpli-</I> <marg>* I $hould have told you, that the true name of this concealed Au- thour is <I>Chri$to- pher Scheinerus,</I> and its title <I>Di$- qui$itiones Ma- thematicæ.</I></marg> <I>cius</I> did bring with him.</P> <P>SIMP. Here is the Book, and this is the place where the Au- thor fir$t briefly de$cribeth the Sy$teme of the world, according to the Hypothe$is of <I>Copernicus,</I> $aying, <I>Terram igitur unà cum Luna, totoque hoc elementari mundo</I> Copernicus, &c.</P> <P>SALV. Forbear a little, <I>Simplicius,</I> for methinks that this Authour, in this fir$t entrance, $hews him$elf to be but very ill ver$t in the Hypothe$is which he goeth about to confute, in re- gard, he $aith that <I>Copernicus</I> maketh the Earth, together with the Moon, to de$cribe the ^{*} grand Orb in a year moving from Ea$t to We$t; a thing that as it is fal$e and impo$$ible, $o was it <marg>* <*> the Ecliptick</marg> never affirmed by <I>Copernicus,</I> who rather maketh it to move the contrary way, I mean from We$t to Ea$t, that is, according to the order of the Signes; whereupon we come to think the $ame to be the annual motion of the Sun, con$tituted immoveable in the centre of the Zodiack. See the too adventurous confidence of this man; to undertake the confutation of anothers Doctrine, and yet to be ignorant of the primary fundamentals; upon which his adver$ary layeth the greate$t and mo$t important part of all the Fabrick. This is a bad beginning to gain him$elf credit with his Reader; but let us go on.</P> <P>SIMP. Having explained the Univer$al Sy$teme, he beginneth to propound his objections again$t this annual motion: and the fir$t are the$e, which he citeth Ironically, and in deri$ion of <marg><I>In$tances of a certain Book Iro- nically propounded again$t</I> Coperni- cus.</marg> <I>Copernicus,</I> and of his followers, writing that in this phanta$tical Hypothe$is of the World one mu$t nece$$arily maintain very gro$$e ab$urdities; namely, that the Sun, <I>Venus,</I> and <I>Mercury</I> are below the Earth; and that grave matters go naturally up- wards, and the light downwards; and that <I>Chri$t,</I> our Lord and Redeemer, a$cended into Hell, and de$cended into Heaven, when he approached towards the Sun, and that when <I>Jo$huah</I> com- manded the Sun to $tand $till, the Earth $tood $till, or the Sun moved a contrary way to that of the Earth; and that when the Sun is in <I>Cancer,</I> the Earth runneth through <I>Capricorn</I>; and that the <I>Hyemal</I> (or Winter) Signes make the Summer, and the <I>Æ$tival</I> Winter; and that the Stars do not ri$e and $et to the Earth, but the Earth to the Stars; and that the Ea$t begin- neth in the We$t, and the We$t in the Ea$t; and, in a word, that almo$t the whole cour$e of the World is inverted.</P> <P>SALV. Every thing plea$eth me, except it be his having inter- <foot>S$ 2 mixed</foot> <p n=>324</p> mixed places out of the $acred Scriptures (alwayes venerable, and to be rever'd) among$t the$e, but two $currilous fooleries, and attempting to wound with holy Weapons, tho$e who Philo$o- phating in je$t, and for diverti$ement, neither affirm nor deny, but, $ome pre$uppo$als and po$itions being a$$umed, do famili- arly argue.</P> <P>SIMP. Truth is, he hath di$plea$ed me al$o, and that not a little; and e$pecially, by adding pre$ently after that, howbeit, the <I>Copernichists</I> an$wer, though but very impertinently to the$e and $uch like other rea$ons, yet can they not reconcile nor an$wer tho$e things that follow.</P> <P>SALV. This is wor$e than all the re$t; for he pretendeth to have things more efficacious and concludent than the Authorities of the $acred Leaves; But I pray you, let us reverence them, and pa$$e on to natural and humane rea$ons: and yet if he give us among$t his natural arguments, things of no more $olidity, than tho$e hitherto alleadged, we may wholly decline this under- taking, for I as to my own parricular, do not think it fit to $pend words in an$wering $uch trifling impertinencies. And as to what he $aith, that the <I>Copernicans</I> an$wer to the$e objections, it is mo$t fal$e, nor may it be thought, that any man $hould $et him $elf to wa$t his time $o unprofitably.</P> <marg><I>Suppo$ing the annual motion to belong to the Earth, it followeth, that one fixed Star, is bigger than the whole grand Orb.</I></marg> <P>SIMP. I concur with you in the $ame judgment; therefore let us hear the other in$tances that he brings, as much $tronger. And ob$erve here, how he with very exact computations conclud- eth, that if the grand Orb of the Earth, or the ecliptick, in which <I>Copernicus</I> maketh it to run in a year round the Sun, $hould be as it were, in$en$ible, in re$pect of the immen$itie of the Starry Sphære, according as the $aid <I>Copernicus,</I> $aith it is to be $up- po$ed, it would be nece$$ary to grant and confirm, that the fixed Stars were remote from us, an unconceivable di$tance, and that the le$$er of them, were bigger than the whole grand Orb afore- $aid, and $ome other much bigger than the whole Sphære of <I>Sa- turn</I>; Ma$$es certainly too exce$$ively va$t, unimaginable, and incredible.</P> <marg>Tycho <I>his Ar- gument grounded upon a fal$e Hypo- the$is.</I></marg> <P>SALV. I have heretofore $een $uch another objection brought by <I>Tycho</I> again$t <I>Copernicus,</I> and this is not the fir$t time that I have di$covered the fallacy, or, to $ay better, the fallacies of this Argumemtation, founded upon a mo$t fal$e Hypothe$is, and upon <marg><I>Litigious Lawyers that are entertain- ed in an ill cau$e, keep clo$e to $ome expre$$ion fallen from the adver$e party at unawares.</I></marg> a Piopo$ition of the $aid <I>Copernicus,</I> under$tood by his adver$a- ries, with too punctual a nicity, according to the practi$e of tho$e pleaders, who finding the flaw to be in the very merit of their cau$e, keep to $ome one word, fallen unawares from the contra- ry partie, and fly out into loud and tedious de$cants upon that. But for your better information; <I>Copernicus</I> having declared <foot>tho$e</foot> <p n=>325</p> tho$e admirable con$equences which are derived from the Earths <marg>* Or progre$$ions.</marg> annual motion, to the other Planets, that is to $ay, of the ^{*} directi- <marg><I>The apparent diver$ity of motion in the Planets, is in$en$ible in the fixed Start.</I></marg> ons and retrogradations of the three uppermo$t in particular; he $ubjoyneth, that this apparent mutation (which is di$cerned more in <I>Mars</I> than in <I>Jupiter,</I> by rea$on <I>Jupiter</I> is more remote, and yet le$$e in <I>Saturn,</I> by rea$on it is more remote than <I>Jupiter</I>) in the fixed Stars, did remain imperceptible, by rea$on of their immen$e remotene$$e from us, in compari$on of the di$tances of <I>Jupiter</I> or <I>Saturn.</I> Here the Adver$aries of this opinion ri$e up, and $uppo$ing that fore-named imperceptibility of <I>Copernicus,</I> as if it had been taken by him, for a real and ab$olute thing of no- thing, and adding, that a fixed Star of one of the le$$er magni- tudes, is notwith$tanding perceptible, $eeing that it cometh un- der the $ence of $eeing, they go on to calculate with the inter- vention of other fal$e a$$umptions, and concluding that it is nece$- $ary by the <I>Copernican</I> Doctrine, to admit, that a fixed Star is much bigger than the whole grand Orb. Now to di$cover the vanity <marg><I>Suppo$ing that a fixed Star of the $ixth magnitude is no bigger than the</I> Sun, <I>the diver$itie which is $o great in the Planets, in the fixed Stars is almost in$en$ible.</I></marg> of this their whole proceeding, I $hall $hew that a fixed Star of the $ixth magnitude, being $uppo$ed to be no bigger than the Sun, one may thence conclude with true demon$trations, that the di- $tance of the $aid fixed Stars from us, cometh to be $o great, that the annual motion of the Earth, which cau$eth $o great and notable variations in the Planets, appears $carce ob$ervable in them; and at the $ame time, I will di$tinctly $hew the gro$s fallacies, in the a$$umptions of <I>Copernicus</I> his Adver$aries.</P> <P>And fir$t of all, I $uppo$e with the $aid <I>Copernicus,</I> and al$o <marg><I>The di$tance of the Sun, containeth</I> 1208 <I>Semid. of the Earth.</I></marg> with his oppo$ers, that the Semidiameter of the grand Orb, which is the di$tance of the Earth from the Sun, containeth 1208 Semi- diameters of the $aid Earth. Secondly, I premi$e with the allow- ance afore$aid, and of truth, that the ^{*} apparent diameter of the <marg>* The Diameter of the Sun, half a degree.</marg> Sun in its mean di$tance, to be about half a degree, that is, 30. <I>min. prim.</I> which are 1800. $econds, that is, 108000. thirds. And becau$e the apparent Diameter of a fixed Star of the fir$t <marg><I>The Diameter of a fixed Star, of the fir$t magni- tude, and of one of the $ixth.</I></marg> magnitude, is no more than 5. $econds, that is, 300. thirds, and the Diameter of a fixed Star of the $ixth magnitude, 50. thirds, (and herein is the greate$t errour of the <I>Anti-Copernicans</I>) There- <marg><I>The apparent Diameter of the Sun, how much it is bigger than that of a fixed $tar.</I></marg> fore the Diameter of the Sun, containeth the Diameter of a fixed Star of the $ixth magnitude 2160 times. And therefore if a fixed Star of the $ixth magnitude, were $uppo$ed to be really equal to the Sun, and not bigger, which is the $ame as to $ay, if the Sun were $o far removed, that its Diameter $hould $eem to be one of the 2160. parts of what it now appeareth, its di$tance ought of nece$$ity to be 2160. times greater than now in effect it is, which is as much as to $ay, that the di$tance of the fixed Stars of the $ixth magnitude, is 2160. Semidiameters of the grand <foot>Orb.</foot> <p n=>326</p> Orb. And becau$e the di$tance of the Sun from the Earth, con- <marg><I>The di$tance of a fixed $tar of the $ixth magnitude, how much it is, the $tar being $uppo$ed to be equal to the Sun.</I></marg> tains by common con$ent 1208. Semidiameters of the $aid Earth, and the di$tance of the fixed Stars (as hath been $aid) 2160. Semediameters of the grand Orb, therefore the Semediameter of the Earth is much greater (that is almo$t double) in compari$on of the grand Orb, than the Semediameter of the grand Orb, in <marg><I>In the fixed $tars the diver$itie of a- $pect, cau$ed by the grand Orb, is little more then that cau$ed by the Earth in the Snn.</I></marg> relation to the di$tance of the Starry Sphære; and therefore the variation of a$pect in the fixed Stars, cau$ed by the Diameter of the grand Orb, can be but little more ob$ervable, than that which is ob$erved in the Sun, occa$ioned by the Semediameter of the Earth.</P> <P>SAGR. This is a great fall for the fir$t $tep.</P> <P>SALV. It is doubtle$$e an errour; for a fixed Star of the $ixth <marg><I>A $tar of the $ixth magnitude, $uppo$ed by</I> Tycho <I>and the Authour of the Book of Con- clu$ions, an hun- dred and $ix m<*>li- ons of times bigger than needs.</I></marg> magnitude, which by the computation of this Authour, ought, for the upholding the propo$ition of <I>Copernicus,</I> to be as big as the whole grand Orb, onely by $uppo$ing it equal to the Sun, which Sun is le$$e by far, than the hundred and $ix milionth part of the $aid grand Orb, maketh the $tarry Sphære $o great and high as $ufficeth to overthrow the in$tance brought again$t the $aid <I>Co- pernicus.</I></P> <P>SAGR. Favour me with this computation.</P> <P>SALV. The $upputation is ea$ie and $hort. The Diameter of the Sun, is eleven $emediameters of the Earth, and the Diameter <marg><I>The computati- on of the magni- tude of the fixed Stars, in re$pect to the grand Orb.</I></marg> of the grand Orb, contains 2416. of tho$e $ame $emediameters, by the a$cent of both parties; $o that the Diameter of the $aid Orb, contains the Suns Diameter 220. times very near. And becau$e the Spheres are to one another, as the Cubes of their Di- ameters, let us make the Cube of 220. which is 106480000. and we $hall have the grand Orb, an hundred and $ix millions, four hundred and eighty thou$and times bigger than the Sun, to which grand Orb, a $tar of the fixth magnitude, ought to be equal, ac- cording to the a$$ertion of this Authour.</P> <P>SAGR. The errour then of the$e men, con$i$teth in being ex- treamly mi$taken, in taking the apparent Diameter of the fixed Stars.</P> <P>SALV. This is one, but not the onely errour of them; and <marg><I>A common er- rour of all the</I> A- $tronomers, <I>touch- ing the magnitude of the $tars.</I></marg> indeed, I do very much admire how $o many <I>A$tronomers,</I> and tho$e very famous, as are <I>Alfagranus, Albategnus, Tebizius,</I> and much more modernly the <I>Tycho's</I> and <I>Clavius's,</I> and in $umm, all the predece$$ors of our <I>Academian,</I> $hould have been $o gro$ly mi$taken, in determining the magnitudes of all the Stars, as well $ixed as moveable, the two Luminaries excepted out of that num- ber; and that they have not taken any heed to the adventitious irradiations that deceitfully repre$ent them an hundred and more times bigger, than when they are beheld, without tho$e capilli- <foot>tious</foot> <p n=>327</p> ous rayes, nor can this their inadvertency be excu$ed, in regard that it was in their power to have beheld them at their plea$ure without tho$e tre$$es, which is done, by looking upon them at their fir$t appearance in the evening, or their la$t occultation in <marg>Venus <I>renders the errour of A$trono- mers in determin- ing the magnitudes of $tars inexcu$a- ble.</I></marg> the comming on of day; and if none of the re$t, yet <I>Venus,</I> which oft times is $een at noon day, $o $mall, that one mu$t $har- pen the $ight in di$cerning it; and again, in the following night, $eemeth a great flake of light, might adverti$e them of their fal- lacy; for I will not believe that they thought the true <I>Di$cus</I> to be that which is $een in the ob$cure$t darkne$$es, and not that which is di$cerned in the luminous <I>Medium</I>: for our lights, which $een by night afar off appear great, and neer at hand $hew their true lu$tre to be terminate and $mall, might have ea$ily have made them cautious; nay, if I may freely $peak my thoughts, I ab$olutely believe that none of them, no not <I>Tycho</I> him$elf, $o accurate in handling A$tronomical In$truments, and that $o great and accurate, without $paring very great co$t in their con$tru- ction, did ever go about to take and mea$ure the apparent dia- meter of any Star, the Sun and Moon excepted; but I think, that arbitrarily, and as we $ay, with the eye, $ome one of the more antient of them pronounced the thing to be $o and $o, and that all that followed him afterwards, without more ado, kept clo$e to what the fir$t had $aid; for if any one of them had ap- plied him$elf to have made $ome new proof of the $ame, he would doubtle$$e have di$covered the fraud.</P> <P>SAGR. But if they wanted the Tele$cope, and you have al- ready $aid, that our <I>Friend</I> with that $ame In$trument came to the knowledge of the truth, they ought to be excu$ed, and not accu$ed of ignorance.</P> <P>SALV. This would hold good, if without the Tele$cope the bu$ine$$e could not be effected. Its true, that this In$trument by $hewing the <I>Di$cus</I> of the Star naked, and magnified an hun- dred or a thou$and times, rendereth the operation much more ea- $ie, but the $ame thing may be done, although not altogether $o exactly, without the In$trument, and I have many times done the $ame, and my method therein was this. I have cau$ed a rope <marg><I>A way to mea- $ure the apparent diameter of a $tar.</I></marg> to be hanged towards $ome Star, and I have made u$e of the Con$tellation, called the <I>Harp,</I> which ri$eth between the North and ^{*} North-ea$t, and then by going towards, and from <marg>* Rendred in Latine <I>Corum,</I> that is to $ay, North- we$t.</marg> the $aid rope, interpo$ed between me and the Star, I have found the place from whence the thickne$$e of the rope hath ju$t hid the Star from me: this done, I have taken the di$tance from the eye to the rope, which was one of the $ides including the angle that was compo$ed in the eye, and ^{*} which in$i$teth upon the <marg>* <I>i.e.</I> Is $ubten- ded by.</marg> thickne$$e of the rope, and which is like, yea the $ame with the <foot>angle</foot> <p n=>328</p> angle in the Starry Sphere, that in$i$teth upon the diameter of the Star, and by the proportion of the ropes thickne$$e to the di$tance from the eye to the rope, by the table of Arches and Chords, I have immediately found the quantity of the angle; u- $ing all the while the wonted caution that is ob$erved in taking angles $o acute, not to forme the concour$e of the vi$ive rayes in the centre of the eye, where they are onely refracted, but beyond the eye, where really the pupils greatne$$e maketh them to concur.</P> <P>SAGR. I apprehend this your cautelous procedure, albeit I have a kind of hæ$itancy touching the $ame, but that which mo$t puzzleth me is, that in this operation, if it be made in the dark of night, methinks that you mea$ure the diameter of the irradia- ted <I>Di$cus,</I> and not the true and naked face of the Star.</P> <P>SALV. Not $o, Sir, for the rope in covering the naked body of the Star, taketh away the rayes, which belong not to it, but to our eye, of which it is deprived $o $oon as the true <I>Di$cus</I> thereof is hid; and in making the ob$ervation, you $hall $ee, how unexpectedly a little cord will cover that rea$onable big body of light, which $eemed impo$$ible to be hid, unle$$e it were with a much broader Screene: to mea$ure, in the next place, and exa- ctly to find out, how many of tho$e thickne$$es of the rope inter- po$e in the di$tance between the $aid rope and the eye, I take not onely one diameter of the rope, but laying many pieces of the $ame together upon a Table, $o that they touch, I take with a pair of Compa$$es the whole $pace occupied by fifteen, or twen- ty of them, and with that mea$ure I commen$urate the di$tance before with another $maller cord taken from the rope to the con- cour$e of the vi$ive rayes. And with this $ufficiently-exact ope- ration I finde the apparent diameter of a fixed Star of the fir$t magnitude, commonly e$teemed to be 2 <I>min. pri.</I> and al$o 3 <I>min. prim.</I> by <I>Tycho</I> in his <I>A$tronomical Letters, cap.</I> 167. to be no <marg><I>The diameter of a fixed $tar of the fir$t magnitude not more than five $ec. min.</I></marg> more than 5 <I>$econds,</I> which is one of the 24. or 36. parts of what they have held it: $ee now upon what gro$$e errours their Do- ctrines are founded.</P> <P>SAGR. I $ee and comprehend this very well, but before we pa$$e any further, I would propound the doubt that ari$eth in me in the finding the concour$e [or inter$ection] of the vi$ual rayes beyond the eye, when ob$ervation is made of objects com- prehended between very acute angles; and my $cruple proceeds from thinking, that the $aid concour$e may be $ometimes more remote, and $ometimes le$$e; and this not $o much, by meanes of the greater or le$$er magnitude of the object that is beheld, as becau$e that in ob$erving objects of the $ame bigne$$e, it $eems to me that the concour$e of the rayes, for certain other re- <foot>$pects</foot> <p n=>329</p> $pects ought to be made more and le$$e remote from the eye.</P> <P>SALV. I $ee already, whither the apprehen$ion of <I>Sagredus,</I> a mo$t diligent ob$erver of Natures $ecrets, tendeth; and I <marg><I>The circle of the pupil of the eye en- largeth and con- tracteth.</I></marg> would lay any wager, that among$t the thou$ands that have ob- $erved Cats to contract and inlarge the pupils of their eyes very much, there are not two, nor haply one that hath ob$erved the like effect to be wrought by the pupils of men in $eeing, whil$t the <I>medium</I> is much or little illumin'd, and that in the open light the circlet of the pupil dimini$heth con$iderably: $o that in loo- king upon the face or <I>Di$cus</I> of the Sun, it is reduced to a $mall- ne$$e le$$er than a grain of ^{*} <I>Panick,</I> and in beholding objects <marg>+ <I>Panicum,</I> a $mall grain like to Mill, I take it to be the $ame with that called Bird Seed.</marg> that do not $hine, and are in a le$$e luminous <I>medium,</I> it is inlar- god to the bigne$$e of a Lintel or more; and in $umme this expan$ion and contraction differeth in more than decuple pro- portion: From whence it is manife$t, that when the pupil is much dilated, it is nece$$ary that the angle of the rayes con- cour$e be more remote from the eye; which happeneth in be- holding objects little luminated. This is a Doctrine which <I>Sa- gredus</I> hath, ju$t now, given me the hint of, whereby, if we were to make a very exact ob$ervation, and of great con$e- quence, we are advertized to make the ob$ervation of that con- cour$e in the act of the $ame, or ju$t $uch another operation; but in this our ca$e, wherein we are to $hew the errour of <I>Astrono- mers,</I> this accuratene$$e is not nece$$ary: for though we $hould, in favour of the contrary party, $uppo$e the $aid concour$e to be made upon the pupil it $elf, it would import little, their mi$take being $o great. I am not certain, <I>Sagredus,</I> that this would have been your objection.</P> <P>SAGR. It is the very $ame, and I am glad that it was not al- together without rea$on, as your concurrence in the $ame a$$u- reth me; but yet upon this occa$ion I would willingly hear what way may be taken to finde out the di$tance of the concour$e of the vi$ual rayes.</P> <P>SALV. The method is very ea$ie, and this it is, I take two long^{*} labels of paper, one black, and the other white, and make <marg>* Stri$ce. <I>How to find the di$tance of the rays concour$e from the pupil.</I></marg> the black half as broad as the white; then I $tick up the white a- gain$t a wall, and far from that I place the other upon a $tick, or other $upport, at a di$tance of fifteen or twenty yards, and rece- ding from this, $econd another $uch a $pace in the $ame right line, it is very manife$t, that at the $aid di$tance the right lines will concur, that departing from the termes of the breadth of the white piece, $hall pa$$e clo$e by the edges of the other label pla- ced in the mid-way; whence it followeth, that in ca$e the eye were placed in the point of the $aid concour$e or inter$ection, the black $lip of paper in the mid$t would preci$ely hide the op- <foot>Tt po$ite</foot> <p n=>330</p> po$ite blank, if the $ight were made in one onely point; but if we $hould find, that the edges of the white cartel appear di$covered, it $hall be a nece$$ary argument that the vi$ual rayes do not i$$ue from one $ole point. And to make the white label to be hid by the black, it will be requi$ite to draw neerer with the eye: Therefore, having approached $o neer, that the intermediate la- bel covereth the other, and noted how much the required ap- proximation was, the quantity of that approach $hall be the cer- tain mea$ure, how much the true concour$e of the vi$ive rayes, is remote from the eye in the $aid operation, and we $hall moreover have the diameter of the pupil, or of that circlet from whence the vi$ive rayes proceed: for it $hall be to the breadth of the black paper, as is the di$tance from the concour$e of the lines, that are produced by the edges of the papers to the place where the eye $tandeth, when it fir$t $eeth the remote paper to be hid by the intermediate one, as that di$tance is, I $ay, to the di$tance that is between tho$e two papers. And therefore when we would, with exactne$$e, mea$ure the apparent diameter of a Star, having made the ob$ervation in manner, as afore$aid, it would be nece$$ary to compare the diameter of the rope to the diameter of the pupil; and having found <I>v.g.</I> the diameter of the rope to be quadruple to that of the pupil, and the di$tance of the eye from the rope to be, for example, thirty yards, we would $ay, that the true concour$e of the lines produced from the ends or extremi- ties of the diameter of the $tar, by the extremities of the dia- meter of the rope, doth fall out to be fourty yards remote from the $aid rope, for $o we $hall have ob$erved, as we ought, the pro- portion between the di$tance of the rope from the concour$e of the $aid lines, and the di$tance from the $aid concour$e to the place of the eye, which ought to be the $ame that is between the diameter of the rope, and diameter of the pupil.</P> <P>SAGR. I have perfectly under$tood the whole bu$ine$$e, and therefore let us hear what <I>Simplicius</I> hath to alledge in defence of the <I>Anti-Copernicans.</I></P> <P>SIMP. Albeit that grand and altogether incredible inconve- nience in$i$ted upon by the$e adver$aries of <I>Copernicus</I> be much moderated and abated by the di$cour$e of <I>Salviatus,</I> yet do I not think it weakened $o, as that it hath not $trength enough left to foil this $ame opinion. For, if I have rightly apprehended the chief and ultimate conclu$ion, in ca$e, the $tars of the $ixth mag- nitude were $uppo$ed to be as big as the Sun, (which yet I can hardly think) yet it would $till be true, that the grand Orb [or Ecliptick] would occa$ion a mutation and variation in the $tarry Sphere, like to that which the $emidiameter of the Earth produ- ceth in the Sun, which yet is ob$ervable; $o that neither that, no <foot>nor</foot> <p n=>331</p> nor a le$$e mutation being di$cerned in the fixed Stars, methinks that by this means the annual motion of the Earth is de$troyed and overthrown.</P> <P>SALV. You might very well $o conclude, <I>Simplicius,</I> if we had nothing el$e to $ay in behalf of <I>Copernicus</I>: but we have many things to alledge that yet have not been mentioned; and as to that your reply, nothing hindereth, but that we may $up- po$e the di$tance of the fixed Stars to be yet much greater than that which hath been allowed them, and you your $elf, and who- ever el$e will not derogate from the propo$itions admitted by <I>Piolomy</I>'s $ectators, mu$t needs grant it as a thing mo$t requi$ite to $uppo$e the Starry Sphere to be very much bigger yet than that which even now we $aid that it ought to be e$teemed. For <marg><I>All Astrono- mers agree that the greater magni- tudes of the O<*>bes is the cau$e of the tardity of the con- ver$ions.</I></marg> all A$tronomers agreeing in this, that the cau$e of the greater tardity of the Revolutions of the Planets is, the majority of their Spheres, and that therefore <I>Saturn</I> is more flow than <I>Ju- piter,</I> and <I>Jupiter</I> than the Sun, for that the fir$t is to de$cribe a greater circle than the $econd, and that than this later, &c. con- $idering that <I>Saturn v.g.</I> the altitude of who$e Orb is nine times higher than that of the Sun, and that for that cau$e the time of one Revolution of <I>Saturn,</I> is thirty times longer than that of a conver$ion of the Sun, in regard that according to the Doctrine of <I>Ptolomy,</I> one conver$ion of the $tarry Sphere is fini$hed in 36000. years, whereas that of <I>Saturn</I> is con$ummate in thirty, and that of the Sun in one, arguing with a like proportion, and <marg><I>By another $up- po$ition taken from A$tronomers, the di$tance of the fix- ed Stars is calcu- lated to be 10800 $emidiameters of the grand Orb.</I></marg> $aying, if the Orb of <I>Saturn,</I> by rea$on it is nine times bigger than that of the Sun, revolves in a time thirty times longer, by conver$ion, how great ought that Orb to be, which revolves 36000. times more $lowly? it $hall be found that the di$tance of the $tarry Sphere ought to be 10800 $emidiameters of the grand Orb, which $hould be full five times bigger than that, which even now we computed it to be, in ca$e that a fixed Star of the $ixth magnitude were equal to the Sun. Now $ee how much le$$er yet, upon this account, the variation occa$ioned in the $aid Stars, by the annual motion of the Earth, ought to appear. And if at the $ame rate we would argue the di$tance of the $tarry Sphere from <marg><I>By the proportion of</I> Jupiter <I>and of</I> Mais, <I>the $tarry Sphere is found to be yet more remote.</I></marg> <I>Jupiter,</I> and from <I>Mars,</I> that would give it us to be 15000. and this 27000 $emidiameters of the grand Orb, to wit, the fir$t $even, and the $econd twelve times bigger than what the mag- nitude of the fixed Star, $uppo$ed equal to the Sun, did make it.</P> <P>SIMP. Methinks that to this might be an$wered, that the mo- tion of the $tarry Sphere hath, $ince <I>Ptolomy,</I> been ob$erved not to be $o $low as he accounted it; yea, if I mi$take. not, I have heard that <I>Copernicus</I> him$elf made the Ob$ervation.</P> <foot>Tt 2 SALV</foot> <p n=>332</p> <P>SALV. You $ay very well; but you alledge nothing in that which may favour the cau$e of the <I>Ptolomœans</I> in the lea$t, who did never yet reject the motion of 36000. years in the $tarry Sphere, for that the $aid tardity would make it too va$t and im- men$e. For if that the $aid immen$ity was not to be $uppo$ed in Nature, they ought before now to to have denied a conver$ion $o $low as that it could not with good proportion adapt it $elf, $ave onely to a Sphere of mon$trous magnitude.</P> <P>SAGR. Pray you, <I>Salviatus,</I> let us lo$e no more time in pro- ceeding, by the way of the$e proportions with people that are apt to admit things mo$t di$-proportionate; $o that its impo$$ible to win any thing upon them this way: and what more di$propor- tionate proportion can be imagined than that which the$e men $wallow down, and admit, in that writing, that there cannot be a more convenient way to di$po$e the Cœle$tial Spheres, in order, than to regulate them by the differences of the times of their pe- riods, placing from one degree to another the more flow above the more $wift, when they have con$tituted the Starry Sphere higher than the re$t, as being the $lowe$t, they frame another higher $till than that, and con$equently greater, and make it re- volve in twenty four hours, whil$t the next below, it moves not round under 36000. years?</P> <P>SALV. I could wi$h, <I>Simplicius,</I> that $u$pending for a time the affection rhat you bear to the followers of your opinion, you would $incerely tell me, whether you think that they do in their minds comprehend that magnitude, which they reject afterwards as uncapable for its immen$ity to be a$cribed to the Univer$e. For I, as to my own part, think that they do not; But believe, <marg><I>Immen$e mag- nitudes and num- bers are incompre- hen$ible by our un- der$tanding.</I></marg> that like as in the apprehen$ion of numbers, when once a man begins to pa$$e tho$e millions of millions, the imagination is con- founded, and can no longer form a conceipt of the $ame, $o it happens al$o in comprehending immen$e magnitudes and di$tan- ces; $o that there intervenes to the comprehen$ion an effect like to that which befalleth the $en$e; For while$t that in a $erene night I look towards the Stars, I judge, according to $en$e, that their di$tance is but a few miles, and that the fixed Stars are not a jot more remote than <I>Jupiter</I> or <I>Saturn,</I> nay than the Moon. But without more ado, con$ider the controver$ies that have pa$t between the A$tronomers and Peripatetick Philo$ophers, upon occa$ion of the new Stars of <I>Ca$$iopeia</I> and of <I>Sagittary,</I> the A- $tronomers placing them among$t the fixed Stars, and the Philo- $ophers believing them to be below the Moon. So unable is our $en$e to di$tingui$h great di$tances from the greate$t, though the$e be in reality many thou$and times greater than tho$e. In a word, I ask of thee, O fooli$h man! Doth thy imagination comprehend <foot>that</foot> <p n=>333</p> that va$t magnitude of the Univer$e, which thou afterwards judg- e$t to be too immen$e? If thou comprehende$t it; wilt thou hold that thy apprehen$ion extendeth it $elf farther than the Di- vine Power? wilt thou $ay, that thou can$t imagine greater things than tho$e which God can bring to pa$$e? But if thou apprehende$t it not, why wilt thou pa$$e thy verdict upon things beyond thy comprehen$ion?</P> <P>SIMP. All this is very well, nor can it be denied, but that Heaven may in greatne$$e $urpa$$e our imagination, as al$o that God might have created it thou$ands of times va$ter than now it is; but we ought not to grant any thing to have been made in vain, and to be idle in the Univer$e. Now, in that we $ee this ad- mirable order of the Planets, di$po$ed about the Earth in di$tan- ces proportionate for producing their effects for our advantage, to what purpo$e is it to interpo$e afterwards between the $ublime Orb of <I>Saturn</I> and the $tarry Sphere, a va$t vacancy, without any $tar that is $uperfluous, and to no purpo$e? To what end? For who$e profit and advantage?</P> <P>SALV. Methinks we arrogate too much to our $elves, <I>Simpli- cius,</I> whil$t we will have it, that the onely care of us, is the ad- æquate work, and bound, beyond which the Divine Wi$dome and Power doth, or di$po$eth of nothing. But I will not con- $ent, that we $hould $o much $horten its hand, but de$ire that we may content our $elves with an a$$urance that God and Nature <marg><I>God & Nature do imploy them- $elves in caring for men, as if they minded nothing el$e.</I></marg> are $o imployed in the governing of humane affairs, that they could not more apply them$elves thereto, although they had no other care than onely that of mankind; and this, I think, I am able to make out by a mo$t pertinent and mo$t noble example, taken from the operation of the Suns light, which while$t it at- <marg><I>An example of Gods care of man- kind taken from the Sun.</I></marg> tracteth the$e vapours, or $corcheth that plant, it attracteth, it $corcheth them, as if it had no more to do; yea, in ripening that bunch of grapes, nay that one $ingle grape, it doth apply it $elf $o, that it could not be more inten$e if the $um of all its bu$ine$s had been the only maturation of that grape. Now if this grape receiveth all that it is po$$ible for it to receive from the Sun, not $uffering the lea$t injury by the Suns production of a thou$and other effects at the $ame time; it would be either envy or folly to blame that grape, if it $hould think or wi$h that the Sun would onely appropriate its rayes to its advantage. I am confident that nothing is omitted by the Divine Providence, of what concernes the government of humane affairs; but that there may not be other things in the Univer$e, that depend upon the $ame infinite Wi$dome, I cannot, of my $elf, by what my rea$on holds forth to me, bring my $elf to believe. However, if it were not $o, yet $hould I not forbear to believe the rea$ons laid before me by <foot>$ome</foot> <p n=>334</p> $ome more $ublime intelligence. In the mean time, if one $hould tell me, that an immen$e $pace interpo$ed between the Orbs of the Planets and the Starry Sphere, deprived of $tars and idle, would be vain and u$ele$$e, as likewi$e that $o great an immen$ity for receipt of the fixed $tars, as exceeds our utmo$t comprehen$ion would be $uperfluous, I would reply, that it is ra$hne$$e to go about to make our $hallow rea$on judg of the Works of God, and to call vain and $uperfluous, what$oever thing in the Univer$e is not $ub$ervient to us.</P> <P>SAGR. Say rather, and I believe you would $ay better, that <marg><I>It is great ra$h- ne$$e to cen$ure that to be $uperflu- ous in the Univer$e, which we do not perceive to be made for us.</I></marg> we know not what is $ub$ervient to us; and I hold it one of the greate$t vanities, yea follies, that can be in the World, to $ay, becau$e I know not of what u$e <I>Jupiter</I> or <I>Saturn</I> are to me, that therefore the$e Planets are $uperfluous, yea more, that there are no $uch things <I>in rerum natura</I>; when as, oh fooli$h man! I know not $o much as to what purpo$e the arteries, the gri$tles, the $pleen, the gall do $erve; nay I $hould not know that I have a gall, $pleen, or kidneys, if in many de$ected Corps, they were not $hewn unto me; and then onely $hall I be able to know what the $pleen worketh in me, when it comes to be taken from me. To be able to know what this or that Cœle$tial body worketh in <marg><I>By depriving Heaven of $ome $tar, one might come to know what influence it hath upon us.</I></marg> me ($eeing you will have it that all their influences direct them- $elves to us) it would be requi$ite to remove that body for $ome time; and then what$oever effect I $hould find wanting in me, I would $ay that it depended on that $tar. Moreover, who will pre- $ume to $ay that the $pace which they call too va$t and u$ele$$e between <I>Saturn</I> and the fixed $tars, is void of other mundane bo- dies? Mu$t it be $o, becau$e we do not $ee them? Then the four <marg><I>Many things may be in Heauen, that are invi$ible to us</I></marg> Medicean Planets, and the companions of <I>Saturn</I> came fir$t in- to Heaven, when we began to $ee them, and not before? And by this rule the innumerable other fixed $tars had no exi$tence before that men did look on them? and the cloudy con$tellati- ons called <I>Nebulo$œ</I> were at fir$t only white flakes, but afterwards with the Tele$cope we made them to become con$tellations of many lucid and bright $tars. Oh pre$umptious, rather oh ra$h ignorance of man!</P> <P>SALV. It's to no purpo$e <I>Sagredus,</I> to $ally out any more into the$e unprofitable exaggerations: Let us pur$ue our intended de$igne of examining the validity of the rea$ons alledged on ei- ther $ide, without determining any thing, remitting the judg- ment thereof when we have done, to $uch as are more knowing. Returning therefore to our natural and humane di$qui$itions, I <marg><I>Great, $mall, immen$e,</I> &c. <I>are relative terms.</I></marg> $ay, that great, little, immen$e, $mall, <I>&c.</I> are not ab$olute, but relative terms, $o that the $elf $ame thing compared with divers others, may one while be called immen$e, and another <foot>while</foot> <p n=>335</p> while imperceptible, not to $ay $mall. This being $o, I demand in relation to what the Starry Sphere of <I>Copernicus</I> may be cal- led over va$t. In my judgment it cannot be compared, or $aid to be $uch, unle$$e it be in relation to $ome other thing of the $ame kind; now let us take the very lea$t of the $ame kind, <marg><I>Vanity of tho$e mens di$cour $ewho judg the $tarry $phere too va$t in the</I> Copernican <I>Hypothe$is.</I></marg> which $hall be the Lunar Orb; and if the Starry Orb may be $o cen$ured to be too big in re$pect to that of the Moon, every o- ther magnitude that with like or greater proportion exceedeth another of the $ame kind, ought to be adjudged too va$t, and for the $ame rea$on to be denied that they are to be found in the World; and thus an Elephant, and a Whale, $hall without more ado be condemned for <I>Chymæra's,</I> and Poetical fictions, be- cau$e that the one as being too va$t in relation to an Ant, which is a Terre$trial animal, and the other in re$pect to the ^{*}Gudgeon, <marg>* <I>Spilloncola,</I> which is here put for the lea$t of Fi$hes.</marg> which is a Fi$h, and are certainly $een to be <I>in rerum natura,</I> would be too immea$urable; for without all di$pute, the Ele- phant and Whale exceed the Ant and Gudgeon in a much great- er proportion than the Starry Sphere doth that of the Moon, although we $hould fancy the $aid Sphere to be as big as the <I>Co- pernican</I> Sy$teme maketh it. Moreover, how hugely big is the <marg><I>The $pace a$- $igned to a fixed $tar, is much ie$$e than that of a Pla- net.</I></marg> Sphere of <I>Jupiter,</I> or that of <I>Saturn,</I> defigned for a receptacle but for one $ingle $tar; and that very $mall in compari$on of one of the fixed? Certainly if we $hould a$$ign to every one of the fixed $tars for its receptacle $o great a part of the Worlds $pace, it would be nece$$ary to make the Orb wherein $uch innumerable multitudes of them re$ide, very many thou$ands of times big- ger than that which $erveth the purpo$e of <I>Copernicus.</I> Be$ides, <marg><I>A $tar is cal- led in re$pect of the $pace that environs it.</I></marg> do not you call a fixed $tar very $mall, I mean even one of the mo$t apparent, and not one of tho$e which $hun our $ight; and do we not call them $o in re$pect of the va$t $pace circumfu$ed? Now if the whole Starry Sphere were one entire lucid body; who <marg><I>The whole $tar- ry $phere beheld from a great di- $tance might ap- pear as $mall as one $ingle $tar.</I></marg> is there, that doth not know that in an infinite $pace there might be a$$igned a di$tance $o great, as that the $aid lucid Sphere might from thence $hew as little, yea le$$e than a fixed $tar, now ap- peareth beheld from the Earth? From thence therefore we $hould <I>then</I> judg that $elf $ame thing to be little, which <I>now</I> from hence we e$teem to be immea$urably great.</P> <P>SAGR. Great in my judgment, is the folly of tho$e who would have had God to have made the World more proportinal to the narrow capacities of their rea$on, than to his immen$e, rather infinite power.</P> <P>SIMP. All this that you $ay is very true; but that upon which the adver$ary makes a $cruple, is, to grant that a fixed $tar $hould be not onely equal to, but $o much bigger than the Sun; when as they both are particular bodies $ituate within the <foot>Starry</foot> <p n=>336</p> Starry Orb: “And indeed in my opinion this Authour very pertinently que$tioneth and asketh: To what end, and for who$e $ake are $uch huge machines made? Were they <marg><I>In$tances of the Authour of the Conclu$ions by way of interrogation.</I></marg> produced for the Earth, for an incon$iderable point? And why $o remote? To the end they might $eem $o very $mall, and might have no influence at all upon the Earth? To <marg>Or Gulph.</marg> what purpo$e is $uch a needle$$e mon$trous ^{*} immen$ity be- tween them and <I>Saturn</I>? All tho$e a$$ertions fall to the ground that are not upheld by probable rea$ons.”</P> <P>SALV. I conceive by the que$tions which this per$on asketh, <marg><I>An$wers to the interrogatories of the $aid Authour.</I></marg> that one may collect, that in ca$e the Heavens, the Stars, and the quantity of their di$tances and magnitudes which he hath hitherto held, be let alone, (although he never certainly fancied to him$elf any conceivable magnitude thereof) he perfectly di$- cerns and comprehends the benefits that flow from thence to the Earth, which is no longer an incon$iderable thing; nor are they any longer $o remote as to appear $o very $mall, but big enough to be able to operate on the Earth; and that the di$tance between them and <I>Saturn</I> is very well proportioned, and that he, for all the$e things, hath very probable rea$ons; of which I would glad- ly have heard $ome one: but being that in the$e few words he <marg><I>The Auihour of the Conclu$i- ons confound and contradicts him- $elfin his interro- gations.</I></marg> confounds and contradicts him$elf, it maketh me think that he is very poor and ill furni$hed with tho$e probable rea$ons, and that tho$e which he calls rea$ons, are rather fallacies, or dreams of an over-weening fancy. For I ask of him, whether the$e Ce- <marg><I>Inter ogatories put to the Au- thour of the Con- clu$ions, by which the weakne$$e of his is made appear.</I></marg> le$tial bodies truly operate on the Earth, and whether for the working of tho$e effects they were produced of $uch and $uch magnitudes, and di$po$ed at $uch and $uch di$tances, or el$e whether they have nothing at all to do with Terrene mattets. If they have nothing to do with the Earth; it is a great folly for us that are Earth-born, to offer to make our $elves arbitrators of their magnitudes, and regulators of their local di$po$itions, $ee- ing that we are altogether ignorant of their whole bu$ine$$e and concerns; but if he $hall $ay that they do operate, and that they are directed to this end, he doth affirm the $ame thing which a little before he denied, and prai$eth that which even now he condemned, in that he $aid, that the Cele$tial bodies $ituate $o far remote as that they appear very $mall, cannot have any in- fluence at all upon the Earth. But, good Sir, in the Starry Sphere pre-e$tabli$hed at its pre$ent di$tance, and which you did ac- knowledg to be in your judgment, well proportioned to have an influence upon the$e Terrene bodies, many $tars appear very $mall, and an hundred times as many more are wholly invi$ible unto us (which is an appearing yet le$$e than very $mall) therefore it is nece$$ary that (contradicting your $elf) you do <foot>now</foot> <p n=>337</p> now deny their operation upon the the Earth; or el$e that ($till contradicting your $elf) you grant that their appearing very $mall doth not in the lea$t le$$en their influence; or el$e that (and this $hall be a more $incere and mode$t conce$$ion) you acknowledg and freely confe$$e, that our pa$$ing judgment upon their mag- nitudes and di$tances is a vanity, not to $ay pre$umption or ra$hne$$e.</P> <P>SIMP. Truth is, I my $elf did al$o, in reading this pa$$age perceive the manife$t contradiction, in $aying, that the Stars. (if one may $o $peak) of <I>Copernicus</I> appearing $o very $mall, could not operate on the Earth, and not perceiving that he had granted an influence upon the Earth to tho$e of <I>Ptolomy,</I> and his $ecta- tors, which appear not only very $mall, but are, for the mo$t part, very invi$ible.</P> <P>SALV. But I proceed to another con$ideration: What is the rea$on, doth he $ay, why the $tars appear $o little? Is it haply, becau$e they $eem $o to us? Doth not he know, that this com- <marg><I>That remote ol- jects appeare $o $mall, is the defect of the eye, as i<*> demon$trated.</I></marg> meth from the In$trument that we imploy in beholding them, to wit, from our eye? And that this is true, by changing In$tru- ment, we $hall $ee them bigger and bigger, as much as we will. And who knows but that to the Earth, which beholdeth them without eyes, they may not $hew very great, and $uch as in reali- ty they are? But it's time that, omitting the$e trifles, we come to things of more moment; and therefore I having already de- mon$trated the$e two things: Fir$t, how far off the Firmament ought to be placed to make, that the grand Orb cau$eth no grea- ter difference than that which the Terre$trial Orb occa$ioneth in the remotene$$e of the Sun; And next, how likewi$e to make that a $tar of the Firmament appear to us of the $ame bigne$$e, as now we $ee it, it is not nece$$ary to $uppo$e it bigger than the Sun; I would know whether <I>Tycho,</I> or any of his adherents hath ever attempted to find out, by any means, whether any appea- rance be to be di$covered in the $tarry Sphere, upon which one may the more re$olutely deny or admit the annual motion of the Earth.</P> <P>SAGR. I would an$wer for them, that there is not, no nor is <marg>Tycho <I>nor his followers ever at- tempted to $ee whe- ther there are any appearances in the Firmament for or against the annual motion.</I></marg> there any need there $hould; $eeing that it is <I>Copernicus</I> him$elf that $aith, that no $uch diver$ity is there: and they, arguing <I>ad hominem,</I> admit him the $ame; and upon this a$$umption they demon$trate the improbability that followeth thereupon, name- ly, that it would be nece$$ary to make the Sphere $o immen$e, that a fixed $tar, to appear unto us as great as it now $eems, ought of nece$$ity to be of $o immen$e a magnitude, as that it would exceed the bigne$$e of the whole grand Orb, a thing, which not- with$tanding, as they $ay, is altogether incredible.</P> <foot>Vv SALV</foot> <p n=>338</p> <P>SALV. I am of the $ame judgment, and verily believe that they argue <I>contra hominem,</I> $tudying more to defend another man, than de$iring to come to the knowledge of the truth. And <marg><I>A $tronomeys, perhaps, have not known what ap- pearances ought to follow upon the an- nual motion of the Earth.</I></marg> I do not only believe, that none of them ever applied them$elves to make any $uch ob$ervation, but I am al$o uncertain, whether any of them do know what alteration the Earths annual motion ought to produce in the fixed $tars, in ca$e the $tarry Sphere were not $o far di$tant, as that in them the $aid diver$ity, by rea$on of its minuity di$-appeareth; for their $urcea$ing that inqui$ition, and referring them$elves to the meer a$$ertion of <I>Copernicus,</I> may very well $erve to convict a man, but not to acquit him of the fact: For its po$$ible that $uch a diver$ity may be, and yet <marg>Copernicus <I>un- der$tood not $ome things for want of In$truments.</I></marg> not have been $ought for; or that either by rea$on of its minui- ty, or for want of exact In$truments it was not di$covered by <I>Co- pernicus</I>; for though it were $o, this would not be the fir$t thing, that he either for want of In$truments, or for $ome other defect hath not known; and yet he proceeding upon other $olid and rational conjectures, affirmeth that, which the things by him not di$covered do $eem to contradict: for, as hath been $aid already, without the Tele$cope, neither could <I>Mars</I> be di$cerned to in- crea$e 60. times; nor <I>Venus</I> 40. more in that than in this po$iti- on; yea, their differences appear much le$$e than really they are: and yet neverthele$$e it is certainly di$covered at length, that tho$e mutations are the $ame, to an hair that the <I>Copernican</I> Sy- <marg>Tycho <I>and o- thers argue a- gain$t the annual motion, from the invariable eleva- tion of the Pole.</I></marg> $teme required. Now it would be very well, if with the greate$t accuratene$$e po$$ible one $hould enquire whether $uch a muta- tion as ought to be di$coverable in the fixed $tars, $uppo$ing the annual motion of the Earth, would be ob$erved really and in effect, a thing which I verily believe hath never as yet been done by any; done, $aid I? no, nor haply (as I $aid before) by many well under$tood how it ought to be done. Nor $peak I this at randome, for I have heretofore $een a certain Manu$cript of one of the$e <I>Anti-Copernicans,</I> which $aid, that there would ne- ce$$arily follow, in ca$e that opinion were true, a continual ri- $ing and falling of the Pole from $ix moneths to $ix moneths, ac- cording as the Earth in $uch a time, by $uch a $pace as is the dia- meter of the grand Orb, retireth one while towards the North, and another while towards the South; and yet it $eemed to him rea$o- nable, yea nece$$ary, that we, following the Earth, when we were towards the North $hould have the Pole more elevated than when we are towards the South. In this very error did one fall that was otherwi$e a very skilful Mathematician, & a follower of <I>Copernic.</I> <marg>* Chri$iophoius Rothmannus.</marg> as <I>Tycho</I> relateth in his ^{*}<I>Progymna$ma. pag</I> 684. which $aid, that he had ob$erved the Polar altitude to vary, and to differ in Summer from what it is in Winter: and becau$e <I>Tycho</I> denieth the merit <foot>of</foot> <p n=>339</p> of the cau$e, but findeth no fault with the method of it; that is, denieth that there is any mutation to be $een in the altitude of the Pole, but doth not blame the inqui$ition, for not being adap- ted to the finding of what is $ought, he thereby $heweth, that he al$o e$tecemed the Polar altitude varied, or not varied every $ix moneths, to be a good te$timony to di$prove or inferre the annual motion of the Earth.</P> <P>SIMP. In truth, <I>Salviatus,</I> my opinion al$o tells me, that the $ame mu$t nece$$arily en$ue: for I do not think that you will de- ny me, but that if we walk only 60. miles towards the North, the Pole will ri$e unto us a degree higher, and that if we move 60. miles farther Northwards, the Pole will be elevated to us a degree more, &c. Now if the approaching or receding 60. miles onely, make $o notable a change in the Polar altitudes, what alteration would follow, if the Earth and we with it, $hould be tran$ported, I will not $ay 60. miles, but 60. thou$and miles that way.</P> <P>SALV. It would follow (if it $hould proceed in the $ame proportion) that the Pole $hall be elevated a thou$and degrees. See, <I>Simplicius,</I> what a long rooted opinion can do. Yea, by rea$on you have fixed it in your mind for $o many years, that it is Heaven, that revolveth in twenty four hours, and not the Earth, and that con$equently the Poles of that Revolution are in Heaven, and not in the Terre$trial Globe, cannot now, in an hours time $hake off this habituated conceipt, and take up the contrary, fancying to your $elf, that the Earth is that which mo- veth, only for $o long time as may $uffice to conceive of what would follow, thereupon $hould that lye be a truth. If the Earth <I>Simplicius,</I> be that which moveth in its $elf in twenty four hours, in it are the Poles, in it is the Axis, in it is the Equinoctial, that is, the grand Circle, de$cribed by the point, equidi$tant from the Poles, in it are the in$inite Parallels bigger and le$$er de$cribed by the points of the $uperficies more and le$$e di$tant from the Poles, in it are all the$e things, and not in the $tarry Sphere, which, as being immoveable, wants them all, and can only by the imagina- tion be conceived to be therein, prolonging the Axis of the Earth $o far, till that determining, it $hall mark out two points placed right over our Poles, and the plane of the Equinoctial being ex- tended, it $hall de$cribe in Heaven a circle like it $elf. Now if the true Axis, the true Poles, the true Equinoctial, do not change in the Earth $o long as you continue in the $ame place of the Earth, and though the Earth be tran$ported, as you do plea$e, yet you $hall not change your habitude either to the Poles, or to the circles, or to any other Earthly thing; and this becau$e, that that tran$po$ition being common to you and to all Terre$trial <foot>Vv 2 things;</foot> <p n=>340</p> things; and that motion where it is common, is as if it never <marg><I>Motion where it is common, is as if it never were.</I></marg> were; and as you change not habitude to the Terre$trial Poles (habitude I $ay, whether that they ri$e, or de$cend) $o neither $hall you change po$ition to the Poles imagined in Heaven; al- wayes provided that by Cele$tial Poles we under$tand (as hath been already defined) tho$e two points that come to be marked out by the prolongation of the Terre$trial Axis unto that length. Tis true tho$e points in Heaven do change, when the Earths tran- $portment is made after $uch a manner, that its Axis cometh to pa$$e by other and other points of the immoveable Cele$tial Sphere, but our habitude thereunto changeth not, $o as that the $econd $hould be more elevated to us than the fir$t. If any one will have one of the points of the Firmament, which do an$wer to the Poles of the Earth to a$cend, and the other to de$cend, he mu$t walk along the Earth towards the one, receding from the other, for the tran$portment of the Earth, and with it us our $elves, (as I told you before) operates nothing at all.</P> <P>SAGR. Permit me, I be$eech you <I>Salviatus,</I> to make this a little more clear by an example, which although gro$$e, is a- commodated to this purpo$e. Suppo$e your $elf, <I>Simplicius,</I> to <marg><I>An example fit- ted to prove that the altitude of the Pole ought not to vary by means of the Earths annual motion.</I></marg> be aboard a Ship, and that $tanding in the Poope, or Hin-deck; you have directed a Quadrant, or $ome other A$tronomical In- $trument, towards the top of the Top-gallant-Ma$t, as if you would take its height, which $uppo$e it were <I>v. gr.</I> 40. degrees, <marg>* <I>Cor$<*>a,</I> the bank or bench on which $laves $it in a Gal- ly.</marg> there is no doubt, but that if you walk along the ^{*} Hatches to- wards the Ma$t 25. or 30. paces; and then again direct the $aid In$trument to the $ame Top-Gallant-Top. You $hall find its ele- vation to be greater, and to be encrea$ed <I>v. gr.</I> 10. degrees; but if in$tead of walking tho$e 25. or 30. paces towards the Ma$t, you $tand $till at the Sterne, and make the whole Ship to move thitherwards, do you believe that by rea$on of the 25. or 30. paces that it had pa$t, the elevation of the Top-Gallant-Top would $hew 10. degrees encrea$ed?</P> <P>SIMP. I believe and know that it would not gain an hairs breadth in the pa$$ing of 30. paces, nor of a thou$and, no nor of an hundred thou$and miles; but yet I believe withal that look- ing through the $ights at the Top and Top-Gallant, if I $hould find a fixed Star that was in the $ame elevation, I believe I $ay, that, holding $till the Quadrant, after I had $ailed towards the $tar 60. miles, the eye would meet with the top of the $aid Ma$t, as before, but not with the $tar, which would be eleva- ted to me one degree.</P> <P>SAGR. Then you do not think that the $ight would fall upon that point of the Starry Sphere, that an$wereth to the direction of the Top-Gallant Top?</P> <foot>SIMP.</foot> <p n=>341</p> <P>SIMP. No: For the point would be changed, and would be beneath the $tar fir$t ob$erved.</P> <P>SAGR. You are in the right. Now like as that which in this example an$wereth to the elevation of the Top-Gallant-Top, is not the $tar, but the point of the Firmament that lyeth in a right line with the eye, and the $aid top of the Ma$t, $o in the ca$e exemplified, that which in the Firmament an$wers to the Pole of the Earth, is not a $tar, or other fixed thing in the Firma- ment; but is that point in which the Axis of the Earth continu- ed $treight out, till it cometh thither doth determine, which point is not fixed, but obeyeth the mutations that the Pole of the Earth doth make. And therefore <I>Tycho,</I> or who ever el$e that <marg><I>Upon the annu- al motion of the Earth, alteration may en$ue in $ome fixed $tar, not in the Pole.</I></marg> did alledg this objection, ought to have $aid that upon that $ame motion of the Earth, were it true, one might ob$erve $ome difference in the elevation and depre$$ion (not of the Pole, but) of $ome fixed $tar toward that part which an$wereth to our Pole.</P> <P>SIMP. I already very well under$tand the mi$take by them committed; but yet therefore (which to me $eems very great) of the argument brought on the contrary is not le$$ened, $uppo- $ing relation to be had to the variation of the $tars, and not of the Pole; for if the moving of the Ship but 60. miles, make a fixed $tar ri$e to me one degree, $hall I not find alike, yea and very much greater mutation, if the Ship $hould $ail towards the $aid $tar for $o much $pace as is the Diameter of the Grand Orb, which you affirm to be double the di$tance that is between the Earth and Sun?</P> <P>SAGR. Herein <I>Simplicius,</I> there is another fallacy, which, <marg><I>The equivoke of tho$e who believe that in the annual motion great mu- tations are to be made about the elevation of a fix- ed $tar, is confu- ted.</I></marg> truth is, you under$tand, but do not upon the $udden think of the $ame, but I will try to bring it to your remembrance: Tell me therefore; if when after you have directed the Quadrant to a fixed $tar, and found <I>v. g.</I> its elevation to be 40. degrees, you $hould without $tirring from the place, incline the $ide of the Ouadrant, $o as that the $tar might remain elevated above that direction, would you thereupon $ay that the $tar had acqui- red greater elevation?</P> <P>SIMP. Certainly no: For the mutation was made in the In- $trument and not in the Ob$erver, that did change place, mo- ving towards the $ame.</P> <P>SAGR. But if you $ail or walk along the $urface of the Terre- $trial Globe, will you $ay that there is no alteration made in the $aid Quadrant, but that the $ame elevarion is $till retained in re- $pect of the Heavens, $o long as you your $elf do not incline it, but let it $tand at its fir$t con$titution?</P> <P>SIMP. Give me leave to think of it. I would $ay without more ado, that it would not retain the $ame, in regard the pro- <foot>gre$$e</foot> <p n=>342</p> gre$$e I make is not <I>in plano,</I> but about the circumference of the Terre$trial Globe, which at every $tep changeth inclination in re$pect to Heaven, and con$equently maketh the $ame change in the In$trument which is erected upon the $ame.</P> <P>SAGR. You $ay very well: And you know withal, that by how much the bigger that circle $hall be upon which you move, $o many more miles you are to walk, to make the $aid $tar to ri$e that $ame degree higher; and that $inally if the motion to- wards the $tar $hould be in a right line, you ought to move yet farther, than if it were about the circumference of never $o great a circle?</P> <marg><I>The right line, and circumference of an infinite cir- cle, are the $ame thing.</I></marg> <P>SALV. True: For in $hort the circumference of an infinite circle, and a right line are the $ame thing.</P> <P>SAGR. But this I do not under$tand, nor as I believe, doth <I>Simplicius</I> apprehend the $ame; and it mu$t needs be concealed from us under $ome mi$tery, for we know that <I>Salviatus</I> never $peaks at random, nor propo$eth any Paradox, which doth not break forth into $ome conceit, not trivial in the lea$t. Therefore in due time and place I will put you in mind to demon$trate this, that the right line is the $ame with the circumference of an infi- nite circle, but at pre$ent I am unwilling that we $hould inter- rupt the di$cour$e in hand. Returning then to the ca$e, I pro- po$e to the con$ideration of <I>Simplicius,</I> how the acce$$ion and rece$$ion that the Earth makes from the $aid fixed $tar which is neer the Pole can be made as it were by a right line, for $uch is the Diameter of the Grand Orb, $o that the attempting to re- gulate the elevation and depre$$ion of the Polar $tar by the mo- tion along the $aid Diameter, as if it were by the motion about the little circle of the Earth, is a great argument of but little judgment.</P> <P>SIMP. But we continue $till un$atisfied, in regard that the $aid $mall mutation that $hould be therein, would not be di$cer- ned; and if this be <I>null,</I> then mu$t the annual motion about the Grand Orb a$cribed to the Earth, be <I>null</I> al$o.</P> <P>SAGR. Here now I give <I>Salviatus</I> leave to go on, who as I believe will not overpa$$e the elevation and depre$$ion of the Polar $tar or any other of tho$e that are fixed as <I>null,</I> although not di$covered by any one, and affirmed by <I>Copernicus</I> him$elf to be, I will not $ay <I>null,</I> but unob$ervable by rea$on of its minuity.</P> <P>SALV. I have already $aid above, that I do not think that <marg><I>Enquiry is made what mutations, & in what $tars, are to be di$covered, by means of the an- nual motion of the Earth.</I></marg> any one did ever $et him$elf to ob$erve, whether in different times of the year there is any mutation to be $een in the fixed $tars, that may have a dependance on the annual motion of the Earth, and added withal, that I doubted lea$t haply $ome might never have <foot>under-</foot> <p n=>343</p> under$tood what tho$e mutations are, and among$t what $tars they $hould be di$cerned; therefore it would be nece$$ary that we in the next place narrowly examine this particular. My ha- <marg><I>A$tronomers ha- ving omitted to in- $tance what alte- rations tho$e are that may be deri- ved from the an- nual motion of the Earth, do thereby te$tifie that they never rightly un- der$tood the $ame.</I></marg> ving onely found written in general terms that the annual moti- on of the Earth about the Grand Orb, ought not to be admit- ted, becau$e it is not probable but that by means of the $ame there would be di$coverd $ome apparent mutation in the fixed $tars, and not hearing $ay what tho$e apparent mutations ought to be in particular, and in what $tars, maketh me very rea$onably to infer that they who rely upon that general po$ition, have not under$tood, no nor po$$ibly endeavoured to under$tand, how the bu$ine$$e of the$e mutations goeth, nor what things tho$e are which they $ay ought to be $een. And to this judgment I am <marg><I>The mutations of the fixed $tars ought to be in $ome greater, in others le$$er, and in others nothing at all.</I></marg> the rather induced, knowing that the annual motion a$cribed by <I>Copernicus</I> to the Earth, if it $hould appear $en$ible in the Starry Sphere, is not to make apparent mutations equal in re- $pect to all the $tars, but tho$e appearances ought to be made in $ome greater, in others le$$er, and in others yet le$$er; and la$tly, in others ab$olutely nothing at all, by rea$on of the va$t magnitude that the circle of this annual motion is $uppo$ed to be of. As for the mutations that $hould b $een, they are of two kinds, one is the $aid $tars changing apparent magnitude, and the other their variation of altitudes in the Meridian. Upon which nece$$arily followeth the mutation of ri$ings and $ettings, and of their di$tances from the Zenith, <I>&c.</I></P> <P>SAGR. Methinks I $ee preparing for me $uch a skean of the$e revolutions, that I wi$h it may never be my task to di$-intangle them, for to confe$$e my infirmity to <I>Salviatus,</I> I have $ome- times thought thereon, but could never find the ^{*} Lay-band of <marg>* <I>Bandola</I> that end of a skeen where with hou$e- wives fa$ten their hankes of yarn, thread or $ilk.</marg> it, and I $peak not $o much of this which pertains to the fixed $tars, as of another more terrible labour which you bring to my remembrance by maintaining the$e Meridian Altitudes, Ortive Latitudes and di$tances from the Vertex, <I>&c.</I> And that which <marg><I>The grand dif- ficulty in</I> Coper- nicus <I>his Doctrine, is that which con- cerns the</I> Phæno- mena <I>of the Sun and fixed $tars.</I></marg> puzzleth my brains, ari$eth from what I am now about to tell you. <I>Copernicus</I> $uppo$eth the Starry Sphere immoveable, and the Sun in the centre thereof immoveable al$o. Therefore eve- ry mutation which $eemeth unto us to be made in the Sun or in the fixed $tars, mu$t of nece$$ity befall the Earth and be ous. But the Sun ri$eth and declineth in our Meridian by a very great arch of almo$t 47. degrees, and by arches yet greater and greatet, varieth its Ortive and Occidual Latitudes in the oblique <marg>* <I>Pettine,</I> it is the $tay in a Wea- vets Loom, that permitteth no knot or $narle to pa$$e it, called by them the Combe of the Loom.</marg> Horizons. Now how can the Earth ever incline and elevate $o notably to the Sun, and nothing at all to the fixed $tars, or $o little, that it is not to be perceived? This is that knot which could never get thorow my ^{*} Loom-Combe; and if you $hall <foot>untie</foot> <p n=>344</p> untie it, I $hall hold you for more than an <I>Alexander.</I></P> <P>SALV. The$e are $cruples worthy of the ingenuity of <I>Sagre- dus,</I> and this doubt is $o intricate, that even <I>Copernicus</I> him$elf almo$t de$paired of being able to explain the $ame, $o as to render it intelligible, which we $ee as well by his own confe$$ion of its ob$curity, as al$o by his, at two $everal times, taking two different wayes to make it out. And, I ingenuou$ly confe$$e that I under$tood not his explanation, till $uch time as another me- thod more plain and manife$t, had rendred it intelligible; and yet neither was that done without a long and laborious applica- tion of my thoughts to the $ame.</P> <P>SIMP. <I>Ari$totle</I> $aw the $ame $cruple, and makes u$e there- <marg>Ari$totles <I>argu- ment again$t the Ancients, who held that the Earth was a Planet.</I></marg> of to oppo$e certain of the Ancients, who held that the Earth was a Planet; again$t whom he argueth, that if it were $o, it would follow that it al$o, as the re$t of the Planets, $hould have a plurality of motions, from whence would follow the$e variati- ons in the ri$ings and $ettings of the fixed $tars, and likewi$e in the Meridian Altitudes. And in regard that he propoundeth the difficulty, and doth not an$wer it, it mu$t needs be, if not im- po$$ible, at lea$t very difficult to be re$olved.</P> <P>SALV. The $tre$$e and $trength of the knot rendereth the $olution thereof more commendable and admirable; but I do not promi$e you the $ame at this time, and pray you to di$pen$e with me therein till too morrow, and for the pre$ent we will go con$idering and explaining tho$e mutations and differences that by means of the annual motion ought to be di$cerned in the fix- ed $tars, like as even now we $aid, for the explication whereof certain preparatory points offer them$elves, which may facili- tate the an$wer to the grand objection. Now rea$$uming the two motions a$cribed to the Earth (two I $ay, for the third is no motion, as in its place I will declare) that is the annual and <marg><I>The annual mo- tion made by the centre of the Earth under the Eclip- tick and the diur- nal motion made by the Earth about its own centre.</I></marg> diurnal, the fir$t is to be under$tood to be made by the centre of the Earth in or about the circumference of the grand Orb, that is of a very great circle de$cribed in the plain of the fixed and immutable Ecliptick; the other, namely the diurnal, is made by the Globe of the Earth in it $elf about its own centre, and own Axis, not erect, but inclined to the Plane of the Ecliptick, with the inclination of 23. degrees and an half, or thereabouts, the which inclination is maintained all the year about, and that which ought e$pecially to be ob$erved, is alwayes $ituate to- wards the $ame point of Heaven: in $o much that the Axis of the <marg><I>The axis of the Earth continueth alwayes parallel to it $elf, and de$cri- beth a Cylindrai- cal $uperficies, in- clining to the grand Orb.</I></marg> diurnal motion doth alwayes remain parallel to it $elf; $o that if we imagine that $ame Axis to be continued out until it reach the fixed $tars, whil$t the centre of the Earth is encircling the whole Ecliptick in a year, the $aid Axis de$cribeth the $uper- <foot>ficies</foot> <p n=>345</p> ficies of an oblique Cylinder, which hath for one of its ba$es the $aid annual circle, and for the other a like circle imagina- rily de$cribed by its extremity, or, (if you will) Pole, among$t the fixed $tars. And this $ame cylinder is oblique to the Plane of the Ecliptick, according to the inclination of the Axis that de- $cribeth it, which we have $aid to be 23 degrees and an half, the which continuing perpetually the $ame ($ave onely, that in many thou$ands of years it maketh $ome very $mall mutation, which nothing importeth in our pre$ent bu$ine$$e) cau$eth that <marg><I>The Orb of the Earth never incli- neth, but is im- mutably the $ame.</I></marg> the Terre$trial Globe doth never more incline or elevate, but $till con$erveth the $ame $tate without mutation. From whence en$ueth, that as to what pertaineth to the mutations to be ob- $erved in the fixed $tars dependant on the $ole annual motion, the $ame $hall happen to any point what$oever of the Earths $urface, as befalleth unto the centre of the Earth it $elf; and therefore in the pre$ent explanations we will make u$e of the centre, as if it were any what$oever point of the $uperficies. And for a more facile under$tanding of the whole, let us de$ign <marg><I>The fixed $tars placed in the E- cliptick never ele- vate nor de$cend, on account of the annual motion, but yet approach and recede.</I></marg> the $ame in lineal figures: And fir$t of all let us de$cribe in the Plane of the Ecliptick the circle A N B O [<I>in Fig.</I> 7.] and let us under$tand the points A and B, to be the extreams towards the North and South; that is, the beginning of [<I>or entrance into</I>] <I>Cancer</I> or <I>Capricorn,</I> and let us prolong the Diameter A B, in- determinately by D and C towards the Starry Sphere. I $ay now in the fir$t place, that none of the fixed $tars placed in the Ecliptick, $hall ever vary elevation, by rea$on of any what$o- ever mutation made by the Earth along the $aid Plane of the Ecliptick, but $hall alwayes appear in the $ame $uperficies, al- though the Earth $hall approach and recede as great a $pace as is that of the diameter of the Grand Orb, as may plainly be $een in the $aid figure. For whether the Earth be in the point A or in B, the $tar C alwayes appeareth in the $ame line A B C; although the di$tance B C, be le$$e than A C, by the whole diameter A B. The mo$t therefore that can be di$covered in the $tar C, and in any other placed in the Ecliptick, is the aug- mented or dimini$hed apparent magnitude, by rea$on of the ap- proximation or rece$$ion of the Earth.</P> <P>SAGR. Stay a while I pray you, for I meet with a certain $cruple, which much troubleth me, and it is this: That the $tar C may be $een by the $ame line A B C, as wel when the Earth is in A, as when it is in B, I under$tand very well, as al$o fur- thermore I apprehend that the $ame would happen in all the <marg><I>Objections again$t the Earths annual motion taken from the fixed stars placed in the E- cliptick.</I></marg> points of the line A B, $o long as the Earth $hould pa$$e from A <*>o B by the $aid line; but it pa$$ing thither, as is to be $uppo$ed, by the arch A N B, it is manife$t that when it $hall be in the <foot>Xx point</foot> <p n=>346</p> point N, and in any other except tho$e two A and B, the $aid $tar $hall no longer be ob$erved in the line A B; but in others. So that, if the appearing under $everal lines ought to cau$e apparent mutations, $ome difference mu$t needs appear in this ca$e. Nay more, I will $peak it with that Philo$ophical freedom, which ought to be allowed among$t Philo$ophick friends, methinks that you, contradicting your $elf, deny that now, which but even now to our admiration, you proved to be really true, and con$iderable; I mean that which happeneth in the Planets, and particularly in the three $uperiour ones, that being con$tantly in the Ecliptick, or very near unto it, do not onely $hew them$elves one while near unto us, and another while remote, but $o deformed in their regular motions, that they $eem $ometimes immoveable, and $ometimes many de- grees retrograde; and all upon no other occa$ion than the an- nual motion of the Earth.</P> <P>SALV. Though by a thou$and accidents I have been hereto- fore a$$ured of the wittine$$e of <I>Sagredus,</I> yet I had a de$ire by this one experiment more to a$certain me of what I may expect from his ingenuity, and all this for my own intere$t, for in ca$e my Propo$itions $tand but proof again$t the hammer and fur- nace of his judgment, I $hall be confident that they will abide <marg>* Or will prove of good alloy.</marg> the ^{*} te$t of all Touch-$tones. I $ay therefore that I had pur- po$ely di$$embled this objection, but yet not with any intent to deceive you, and to put any fal$hood upon you, as it might have happened if the objection by me di$gui$ed, and by you o- ver-lookt, had been the $ame in effect as it $eemed to be in ap- pearance, that is, really valid and conclu$ive; but it is not $o; nay I rather $u$pect that to try me, you make as if you did not $ee its nullity. But I will herein be too hard for you, and force from your tongue, that which you would $o artificially conceal; and therefore tell me, what that thing $hould be, whereby you come to know the $tation and retrogradation of the Planets, which is derived from the annual motion, aud which is $o great, that at lea$t $ome foot-$teps of $uch an effect ought to appear in the $tars of the Ecliptick?</P> <P>SAGR. This demand of yours containeth two que$tions, to which it is nece$$ary that I make reply; the fir$t relates to the imputation which you lay upon me of a Di$$embler; the other concerneth that which may appear in the $tars, <I>&c.</I> As to the fir$t, I will $ay with your permi$$ion, that it is not true, that I have di$$embled my knowing the nullity of that objection; and to a$$ure you of the $ame, I now tell you that I very well under- $tand the nullity thereof.</P> <P>SALV. But yet I do not under$tand how it can be, that you <foot>$pake</foot> <p n=>347</p> $pake not friendly, when you $aid you did not know that $ame fallacy which you now confe$$e that you know very well.</P> <P>SAGR. The very confe$$ion of knowing it may a$$ure you that I did not di$$emble, when I $aid that I did not under$tand it; for if I had had a mind, and would di$$emble, who could hin- der me from continuing in the $ame $imulation, and denying $till that I under$tand the fallacy? I $ay therefore that I under$tood not the $ame, at that time, but that I do now at this pre$ent ap- prehend it, for that you have prompted my intellect, fir$t by telling me re$olutely that it is <I>null,</I> and then by beginning to que$tion me $o at large what thing that might be, whereby I might come to know the $tation and retrogradation of the Pla- <marg><I>The $tation, di- rection and retro- gradation of the Planets is known, in relation to the fixed $tars.</I></marg> nets; and becau$e this is known by comparing them with the fix- ed $tars, in relation to which, they are $een to vary their mo- tions, one while towards the We$t, and another towards the Ea$t, and $ometimes to abide immoveable; and becau$e there is not any thing above the Starry Sphere, immen$ely more remote from us, and vi$ible unto us, wherewith we may compare our fixed $tars, therefore we cannot di$cover in the fixed $tars any foot-$teps of what appeareth to us in the Planets. This I believe is the $ub$tance of that which you would force from me.</P> <P>SALV. It is $o, with the addition moreover of your admi- <marg><I>An Indice in the fixed $tars like to that which is $een in the Pla- nets, is an argu- ment of the Earths annual motion.</I></marg> rable ingenuity; and if with half a word I did open your eyes, you by the like have remembred me that it is not altogether im- po$$ible, but that $ometime or other $omething ob$ervable may be found among$t the fixed $tars, by which it may be gathered wherein the annual conver$ion re$ides, $o as that they al$o no le$$e than the Planets and Sun it $elf, may appear in judgment to bear witne$$e of that motion, in favour of the Earth; for I do not think that the $tas are $pread in a $pherical $uperficies equally re- mote from a common centre, but hold, that their di$tances from us are $o various, that $ome of them may be twice and thrice as remote as others; $o that if with the Tele$cope one $hould ob- $erve a very $mall $tar neer to one of the bigger, and which therefore was very exceeding high, it might happen that $ome $en$ible mutation might fall out between them, corre$pondent to that of the $uperiour Planets. And $o much $hall $erve to have $poken at this time touching the $tars placed in the Ecliptick. <marg><I>The fixed $tars without the Eclip- tick elevate and de$cend more or le$$e, according to their di$tance from the $aid Ecliptick.</I></marg> Let us now come to the fixed $tars, placed out of the Ecliptick, and let us $uppo$e a great circle erect upor [<I>i. e. at right angles to</I>] the Plane of the ^{*} $ame; and let it, for example, be a cir- cle that in the Starry Sphere an$wers to the Sol$titial Colure, <marg>* <I>i. e.</I> of the E- cliptick.</marg> and let us mark it C E H [<I>in Fig.</I> 8.] which $hall happen to be withal a Meridian, and in it we will take a $tar without the Eclip- tick, which let be E. Now this $tar will indeed vary its elevati- <foot>Xx 2 on</foot> <p n=>348</p> on upon the Earths motion; for from the Earth in A it $hall be $een according to the ray A E, with the elevation of the angle E A C; but from the Earth placed in B, it $hall be $een ac- cording to the ray B E, with the elevation of the angle E B C, bigger than the other E A C, that being extern, and this in- tern and oppo$ite in the triangle E A B, the di$tance therefore of the $tar E from the Ecliptick, $hall appear changed; and likewi$e its altitude in the Meridian $hall become greater in the po$ition B, than in the place A, according as the angle E B C exceeds the angle E A C, which exce$$e is the quantity of the angle A E B: For in the triangle E A B, the $ide A B being continued to C, the exteriour angle E B C (as being equal to the two interiour and oppo$ite E and A) exceedeth the $aid an- gle A, by the quantity of the angle <I>E.</I> And if we $hould take another $tar in the $ame Meridian, more remote from the Ecli- ptick, as for in$tance the $tar H, the diver$ity in it $hall be greater by being ob$erved from the two $tations A and B, accor- ding as the angle A H B is greater than the other <I>E</I>; which an- gle $hall encrea$e continually according as the ob$erved $tar $hall be farther and farther from the Ecliptick, till that at la$t the greate$t mutation will appear in that $tar that $hould be placed in the very Pole of the Ecliptick. As for a full under$tanding there- of we thus demon$trate. Suppo$e the diameter of the Grand Orb to be A B, who$e centre [<I>in the $ame Figure</I>] is G, and let it be $uppo$ed to be continued out as far as the Starry Sphere in the points D and C, and from the centre G let there be erected the Axis of the Ecliptick G F, prolonged till it arrive at the $aid Sphere, in which a Meridian D F C is $uppo$ed to be de$cribed, that $hall be perpendicular to the Plane of the Ecliptick; and in the arch F C any points H and <I>E,</I> are imagined to be taken, as places of fixed $tars: Let the lines F A, F B, A H, H G, H B, A <I>E,</I> G <I>E,</I> B <I>E,</I> be conjoyned. And let the angle of dif- ference, or, if you will, the Parallax of the $tar placed in the Pole F, be A F B, and let that of the $tar placed in H, be the angle A H <I>B,</I> and let that of the $tar in <I>E,</I> be the angle A <I>E</I> B. I $ay, that the angle of difference of the Polar $tar F, is the greate$t, and that of the re$t, tho$e that are nearer to the greate$t are bigger than the more remote; that is to $ay, that the angle F is bigger than the angle H, and this bigger than the angle <I>E.</I> Now about the triangle F A B, let us $uppo$e a circle to be de- $cribed. And becau$e the angle F is acute, (by rea$on that its ba$e AB is le$$e than the diameter DC, of the $emicircle D F C) it $hall be placed in the greater portion of the circum$cribed circle cut by the ba$e A B. And becau$e the $aid A B is divided in the mid$t, and at right angles by F G, the centre of the circum$cri- <foot>bed</foot> <p n=>349</p> bed circle $hall be in the line F G, which let be the point I; and becau$e that of $uch lines as are drawn from the point G, which is not the centre, unto the circumference of the circum$cribed circle, the bigge$t is that which pa$$eth by the centre, G F $hall be bigger than any other that is drawn from the point G, to the circumference of the $aid circle; and therefore that circumfe- rence will cut the line G H (which is equal to the line G F) and cutting G H, it will al$o cut A H. Let it cut it in L, and con- joyn the line L B. The$e two angles, therefore, A F B and A L B $hall be equal, as being in the $ame portion of the circle cir- cum$cribed. But A L B external, is bigger than the internal H; therefore the angle F is bigger than the angle H. And by the $ame method we might demon$trate the angle H to be bigger than the angle E, becau$e that of the circle de$cribed about the triangle A H B, the centre is in the perpendicular G F, to which the line G H is nearer than the line G E, and therefore the cir- cumference of it cutteth G E, and al$o A E, whereupon the pro- po$ition is manife$t. We will conclude from hence, that the dif- ference of appearance, (which with the proper term of art, we might call the Parallax of the fixed $tars) is greater, or le$$e, ac- cording as the Stars ob$erved are more or le$$e adjacent to the Pole of the Ecliptick, $o that, in conclu$ion of tho$e Stars that are in the Ecliptick it $elf, the $aid diver$ity is reduced to nothing. In the next place, as to the Earths acce$$ion by that motion to, <marg><I>The Earth ap- proacheth or rece- deth from the fix- ed $tars of the E- cliptick, the quan- tity of the Dinme- ter of the Grand Orb.</I></marg> or rece$$ion from the Stars, it appeareth to, and recedeth from tho$e that are in the Ecliptick, the quantity of the whole diame- ter of the grand Orb, as we did $ee even now, but that acce$$ion or rece$$ion to, or from the $tars about the Pole of the Ecliptick, is almo$t nothing; and in going to and from others, this diffe- rence groweth greater, according as they are neerer to the Eclip- tick. We may, in the third place, know, that the $aid difference <marg><I>The $tars near- er to us make greater differences than the more re- more.</I></marg> of A$pect groweth greater or le$$er, according as the Star ob$er- ved $hall be neerer to us, or farther from us. For if we draw a- nother Meridian, le$$e di$tant from the Earth; as for example, this D F I [<I>in Fig.</I> 7.] a Star placed in F, and $een by the $ame ray A F E, the Earth being in A, would, in ca$e it $hould be ob- $erved from the Earth in B, appear according to the ray B F, and would make the angle of difference, namely, B F A, bigger than the former A E B, being the exteriour angle of the trian- gle B F E.</P> <P>SAGR. With great delight, and al$o benefit have I heard your di$cour$e; and that I may be certain, whether I have right- <marg><I>The Epilogue of the</I> Phænomena <I>of the fixed $tars cau$ed by the an- nual motion of the Earth.</I></marg> ly under$tood the $ame, I $hall give you the $umme of the Con- clu$ions in a few words. As I take it, you have explained to us the different appearances, that by means of the Earths annual mo- <foot>tion,</foot> <p n=>350</p> tion, may be by us ob$erved in the fixed $tars to be of two kinds: The one is, that of their apparent magnitudes varied, ac- cording as we, tran$ported by the Earth, approach or recede from the $ame: The other (which likewi$e dependeth on the $ame acce$$ion and reee$$ion) their appearing unto us in the $ame Meridian, one while more elevated, and another while le$$e. Moreover, you tell us (and I under$tand it very well) that the one and other of the$e mutations are not made alike in all the $tars, but in $ome greater, and in others le$$er, and in others not at all. The acce$$ion and rece$$ion whereby the $ame $tar ought to appear, one while bigger, and another while le$$er, is in$en$i- ble, and almo$t nothing in the $tars neer unto the pole of the E- cliptick, but is greate$t in the $tars placed in the Ecliptick it $elf, and indifferent in the intermediate: the contrary happens in the other difference, that is, the elevation or depre$$ion of the $tars placed in the Ecliptick is nothing at all, greate$t in tho$e neere$t to the Pole of the $aid Ecliptick, and indifferent in the interme- diate. Be$ides, both the$e differences are more $en$ible in the Stars neere$t to us, in the more remote le$$e $en$ible, and in tho$e that are very far di$tant wholly di$appear. This is, as to what concerns my $elf; it remaineth now, as I conceive, that $omething be $aid for the $atisfaction of <I>Simplicius,</I> who, as I believe, will not ea$ily be made to over-pa$$e tho$e differences, as in$en$ible that are derived from a motion of the Earth $o va$t, and from a mutation that tran$ports the Earth into places twice as far di$tant from us as the Sun.</P> <P>SIMP. Truth is, to $peak freely, I am very loth to confe$$e, that the di$tance of the fixed Stars ought to be $uch, that in them the fore-mentioned differences $hould be wholly imperceptible.</P> <P>SALV. Do notthrow your $elf into ab$olute de$pair, <I>Simpli- cius,</I> for there may perhaps yet $ome qualification be found for your difficulties. And fir$t, that the apparent magnitude of the $tars is not $een to make any $en$ible alteration, ought not to be judged by you a thing improbable, in regard you $ee the gue$$es of men in this particular to be $o gro$$ely erroneous, e$pecially in looking upon $plendid objects; and you your $elf beholding <I>v. g.</I> a lighted Torch at the di$tance of 200 paces, if it ap- <marg><I>In objects far remote, and lumi- nous, a $mall ap- proach or rece$$ion is imperceptible.</I></marg> proach nearer to you 3. or 4. yards, do you think that it will $hew any whit encrea$ed in magnitude? I for my part $hould not perceive it certainly, although it $hould approach 20. or 30. yards nearer; nay it hath $ometimes happened that in $eeing $uch a light at that di$tance I know not how to re$olve whether it came towards me, or retreated from me, when as it did in reality approach nearer to me. But what need I $peak of this? If the $elf $ame acce$$ion and rece$$ion (I $peak of a di$tance <foot>twice</foot> <p n=>351</p> twice as great as that from the Sun to us) in the $tar of <I>Saturn</I> is almo$t totally imperceptible, and in <I>Jupiter</I> not very ob$erva- ble, what $hall we think of the fixed $tars, which I believe you will not $cruple to place twice as far off as <I>Saturn</I>? In <I>Mars,</I> which for that it is nearer to us -------</P> <P>SIMP. Pray Sir, put your $elf to no farther trouble in this particular, for I already conceive that what hath been $poken touching the unaltered apparent magnitude of the fixed $tars may very well come to pa$$e, but what $hall we $ay of the other dif- ficulty that proceeds from not perceiving any variation in the mutation of a$pect?</P> <P>SALV. We will $ay that which peradventure may $atisfie you al$o in this particular. And to make $hort, would you not be $atisfied if there $hould be di$covered in the $tars face muta- tions that you think ought to be di$covered, in ca$e the annual motion belonged to the Earth?</P> <P>SIMP. I $hould $o doubtle$$e, as to what concerns this par- ticular.</P> <P>SALV. I could wi$h you would $ay that in ca$e $uch a diffe- <marg><I>If in the fixed $tars one $hould di$cover any an- nual mutation, the motion of the Earth would be undeniable.</I></marg> rence were di$covered, nothing more would remain behind, that might render the mobility of the Earth que$tionable. But al- though yet that $hould not $en$ibly appear, yet is not its mo- bility removed, nor its immobility nece$$arily proved, it being po$$ible, (as <I>Copernicus</I> affirmeth) that the immen$e di$tance of the Starry Sphere rendereth $uch very $mall <I>Phænomena</I> unob$er- vable; the which as already hath been $aid, may po$$ibly not have been hitherto $o much as $ought for, or if $ought for, yet not $ought for in $uch a way as they ought, to wit, with that <marg><I>It is proved what $mall credit is to be given to A$trono- mical In$truments in minute ob$erva- tions.</I></marg> exactne$$e which to $o minute a punctuality would be nece$$ary; which exactne$$e is very difficult to obtain, as well by rea$on of the deficiency of A$ttonomical In$truments, $ubject to many altera- tions, as al$o through the fault of tho$e that manage them with le$s diligence then is requi$ite. A nece$$ary argument how little cre- dit is to be given to tho$e ob$ervations may be deduced from the differences which we find among$t A$tronomers in a$$igning the places, I will not $ay, of the new Stars or Comets, but of the fixed $tars them$elves, even to the altitudes of the very Poles, in which, mo$t an end, they are found to differ from one another many minutes. And to $peak the truth, who can in a Quadrant, or Sextant, that at mo$t $hall have its $ide ^{*} 3. or 4. yards long, <marg>* Braceia Italian.</marg> a$certain him$elf in the incidence of the perpendicular, or in the direction of the $ights, not to erre two or three minutes, which in its circumference $hall not amount to the breadth of a grain of ^{*}<I>Mylet</I>? Be$ides that, it is almo$t impo$$ible, that the In$trument <marg>* Or Mill.</marg> $hould be made, and kept with ab$olute exactne$$e. <I>Ptolomey</I> <foot>$heweth</foot> <p n=>352</p> <marg>Ptolomy <I>did not tru$t to an In$tru- ment made by</I> Ar- chimedes.</marg> $heweth his di$tru$t of a Spherical In$trument compo$ed by <I>Ar- chimedes</I> hi$melf to take the Suns ingre$$ion into the Æqui- noctial.</P> <marg><I>In$truments of</I> Tycho <I>made with great expence.</I></marg> <P>SIMP. But if the In$truments be $o $u$pitious, and the ob$er- vations $o dubious, how can we ever come to any certainty of things, or free our $elves from mi$takes? I have heard $trange things of the In$truments of <I>Tycho</I> made with extraordinary co$t, and of his $ingular diligence in ob$ervations.</P> <P>SALV. All this I grant you; but neither one nor other of the$e is $ufficient to a$certain us in a bu$ine$$e of this importance. <marg><I>What In$tru- ments are apt for mo$t exact ob$er- vation.</I></marg> I de$ire that we may make u$e of In$truments greater by far, and by far certainer than tho$e of <I>Tycho,</I> made with a very $mall charge; the $ides of which are of 4. 6. 20. 30. and 50. miles, $o <marg>* Italian braces.</marg> as that a degree is a mile broad, a minute prim. 50 ^{*} yards, a $econd but little le$$e than a yard, and in $hort we may without a farthing expence procure them of what bigne$$e we plea$e. I <marg><I>An exqui$ite ob$ervation of the approach and de- parture of the Sun from the Summer Sol$tice.</I></marg> being in a Countrey Seat of mine near to <I>Florence,</I> did plainly ob$erve the Suns arrival at, and departure from the Summer Sol$tice, whil$t one Evening at the time of its going down it ap- peared upon the top of a Rock on the Mountains of <I>Pictrapana,</I> about 60. miles from thence, leaving di$covered of it a $mall $treak or filament towards the North, who$e breadth was not the hundredth part of its Diameter; and the following Evening at the like $etting, it $hew'd $uch another part of it, but notably more $mall, a nece$$ary argument, that it had begun to recede from the Tropick; and the regre$$ion of the Sun from the fir$t to the $econd ob$ervation, doth not import doubtle$$e a $econd mi- <marg><I>A place aecom- modated for the ob$<*> of the fixed $tars, as <*>o what concers the annual motion of the Earth.</I></marg> nute in the Ea$t. The ob$ervation made afterwards with an ex- qui$ite Tele$cope, and that multiplyeth the <I>Di$cus</I> of the Sun more than a thou$and times, would prove ea$ie, and with all delightful. Now with $uch an In$trument as this, I would have ob$ervations to be made in the fixed $tars, making u$e of $ome of tho$e wherein the mutation ought to appear more con$picu- ous, $uch as are (as hath already been declared) the more re- mote from the Ecliptick, among$t which the Harp a very great $tar, and near to the Pole of the Ecliptick, would be very pro- per in Countries far North, proceeding according to the man- ner that I $hall $hew by and by, but in the u$e of another $tar; and I have already fancied to my $elf a place very well adapted for $uch an ob$ervation. The place is an open Plane, upon which towards the North there ri$eth a very eminent Mountain, in the apex or top whereof is built a little Chappel, $ituate Ea$t and We$t, $o as that the ridg of its Roof may inter$ect at right angles, the meridian of $ome building $tanding in the Plane. I will place a beam parallel to the $aid ridg, or top of the Roof, <foot>and</foot> <p n=>353</p> and di$tant from it a yard or thereabouts. This being placed, I will $eek in the Plain the place from whence one of the $tars of <I>Charls's</I> Waine, in pa$$ing by the Meridian, cometh to hide it $elf behind the beam $o placed, or in ca$e the beam $hould not be $o big as to hide the $tar, I will finde a $tation where one may $ee the $aid beam to cut the $aid $tar into two equal parts; an effect that with an ^{*} exqui$ite Tele$cope may be perfectly di$cerned. And if in the place where the $aid accident is di$cover- ed, there were $ome building, it will be the more commodious; but if not, I will cau$e a Pole to be $tuck very fa$t in the ground, with $ome $tanding mark to direct where to place the eye anew, when ever I have a mind to repeat the ob$ervation. The fir$t of which ob$ervations I will make about the Summer Sol$tice, to continue afterwards from Moneth to Moneth, or when I $hall $o plea$e, to the other Sol$tice; with which ob$er- vation one may di$cover the elevation and depre$$ion of the $tar, though it be very $mall. And if in that operation it $hall hap- pen, that any mutation $hall di$cover it $elf, what and how great benefit will it bring to A$tronomy? Seeing that thereby, be$ides our being a$$ured of the annual motion, we may come to know the grandure and di$tance of the $ame $tar.</P> <P>SAGR. I very well comprehend your whole proceedings; and the operation $eems to me $o ea$ie, and $o commodious for the purpo$e, that it may very rationally be thought, that either <I>Copernicus</I> him$elf, or $ome other A$tronomer had made trial of it.</P> <P>SALV. But I judg the quite contrary, for it is not probable, that if any one had experimented it, he would not have men- tioned the event, whether it fell out in favour of this, or that opinion; be$ides that, no man that I can find, either for this, or any other end, did ever go about to make $uch an Ob$ervati- on; which al$o without an exact Tele$cope could but badly be effected.</P> <P>SIMP. I am fully $atisfied with what you $ay. But $eeing that it is a great while to night, if you defire that I $hall pa$$e the $ame quietly, let it not be a trouble to you to explain unto us tho$e Problems, the declaration whereof you did even now reque$t might be deferred until too morrow. Be plea$ed to grant us your promi$ed indulgence, and, laying a$ide all other di$cour- $es, proceed to $hew us, that the motions which <I>Copernicus</I> a$$igns to the Earth being taken for granted, and $uppo$ing the Sun and fixed $tars immoveable, there may follow the $ame acci- dents touching the elevations and depre$$ions of the Sun, touch- ing the mutations of the Sea$ons, and the inequality of dayes and nights, <I>&c.</I> in the $elf $ame manner, ju$t as they are with <foot>Yy fa-</foot> <p n=>345</p> facility apprehended in the <I>Prolomaick</I> Sy$teme.</P> <P>SALV. I neither ought, nor can deny any thing that <I>Sagredus</I> $hall reque$t: And the delay by me de$ired was to no other end, $ave only that I might have time once again to methodize tho$e prefatory points, in my fancy, that $erve for a large and plain de- claration of the manner how the forenamed accidents follow, as well in the <I>Copernican</I> po$ition, as in the <I>Ptolomaick</I>: nay, with <marg><I>The</I> Coperni- can <I>Sy$teme diffi- cult to be under- $tood, but ea$ie to be effected.</I></marg> much greater facility and $implicity in that than in this. Whence one may manife$tly conceive that Hypothe$is to be as ea$ie to be effected by nature, as difficult to be apprehended by the under- $tanding: yet neverthele$$e, I hope by making u$e of another <marg><I>Nece$$ary pre- po$itions for the better conceiving of the con$equences of the Earths mo- tion.</I></marg> kind of explanation, than that u$ed by <I>Copernicus,</I> to render like- wi$e the apprehending of it $omewhat le$$e ob$cure. Which that I may do, I will propo$e certain $uppo$itions of them$elves known and manife$t, and they $hall be the$e that follow.</P> <P>Fir$t, I $uppo$e that the Earth is a $pherical body, turning round upon its own Axis and Poles, and that each point a$$igned in its $uperficies, de$cribeth the circumference of a circle, great- er or le$$er, according as the point a$$igned $hall be neerer or farther from the Poles: And that of the$e circles the greate$t is that which is de$cribed by a point equidi$tant from the $aid Poles; and all the$e circles are parallel to each other; and <I>Parallels</I> we will call them.</P> <P>Secondly, The Earth being of a Spherical Figure, and of an o- pacous $ub$tance, it is continually illuminated by the Sun, accor- ding to the half of its $urface, the other half remaining ob$cure, and the boundary that di$tingui$heth the illuminated part from the dark being a grand circle, we will call that circle the <I>termi- nator of the light.</I></P> <P>Thirdly, If the Circle that is terminator of the light $hould pa$$e by the Poles of the Earth, it would cut (being a grand and principal circle) all the parallels into equal parts; but not pa$$ing by the Poles, it would cut them all in parts unequal, ex- cept only the circle in the middle, which, as being a grand circle will be cut into equal parts.</P> <P>Fourthly, The Earth turning round upon its own Poles, the quantities of dayes and nights are termined by the arches of the Parallels, inter$ected by the circle, that is, the terminator of the light, and the arch that is $cituate in the illuminated Hemi$phere pre$cribeth the length of the day, and the remainer is the quan- tity of the night.</P> <P>The$e things being pre$uppo$ed, for the more clear under- <marg><I>A plain Scheme repre$enting the</I> Copernican <I>Hypo- the$is, and its con- $equences.</I></marg> $tanding of that which remaines to be $aid, we will lay it down in a Figure. And fir$t, we will draw the circumference of a circle, that $hall repre$ent unto us that of the grand Orb de$cri- <foot>bed</foot> <p n=>355</p> bed in the plain of the Ecliptick, and this we will divide into four equal parts with the two diameters <I>Capricorn Cancer,</I> and <I>Libra Aries,</I> which, at the $ame time, $hall repre$ent unto us the four Cardinal points, that is, the two Sol$tices, and the two E- quinoctials; and in the centre of that circle we will place the Sun O, fixed and immoveable.</P> <fig> <P>Let us next draw about the four points, Capricorn, Cancer, Libra and Aries, as centres, four equal circles, which repre$ent unto us the Earth placed in them at four $everal times of the year. The which, with its centre, in the $pace of a year, pa$$eth through the whole circumference, Capricorn, Aries, Cancer, Li- bra, moving from Ea$t to We$t, that is, according to the order of the Signes. It is already manife$t, that whil$t the Earth is in Capricorn, the Sun will appear in Cancer, and the Earth moving <marg><I>The Suns an- nual motion, how it comes to pa$$e, according to</I> Co- pernicus.</marg> along the arch Capricorn Aries, the Sun will $eem to move along the arch Cancer Libra, and in $hort, will run thorow the Zodiack according to the order of the Signes, in the $pace of a year; and by this fir$t a$$umption, without all que$tion, full $atisfaction is given for the Suns apparent annual motion under the Ecliptick. Now, coming to the other, that is, the diurnal motion of the Earth in it $elf, it is nece$$ary to e$tabli$h its Poles and its Axis, the which mu$t be under$tood not to be erect perpendicularly upon the plain of the Ecliptick, that is, not to be parallel to the Axis of the grand Orb, but declining from a right angle 23 de- grees and an half, or thereabouts, with its North Pole towards <foot>Yy 2 the</foot> <p n=>356</p> the Axis of the grand Orb, the Earths centre being in the Sol$ti- tial point of Capricorn. Suppo$ing therefore the Terre$trial Globe to have its centre in the point Capricorn, we will de$cribe its Poles and Axis A B, inclined upon the diameter Capricorn Cancer 23 degrees and an half; $o that the angle A Capricorn Cancer cometh to be the complement of a Quadrant or Radius, that is, 66 degrees and an half; and this inclination mu$t be un- der$tood to be immutable, and we will $uppo$e the $uperiour Pole A to be Boreal, or North, and the other Au$tral, or South. Now imagining the Earth to revolve in it $elf about the Axis A B in twenty four hours, from We$t to Ea$t, there $hall by all the points a$$igned in its $uper$icies, be circles de$cribed parallel to each other. We will draw, in this fir$t po$ition of the Earth, the greate$t C D, and tho$e two di$tant from it <I>gr.</I> 23. and an half, E F above, and G M beneath, and the other two extream ones I K and L M remote, by tho$e intervals from the Poles A and B; and as we have marked the$e five, $o we may imagine in- numerable others, parallel to the$e, de$cribed by the innumera- ble points of the Terre$trial $urface. Next let us $uppo$e the Earth, with the annual motion of its centre, to transferre it $elf into the other places already marked; but to pa$$e thither in $uch a manner, that its own Axis A B $hall not only not change incli- nation upon the plain of the Ecliptick, but $hall al$o never vary direction; $o that alwayes keeping parallel to it $elf, it may continually tend towards the $ame part of the Univer$e, or, if you will, of the Firmament, whereas, if we do but $uppo$e it prolonged, it will, with its extream termes, de$igne a Circle pa- rallel and equal to the grand Orb, Libra Capricorn Aries Cancer, as the $uperiour ba$e of a Cylinder de$cribed by it $elf in the an- nual motion above the inferiour ba$e, Libra Capricorn Aries Cancer. And therefore this immutability of inclination conti- nuing, we will de$ign the$e other three figures about the centres Aries, Cancer, and Libra, alike in every thing to that fir$t de- $cribed about the centre Capricorn. Now we will con$ider the fir$t figure of the Earth, in which, in regard the Axis A B is de- clined from perpendicularity upon the diameter Capricorn Can- cer 23 degrees and an half towards the Sun O, and the arch A I being al$o 23 degrees and an half, the illumination of the Sun will illu$trate the Hemi$phere of the Terre$trial Globe expo$ed towards the Sun (of which, in this place, half is to be $een) di- vided from the ob$cure part by the Terminator of the light I M, by which the parallel C D, as being a grand circle, $hall come to be divided into equal parts, but all the re$t into parts un- equal; being that the terminator of the light I M pa$$eth not by their Poles A B, and the parallel I K, together with all the re$t <foot>de$cribed</foot> <p n=>357</p> de$cribed within the $ame, and neerer to the pole A, $hall wholly be included in the illuminated part; as on the contrary, the op- po$ite ones towards the Pole B, contained within the paral- lel L M, $hall remain in the dark. Moreover, the arch A I be- ing equal to the arch F D, and the arch A F, common to them both, the two arches I K F and A F D $hall be equal, and each a quadrant or 90 degrees. And becau$e the whole arch I F M is a $emicircle, the arch F M $hall be a quadrant, and equal to the other F K I; and therefore the Sun O $hall be in this $tate of the Earth vertical to one that $tands in the point F. But by the revolution diurnal about the $tanding Axis A B, all the points of the parallel E F pa$$e by the $ame point F: and therefore in that $ame day the Sun, at noon, $hall be vertical to all the inha- bitants of the Parallel E F, and will $eem to them to de$cribe in its apparent motion the circle which we call the Tropick of Cancer. But to the inhabitants of all the Parallels that are above the pa- rallel E F, towards the North pole A, the Sun declineth from their <I>Vertex</I> or Zenith towards the South; and on the contrary, to all the inhabitants of the Parallels that are beneath E F, to- wards the Equinoctial C D, and the South Pole B, the Meridian Sun is elevated beyond their <I>Vertex</I> towards the North Pole A. Next, it is vi$ible that of all the Parallels, only the greate$t C D is cut in equal parts by the Terminator of the light I M. But the re$t, that are beneath and above the $aid grand circle, are all inter$ected in parts unequal: and of the $uperiour ones, the $e- midiurnal arches, namely tho$e of the part of the Terre$trial $ur- face, illu$trated by the Sun, are bigger than the $eminocturnal ones that remain in the dark: and the contrary befalls in the remainder, that are under the great one C D, towards the pole B, of which the $emidiurnal arches are le$$er than the $eminocturnal, It is likewi$e apparently manife$t, that the differences of the $aid arches go augmenting, according as the Parallels are neerer to the Poles, till $uch time as the parallel I K comes to be wholly in the part illuminated, and the inhabitants thereof have a day of twenty four hours long, without any night; and on the contrary, the Parallel L M, remaining all in ob$curity, hath a night of twenty four hours, without any day. Come we next to the third Figure of the Earth, placed with its centre in the point Cancer, where the Sun $eemeth to be in the fir$t point of Ca- pricorn. We have already $een very manife$tly, that by rea$on the Axis A B doth not change inclination, but continueth paral- lel to it $elf, the a$pect and $ituation of the Earth is the $ame to an hair with that in the fir$t Figure; $ave onely that that Hemi- $phere which in the fir$t was illuminated by the Sun, in this re- maineth obtenebrated, and that cometh to be luminous, which in <foot>the</foot> <p n=>358</p> the fir$t was tenebrous: whereupon that which happened before concerning the differences of dayes and nights, touching the dayes being greater or le$$er than the nights, now falls out quite contrary. And fir$t, we $ee, that whereas in the fir$t Figure the circle I K was wholly in the light, it is now wholly in the dark; and the oppo$ite arch L M is now wholly in the light, which was before wholly in the dark. Of the parallels between the grand circle C D, and the Pole A, the $emidiurnal arches are now le$$er than the $eminocturnal, which before were the contrary. Of the others likewi$e towards the Pole B, the $emidiurnal arch- es are now bigger than the $eminocturnal, the contrary to what happened in the other po$ition of the Earth. We now $ee the Sun made vertical to the inhabitants of the Tropick G N, and to be depre$$ed towards the South, with tho$e of the Parallel E F, by all the arch E C G, that is, 47 degrees; and in $umme, to have pa$$ed from one to the other Tropick, traver$ing the Equinoctial, elevating and declining in the Meridians the $aid $pace of 47 de- grees. And all this mutation is derived not from the inclination or elevation of the Earth, but on the contrary, from its not in- clining or elevating at all; and in a word, by continuing always in the $ame po$ition, in re$pect of the Univer$e, onely with turn- ing about the Sun $ituate iu the mid$t of the $aid plane, in which it moveth it $elf about circularly with its annual motion. And <marg><I>An admirable accident depending on the not inclining of the Earths axis</I></marg> here is to be noted an admirable accident, which is, that like as the Axis of the Earth con$erving the $ame direction towards the Univer$e, or we may $ay, towards the highe$t Sphere of the fixed $tars, cau$eth the Sun to appear to elevate and incline $o great a $pace, namely, for 47 degrees, and the fixed Stars to incline or e- levate nothing at all; $o, on the contrary, if the $ame Axis of the Earth $hould maintain it $elf continually in the $ame inclina- tion towards the Sun, or, if you will, towards the Axis of the Zodiack, no mutation would appear to be made in the Sun about its elevating or declining, whereupon the inhabitants of one and the $ame place would alwayes have one and the $ame difference of dayes and nights, and one and the $ame con$titution of Sea- $ons, that is, $ome alwayes Winter, others alwayes Summer, others Spring, &c. but, on the contrary, the alterations in the fixed Stars would appear very great, as touching their elevation, and inclination to us, which would amount to the $ame 47 de- grees. For the under$tanding of which let us return to con$ider the po$ition of the Earth, in its fir$t Figure, where we $ee the Axis A B, with the $uperiour Pole A, to incline towards the Sun; but in its third Figure, the $ame Axis having kept the $ame dire- ction towards the highe$t Sphere, by keeping parallel to it $elf, inclines no longer towards the Sun with its $uperiour Pole A, but <foot>on</foot> <p n=>359</p> on the contrary reclines from its former po$ition <I>gr.</I> 47. and in- clineth towards the oppo$ite part, $o that to re$tore the $ame in- clination of the $aid Pole A towards the Sun, it would be requi- $ite by turning round the Terre$trial Globe, according to the circumference A C B D, to tran$port it towards E tho$e $ame <I>gr.</I> 47. and for $o many degrees, any what$oever fixed $tar ob- $erved in the Meridian, would appear to be elevated, or inclined. Let us come now to the explanation of that which remains, and let us con$ider the Earth placed in the fourth Figure, that is, with its centre in the fir$t point of Libra; upon which the Sun will appear in the beginning of Aries. And becau$e the Axis of <fig> the Earth, which in the fir$t Figure is $uppo$ed to be inclined up- on the diameter Capricorn Cancer, and therefore to be in that $ame plane, which cutting the plane of the grand Orb, accor- ding to the line Capricorn Cancer, was erected perpendicularly upon the $ame, tran$po$ed into the fourth Figure, and maintai- ned, as hath alwayes been $aid, parallel to it $elf, it $hall come to be in a plane in like manner erected to the $uperficies of the Grand Orbe, and parallel to the plane, which at right angles cuts the $ame $uperficies, according to the diameter Ca- pricorn Cancer. And therefore the line which goeth from the centre of the Sunne to the centre of the Earth, that is, O Libra, $hall be perpendicular to the Axis BA: but the $ame line which goeth from the centre of the Sunne to the centre of the Earth, is al$o alwayes perpendicular to the <foot>circle</foot> <p n=>360</p> circle that is the Terminator of the light; therefore this $ame circle $hall pa$$e by the Poles A B in the fourth figure, and in its plain the Axis A B $hall fall, but the greate$t circle pa$$ing by the Poles of the Parallels, divideth them all in equal parts; therefore the arches I K, E F, C D, G N, L M, $hall be all $emicircles, and the illumin'd Hemi$phere $hall be this which looketh towards us, and the Sun, and the Terminator of the light $hall be one and the $ame circle A C B D, and the Earth being in this place $hall make it Equinoctial to all its Inhabitants. And the $ame happeneth in the $econd figure, where the Earth having its illuminated Hemi$phere towards the Sun, $heweth us the other that is ob$cure, with its nocturnal arches, which in like manner are all $emicircles, and con$equently, here al$o it maketh the Equinoctial. And la$tly in regard that the line pro- duced from the centre of the Sun to the centre of the Earth, is perpendicular to the Axis A B, to which the greate$t circle of the parallels C D, is likewi$e erect, the $aid line O <I>Libra</I> $hall pa$$e of nece$$ity by the $ame Plain of the parallel C D, cutting its circumference in the mid$t of the diurnal arch C D; and therefore the Snn $hall be vertical to any one that $hall $tand where that inter$ection is made; but all the Inhabitants of that Parallel $hall pa$$e the $ame, as being carried about by the Earths diurnal conver$ion; therefore all the$ upon that day $hall have the Meridian Sun in their vertex. And the Sun at the $ame time to all the Inhabitants of the Earth $hall $eem to de- $cribe the Grand Parallel called the Equinoctial. Furthermore, fora$much as the Earth being in both the Sol$titial points of the Polar circles I K and L M, the one is wholly in the light, and the other wholly in the dark; but when the Earth is in the Equi- noctial points, the halves of tho$e $ame polar circles are in the light, the remainder of them being in the dark; it $hould not be hard to under$tand, how that the Earth <I>v. gr.</I> from <I>Cancer</I> (where the parallel I K is wholly in the dark) to <I>Leo,</I> one part of the parallel towards the point I, beginneth to enter into the light, and that the Terminator of the light I M beginneth to retreat to- wards the Pole AB, inter$ecting the circle ACBD nolonger in IM, but in two other points falling between the terms I A and MB, of the arches IA and M B; whereupon the Inhabitants of the circle begin to enjoy the light, and the other Inhabitants of the circle L M to partake of night. And thus you $ee that by two $imple motions made in times proportionate to their bigne$$es, and not contrary to one another, but performed, as all others that be- long to moveable mundane bodies, from We$t to Ea$t a$$igned to the Terre$trial Globe, adequate rea$ons are rendred of all tho$e <I>Phænomena</I> or appearances, for the accommodating of <foot>which</foot> <p n=>361</p> which to the $tability of the Earth it is nece$$ary (for$aking that Symetry which is ob$erved to be between the velocities and mag- nitudes of moveables) to a$cribe to a Sphere, va$t above all others, an unconceiveable celerity, whil$t the other le$$er Spheres move extream $lowly; and which is more, to make that motion contrary to all their motions; and, yet again to adde to the improbability, to make that $uperiour Sphere forcibly to tran$port all the inferionr ones along with it contrary to their proper inclination. And here I refer it to your judgment to de- termine which of the two is the mo$t probable.</P> <P>SAGR. To me, as far as concerneth $en$e, there appeareth no $mall difference betwixt the $implicity and facility of opera- ting effects by the means a$$igned in this new con$titution, and the multiplicity, con$ufion, and difficulty, that is found in the ancient and commonly received Hypothe$is. For if the Univer$e were di$po$ed according to this multiplicity, it would be ne- ce$$ary to renounce many Maximes in Philo$ophy commonly re- <marg><I>Axiomes com- monly admitted by all Philo$ophers.</I></marg> ceived by Philo$ophers, as for in$tance, That Nature doth not multiply things without nece$$ity; and, That She makes u$e of the mo$t facile and $imple means in producing her effects; and, That She doth nothing in vain, and the like. I do confe$$e that I never heard any thing more admirable than this, nor can I believe that Humane Under$tanding ever penetrated a more $ublime $peculation. I know not what <I>Simplicius</I> may think of it.</P> <P>SIMP. The$e (if I may $peak my judgment freely) do $eem <marg>Ari$totle <I>tax- eth</I> Plato <I>for being too $tudious of Ge- ometry.</I></marg> to me $ome of tho$e Geometrical $ubtilties which <I>Ari$totle</I> finds fault with in <I>Plato,</I> when he accu$eth him that by his too much $tudying of Geometry he for$ook $olid Philo$ophy; and I have known and heard very great <I>Peripatetick</I> Philo$ophers to di$$wade their Scholars from the Study of the Mathematicks, as tho$e that render the wit cavilous, and unable to philo$ophate well; an In$titute diametrically contrary to that of <I>Plato,</I> who admitted uone to Philo$ophy, unle$$e he was fir$t well entered in Geometry.</P> <P>SALV. I commend the policy of the$e your <I>Peripateticks,</I> in <marg>Peripatetick <I>Phi- lo$ophers condemn the Study of Geo- metry, and why.</I></marg> dehorting their Di$ciples from the Study of Geometry, for that there is no art more commodious for detecting their fallacies; but $ee how they differ from the Mathematical Philo$ophers, who much more willingly conver$e with tho$e that are well ver$t in the commune Peripatetick Philo$ophy, than with tho$e that are de$titute of that knowledg, who for want thereof cannot di- $tingui$h between doctrine and doctrine. But pa$$ing by this, tell me I be$eech you, what are tho$e extravagancies and tho$e too affected $ubtilties that make you think this <I>Copernican</I> Sy$teme the le$$e plau$ible?</P> <foot>Zz SIMP.</foot> <p n=>362</p> <P>SIMP. To tell you true, I do not very well know; perhaps, becau$e I have not $o much as learnt the rea$ons that are by <I>Ftolo- my</I> produced, of tho$e effects, I mean of tho$e $tations, retrogra- dations, acce$$ions, rece$$ions of the Planets; lengthenings and $hortnings of dayes, changes of $ea$ons, &c. But omitting the con$equences that depend on the fir$t $uppo$itions, I find in the $uppo$itions them$elves no $mall difficulties; which $uppo$itions, if once they be overthrown, they draw along with them the ruine of the whole fabrick. Now fora$much as becau$e the whole module of <I>Copernicus</I> $eemeth in my opinion to be built upon in- firm foundations, in that it relyeth upon the mobility of the earth, if this $hould happen to be di$proved, there would be no need of farther di$pute. And to di$prove this, the Axiom of <I>Ari$totle</I> is in my judgment mo$t $ufficient, That of one $imple body, one $ole $imple motion can be natural: but here in this ca$e, to <marg><I>Four $everal motions a$$igned to the Earth.</I></marg> the Earth, a $imple body, there are a$$igned 3. if not 4. motions, and all very different from each other. For be$ides the light motion, as a grave body towards its centre, which cannot be de- nied it, there is a$$igned to it a circular motion in a great circle about the Sun in a year, and a vertiginous conver$ion about its own centre in twenty four hours. And that in the next place which is more exorbitant, & which happly for that rea$on you pa$s over in $ilence, there is a$cribed to it another revolution about its own centre, contrary to the former of twenty four hours, and which fini$heth its period in a year. In this my under$tand- ing apprehendeth a very great contradiction.</P> <marg><I>The motion of de$cent belongs not to the terre$trial Globe, but to its parts.</I></marg> <P>SALV. As to the motion of de$cent, it hath already been con- cluded not to belong to the Terre$trial Globe which did never move with any $uch motion, nor never $hall do; but is (if there be $uch a thing) that propen$ion of its parts to reunite them$elves to their whole. As, in the next place, to the Annual motion, <marg><I>The annual and diurnal motion are compatible in the Earth.</I></marg> and the Diurnal, the$e being both made towards one way, are very compatible, in the $ame manner ju$t as if we $hould let a Ball trundle downwards upon a declining $uperficies, it would in its de$cent along the $ame $pontaneou$ly revolve in it $elf. As to the third motion a$$igned it by <I>Copernicus,</I> namely about it $elf in a year, onely to keep its Axis inclined and directed towards the $ame part of the Firmament, I will tell you a thing worthy of great con$ideration: namely <I>ut tantum abe$t</I> (although it be made contrary to the other annual) it is $o far from having any repugnance or difficulty in it, that naturally and without any <marg><I>Every pen$il and librated, body car- ryed round in the circumference of a circle, acquireth of it $elf a motion in it $elf contrary to that.</I></marg> moving cau$e, it agreeth to any what$oever $u$pended and libra- ted body, which if it $hall be carried round in the circumference of a circle, immediate of it $elf, it acquireth a conver$ion about its own centre, contrary to that which carrieth it about, and of <foot>$uch</foot> <p n=>363</p> $uch velocity, that they both fini$h one revolution in the $ame time preci$ely. You may $ee this admirable, and to our pur- <marg><I>An Experiment which $en$ibly $hews that two con- trary motions may naturally agree i<*> the $ame move- able.</I></marg> po$e accommodate experience, if putting in a Ba$on of water a Ball that will $wim; and holding the Ba$on in your hand, you turn round upon your toe, for you $hall immediatly $ee the Ball begin to revolve in it $elf with a motion, contrary to that of the Ba$on, and it $hall fini$h its revolution, when that of the Ba$on it $hall fini$h. Now what other is the Earth than a pen$il Globe librated in tenuous and yielding aire, which being carried a- bout in a year along the circumference of a great circle, mu$t <marg><I>The third motion a$cribed to the Earth is rather re$ting immove- able.</I></marg> needs acquire, without any other mover, a revolution about its own centre, annual, and yet contrary to the other motion in like manner annual? You $hall $ee this effect I $ay, but if afterwards you more narrowly con$ider it, you $hall find this to be no real thing, but a meer appearance; and that which you think to be a revolution in it $elf, you will find to be a not moving at all, but a continuing altogether immoveable in re$pect of all that which without you, and without the ve$$el is immoveable: for if in that Ball you $hall make $ome mark, and con$ider to what part of the Room where you are, or of the Field, or of Heaven it is $ituate, you $hall $ee that mark in yours, and the ve$$els revolu- tion to look alwayes towards that $ame part; but comparing it to the ve$$el and to your $elf that are moveable, it will appear to go altering its direction, and with a motion contrary to yours, and that of the ve$$el, to go $eeking all the points of its circumgyra- tion; $o that with more rea$on you and the ba$on may be $aid to turn round the immoveable Ball, than that it moveth round in the ba$on. In the $ame manner the Earth $u$pended and li- brated in the circumference of the Grand Orbe, and $cituate in $uch $ort that one of its notes, as for example, its North Pole, loo- keth towards $uch a Star or other part of the Firmament, it always keepeth directed towards the $ame, although carried round by the annual motion about the circumference of the $aid Grand Orbe. This alone is $ufficient to make the Wonder cea$e, and to remove all difficulties. But what will <I>Simplicius</I> $ay, if to this non-indigence of the co-operating cau$e we $hould adde an admirable intrin$ick vertue of the Terre$trial Globe, of look- <marg><I>An admirable intern vertœe of the terre$trial Globe of alwayes beholding the $ame part of Heaven.</I></marg> ing with its determinate parts towards determinate parts of the Firmament, I $peak of the Magnetick vertue con$tantly partici- pated by any what$oever piece of Loade-$tone. And if every minute particle of that S one have in it $uch a vertue, who will <marg><I>The terre$triæl Globe made of Loade-$tone.</I></marg> que$tion but that the $ame more powerfully re$ides in this whole Terre$trial Globe, abounding in that Magnetick matter, and which happily it $elf, as to its internal and primary $ub$tance, is nothing el$e but a huge ma$$e of Loade-$tone.</P> <foot>Zz 2 SIMP.</foot> <p n=>364</p> <P>SIMP. Then you are one of tho$e it $eems that hold the Mag- <marg>An eminent Doctor of Phy$ick, our Countreyman, born at <I>Colohe$ter,</I> and famous for this his learned Trea- ti$e, publi$hed a- bout 60 years $ince at <I>London, The Magnetick Phi- lo$ophy of</I> William Gilbert.</marg> netick Phylo$ophy <I>William</I> ^{*} <I>Gilbert.</I></P> <P>SALV. I am for certain, and think that all tho$e that have $eriou$ly read his Book, and tried his experiments, will bear me company therein; nor $hould I de$pair, that what hath befallen me in this ca$e, might po$$ibly happen to you al$o, if $o be a cu- <*>io$ity, like to mine, and a notice that infinite things in Nature are $till conceal'd from the wits of mankind, by delivering you from being captivated by this or that particular writer in natural things, $hould but $lacken the reines of your Rea$on, and mol- lifie the contumacy and tenaceou$ne$$e of your $en$e; $o as that they would not refu$e to hearken $ometimes to novelties never <marg><I>The Pu$illani- mity of Popular Wits.</I></marg> before $poken of. But (permit me to u$e this phra$e) the pu$illa- nimity of vulgar Wits is come to that pa$$e, that not only like blind men, they make a gift, nay tribute of their own a$$ent to what$oever they find written by tho$e Authours, which in the infancy of their Studies were laid before them, as authentick by their Tutors, but refu$e to hear (not to $ay examine) any new Propo$ition or Probleme, although it not only never hath been confuted, but not $o much as examined or con$idered by their Authours. Among$t which, one is this, of inve$tigating what is the true, proper, primary, interne, and general matter and $ub- $tance of this our Terre$trial Globe; For although it never came into the mind either of <I>Ari$totle,</I> or of any one el$e, before <I>Wil- liam Gilbert</I> to think that it might be a Magnet, $o far are <I>Ari- $totle</I> and the re$t from confuting this opinion, yet neverthele$$e I have met with many, that at the very fir$t mention of it, as a Hor$e at his own $hadow, have $tart back, and refu$ed to di$- cour$e thereof, and cen$ured the conceipt for a vain <I>Chymæra,</I> yea, for a $olemn madne$$e: and its po$$ible the Book of <I>Gilbert</I> had never come to my hands, if a Peripatetick Philo$opher, of great fame, as I believe, to free his Library from its contagion, had not given it me.</P> <P>SIMP. I, who ingenuou$ly confe$$e my $elf to be one of tho$e vulgar Wits, and never till within the$e few dayes that I have been admitted to a $hare in your conferences, could I pre- tend to have in the lea$t withdrawn from tho$e trite and popu- lar paths, yet, for all that, I think I have advantaged my $elf $o much, as that I could without much trouble or difficulty, ma$ter the roughne$$es of the$e novel and fanta$tical opinions.</P> <P>SALV. If that which <I>Gilbert</I> writeth be true, then is it no o- pinion, but the $ubject of Science; nor is it new, but as antient as the Earth it $elf; nor can it (being true) be rugged or diffi- cult, but plain and ea$ie; and when you plea$e I $hall make you feel the $ame in your hand, for that you of your $elf fancy it to <foot>be</foot> <p n=>365</p> be a Gho$t, and $tand in fear of that which hath nothing in it of dreadfull, like as a little child doth fear the Hobgoblin, without knowing any more of it, $ave the name; as that which be$ides the name is nothing.</P> <P>SIMP. I $hould be glad to be informed, and reclaimed from an errour.</P> <P>SALV. An$wer me then to t<*> que$tions that I $hall ask you. And fir$t of all, Tell me whether you believe, that this our Globe, which we inhabit and call Earth, con$i$teth of one $ole and $im- ple matter, or el$e that it is an aggregate of matters different from each other.</P> <P>SIMP. I $ee it to be compo$ed of $ub$tances and bodies very <marg><I>The</I> Terre$trial Globe <I>compo$ed of $undry matters.</I></marg> different; and fir$t, for the greate$t parts of the compo$ition, I $ee the Water and the Earth, which extreamly differ from one another.</P> <P>SAIV. Let us, for this once, lay a$ide the Seas and other Wa- ters, and let us con$ider the $olid parts, and tell me, if you think them one and the $ame thing, or el$e different.</P> <P>SIMP. As to appearance, I $ee that they are different things, there being very great heaps of unfruitful $ands, and others of fruitful $oiles; There are infinite $harp and $teril mountains, full of hard $tones and quarries of $everal kinds, as Porphyre, Ala- bla$ter, Ja$per, and a thou$and other kinds of Marbles: There are va$t Minerals of $o many kinds of metals; and in a word, $uch varieties of matters, that a whole day would not $uffice on- ly to enumerate them.</P> <P>SALV. Now of all the$e different matters, do you think, that in the compo$ition of this grand ma$$e, there do concur por- tions, or el$e that among$t them all there is one part that far ex- ceeds the re$t, and is as it were the matter and $ub$tance of the immen$e lump?</P> <P>SIMP. I believe that the Stones, Marbles, Metals, Gems, and the $o many other $everal matters are as it were Jewels, and ex- teriour and $uperficial Ornaments of the primary Globe, which in gro$$e, as I believe, doth without compare exceed all the$e things put together.</P> <P>SALV. And this principal and va$t ma$$e, of which tho$e things above named are as it were excre$$ences and ornaments, of what matter do you think that it is compo$ed?</P> <P>SIMP. I think that it is the $imple, or le$$e impure element of Earth.</P> <P>SALV. But what do you under$tand by Earth? Is it haply that which is di$per$ed all over the fields, which is broke up with Mattocks and Ploughs, wherein we $owe corne, and plant fruits, and in which great bo$cages grow up, without the help of cul- <foot>ture</foot> <p n=>366</p> ture, and which is, in a word, the habitation of all animals, and the womb of all vegetables?</P> <P>SIMP. Tis this that I would affirm to be the $ub$tance of this our Globe.</P> <P>SALV. But in this you do, in my judgment, affirm that which is not right: for this Earth which is broke up, is $owed, and is fertile, is but one part, and that very $mall of the $urface of the Globe, which doth not go very deep, yea, its depth is very $mall, in compari$on of the di$tance to the centre: and experience $heweth us, that one $hall not dig very low, but one $hall finde matters very different from this exteriour $curf, more $olid, and not good for the production of vegetables. Be$ides the interne parts, as being compre$$ed by very huge weights that lie upon them, are, in all probability, $lived, and made as hard as any hard rock. One may adde to this, that fecundity would be in vain conferred upon tho$e matters which never were de$igned to bear fruit, but to re$t eternally buried in the profound and dark aby$$es of the Earth.</P> <P>SIMP. But who $hall a$$ure us, that the parts more inward and near to the centre are unfruitful? They al$o may, perhaps, have their productions of things unknown to us?</P> <P>SALV. You may a$well be a$$ured thereof, as any man el$e, as being very capable to comprehend, that if the integral bodies of the Univer$e be produced onely for the benefit of Mankind, this above all the re$t ought to be de$tin d to the $ole convenien- ces of us its inhabitants. But what bene$it can we draw from matters $o hid and remote from us, as that we $hall never be a- <marg><I>The interne parts of the terre$trial Globe mu$t of ne- ce$$ity be $olid.</I></marg> ble to make u$e of them? Therefore the interne $ub$tance of this our Globe cannot be a matter frangible, di$$ipable, and non- coherent, like this $uperficial part which we call ^{*} EARTH: but <marg>* Or MOULD.</marg> it mu$t, of nece$$ity, be a mo$t den$e and $olid body, and in a word, a mo$t hard $tone. And, if it ought to be $o, what rea$on is there that $hould make you more $crupulous to believe that it is a Load$tone than a Porphiry, a Ja$per, or other hard Mar- ble? Happily if <I>Gilbert</I> had written, that this Globe is all com- <marg>Of which with the Latin tran$la- tour, I mu$t once more profe$$e my $elf ignorant.</marg> pounded within of ^{*} <I>Pietra Serena,</I> or of <I>Chalcedon,</I> the paradox would have $eemed to you le$$e exorbitant?</P> <P>SIMP. That the parts of this Globe more intern are more compre$$ed, and $o more $lived together and $olid, and more and more $o, according as they lie lower, I do grant, and $o likewi$e doth <I>Ari$totle,</I> but that they degenerate and become other than Earth, of the $ame $ort with this of the $uperficial parts, I $ee nothing that obliege h me to believe.</P> <P>SALV. I undertook not this di$cour$e with an intent to prove demon$tratively that the primary and real $ub$tance of this our <foot>Globe</foot> <p n=>367</p> Globe is Load-$tone; but onely to $hew that no rea$on could be given why one $hould be more unwilling to grant that it is of Load-$tone, than of $ome other matter. And if you will but <marg><I>Our Globe would have been called $tone, in $tead of Earth, if that name had been gi- uen it in the be- ginning.</I></marg> $eriou$ly con$ider, you $hall find that it is not improbable, that one $ole, pure, and arbitrary name, hath moved men to think that it con$i$ts of Earth; and that is their having made u$e com- monly from the beginning of this word Earth, as well to $igni- $ie that matter which is plowed and $owed, as to name this our Globe. The denomination of which if it had been taken from $tone, as that it might as well have been taken from that as from the Earth; the $aying that its primary $ub$tance was $tone, would doubtle$$e have found no $cruple or oppo$ition in any man. And is $o much the more probable, in that I verily be- lieve, that if one could but pare off the $curf of this great Globe, taking away but one full thou$and or two thou$and yards; and afterwards $eperate the Stones from the Earth, the accumulati- on of the $tones would be very much biger than that of the fer- tile Mould. But as for the rea$ons which concludently prove <I>de facto,</I> that is our Globe is a Magnet, I have mentioned none of them, nor is this a time to alledg them, and the rather, for that to your benefit you may read them in <I>Gilbert</I>; onely to encou- rage you to the peru$al of them, I will $et before you, in a $imi- <marg><I>The method of</I> Gilbert <I>in his Phi- lo$ophy.</I></marg> litude of my own, the method that he ob$erved in his Philo$o- phy. I know you under$tand very well how much the know- ledg of the accidents is $ub$ervient to the inve$tigation of the $ub$tance and e$$ence of things; therefore I de$ire that you would take pains to informe your $elf well of many accidents and properties that are found in the Magnet, and in no other $tone, <marg><I>Many proper- ties in the Mag- net.</I></marg> or body; as for in$tance of attracting Iron, of conferring up- on it by its $ole pre$ence the $ame virtue, of communicating likewi$e to it the property of looking towards the Poles, as it al$o doth it $elf; and moreover endeavour to know by trial, that it containeth in it a virtue of conferring upon the magnetick needle not onely the direction under a Meridian towards the Poles, with an Horizontal motion, (a property a long time ago known) but a new found accident, of declining (being ballanced under the Meridian before marked upon a little $pherical Mag- net) of declining I $ay to determinate marks more or le$$e, ac- cording as that needle is held nearer or farther from the Pole, till that upon the Pole it $elf it erecteth perpendicularly, where- as in the middle parts it is parallel to the Axis. Furthermore pro- cure a proof to be made, whether the virtue of attracting Iron, re$iding much more vigorou$ly about the Poles, than about the middle parts, this force be not notably more vigorous in one Pole than in the other, and that in all pieces of Magnet; the <foot>$tronger</foot> <p n=>368</p> $tronger of which Poles is that which looketh towards the South. Ob$erve, in the next place, that in a little Magnet this South and more vigorous Pole, becometh weaker, when ever it is to take up an iron in pre$ence of the North Pole, of another much big- ger Magnet: and not to make any tedious di$cour$e of it, a$$er- tain your $elf, by experience, of the$e and many other properties de$cribed by <I>Gilbert,</I> which are all $o peculiar to the Magnet, as <marg><I>An Argument proving the terre- $trial Globe to be a</I> Magnet.</marg> that none of them agree with any other matter. Tell me now, <I>Simplicius,</I> if there were laid before you a thou$and pieces of $everal matters, but all covered and concealed in a cloth, under which it is hid, and you were required, without uncovering them, to make a gue$$e, by external $ignes, at the matter of each of them, and that in making trial, you $hould hit upon one that $hould openly $hew it $elf to have all the properties by you alrea- dy acknowledged to re$ide onely in the Magnet, and in no other matter, what judgment would you make of the e$$ence of $uch a body? Would you $ay, that it might be a piece of Ebony, or Alabla$ter, or Tin.</P> <P>SIMP. I would $ay, without the lea$t hæ$itation, that it was a piece of Load-$tone.</P> <P>SAL<*>. If it be $o, $ay re$olutely, that under this cover and $curf of Earth, $tones, metals, water, &c. there is hid a great Magnet, fora$much as about the $ame there may be $een by any one that will heedfully ob$erve the $ame, all tho$e very accidents that agree with a true and vi$ible Globe of Magnet; but if no more were to be $een than that of the Declinatory Needle, which being carried about the Earth, more and more inclineth, as it ap- proacheth to the North Pole, and declineth le$$e towards the E- quinoctial, under which it finally is brought to an <I>Æquilibrium,</I> it might $erve to per$wade even the mo$t $crupulous judgment. I forbear to mention that other admirable effect, which is $en$ibly ob$erved in every piece of Magnet, of which, to us inhabitants of the Northern Hemi$phere, the Meridional Pole of the $aid Mag- net is more vigorous than the other; and the difference is found greater, by how much one recedeth from the Equinoctial; and under the Equinoctial both the parts are of equal $trength, but notably weaker. But, in the Meridional Regions, far di$tant from the Equinoctial, it changeth nature, and that part which to us was more weak, acquireth more $trength than the other: and all this I confer with that which we $ee to be done by a $mall piece of Magnet, in the pre$ence of a great one, the vertue of which $uperating the le$$er, maketh it to become obedient to it, and according as it is held, either on this or on that $ide the Equi- noctial of the great one, maketh the $elf $ame mutations, which I have $aid are made by every Magnet, carried on this <foot>$ide</foot> <p n=>369</p> $ide, or that $ide of the Equinoctiall of the Earth.</P> <P>SAGR. I was per$waded, at the very fir$t reading of the Book of <I>Gilbertus</I>; and having met with a mo$t excellent piece of <marg><I>|The Magnet armed takes up much more Iron, than when unar- med.</I></marg> Magnet, I, for a long time, made many Ob$ervations, and all worthy of extream wonder; but above all, that $eemeth to me very $tupendious of increa$ing the faculty of taking up Iron $o much by arming it, like as the $aid Authour teacheth; and with arming that piece of mine, I multiplied its force in octuple propor- tion; and whereas unarmed it $carce took up nine ounces of Iron, it being armed did take up above $ix pounds: And, it may be, you have $een this Load$tone in the ^{*} Gallery of your <marg>+ Or Clo$et of rarities.</marg> <I>Mo$t Serene Grand Duke</I> (to whom I pre$ented it) upholding two little Anchors of Iron.</P> <P>SALV. I $aw it many times, and with great admiration, till that a little piece of the like $tone gave me greater cau$e of won- der, that is in the keeping of our Academick, which being no more than of $ix ounces weight, and $u$taining, when unarmed, hardly two ounces, doth, when armed, take up 160. ounces, $o as that it is of 80. times more force armed than unarmed, and takes up a weight 26. times greater than its own; a much greater wonder than <I>Gilbert</I> could ever meet with, who writeth, that he could never get any Load$tone that could reach to take up four times its own weight.</P> <P>SAGR. In my opinion, this Stone offers to the wit of man a large Field to Phylo$ophate in; and I have many times thought with my $elf, how it can be that it conferreth on that Iron, which armeth it, a $trength $o $uperiour to its own; and finally, I finde nothing that giveth me $atisfaction herein; nor do I find any thing extraordinary in that which <I>Gilbert</I> writes about this parti- cular; I know not whether the $ame may have befallen you.</P> <P>SALV. I extreamly prai$e, admire, and envy this Authour, for that a conceit $o $tupendious $hould come into his minde, touching a thing handled by infinite $ublime wits, and hit upon by none of them: I think him moreover worthy of extraordi- nary applau$e for the many new and true Ob$ervations that he made, to the di$grace of $o many fabulous Authours, that write not only what they do not know, but what ever they hear $po- ken by the fooli$h vulgar, never $eeking to a$$ure them$elves of the $ame by experience, perhaps, becau$e they are unwilling to dimini$h the bulk of their Books. That which I could have de- $ired in <I>Gilbert,</I> is, that he had been a little greater Mathematici- an, and particularly well grounded in <I>Geometry,</I> the practice whereof would have rendered him le$s re$olute in accepting tho$e rea$ons for true Demon$trations, which he produceth for true <foot>Aaa cau$es</foot> <p n=>370</p> cau$es of the true conclu$ions ob$erved by him$elf. Which rea- $ons (freely $peaking) do not knit and bind $o fa$t, as tho$e un- doubtedly ought to do, in that of natural, nece$$ary, and la$ting conclu$ions may be alledged. And I doubt not, but that in pro- ce$$e of time this new Science will be perfected with new ob$er- vations, and, which is more, with true and nece$$ary Demon$tra- <marg><I>The fir$t ob$er- vers and inventers of things ought to be admired.</I></marg> tions. Nor ought the glory of the fir$t Inventor to be thereby dimini$hed, nor do I le$$e e$teem, but rather more admire, the Inventor of the Harp (although it may be $uppo$ed that the In- $trument at fir$t was but rudely framed, and more rudely finger- ed) than an hundred other Arti$ts, that in the in$uing Ages redu- ced that profe$$ion to great perfection. And methinks, that An- tiquity had very good rea$on to enumerate the fir$t Inventors of the Noble Arts among$t the Gods; $eeing that the common wits have $o little curio$ity, and are $o little regardful of rare and ele- gant things, that though they $ee and hear them exercirated by the exquifite profe$$ors of them, yet are they not thereby per- $waded to a de$ire of learning them. Now judge, whether Capa- cities of this kind would ever have attempted to have found out the making of the Harp, or the invention of Mu$ick, upon the hint of the whi$tling noi$e of the dry $inews of a Tortois, or from the $triking of four Hammers. The application to great inventions moved by $mall hints, and the thinking that under a primary and childi$h appearance admirable Arts may lie hid, is not the part of a trivial, but of a $uper-humane $pirit. Now an- $wering to your demands, I $ay, that I al$o have long thought upon what might po$$ibly be the cau$e of this $o tenacious and potent union, that we $ee to be made between the one Iron that armeth the Magnet, and the other that conjoyns it $elf unto it. <marg><I>The true cau$e of the multiplica- tion of vertue in the Magnet, by means of the ar- ming.</I></marg> And fir$t, we are certain, that the vertue and $trength of the $tone doth not augment by being armed, for it neither attracts at greater di$tance, nor doth it hold an Iron the fa$ter, if between it, and the arming or cap, a very fine paper, or a leaf of beaten gold, be interpo$ed; nay, with that interpo$ition, the naked $tone takes up more Iron than the armed. There is therefore no alte- ration in the vertue, and yet there is an innovation in the effect. <marg><I>Of a new effect its nece$$ary that the cau$e be like- wi$e new.</I></marg> And becau$e its nece$$ary, that a new effect have a new cau$e, if it be inquired what novelty is introduced in the act of taking up with the cap or arming, there is no mutation to be di$covered, but in the different contact; for whereas before Iron toucht Load- $tone, now Iron toucheth Iron. Therefore it is nece$$ary to con- clude, that the diver$ity of contacts is the cau$e of the diver$ity <marg><I>It is proved, that Iron con$ists of parts more $ub- til, pure, and com- pact than the mag- net.</I></marg> of effects. And for the difference of contacts it cannot, as I $ee, be derived from any thing el$e, $ave from that the $ub$tance of the Iron is of parts more $ubtil, more pure, and more compact- <foot>ed</foot> <p n=>371</p> ed than tho$e of the Magnet, which are more gro$$e, impure, and rare. From whence it followeth, that the $uperficies of two I- rons that are to touch, by being exqui$itely plained, filed, and burni$hed, do $o exactly conjoyn, that all the infinite points of the one meet with the infinite points of the other; $o that the filaments, if I may $o $ay, that collegate the two Irons, are many more than tho$e that collegate the Magnet to the Iron, by rea$on that the $ub$tance of the Magnet is more porous, and le$$e com- pact, which maketh that all the points and filaments of the Load- $tone do not clo$e with that which it unites unto. In the next place, that the $ub$tance of Iron (e$pecially the well refined, as namely, the pure$t $teel) is of parts much more den$e, $ubtil, and pure than the matter of the Load$tone, is $een, in that one may bring its edge to an extraordinary $harpne$$e, $uch as is that of the Ra$or, which can never be in any great mea$ure effected in a piece of Magnet. Then, as for the impurity of the Magnet, and <marg><I>A $en$ible proof of the impurity of the Magnet.</I></marg> its being mixed with other qualities of $tone, it is fir$t $en$ibly di$covered by the colour of $ome little $pots, for the mo$t part white; and next by pre$enting a needle to it, hanging in a thread, which upon tho$e $tonyne$$es cannot find repo$e, but being attracted by the parts circumfu$ed, $eemeth to fly from <marg>* The hereby me<*> that the $ton<*> doth not all con- $i$t of magnetick matter, but that the whiter $pecks being weak, tho$e other parts of the Load$tone of a more dark & con- $tant colour, con- tain all that vertue wherewith bodies are attracted.</marg> ^{*} <I>tho$e,</I> and to leap upon the Magnet contiguous to <I>them:</I> and as $ome of tho$e Heterogeneal parts are for their magnitude ve- ry vi$ible, $o we may believe, that there are others, in great a- bundance, which, for their $mallne$$e, are imperceptible, that are di$$eminated throughout the whole ma$$e. That which I $ay, (namely, that the multitude of contacts that are made between Iron and Iron, is the cau$e of the $o $olid conjunction) is con- firmed by an experiment, which is this, that if we pre$ent the $harpned point of a needle to the cap of a Magnet, it will $tick no fa$ter to it, than to the $ame $tone unarmed: which can proceed from no other cau$e, than from the equality of the con- tacts that are both of one $ole point. But what then? Let a ^{*} Needle be taken and placed upon a Magnet, $o that one of its <marg>* A common $ewing needle.</marg> extremities hang $omewhat over, and to that pre$ent a Nail; to which the Needle will in$tantly cleave, in$omuch that withdraw- ing the Nail, the Needle will $tand in $u$pen$e, and with its two ends touching the Magnet and the Iron; and withdrawing the Nail yet a little further, the Needle will for$ake the Magnet; provided that the eye of the Needle be towards the Nail, and the point towards the Magnet; but if the eye be towards the Load$tone, in withdrawing the Nail the Needle will cleave to the Magnet; and this, in my judgment, for no other rea$on, $ave onely that the Needle, by rea$on it is bigger towards the eye, toucheth in much more points than its $harp point doth.</P> <foot>Aaa 2 SAGR.</foot> <p n=>372</p> <P>SAGR. Your whole di$cour$e hath been in my judgment very concluding, and this experiment of the Needle hath made me think it little inferiour to a Mathematical Demon$tration; and I ingenuou$ly confe$$e, that in all the Magnetick Philo$ophy, I never heard or read any thing, that with $uch $trong rea$ons gave account of its $o many admirable accidents, of which, if the cau$es were with the $ame per$picuity laid open, I know not what $weeter food our Intellects could de$ire.</P> <P>SALV. In $eeking the rea$ons of conclu$ions unknown unto us, it is requi$ite to have the good fortune to direct the di$- cour$e from the very beginning towards the way of truth; in which if any one walk, it will ea$ily happen, that one $hall meet with $everal other Propo$itions known to be true, either by di$- putes or experiments, from the certainty of which the truth of ours acquireth $trength and evidence; as it did in every re$pect happen to me in the pre$ent Probleme, for being de$irous to a$- $ure my $elf, by $ome other accident, whether the rea$on of the Propo$ition, by me found, were true; namely, whether the $ub- $tance of the Magnet were really much le$$e continuate than that of Iron or of Steel, I made the Arti$ts that work in the Gallery of my Lord the Grand Duke, to $mooth one $ide of that piece of Magnet, which formerly was yours, and then to poli$h and burni$h it; upon which to my $atisfaction I found what I de$ired. For I di$covered many $pecks of colour different from the re$t, but as $plendid and bright, as any of the harder $ort of $tones; the re$t of the Magnet was polite, but to the tact onely, not being in the lea$t $plendid; but rather as if it were $meered over with $oot; and this was the $ub$tance of the Load $tone, and the $hining part was the fragments of other $tones intermixt therewith, as was $en$ibly made known by pre$enting the face thereof to filings of Iron, the which in great number leapt to the Load-$tone, but not $o much as one grain did $tick to the $aid $pots, which were many, $ome as big as the fourth part of the nail of a mans finger, others $omewhat le$$er, the lea$t of all very many, and tho$e that were $carce vi$ible almo$t innu- merable. So that I did a$$ure my $elf, that my conjecture was true, when I fir$t thought that the $ub$tance of the Magnet was not clo$e and compact, but porous, or to $ay better, $pon- gy; but with this difference, that whereas the $ponge in its cavities and little cels conteineth Air or Water, the Magnet hath its pores full of hard and heavy $tone, as appears by the exqui- $ite lu$tre which tho$e $pecks receive. Whereupon, as I have $aid from the beginning, applying the $urface of the Iron to the $u- perficies of the Magnet the minute particles of the Iron, though perhaps more continuate than the$e of any other body (as its <foot>$hining</foot> <p n=>373</p> $hining more than any other matter doth $hew) do not all, nay but very few of them incounter pure Magnet; and the contacts being few, the union is but weak. But becau$e the cap of the Load-$tone, be$ides the contact of a great part of its $uperficies, inve$ts its $elf al$o with the virtue of the parts adjoyning, al- though they touch not; that $ide of it being exactly $moothed to which the other face, in like manner well poli$ht of the Iron to be attracted, is applyed, the contact is made by innumera- ble minute particles, if not haply by the infinite points of both the $uperficies, whereupon the union becometh very $trong. This ob$ervation of $moothing the $urfaces of the Irons that are to touch, came not into the thoughts of <I>Gilbert,</I> for he makes the Irons convex, $o that their contact is very $mall; and there- upon it cometh to pa$$e that the tenacity, wherewith tho$e Irons conjoyn, is much le$$er.</P> <P>SAGR. I am, as I told you before, little le$$e $atisfied with this rea$on, that if it were a pure Geometrical Demon$tration; and becau$e we $peak of a Phy$ical Problem, I believe that al$o <I>Simplicius</I> will find him$elf $atisfied as far as natural $cience ad- mits, in which he knows that Geometrical evidence is not to be required.</P> <P>SIMP. I think indeed, that <I>Salviatus</I> with a fine circumlo- <marg>Sympathy <I>and</I> Antipathy, <I>terms u$ed by Philo$o- phers to give a rea- $on ea$ily of ma- ny narural effests.</I></marg> cution hath $o manife$tly di$played the cau$e of this effect, that any indifferent wit, though not ver$t in the Sciences, may ap- prehend the $ame; but we, confining our $elves to the terms of Art, reduce the cau$e of the$e and other the like natural effects to <I>Sympathy,</I> which is a certain agreement and mutual appetite which ari$eth between things that are $emblable to one another in qualities; as likewi$e on the contrary that hatred & enmity for which other things $hun & abhor one another we call <I>Antipathy.</I></P> <P>SAGR. And thus with the$e two words men come to render rea$ons of a great number of accidents and effects which we $ee not without admiration to be produced in nature. But this kind of philo$ophating $eems to me to have great $ympathy with a <marg><I>A plea$ant ex- ampleaeclaring the invalidity of $ome Phylo$ophical ar- gumentations.</I></marg> certain way of Painting that a Friend of mine u$ed, who writ upon the <I>Tele</I> or Canva$$e in chalk, here I will have the Foun- tain with <I>Diana</I> and her Nimphs, there certain Hariers, in this corner I will have a Hunt$-man with the Head of a Stag, the re$t $hall be Lanes, Woods, and Hills; and left the remainder for the Painter to $et forth with Colours; and thus he per$waded him$elf that he had painted the Story of <I>Acteon,</I> when as he had contributed thereto nothing of his own more than the names. But whether are we wandred with $o long a digre$$ion, contrary to our former re$olutions? I have almo$t forgot what the point was that we were upon when we fell into this magnetick di$- <foot>cour$e;</foot> <p n=>374</p> cour$e; and yet I had $omething in my mind that I intended to have $poken upon that $ubject.</P> <P>SALV. We were about to demon$trate that third motion a- $cribed by <I>Copernicus</I> to the Earth to be no motion but a quie- $cence and maintaining of it $elf immutably directed with its de- terminate parts towards the $ame & determinate parts of the Uni- ver$e, that is a perpetual con$ervation of the Axis of its diurnal revolution parallel to it $elf, and looking towards $uch and $uch fixed $tars; which mo$t con$tant po$ition we $aid did naturally agree with every librated body $u$pended in a fluid and yielding <I>medium,</I> which although carried about, yet did it not change di- rectionin re$pect of things external, but onely $eemed to revolve in its $elf, in re$pect of that which carryed it round, and to the ve$$el in which it was tran$ported. And then we added to this $imple and natural accident the magnetick virtue, whereby the $elf Terre$trial Globe might $o much the more con$tantly keep it immutable, -----</P> <P>SAGR. Now I remember the whole bu$ine$$e; and that which then came into my minde, & which I would have intimated, was a certain con$ideration touching the $cruple and objection of <I>Sim- plicius,</I> which he propounded again$t the mobility of the Earth, <marg><I>The $everal na- tural motions of the Magnet.</I></marg> taken from the multiplicity of motions, impo$$ible to be a$$igned to a $imple body, of which but one $ole and $imple motion, ac- cording to the doctrine of <I>Ari$totle,</I> can be natural; and that which I would have propo$ed to con$ideration, was the Magnet, to which we manife$tly $ee three motions naturally to agree: one towards the centre of the Earth, as a <I>Grave</I>; the $econd is the circular Horizontal Motion, whereby it re$tores and con- $erves its Axis towards determinate parts of the Univer$e; and the third is this, newly di$covered by <I>Gilbert,</I> of inclining its Axis, being in the plane of a Meridian towards the $urface of the Earth, and this more and le$$e, according as it $hall be di$tant from the Equinoctial, under which it is parallel to the Axis of the Earth. Be$ides the$e three, it is not perhaps improbable, but that it may have a fourth, of revolving upon its own Axis, in ca$e it were librated and $u$pended in the air or other fluid and yielding <I>Medium,</I> $o that all external and accidental impediments were removed, and this opinion <I>Gilbert</I> him$elf $eemeth al$o to applaud. So that, <I>Simplicius,</I> you $ee how tottering the Axiome of <I>Ari$totle</I> is.</P> <P>SIMP. This doth uot only not make again$t the Maxime, but not $o much as look towards it: for that he $peaketh of a fimple body, and of that which may naturally con$i$t therewith; but you propo$e that which befalleth a mixt body; nor do you tell us of any thing that is new to the doctrine of <I>Ari$totle,</I> for that <foot>he</foot> <p n=>375</p> he likewi$e granteth to mixt bodies compound motions by -----</P> <P>SAGR. Stay a little, <I>Simplicius,</I> & an$wer me to the que$tions I $hall ask you. You $ay that the Load-$tone is no $imple body, <marg>Ari$tole <I>grants a compound motion to mixt bodies.</I></marg> now I defire you to tell me what tho$e $imple bodies are, that mingle in compo$ing the Load-$tone.</P> <P>SIMP. I know not how to tell you th'ingredients nor $imples preci$ely, but it $ufficeth that they are things elementary.</P> <P>SALV. So much $ufficeth me al$o. And of the$e $imple ele- mentary bodies, what are the natural motions?</P> <P>SIMP. They are the two right and $imple motions, <I>$ur$um</I> and <I>deor$um.</I></P> <P>SAGR. Tell me in the next place? Do you believe that the motion, that $hall remain natural to that $ame mixed body, $hould be one that may re$ult from the compo$ition of the two $imple natural motions of the $imple bodies compounding, or that it may be a motion impo$$ible to be compo$ed of them.</P> <marg><I>The motion of mixt bodies ought to be $uch as may re$ult from the compo$ition of the motions of the $im- ple bodies com- pounding.</I></marg> <P>SIMP. I believe that it $hall move with the motion re$ulting from the compo$ition of the motions of the $imple bodies com- pounding, and that with a motion impo$$ible to be compo$ed of the$e, it is impo$$ible that it $hould move.</P> <P>SAGR. But, <I>Simplicius,</I> with two right and $imple motions, you $hall never be able to compo$e a circular motion, $uch as are the <marg><I>With two right motions one cannot compo$e circular motions.</I></marg> two, or three circular motions that the magnet hath: you $ee then into what ab$urdities evil grounded Principles, or, to $ay <marg><I>Philo$ophers are forced to confe$$e that the Magnet is compounded of cœle$tial $ub$tan- ces, and of elemen- tary.</I></marg> better, the ill-inferred con$equences of good Principles carry a man; for you are now forced to $ay, that the Magnet is a mix- ture compounded of $ub$tances elementary and cœle$tial, if you will maintain that the $traight motion is a peculiar to the Ele- ments, and the circular to the cœle$tial bodies. Therefore if you will more $afely argue, you mu$t $ay, that of the integral bodies of the Univer$e, tho$e that are by nature moveable, do all move circularly, and that therefore the Magnet, as a part of the <marg><I>The errour of tho$e who call the Magnet a mixt body, and the ter- re$trial Globe $imble body.</I></marg> true primary, and integral $ub$tance of our Globe, pertaketh of the $ame qualities with it. And take notice of this your fallacy, in calling the Magnet a mixt body, and the Terre$trial Globe a $imple body, which is $en$ibly perceived to be a thou$and times more compound: for, be$ides that it containeth an hundred an hundred matters, exceeding different from one another, it con- taineth great abundance of this which you call mixt, I mean of the Load-$tone. This $eems to me ju$t as if one $hould call <marg>* Ogliopotrida <I>a Spani$h di$h of many ingredients boild together.</I></marg> bread a mixt body, and ^{*} <I>Pannada</I> a $imple body, in which there is put no $mall quantity of bread, be$ides many other things edi- ble. This $eemeth to me a very admirable thing, among$t others <foot>of</foot> <p n=>376</p> <marg><I>The Di$cour$es of Peripateticks, full of errours and contradictions.</I></marg> of the Peripateticks, who grant (nor can it be denied) that our Terre$trial Globe is, <I>de facto,</I> a compound of infinite different matters; and grant farther that of compound bodies the motion ought to be compound: now the motions that admit of compo- $ition are the right and circular: For the two right motions, as being contrary, are incompatible together, they affirm, that the pure Element of Earth is no where to be found; they confe$$e, that it never hath been moved with a local motion; and yet they will introduce in Nature that body which is not to be found, and make it move with that motion which it never exerci$ed, nor ne- ver $hall do, and to that body which hath, and ever had a being, they deny that motion, which before they granted, ought natu- rally to agree therewith.</P> <P>SALV. I be$eech you, <I>Sagredus,</I> let us not weary our $elves any more about the$e particulars, and the rather, becau$e you know that our purpo$e was not to determine re$olutely, or to accept for true, this or that opinion, but only to propo$e for our diverti$ement $uch rea$ons, and an$wers as may be alledged on the one $ide, or on the other; and <I>Simplicius</I> maketh this an- $wer, in defence of his Peripateticks, therefore let us leave the judgment in $u$pen$e, and remit the determination into the hands of $uch as are more known than we. And becau$e I think that we have, with $ufficient prolixity, in the$e three dayes, di$- cour$ed upon the Sy$teme of the Univer$e, it will now be $ea$o- nable, that we proceed to the grand accident, from whence our Di$putations took beginning, I mean, of the ebbing and flowing of the Sea, the cau$e whereof may, in all probability, be referred to the motion of the Earth. But that, if you $o plea$e, we will re$erve till to morrow. In the mean time, that I may not forget it, I will $peak to one particular, to which I could have wi$hed, that <I>Gilbert</I> had not lent an ear; I mean that of admitting, that <marg><I>An improba- ble effect admired by</I> Gilbertus <I>in the Load$tone.</I></marg> in ca$e a little Sphere of Load$tone might be exactly librated, it would revolve in it $elf; becau$e there is no rea$on why it $hould do $o; For if the whole Terre$trial Globe hath a natural facul- ty of revolving about its own centre in twenty four hours, and that all its parts ought to have the $ame, I mean, that faculty of turning round together with their <I>whole,</I> about its centre in twen- ty four hours; they already have the $ame in effect, whil$t that, being upon the Earth, they turn round along with it: And the a$$igning them a revolution about their particular centres, would be to a$cribe unto them a $econd motion much different from the fir$t; for $o they would have two, namely, the revolving in twen- ty four hours about the centre of their <I>whole</I>; and the turning about their own: now this $econd is arbitrary, nor is there any <foot>rea-</foot> <p n=>377</p> rea$on for the introducing of it: If by pluoking away a piece of Load$tone from the whole natural ma$$d, it were deprived of the faculty of following it, as it did, whil$t it was unitedy thereto, $o that it is thereby deprived of the revodution about the univer- $al centre of the Terre$trial Globe, it might Chaply, with $ome- what greater probability be thought by $ome, that the $aid Mag- net was to appropriate to it $elf a new conver$ion about its parti- cular centre; but if it do no le$$e, when $eparated, than when conjoyned, continue always to pur$ue its fir$t, eternal, and natu- ral cour$e, to what purpo$e $hould we go about to obtrude upon it another new one?</P> <P>SAGR. I under$tand you very well, and this puts me in mind of a Di$cour$e very like to this for the vanity of it, falling from <marg><I>The vain <*> mentation of $ome to prove the Ele- ment of Water to be of a Spherical $uper ficies.</I></marg> certain Writers upon the Sphere, and I think, if I well remem- ber, among$t others from <I>Sacrobo$co,</I> who, to $hew how the E- lement of Water, doth, together with the Earth, make a com- pleat Spherical Figure, and $o between them both compo$e this our Globe, writeth, that the $eeing the $mall ^{*} particles of water $hape them$elves into rotundity, as in the drops, and in the dew daily apparent upon the leaves of $everal herbs, is a $trong ar- gument; and becau$e, according to the trite Axiome, there is the $ame rea$on for the whole, as for the parts, the parts affecting that $ame figure, it is nece$$ary that the $ame is proper to the whole Element: and truth is, methinks it is a great over$ight that the$e men $hould not perceive $o apparent a vanity, and con- $ider that if their argument had run right, it would have follow- ed, that not only the $mall drops, but that any what$oever greater quantity of water $eparated from the whole Element, $hould be re- duced into a Globe: Which is not $een to happen; though indeed the Sen$es may $ee, and the Under$tanding perceive that the E- lement of Water loving to form it $elf into a Spherical Figure about the common centre of gravity, to which all grave bo- dies tend (that is, the centre of the Terre$trial Globe) it therein is followed by all its parts, according to the Axiome; $o that all the $urfaces of Seas, Lakes, Pools, and in a word, of all the parts of Waters conteined in ve$$els, di$tend them$elves into a Spherical Figure, but that Figure is an arch of that Sphere that hath for its centre the centre of the Ter- re$trial Globe, and do not make particular Spheres of them- $elves.</P> <P>SALV. The errour indeed is childi$h; and if it had been onely the $ingle mi$take of <I>Sacrobo$co,</I> I would ea$ily have allowed him in it; but to pardon it al$o to his Com- mentators, and to other famous men, and even to <I>Ptolomy</I> <foot>Bbb him-</foot> <p n=>378</p> him$elfe; this I cannot do, without blu$hing for their repu- tation. But it is high time to take leave, it row being very late, and we being to meet again to morrow, at the u$ual hour, to bring all the foregoing Di$cour$es to a final conclu$ion.</P> <fig> <pb> <fig> <cap><I>Place this Plate at the end of the third</I></cap> <cap>Dialogue</cap> <p n=>379</p> <head>GALILÆUS Gailæus Lyncæus, HIS SYSTEME OF THE WORLD.</head> <head>The Fourth Dialogue.</head> <head><I>INTERLOCVTORS.</I></head> <head>SALVIATUS, SAGREDUS, & SIMPLICIUS.</head> <P>SAGR. I know not whether your return to our accu$tomed conferences hath really been later than u$ual, or whether the de$ire of hearing the thoughts of <I>Salviatus,</I> touching a matter $o curious, hath made me think it $o: But I have tar- ried a long hour at this window, expe- cting every moment when the <I>Gondola</I> would appear that I $ent to fetch you.</P> <P>SALV. I verily believe that your imagination more than our tarriance hath prolonged the time: and to make no longer de- <marg><I>Nature in $port maketh the ebbing and flowing of the Sea, to approve the Earths mobility.</I></marg> murre, it would be well, if without interpo$ing more words, we came to the matter it $elf; and did $hew, that nature hath per- mitted (whether the bu$ine$s <I>in rei veritate</I> be $o, or el$e to play <foot>Bbb 2 and</foot> <p n=>380</p> and $port with our Fancies) hath, I $ay, hath permitted that the <marg><I>The tide, and mobility of the Earth mutually confirm each other</I></marg> motions for every other re$pect, except to re$olve the ebbing and flowing of the Sea, a$$igned long $ince to the earth, $hould be found now at la$t to an$wer exactly to the cau$e thereof; and, as it <marg><I>All terrene ef- fects, indifferently confirm the motion or re$t of the Earth, except the ebbing and flowing of the Sea.</I></marg> were, with mutual a emulation, the $aid ebbing and flowing to appear in confirmation of the Terre$trial motion: the <I>judices</I> whereof have hitherto been taken from the cœle$tial Phænomena, in regard that of tho$e things that happen on Earth, not any one was of force to prove one opinion more than another, as we al- ready have at large proved, by $hewing that all the terrene occur- rences upon which the $tability of the Earth and mobility of the Sun and Firmament is commonly inferred, are to $eem to us per- formed in the $ame manner, though we $uppo$ed the mobility of the Earth, and the immobility of them. The Element of Wa- ter onely, as being mo$t va$t, and which is not annexed and con- catenated to the Terre$trial Globe as all its other $olid parts are; yea, rather which by rea$on of its fluidity remaineth apart <I>$ui juris,</I> and free, is to be ranked among$t tho$e $ublunary things, from which we may collect $ome hinte and intimation of what the Earth doth in relation to motion and re$t. After I had many and many a time examined with my $elf the effects and accidents, partly $een and partly under$tood from others, thar are to be ob- $erved in the motions of waters: and moreover read and heard the great vanities produced by many, as the cau$es of tho$e acci- dents, I have been induced upon no $light rea$ons to omit the$e <marg><I>The fir$t gene- ral conclu$ion of the impo$$ibility of the ebbing and flowing the immo- bility of the terre- $trial Globe being granted.</I></marg> two conclu$ions (having made withal the nece$$ary pre$uppo- $als) that in ca$e the terre$trial Globe be immoveable, the flux and reflux of the Sea cannot be natural; and that, in ca$e tho$e motions be conferred upon the $aid Globe, which have been long $ince a$$igned to it, it is nece$$ary that the Sea be $ubject to eb- bing and flowing, according to all that which we ob$erve to hap- pen in the $ame.</P> <P>SAGR. The Propo$ition is very con$iderable, as well for it $elf as for what followeth upon the $ame by way of con$equence, $o that I $hall the more inten$ly hearken to the explanation and confirmation of it.</P> <marg><I>The knowledge of the offests con- tributes to the in- ve$tigation of the cau$es.</I></marg> <P>SALV. Becau$e in natural que$tions, of which number this which we have in hand is one, the knowledge of the effects is a means to guide us to the inve$tigation and di$covery of the cau- $es, and without which we $hould walk in the dark, nay with more uncertainty, for that we know not whither we would go, whereas the blind, at lea$t, know where they de$ire to arrive; there- fore fir$t of all it is nece$$ary to know the effects whereof we en- quire the cau$es: of which effects you, <I>Sagredus,</I> ought more abundantly and more certainly to be informed than I am, <foot>as</foot> <p n=>381</p> as one, that be$ides your being born, and having, for a long time, dwelt in <I>Venice,</I> where the Tides are very notable for their greatne$$e, have al$o $ailed into <I>Syria,</I> and, as an ingenuous and apprehen$ive wit, mu$t needs have made many Ob$ervations up- on this $ubject: whereas I, that could onely for a time, and that very $hort, ob$erve what happened in the$e extream parts of the <I>Adriatick</I> Gulph, and in our Seas below about the <I>Tyrrhene</I> $hores, mu$t needs take many things upon the relation of o- thers, who, for the mo$t part, not very well agreeing, and con- $equently being very uncertain, contribute more of confu$ion than confirmation to our $peculations. Neverthele$$e, from tho$e that we are $ure of, and which are the principal, I think I am a- ble to attain to the true and primary cau$es; not that I pretend to be able to produce all the proper and adequate rea$ons of tho$e effects that are new unto me, and which con$equently I could never have thought upon. And that which I have to $ay, I propo$e only, as a key that openeth the door to a path never yet trodden by any, in certain hope, that $ome wits more $pecu- lative than mine, will make a further progre$$e herin, and pene- trate much farther than I $hall have done in this my fir$t Di$co- very: And although that in other Seas, remote from us, there may happen $everal accidents, which do not happen in our Mediter- ranean Sea, yet doth not this invalidate the rea$on and cau$e that I $hall produce, if $o be that it veri$ie and fully re$olve the ac- cidents which evene in our Sea: for that in conclu$ion there can be but one true and primary cau$e of the effects that are of the $ame kind. I will relate unto you, therefore, the effects that I know to be true, and a$$igne the cau$es thereof that I think to be true, and you al$o, Gentlemen, $hall produce $uch others as are known to you, be$ides mine, and then we will try whether the cau$e, by me alledged, may $atisfie them al$o.</P> <marg><I>Three Periods of ebbings and flowings, diurnal, monethly, and an- nual.</I></marg> <P>I therefore affirm the periods that are ob$erved in the fluxes and refluxes of the Sea-waters to be three: the fir$t and princi- pal is this great and mo$t obvious one; namely, the diurnal, accor- ding to which the intervals of $ome hours with the waters flow and ebbe; and the$e intervals are, for the mo$t part, in the Mediter- rane from $ix hours to $ix hours, or thereabouts, that is, they for $ix hours flow, and for $ix hours ebbe. The $econd period is monethly, and it $eemes to take its origen from the motion of the Moon, not that it introduceth other motions, but only al- tereth the greatne$$e of tho$e before mentioned, with a notable difference, according as it $hall wax or wane, or come to the Quadrature with the Sun. The third Period is annual, and is $een to depend on the Sunne, and onely altereth the diurnal <foot>motions,</foot> <p n=>382</p> motions, by making them different in the times of the Sol- $tices, as to greatne$$e, from what they are in the Equinoxes.</P> <P>We will $peak (in the fir$t place, of the diurnal motion, as being the principal, and upon which the Moon and Sun $eem to exerci$e their power $econdarily, in their monethly and annual <marg><I>Varieties that happen in the diur- nal period.</I></marg> alterations. Three differences are ob$ervable in the$e horary mutations; for in $ome places the waters ri$e and fall, without making any progre$$ive motion; in others, without ri$ing or fal- ling they run one while towards the Ea$t, and recur another while towards the We$t; and in others they vary the heights and cour$e al$o, as happeneth here in <I>Venice,</I> where the Tides in coming in ri$e, and in going out fall; and this they do in the ex- termities of the lengths of Gulphs that di$tend from We$t to Ea$t, and terminate in open $hores, up along which $hores the Tide at time of flood hath room to extend it $elf: but if the courfe of the Tide were iutercepted by Cliffes and Banks of great height and $teepne$$e, there it will flow and ebbe without any progre$$ive motion. Again, it runs to and again, without changing height in the middle parts of the Mediterrane, as nota- <marg>* A Strait, $o called.</marg> bly happeneth in the ^{*} <I>Faro de Me$$ina,</I> between <I>Scylla</I> and <I>Ca- rybdis,</I> where the Currents, by rea$on of the narrowne$$e of the Channel, are very $wift; but in the more open Seas, and about the I$les that $tand farther into the Mediterranean Sea, as <marg>* Or Ilva.</marg> the <I>Baleares, Cor$ica, Sardignia, ^{*} Elba, Sicily</I> towards the <I>Affrican</I> <marg>* Or Creta.</marg> Coa$ts, <I>Malta, ^{*} Candia, &c.</I> the changes of watermark are very $mall; but the currents indeed are very notable, and e$pe- cially when the Sea is pent between I$lands, or between them and the Continent.</P> <P>Now the$e onely true and certain effects, were there no more to be ob$erved, do, in my judgment, very probably per$wade any man, that will contain him$elf within the bounds of natu- ral cau$es, to grant the mobility of the Earth: for to make the ve$$el (as it may be called) of the Mediterrane $tand $till, and to make the water contained therein to do, as it doth, exceeds my imagination, and perhaps every mans el$e, who will but pierce beyond the rinde in the$e kind of inquiries.</P> <P>SIMP. The$e accidents, <I>Salviatus,</I> begin not now, they are mo$t ancient, and have been ob$erved by very many, and $everal have attempted to a$$igne, $ome one, $ome another cau$e for the $ame: and there dwelleth not many miles from hence a famous Peripatetick, that alledgeth a cau$e for the $ame newly fi$hed out <marg><I>The cau$e of the abbing and flowing alledged by a cer- tain modern Phi- lo$opher.</I></marg> of a certain Text of <I>Ari$totle,</I> not well under$tood by his Ex- po$itors, from which Text he collecteth, that the true cau$e of the$e motions doth only proceed from the different profundities of Seas: for that the waters of greate$t depth being greater in <foot>abun-</foot> <p n=>383</p> abundance, and therefore more grave, drive back the Waters of le$$e depth, which being afterwards rai$ed, de$ire to de- $cend, and from this continual colluctation or conte$t proceeds <marg><I>The cau$e of the ebbing and flowing a$cribed to the Moon by a certain Prelate.</I></marg> the ebbing and flowing. Again tho$e that referre the $ame to the Moon are many, $aying that $he hath particular Dominion over the Water; and at la$t a certain Prelate hath publi$hed a little Treati$e, wher in he $aith that the Moon wandering too and fro in the Heavens attracteth and draweth towards it a Ma$$e of Water, which goeth continually following it, $o that it is full Sea alwayes in that part which lyeth under the Moon; and becau$e, that though $he be under the Horizon, yet neverthele$$e the Tide returneth, he $aith that no more can be $aid for the $alving of that particular, $ave onely, that the Moon doth not onely naturally retain this faculty in her $elf; but in this ca$e hath power to con- fer it upon that degree of the Zodiack that is oppo$ite unto it. Others, as I believe you know, do $ay that the Moon is able <marg>Hieronymus Bor- rius <I>and other</I> Pe- ripateticks <I>refer it to the temperate heat of the Moon.</I></marg> with her temperate heat to rarefie the Water, which being ra- refied, doth thereupon flow. Nor hath there been wanting $ome that ----</P> <P>SAGR. I pray you <I>Simplicius</I> let us hear no more of them, for I do not think it is worth the while to wa$t time in relating them, or to $pend our breath in confuting them; and for your part, if you gave your a$$ent to any of the$e or the like foole- ries, you did a great injury to your judgment, which neverthe- le$$e I acknowledg to be very piercing.</P> <P>SALV. But I that am a little more flegmatick than you, <I>Sagre-</I> <marg><I>An$wers to the vanities alledged as cau$es of the eb- bing and flowing.</I></marg> <I>dus,</I> will $pend a few words in favour of <I>Simplicius,</I> if haply he thinks that any probability is to be found in tho$e things that he hath related. I $ay therefore: The Waters, <I>Simplicius,</I> that have their exteriour $uperficies higher, repel tho$e that are infe- riour to them, and lower; but $o do not tho$e Waters that are of greate$t profundity; and the higher having once driven back the lower, they in a $hort time grow quiet and ^{*} level. This <marg>+ Or rather $mooth.</marg> your <I>Peripatetick</I> mu$t needs be of an opinion, that all the Lakes in the World that are in a calme, and that all the Seas where the ebbing and flowing is in$en$ible, are level in their bottoms; but I was $o $imple, that I per$waded my $elf that had we no o- <marg><I>The I$les are to- kens of the une- venne$$e of the bottomes of Seas.</I></marg> ther plummet to $ound with, the I$les that advance $o high a- bove Water, had been a $ufficient evidence of the unevenne$$e of their bottomes. To that Prelate I could $ay that the Moon runneth every day along the whole Mediterrane, and yet its Waters do not ri$e thereupon $ave onely in the very extream bounds of it Ea$tward, and here to us at <I>Venice.</I> And for tho$e that make the Moons temperate heat able to make the Water $well, bid them put fire under a Kettle full of Water, and hold <foot>their</foot> <p n=>384</p> their right hand therein till that the Water by rea$on of the heat do ri$e but one $ole inch, and then let them take it out, and write off the tumefaction of the Sea. Or at lea$t de$ire them to $hew you how the Moon doth to rarefie a certain part of the Waters, and not the remainder; as for in$tance, the$e here of <I>Venice,</I> and not tho$e of <I>Ancona, Naples, Genova</I>: the truth is <marg><I>Poetick wits of two kinds.</I></marg> Poetick Wits are of two kinds, $ome are ready and apt to invent Fables, and others di$po$ed and inclined to believe them.</P> <P>SIMP. I believe that no man believeth Fables, $o long as he knows them to be $o; and of the opinions concerning the cau$es of ebbing and flowing, which are many, becau$e I know that of one $ingle effect there is but one $ingle cau$e that is true and pri- mary, I under$tand very well, and am certain that but one alone at the mo$t can be true, and for all the re$t I am $ure that they are fabulous, and fal$e; and its po$$ible that the true one may not be among tho$e that have been hitherto produced; nay I verily be- lieve that it is not, for it would be very $trange that the truth <marg><I>Truth hath not $o little light as not to be di$cover- ed amid$t the um- brages of fal- $hoods.</I></marg> $hould have $o little light, as that it $hould not be vi$ible among$t the umbrages of $o many fal$hoods. But this I $hall $ay with the liberty that is permitted among$t us, that the introduction of the Earths motion, and the making it the cau$e of the ebbing and flowing of Tides, $eemeth to me as yet a conjecture no le$$e fa- bulous than the re$t of tho$e that I have heard; and if there $hould not be propo$ed to me rea$ons more conformable to natu- ral matters, I would without any more ado proceed to believe this to be a $upernatural effect, and therefore miraculous, and un$earchable to the under$tandings of men, as infinite others there are, that immediately depend on the Omnipotent hand of God.</P> <marg>Ari$totle <I>holdeth tho$e effects to be miraculous, of which the cau$es are unknown.</I></marg> <P>SAGR. You argue very prudently, and according to the Doctrine of <I>Ari$totle,</I> who you know in the beginning of his mechanical que$tions referreth tho$e things to a Miracle, the cau$es whereof are occult. But that the cau$e of the ebbing and flowing is one of tho$e that are not to be found out, I believe you have no greater proof than onely that you $ee, that among$t all tho$e that have hitherto been produced for true cau$es there- of, there is not one wherewith, working by what artifice you will, we are able to repre$ent $uch an effect; in regard that nei- ther with the light of the Moon nor of the Sun, nor with temperate heats, nor with different profundities, $hall one ever artificially make the Water conteined in an immoveable Ve$$el to run one way or another, and to ebbe and flow in one place, and not in another. But if without any other artifice, but with the onely moving of the Ve$$el, I am able punctually to repre- $ent all tho$e mutations that are ob$erved in the Sea Water, why will you refu$e this rea$on and run to a Miracle?</P> <foot>SIMP</foot> <p n=>385</p> <P>SIMP. I will run to a Miracle $till, if you do not with $ome other natural cau$es, be$ides that of the motion of the Ve$$els of the Sea-water di$$wade me from it; for I know that tho$e Ve$$els move not, in regard that all the entire Terre$trial Globe is natu- rally immoveable.</P> <P>SALV. But do not you think, that the Terre$trial Globe might $upernaturally, that is, by the ab$olute power of God, be made moveable? SIMP. Who doubts it?</P> <P>SALV. Then <I>Simplicius,</I> $eeing that to make the flux and reflux of the Sea, it is nece$$ary to introduce a Miracle, let us $uppo$e the Earth to move miraculou$ly, upon the motion of which the Sea moveth naturally: and this effect $hall be al$o the more $imple, and I may $ay natural, among$t the miraculous o- perations, in that the making a Globe to move round, of which kind we $ee many others to move, is le$$e difficult than to make an immen$e ma$$e of water go forwards and backwards, in one place more $wiftly, and in another le$$e, and to ri$e and fall in $ome places more; in $ome le$$e, and in $ome not at all: and to work all the$e different effects in one and the $ame Ve$$el that containeth it: be$ides, that the$e are $everal Miracles, and that is but one onely. And here it may be added, that the Miracle of making the water to move is accompanied with another, namely, the holding of the Earth $tedfa$t again$t impetuosities of the water, able to make it $wage $ometimes one way, and $ometimes another, if it were not miraculou$ly kept to rights.</P> <P>SAGR. Good <I>Simplicius,</I> let us for the pre$ent $u$pend our judgement about $entencing the new opinion to be vain that <I>Sal- viatus</I> is about to explicate unto us, nor let us $o ha$tily flye out into pa$$ion like the $colding overgrown Haggs: and as for the Miracle, we may as well recurre to it when we have done hea- ring the Di$cour$es contained within the bounds of natural cau- $es: though to $peak freely, all the Works of nature, or rather of God, are in my judgement miraculous.</P> <P>SALV. And I am of the $ame opinion; nor doth my $aying, that the motion of the Earth is the Natural cau$e of the ebbing and flowing, hinder, but that the $aid motion of the Earth may be miraculous. Now rea$$uming our Argument, I apply, and once again affirm, that it hath been hitherto unknown how it might be that the Waters contained in our Mediterranean Straights $hould make tho$e motions, as we $ee it doth, if $o be the $aid Straight, or containing Ve$$el were immoveable. And that which makes the difficulty, and rendreth this matter inextri- cable, are the things which I am about to $peak of, and which are daily ob$erved. Therefore lend me your attention.</P> <P>We are here in <I>Venice,</I> where at this time the Waters are low, <foot>Ccc the</foot> <p n=>386</p> <marg><I>It is proved impo$$ible that there $hould natu- rally be any ebbing and flowing, the Earth being im- moveable.</I></marg> the Sea calm, the Air tranquil; $uppo$e it to be young flood, and that in the term of five or $ix hours the water do ri$e ten ^{*} hand breadths and more; that ri$e is not made by the fir$t water, which was $aid to be rarefied, but it is done by the acce$- $ion of new Water: Water of the $ame $ort with the former, <marg>* Palms.</marg> of the $ame bracki$hne$s, of the $ame den$ity, of the $ame weight: Ships, <I>Simplicius,</I> float therein as in the former, with- out drawing an hairs breadth more water; a Barrel of this $econd doth not weigh one $ingle grain more or le$s than $uch another quantity of the other, and retaineth the $ame coldne$s without the lea$t alteration: And it is, in a word, Water newly and vi$i- <marg>+ <I>Lio</I> is a fair Port in the Vene- tian Gulph, lying N. E. from the City.</marg> bly entred by the Channels and Mouth of the ^{*} <I>Lio.</I> Con$ider now, how and from whence it came thither. Are there happly hereabouts any Gulphs or Whirle pools in the bottom of the Sea, by which the Earth drinketh in and $pueth out the Water, breathing as it were a great and mon$truous Whale? But if this be $o, how comes it that the Water doth not flow in the $pace of $ix hours in <I>Ancona,</I> in ^{*} <I>Ragu$a,</I> in <I>Corfu,</I> where the Tide is ve- ry $mall, and happly unob$ervable? Who will invent a way to pour new Water into an immoveable Ve$$el, and to make that it ri$e onely in one determinate part of it, and in other places not? Will you $ay, that this new Water is borrowed from the Ocean, being brought in by the Straight of <I>Gibraltar</I>? This will not remove the doubt afore$aid, but will beget a greater. And fir$t tell me what ought to be the current of that Water, that entering at the Straights mouth, is carried in $ix hours to the remote$t Creeks of the Mediterrane, at a di$tance of two or three thou$and Miles, and that returneth the $ame $pace again in a like time at its going back? What would Ships do that lye out at Sea? What would become of tho$e that $hould be in the Straights-mouth in a continual precipice of a va$t accumulation of Waters, that entering in at a Channel but eight Mile, broad, is to give admittance to $o much Water as in $ix hours over-floweth a tract of many hundred Miles broad, & thou$ands in length? What Tygre, what Falcon runneth or flyeth with $o much $wiftne$s? With the $wiftne$s, I $ay, of above 400 Miles an hour. The cur- rents run (nor can it be denied) the long-wayes of the Gulph, but $o $lowly, as that a Boat with Oars will out-go them, though in- deed not without defalking for their wanderings. Moreover, if this Water come in at the Straight, the other doubt yet remaineth, namely, how it cometh to flow here $o high in a place $o remote, without fir$t ri$ing a like or greater height in the parts more adja- cent? In a word, I cannot think that either ob$tinacy, or $harpne$s of wit can ever find an an$wer to the$e Objections, nor con$e- quently to maintain the $tability of the Earth again$t them, keep- ing within the bounds of Nature.</P> <foot>SAGR.</foot> <p n=>387</p> <P>SAGR. I have all the while perfectly apprehended you in this; and I $tand greedily attending to hear in what manner the$e won- ders may occur without ob$truction from the motion already a$- $igned to the Earth.</P> <P>SALV. The$e effects being to en$ue in con$equence of the mo- tions that naturally agree with the Earth, it is nece$$ary that they not onely meet with no impediment or ob$tacle, but that they do follow ea$ily, & not onely that they follow with facility, but with nece$$ity, $o as that it is impo$$ible that it $hould $ucceed otherwi$e, for $uch is the property & condition of things natural & true. Ha- <marg><I>True and natu- ral effects follow without difficulty.</I></marg> ving therefore $hewen the impo$$ibility of rendring a rea$on of the motions di$cerned in the Waters, & at the $ame time to maintain the immobility of the ve$$el that containeth them: we may proceed to enquire, whether the mobility of the Container may produce the required effect, in the manner that it is ob$erved to evene.</P> <P>Two kinds of motions may be conferred upon a Ve$$el, where- <marg><I>Two $orts of motions of the con- taining Ve$$el, may make the contai- ned water to ri$e and fall.</I></marg> by the Water therein contained, may acquire a faculty of flu- ctuating in it, one while towards one $ide, and another while towards another; and there one while to ebbe, and another while to flow. The fir$t is, when fir$t one, and then another of tho$e $ides is declined, for then the Water running towards the inclining $ide, will alternately be higher and lower, $ometimes on one $ide, and $ometimes on another. But becau$e that this ri$ing and abating is no other than a rece$$ion and acce$$ion to the centre of the Earth, $uch a motion cannot be a$cribed to the Cavi- ties of the $aid Earth, that are the Ve$$els which contain the Wa- ters; the parts of which Ve$$el cannot by any what$oever motion a$$igned to the Earth, be made to approach or recede from the <marg><I>The Cavities of the Earth cannot approach or go far- ther from the cen- tre of the $ame.</I></marg> centre of the $ame: The other $ort of motion is, when the Ve$$el moveth (without inclining in the lea$t) with a progre$$ive motion, not uniform, but that changeth velocity, by $ometimes accellerating, and other times retarding: from which di$parity <marg><I>The progpe$$ive and uneven motion may make the wa- ter contained in a Ve$$el to run to and fro.</I></marg> it would follow, that the Water contained in the Ve$$el its true, but not fixed fa$t to it, as its other $olid parts, but by rea$on of its fluidity, as if it were $eparated and at liberty, and not obli- ged to follow all the mutations of its Container, in the retardation of the Ve$$el, it keeping part of the <I>impetus</I> before conceived, would run towards the the preceding part, whereupon it would of nece$$ity come to ri$e; and on the contrary, if new velocity $hould be added to the Ve$$el, with retaining parts of its tardity, $taying $omewhat behind, before it could habituate it $elf to the new <I>impetus,</I> it would hang back towards the following part, where it would come to ri$e $omething. The which effects we may plainly declare and make out to the Sen$e by the example of one of tho$e $ame Barks yonder, which continually come from <foot>Ccc 2 ^{*} <I>Lizza-</I></foot> <p n=>388</p> <marg>+ A Town ly- ing S. E. of <I>Venice</I></marg> ^{*} <I>Lizza-Fu$ina,</I> laden with fre$h water, for the $ervice of the City. Let us therefore fancy one of tho$e Barks, to come from thence with moderate velocity along the Lake, carrying the water gently, of which it is full: and then either by running a ground, or by $ome other impediment that it $hall meet with, let it be notably retarded. The water therein contained $hall not, by that means, lo$e, as the Bark doth, its pre-conceived <I>impetus,</I> but retaining the $ame, $hall run forwards towards the prow, where it $hall ri$e notably, falling as much a $tern. But if, on the contrary, the $aid Bark, in the mid$t of its $mooth cour$e, $hall have a new velocity, with notable augmentation added to it, the water con- tained before it can habituate it $elf thereto, continuing in its tardity, $hall $tay behinde, namely a $tern, where of con$e- quence it $hall mount, and abate for the $ame at the prow. This effect is undoubted and manife$t, and may hourly be experimen- ted; in which I de$ire that for the pre$ent three particulars may be noted. The flr$t is, that to make the water to ri$e on one $ide of the ve$$el, there is no need of new water, nor that it run thither, for$aking the other $ide. The $econd is, that the water in the middle doth not ri$e or fall notably, unle$$e the cour$e of the Bark were not before that very $wift, and the $hock or other arre$t that held it exceeding $trong and $udden, in which ca$e its po$$ible, that not only all the water might run forwards, but that the greater part thereof might i$$ue forth of the Bark: and the $ame al$o would en$ue, whil$t that being under $ail in a $mooth cour$e, a mo$t violent <I>impetus</I> $hould, upon an in$tant, overtake it: But when to its calme motion there is added a mo- derate retardation or incitation, the middle parts (as I $aid) un- ob$ervedly ri$e and fall: and the other parts, according as they are neerer to the middle, ri$e the le$$e; and the more remote, more. The third is, that whereas the parts about the mid$t do make little alteration in ri$ing and falling, in re$pect of the wa- ters of the $ides; on the contrary, they run forwards and back- wards very much, in compari$on of the extreams. Now, my Ma$ters, that which the Bark doth, in re$pect of the water by it contained, and that which the water contained doth, in re- $pect of the Bark its container, is the $elf-$ame, to an hair, with that which the Mediterranean Ve$$el doth, in re$pect of the wa- ters in it contained, and that which the waters contained do, in <marg><I>The parts of the terre$trial Globe accelerate and re- <*>ard in their moti- on,</I></marg> re$pect of the Mediterranean Ve$$el their container. It follow- eth now that we demon$trate how, and in what manner it is true, that the Mediterrane, and all the other Straits; and in a word, all the parts of the Earth do all move, with a motion notably uneven, though no motion that is not regular and uniforme, is thereby a$$igned to all the $aid Globe taken collectively.</P> <foot>SIMP.</foot> <p n=>389</p> <P>SIMP. This Propo$ition, at fir$t $ight to me, that am neither Geometrician nor A$tronomer, hath the appearance of a very great Paradox; and if it $hould be true, that the motion of the <I>whole,</I> being regular, that of the parts, which are all united to their <I>whole,</I> may be irregular, the Paradox will overthrow the Axiome that affirmeth, <I>Eandem e$$e rationem totius & par- tium.</I></P> <P>SALV. I will demon$trate my Paradox, and leave it to your care, <I>Simplicius,</I> to defend the Axiome from it, or el$e to re- concile them; and my demon$tration $hall be $hort and fa- miliar, depending on the things largely handled in our prece- dent conferences, without introducing the lea$t $yllable, in fa- vour of the flux and reflux.</P> <P>We have $aid, that the motions a$$igned to the Terre$trial <marg><I>Demon$trations how the parts of the terre$triall Globe accelerats and ratard.</I></marg> Globe are two, the fir$t Annual, made by its centre about the circumference of the Grand Orb, under the Ecliptick, according to the order of the Signes, that is, from We$t to Ea$t; the other made by the $aid Globe revolving about its own centre in twenty four hours; and this likewi$e from We$t to Ea$t: though a- bout an Axis $omewhat inclined, and not equidi$tant from that of the Annual conver$ion. From the mixture of the$e two mo- tions, each of it $elf uniform, I $ay, that there doth re$ult an uneven and deformed motion in the parts of the Earth. Which, that it may the more ea$ily be under$tood, I will explain, by drawing a Scheme thereof. And fir$t, about the centre A [<I>in Fig. 1. of this Dialogue</I>] I will de$cribe the circumference of <marg><I>The parts of a Circle regularly moved about its own centre move in divers times with contrary motions.</I></marg> the Grand Orb B C, in which any point being taken, as B, about it as a centre we will de$cribe this le$$er circle D E F G, repre$enting the Terre$trial Globe; the which we will $uppo$e to run thorow the whole circumference of the Grand Orb, with its centre B, from the We$t towards the Ea$t, that is, from the part B towards C; and moreover we will $uppo$e the Terre- $trial Globe to turn about its own centre B likewi$e from We$t to Ea$t, that is, according to the $ucce$$ion of the points D E F G, in the $pace of twenty four hours. But here we ought carefully to note, that a circle turning round upon its own centre, each part of it mu$t, at different times, move with contrary motions: the which is manife$t, con$idering that whil$t the parts of the circumference, about the point D move to the left hand, that is, towards E, the oppo$ite parts that are about F, approach to the right hand, that is, towards G; $o that when the parts D $hall be in F, their motion $hall be contrary to what it was before. when it was in D. Furthermore, the $ame time that the parts E de$cend, if I may $o $peak, towards F, tho$e in G a$cend towards D. It being therefore pre$uppo$ed, that <foot>there</foot> <p n=>390</p> <marg><I>The mixture of the two motions annnal and diur- nal, cau$eth the inequality in the motion of the parts of the terre$trial Globe.</I></marg> there are $uch contrarieties of motions in the parts of the Terre- $trial Surface, whil$t it turneth round upon its own centre, it is nece$$ary, that in conjoyning this Diurnal Motion, with the other Annual, there do re$ult an ab$olute motion for the parts of the $aid Terre$trial Superficies, one while very accelerate, and ano- ther while as $low again. The which is manife$t, con$idering fir$t the parts about D, the ab$olute motion of which $hall be extream $wift, as that which proceedeth from two motions made both one way, namely, towards the left hand; the fir$t of which is part of the Annual Motion, common to all the parts of the Globe, the other is that of the $aid point D., carried likewi$e to the left, by the Diurnal Revolution; $o that, in this ca$e, the Diurnal motion increa$eth and accelerateth the Annual. The contrary to which happeneth in the oppo$ite part F, which, whil$t it is by the common annual motion carried, together with the whole Globe, towards the left, it happeneth to be carried by the Diurnal conver$ion al$o towards the right: $o that the Diur- nal motion by that means detracteth from the Annual, where- upon the ab$olute motion, re$ulting from the compo$ition of both the other, is much retarded. Again, about the points E and G, the ab$olute motion becometh in a manner equal to the $imple Annual one, in regard that little or nothing increa$eth or dimi- ni$heth it, as not tending either to the left hand, or to the right, but downwards and upwards. We will conclude therefore, that like as it is true, that the motion of the whole Globe, and of each of its parts, would be equal and uniforme, in ca$e they did move with one $ingle motion, whether it were the meer Annual, or the $ingle Diurnal Revolution, $o it is requi$ite, that mixing tho$e two motions together, there do re$ult thence for the parts of the $aid Globe irregular motions, one while accelerated, and another while retarded, by means of the additions or $ub$tracti- ons of the Diurnal conver$ion from the annual circulation. So that, if it be true (and mo$t true it is, as experience proves) that the acceleration and retardation of the motion of the Ve$- $el, makes water contained therein to run to and again the long waves of it, and to ri$e and fall in its extreames, who will make $cruple of granting, that the $aid effect may, nay ought to $uc- ceed in the Sea-waters, contained within their Ve$$els, $ubject to $uch like alterations, and e$pecially in tho$e that di$tend them- $elves long-wayes from We$t to Ea$t, which is the cour$e that <marg><I>The mo$t potent and primary cau$e of the ebbing and flowing.</I></marg> the motion of tho$e $ame Ve$$els $teereth? Now this is the mo$t potent and primary cau$e of the ebbing and flowing, with- out the which no $uch effect would en$ue. But becau$e the par- ticular accidents are many and various, that in $everal places and times are ob$erved, which mu$t of nece$$ity have dependance <foot>on</foot> <p n=>391</p> on other different concomitant cau$es, although they ought all to have connexion with the primary; therefore it is convenient that we propound and examine the $everal accidents that may be the cau$es of $uch different effects.</P> <P>The fir$t of which is, that when ever the water, by means of a <marg><I>Sundry accidents that happen in the ebbings & flowings</I></marg> notable retardation or acceleration of the motion of the Ve$$el, its container, $hall have acquired a cau$e of running towards this <marg><I>The first acci- dent.</I></marg> or that extream, and $hall be rai$ed in the one, and abated in the <marg><I>The Water rai- $ed in one end of the Ve$$el return- eth of its $<*>f to</I> Æquilibrium.</marg> other, it $hall not neverthele$$e continue, for any time in that $tate, when once the primary cau$e is cea$ed: but by vertue of its own gravity and natural inclination to level and grow, even it $hall $peedily return backwards of its own accord, and, as being grave and fluid, $hall not only move towards <I>Æquilibrium</I>; but being impelled by its own <I>impetus,</I> $hall go beyond it, ri$ing in the part, where before it was lowe$t; nor $hall it $tay here, but returning backwards anew, with more reiterated reciprocations of its undulations, it $hall give us to know, that it will not from a velocity of motion, once conceived, reduce it $elf, in an in$tant, to the privation thereof, and to the $tate of re$t, but will $ucce$- $ively, by decrea$ing a little and a little, reduce it $elf unto the $ame, ju$t in the $ame manner as we $ee a weight hanging at a cord, after it hath been once removed from its $tate of re$t, that is, from its perpendicularity, of its own accord, to return thither and $ettle it $elf, but not till $uch time as it $hall have often pa$t to one $ide, and to the other, with its reciprocall vi- brations.</P> <P>The $econd accident to be ob$erved is, that the before- <marg><I>In the $horter Vi$$els the undula- tions of waters are more frequent.</I></marg> declared reciprocations of motion come to be made and repeated with greater or le$$er frequency, that is, under $horter or longer times, according to the different lengths of the Ve$$els contain- ing the waters; $o that in the $horter $paces the reciprocati- ons are more frequent, and in the longer more rare: ju$t as in the former example of pendent bodies, the vibrations of tho$e that are hanged to longer cords are $een to be le$$e frequent, than tho$e of them that hang at $horter $trings.</P> <P>And here, for a third ob$ervation, it is to be noted, that not <marg><I>The greater profundity maketh the undulations of waters more fre- quent.</I></marg> onely the greater or le$$er length of the Ve$$el is a cau$e that the water maketh its reciprocations under different times; but the greater or le$$er profundity worketh the $ame effect. And it happeneth, that of waters contained in receptacles of equall length, but of unequal depth, that which $hall be the deepe$t, maketh its undulations under $horter times, and the reciprocati- ons of the $hallower waters are le$$e frequent.</P> <P>Fourthly, there are two effects worthy to be noted, and di- ligently ob$erved, which the water worketh in tho$e its vibra- <foot>tions;</foot> <p n=>392</p> <marg><I>W<*> ri$eth & falleth in the ex- tream parts of the Ve$$el, and runneth to and fro in the midst.</I></marg> tions; the one is its ri$ing and falling alternately towards the one and other extremity; the other is its moving and running, to $o $peak, Horizontally forwards and backwards. Which two dif- ferent motions differently re$ide in divers parts of the Water: for its extream parts are tho$e which mo$t eminently ri$e and fall; tho$e in the middle never ab$olutely moving upwards and down- wards, of the re$t $ucce$$ively tho$e that are neere$t to the ex- treams ri$e and fall proportionally more than the remote: but on the contrary, touching the other progre$$ive motion forwards and backwards, the middle parts move notably, going and re- turning, and the waters that are in the extream parts gain no ground at all; $ave onely in ca$e that in their ri$ing they over- flow their banks, and break forth of their fir$t channel and re- ceptacle; but where there is the ob$tacle of banks to keep them in, they onely ri$e and fall; which yet hindereth not the waters in the middle from fluctuating to and again; which likewi$e the other parts do in proportion, undulating more or le$$e, according as they are neerer or more remote from the middle.</P> <marg><I>An accident of the Earths motions impo$$ible to be re- duced to practice by art.</I></marg> <P>The fifth particular accident ought the more attentively to be con$idered, in that it is impo$$ible to repre$ent the effect there- of by an experiment or example; and the accident is this. In the ve$$els by us framed with art, and moved, as the above- named Bark, one while more, and another while le$$e $wiftly, the acceleration and retardation is imparted in the $ame manner to all the ve$$el, and to every part of it; $o that whil$t <I>v. g.</I> the Bark forbeareth to move, the parts precedent retard no more than the $ub$equent, but all equally partake of the $ame re- tardment; and the $elf-$ame holds true of the acceleration, namely, that conferring on the Bark a new cau$e of grea- ter velocity, the Prow and Poop both accelerate in one and the $ame manner. But in huge great ve$$els, $uch as are the very long bottomes of Seas, albeit they al$o are no other than cer- tain cavities made in the $olidity of the Terre$trial Globe, it alwayes admirably happeneth, that their extreams do not unitedly equall, and at the $ame moments of time increa$e and dimini$h their motion, but it happeneth that when one of its extreames hath, by vertue of the commixtion of the two Motions, Diurnal, and Annual, greatly retarded its velocity, the other extream is animated with an extream $wift motion. Which for the better under$tanding of it we will explain, rea$- $uming a Scheme like to the former; in which if we do but $up- po$e a tract of Sea to be long, <I>v. g.</I> a fourth part, as is the arch B C [<I>in Fig.</I> 2.] becau$e the parts B are, as hath been already declared, very $wift in motion, by rea$on of the union of the two motions diurnal and annual, towards one and the $ame way, <foot>but</foot> <p n=>393</p> but the part C at the $ame time is retarded in its motion, as be ing deprived of the progre$$ion dependant on the diurnal motion: If we $uppo$e, I $ay, a tract of Sea as long as the arch B C, we have already $een, that its extreams $hall move in the $ame time with great inequality. And extreamly different would the velo- cities of a tract of Sea be that is in length a $emicircle, and pla- ced in the po$ition B C D, in regard that the extream B would be in a mo$t accelerate motion, and the other D, in a mo$t $low one; and the intermediate parts towards C, would be in a moderate motion. And according as the $aid tracts of Sea $hall be $horter, they $hall le$$e participate of this extravagant acci- dent, of being in $ome hours of the day with their parts diver$ly affected by velocity and tardity of motion. So that, if, as in the fir$t ca$e, we $ee by experience that the acceleration and retardation, though equally imparted to all the parts of the conteining Ve$$el, is the cau$e that the water contained, fluctuates too and again, what may we think would happen in a Ve$$el $o admirably di$po$ed, that retardation and acceleration of motion is very unequally contributed to its parts? Certainly we mu$t needs grant that greater and more wonderful cau$es of the commotions in the Water ought to be looked for. And though it may $eem im- po$$ible to $ome, that in artificial Machines and Ve$$els we $hould be able to experiment the effects of $uch an accident; yet ne- verthele$$e it is not ab$olutely impo$$ible to be done; and I have by me the model of an Engine, in which the effect of the$e admi- rable commixtions of motions may be particularly ob$erved. But as to what concerns our pre$ent purpo$e, that which you may have hitherto comprehended with your imagination may $uf- fice.</P> <P>SAGR. I for my own particular very well conceive that this admirable accident ought nece$$arily to evene in the Straights of Seas, and e$pecially in tho$e that di$tend them$elves for a great length from We$t to Ea$t; namely according to the cour$e of the motions of the Terre$trial Globe; and as it is in a certain manner unthought of, and without a pre$ident among the moti- ons po$$ible to be made by us, $o it is not hard for me to believe, that effects may be derived from the $ame, which are not to be i- mitated by our artificial experiments.</P> <P>SALV. The$e things being declared, it is time that we pro- ceed to examine the particular accidents, which, together with their diver$ities, are ob$erved by experience in the ebbing and flowing of the waters. And fir$t we need not think it hard to <marg><I>Rea$ons renew- ed of the particu- lar accidents ob- $erved in the eb- bings and flowings.</I></marg> gue$$e whence it happeneth, that in Lakes, Pooles, and al$o in the le$$er Seas there is no notable flux and reflux; the which hath two very $olid rea$ons. The one is, that by rea$on of the $hort- <foot>Ddd ne$$e</foot> <p n=>394</p> <marg><I>Second cau$es why in $mall Seas and in Lakes there are no ebbings and flowings.</I></marg> ne$$e of the Ve$$el, in its acquiring in $everal hours of the day $everal degrees of velocity, they are with very little difference acquired by all its parts; for as well the precedent as the $ub$e- quent, that is to $ay, both the Ea$tern and We$tern parts, do accelerate and retard almo$t in the $ame manner; and withal making that alteration by little and little, and not by giving the motion of the conteining Ve$$el a $udden check, and retard- ment, or a $udden and great impul$e or acceleration; both it and all its parts, come to be gently and equally impre$$ed with the $ame degrees of velocity; from which uniformity it follow- eth, that al$o the conteined water with but $mall re$i$tance and oppo$ition, receiveth the $ame impre$$ions, and by con$equence doth give but very ob$cure $ignes of its ri$ing or falling, or of its running towards one part or another. The which effect is likewi$e manife$tly to be $een in the little artificial Ve$$els, wherein the contained water doth receive the $elf $ame impre$$ions of veloci- ty; when ever the acceleration and retardation is made by gentle and uniform proportion. But in the Straights and Bays that for a great length di$tend them$elves from Ea$t to We$t, the accele- ration and retardation is more notable and more uneven, for that one of its extreams $hall be much retarded in motion, and the other $hall at the $ame time move very $wiftly: The reci- procal libration or levelling of the water proceeding from the <I>im- petus</I> that it had conceived from the motion of its container. The which libration, as hath been noted, hath its undulations very frequent in $mall Ve$$els; from whence en$ues, that though there do re$ide in the Terre$trial motions the cau$e of confer- ring on the waters a motion onely from twelve hours to twelve hours, for that the motion of the conteining Ve$$els do ex- treamly accelerate and extreamly retard but once every day, and no more; yet neverthele$$e this $ame $econd cau$e depend- ing on the gravity of the water which $triveth to reduce it $elf to equilibration, and that according to the $hortne$$e of the Ve$- $el hath its reciprocations of one, two, three, or more hours, this intermixing with the fir$t, which al$o it $elf in $mall Ve$$els is very little, it becommeth upon the whole altogether in$en$ible. For the primary cau$e, which hath the periods of twelve hours, having not made an end of imprinting the precedent commoti- on, it is overtaken and oppo$ed by the other $econd, depen- dant on the waters own weight, which according to the brevity and profundity of the Ve$$el, hath the time of its undulations of one, two, three, four, or more hours; and this contending with the other former one, di$turbeth and removeth it, not per- mitting it to come to the height, no nor to the half of its moti- on; and by this conte$tation the evidence of the ebbing and <foot>flow-</foot> <p n=>395</p> flowing is wholly annihilated, or at lea$t very much ob$cured. I pa$$e by the continual alteration of the air, which di$quieting the water, permits us not to come to a certainty, whether any, though but $mall, encrea$e or abatement of half an inch, or le$$e, do re$ide in the Straights, or receptacles of water not a- bove a degree or two in length.</P> <P>I come in the $econd place to re$olve the que$tion, why, there <marg><I>The rea$on gi- ven, why the eb- bings and flowings, for the mo$t part, are every $ix hours.</I></marg> not re$iding any vertue in the primary principle of commoving the waters, $ave onely every twelve hours, that is to $ay, once by the greate$t velocity, and once by the greate$t tardity of motion; the ebbings and flowings $hould yet neverthele$$e ap- pear to be every $ix hours. To which is an$wered, that this de- termination cannot any wayes be taken from the primary cau$e onely; but there is a nece$$ity of introducing the $econdary cau- $es, as namely the greater or le$$e length of the Ve$$els, and the greater or le$$e depth of the waters in them conteined. Which cau$es although they have not any operation in the moti- ons of the waters, tho$e operations belonging to the $ole prima- ry cau$e, without which no ebbing or flowing would happen, yet neverthele$$e they have a principal $hare in determining the times or periods of the reciprocations, and herein their influ- ence is $o powerful, that the primary cau$e mu$t of force give way unto them. The period of $ix hours therefore is no more proper or natural than tho$e of other intervals of times, though indeed its the mo$t ob$erved, as agreeing with our Mediterrane, which was the onely Sea that for many Ages was navigated: though neither is that period ob$erved in all its parts; for that in $ome more angu$t places, $uch as are the <I>Helle- $pont,</I> and the <I>Ægean</I> Sea, the periods are much $horter, and al$o very divers among$t them$elves; for which diver- $ities, and their cau$es incomprehen$ible to <I>Ari$totle,</I> $ome $ay, that after he had a long time ob$erved it upon $ome cliffes of <I>Negropont,</I> being brought to de$peration, he threw him$elf into the adjoyning <I>Euripus,</I> and voluntarily drowned him$elf.</P> <P>In the third place we have the rea$on ready at hand, whence <marg><I>The cau$e why $ome Seas, though very long, $uffer no ebbing and flowing.</I></marg> it commeth to pa$$e, that $ome Seas, although very long, as is the Red Sea, are almo$t altogether exempt from Tides, which happeneth becau$e their length extendeth not from Ea$t to We$t, but rather tran$ver$ly from the Southea$t to the North- we$t; but the motions of the Earth going from We$t to Ea$t; the impul$es of the water, by that means, alwayes happen to fall in the Meridians, and do not move from parallel to parallel; in$omuch that in the Seas that extend them$elves athwart to- wards the Poles, and that the contrary way are narrow, there is <foot>Ddd 2 no</foot> <p n=>396</p> no cau$e of ebbing and flowing, $ave onely by the participation of another Sea, wherewith it hath communication, that is $ub- ject to great commotions.</P> <marg><I>Ebbings and flowings why grea- te$t in the extre- mities of gulphs, and lea$t in the middle parts.</I></marg> <P>In the fourth place we $hall very ea$ily find out the rea$on why the fluxes and refluxes are greate$t, as to the waters ri$ing and falling in the utmo$t extremities of Gulphs, and lea$t in the intermediate parts; as daily experience $heweth here in <I>Venice,</I> lying in the farther end of the <I>Adriatick</I> Sea, where that diffe- rence commonly amounts to five or $ix feet; but in the places of the Mediterrane, far di$tant from the extreams, that mutati- on is very $mall, as in the I$les of <I>Cor$ica</I> and <I>Sardinnia,</I> and in the Strands of <I>Rome</I> and <I>Ligorne,</I> where it exceeds not half a foot; we $hall under$tand al$o, why on the contrary, where the ri$ings and fallings are $mall, the cour$es and recour$es are great: I $ay it is an ea$ie thing to under$tand the cau$es of the$e accidents, $eeing that we meet with many manife$t occurrences of the $ame nature in every kind of Ve$$el by us artificially com- po$ed, in which the $ame effects are ob$erved naturally to fol- low upon our moving it unevenly, that is, one while fa$ter, and another while $lower.</P> <marg><I>Why in narrow places the cour$e of the waters is more $wift than in larger.</I></marg> <P>Moreover, con$idering in the fifth place, that the $ame quantity of Water being moved, though but gently, in a $patious Channel, comming afterwards to go through a narrow pa$$age, will of nece$$ity run, with great violence, we $hall not finde it hard to comprehend the cau$e of the great Currents that are made in the narrow Channel that $eparateth <I>Calabria</I> from <I>Sicilia:</I> for that all the Water that, by the $paciou$ne$$e of the I$le, and by the <I>Ionick</I> Gulph, happens to be pent in the Ea$tern part of the Sea, though it do in that, by rea$on of its largene$s, gently de$cend towards the We$t, yet neverthele$$e, in that it is pent up in the <I>Bo$phorus,</I> it floweth with great violence be- tween <I>Scilla</I> and <I>Caribdis,</I> and maketh a great agitation. Like to which, and much greater, is $aid to be betwixt <I>Africa</I> and the great I$le of St. <I>Lorenzo,</I> where the Waters of the two va$t Seas, <I>Indian</I> and <I>Ethiopick,</I> that lie round it, mu$t needs be $traightned into a le$$e Channel between the $aid I$le and the <I>Ethiopian</I> Coa$t. And the Currents mu$t needs be very great in the Straights of <I>Magellanes,</I> which joyne together the va$t Oceans of <I>Ethiopia,</I> and <I>Del Zur,</I> called al$o the <I>Pacifick</I> Sea.</P> <marg><I>A di$cu$$ion of $ome more ab$tru$e accidents ob$erved in the ebbing and flowing.</I></marg> <P>It follows now, in the $ixth place, that to render a rea$on of $ome more ab$tru$e and incredible accidents, which are ob$er- ved upon this occa$ion, we make a con$iderable reflection upon the two principal cau$es of ebbings and flowings, afterwards compounding and mixing them together. The fir$t and $imple$t <foot>of</foot> <p n=>397</p> of which is (as hath often been $aid) the determinate accelera- tion and retardation of the parts of the Earth, from whence the Waters have a determinate period put to their decur$ions towards the Ea$t, and return towards the We$t, in the time of twenty $our hours. The other is that which dependeth on the pro- per gravity of the Water, which being once commoved by the primary cau$e, $eeketh, in the next place, to reduce it $elf to <I>Æ- quilibrium,</I> with iterated reciprocations; which are not deter- mined by one $ole and prefixed time; but have as many varie- ties of times as are the different lengths and profundities of the receptacles, and Straights of Seas; and by what dependeth on this $econd principle, they would ebbe. and flow, $ome in one hour, others in two, in four, in $ix, in eight, in ten, &c. Now if we begin to put together the fir$t cau$e, which hath its $et Period from twelve hours to twelve hours, with $ome one of the $econ- dary, that hath its Period <I>verb. grat.</I> from five hours to five hours, it would come to pa$$e, that at $ometimes the primary cau$e and $econdary would accord to make impul$es both one and the $ame way; and in this concurrency, and (as one may call it) unanimous con$piration the flowings $hall be great. At other times it happening that the primary impul$e doth, in a certain manner, oppo$e that which the $econdary Period would make, and in this conte$t one of the Principles being taken away, that which the other would give, will weaken the commotion of the Waters, and the Sea will return to a very tranquil State, and almo$t immoveable. And at other times, according as the two afore$aid Principles $hall neither altogether conte$t, nor altoge- ther concur, there $hall be other kinds of alterations made in the increa$e and diminution of the ebbing and flowing. It may likewi$e fall out that two Seas, con$iderably great and which communicate by $ome narrow Channel, may chance to have, by rea$on of the mixtion of the two Principles of motion, one cau$e to flow at the time that the other hath cau$e to move a contrary way; in which ca$e in the Channel, whereby they di$- imbogue them$elves into each other, there do extraordinary conturbations in$ue, with oppo$ite and vortick motions, and mo$t dangerous boilings and breakings, as frequent relations and experiences do a$$ure us. From $uch like di$cordant moti- ons, dependent not onely on the differenr po$itions and longi- tudes, but very much al$o upon the different profundities of the Seas, which have the $aid intercour$e there do happen at $ome- times different commotions in the Waters, irregular, and that can be reduced to no rules of ob$ervation, the rea$ons of which have much troubled, and alwayes do trouble Mariners, for that they meet with them without $eeing either impul$e of winds, or <foot>other</foot> <p n=>398</p> other eminent aereal alteration that might occa$ion the $ame; of which di$turbance of the Air we ought to make great account in other accidents, and to take it for a third and accidental cau$e, able to alter very much the ob$ervation of the effects de- pending on the $econdary and more e$$ential cau$es. And it is not to be doubted, but that impetuous windes, continuing to blow, for example, from the Ea$t, they $hall retein the Waters and prohibit the reflux or ebbing; whereupon the $econd and third reply of the flux or tide overtaking the former, at the hours prefixed, they will $well very high; and being thus born up for $ome dayes, by the $trength of the Winds, they $hall ri$e more than u$ual, making extraordinary inundations.</P> <P>We ought al$o, (and this $hall $erve for a $eventh Probleme) to take notice of another cau$e of motion dependant on the great abundance of the Waters of great Rivers that di$charge <marg><I>The cau$e why, in $ome narrow Channels, we $ee the Sea-waters run alwayes one way.</I></marg> them$elves into Seas of no great capacity, whereupon in the Straits or <I>Bo$phori</I> that communicate with tho$e Seas, the Waters are $een to run always one way: as it happeneth in the <I>Thraci- an Bo$phorus</I> below <I>Con$tantinople,</I> where the water alwayes runneth from the <I>Black-Sea,</I> towards the <I>Propontis</I>: For in the $aid <I>Black-Sea</I> by rea$on of its $hortne$$e, the principal cau$es of ebbing and flowing are but of $mall force. But, on the con- trary, very great Rivers falling into the $ame, tho$e huge de- fluxions of water being to pa$$e and di$gorge them$elves by the <marg>* Or current.</marg> the Straight, the ^{*}cour$e is there very notable and alwayes to- wards the South. Where moreover we ought to take notice, that the $aid Straight or Channel, albeit very narrow, is not $ubject to perturbations, as the Straight of <I>Soilla</I> and <I>Carybdis</I>; for that that hath the <I>Black-Sea</I> above towards the North, and the <I>Pro- pontis,</I> the <I>Ægean,</I> and the <I>Mediterranean</I> Seas joyned unto it, though by a long tract towards the South; but now, as we have ob$erved, the Seas, though of never $o great length, lying North and South, are not much $ubject to ebbings and flowings; but becau$e the <I>Sicilian</I> Straight is $ituate between the parts of the Mediterrane di$tended for a long tract or di$tance from We$t to Ea$t, that is, according to the cour$e of the fluxes and refluxes, therefore in this the agitations are very great; and would be much more violent between <I>Hercules Pillars,</I> in ca$e the Straight of <I>Gibraltar</I> did open le$$e; and tho$e of the Straight of <I>Magellanes</I> are reported to be extraordinary violent.</P> <P>This is what, for the pre$ent, cometh into my mind to $ay unto you about the cau$es of this fir$t period diurnal of the Tide, and its various accidents, touching which, if you have any thing to offer, you may let us hear it, that $o we may afterwards pro- ceed to the other two periods, monethly and annual.</P> <foot>SIMP.</foot> <p n=>399</p> <P>SIMP. In my opinion, it cannot be denied, but that your di$- cour$e carrieth with it much of probability, arguing, as we $ay, <I>ex $uppo$itione,</I> namely, granting that the Earth moveth with the two motions a$$igned it by <I>Copernicus</I>: but if that motion <marg><I>The Hypothe$ir of the Earths mo- bility taken in fa- vour of the Tide, oppo$ed.</I></marg> be di$proved, all that you have $aid is vain, and in$ignificant: and for the di$proval of that <I>Hypothe$is,</I> it is very manife$tly hinted by your Di$cour$e it $elf. You, with the $uppo$ition of the two Terre$trial motions, give a rea$on of the ebbing and flowing; and then again, arguing circularly, from the ebbing and flowing, draw the rea$on and confirmation of tho$e very motions; aud $o proceeding to a more $pecious Di$cour$e, you $ay that the Water, as being a fluid body, and not tenaciou$ly annexed to the Earth, is not con$trained punctually to obey eve- ry of its motions, from which you afterwards infer its ebbing and flowing, Now I, according to your own method, argue the quite contrary, and $ay; the Air is much more tenuous, and fluid than the Water, and le$$e annexed to the Earths $uperfici- es, to which the Water, if it be for nothing el$e, yet by rea$on of its gravity that pre$$eth down upon the $ame more than the light Air, adhereth; therefore the Air is much obliged to fol- low the motions of the Earth: and therefore were it $o, that the Earth did move in that manner, we the inhabitants of it, and carried round with like velocity by it, ought perpetually to feel a Winde from the Ea$t that beateth upon us with intolerable force. And that $o it ought to fall out, quotidian experience a$- $ureth us: for if with onely riding po$t, at the $peed of eight or ten miles an hour in the tranquil Air, the incountering of it with our face $eemeth to us a Winde that doth not lightly blow upon us, what $hould we expect from our rapid cour$e of 800. or a thou$and miles an hour, again$t the Air, that is, free from that motion? And yet, notwith$tanding we cannot perceive any thing of that nature.</P> <P>SALV. To this objection that hath much of likelihood in it, I <marg><I>The an$wer to the objections made again$t the motion of the Ter- re$trial Globe.</I></marg> reply, that its true, the Air is of greater tenuity and levity; and, by rea$on of its levity, le$$e adherent to the Earth than Water $o much more grave and ^{*}bulky; but yet the con$equence is fal$e that you infer from the$e qualities; namely, that upon account <marg>+ Corpulenta.</marg> of that its levity, tenuity, and le$$e adherence to the Earth, it $hould be more exempt than the Water from following the Terre$trial Motions; $o as that to us, who ab$olutely pertake of of them, the $aid exemption $hould be $en$ible and manife$t; nay, it happeneth quite contrary; for, if you well remember, the cau$e of the ebbing and flowing of the Water a$$igned by us, con$i$teth in the Waters not following the unevenne$$e of the motion of its Ve$$el, but retaining the <I>impetus</I> conceived before, <foot>without</foot> <p n=>400</p> without dimini$hing or increa$ing it according to the preci$e rate of its dimini$hing or increa$ing in its Ve$$el. Becau$e therefore <marg><I>The Water more apt to con$erve an</I> impetus <I>conceived, then the Air.</I></marg> that in the con$ervation and retention of the <I>impetus</I> before con- ceived, the di$obedience to a new augmentation or diminution of motion con$i$teth, that moveable that $hall be mo$t apt for $uch a retention, $hall be al$o mo$t commodious to demon$trate the effect that followeth in con$equence of that retention. Now how much the Water is di$po$ed to maintain $uch a conceived agita- tion; though the cau$es cea$e that impre$s the $ame, the experi- ence of the Seas extreamly di$turbed by impetuous Winds $hew- eth us; the Billows of which, though the Air be grown calm, and the Wind laid, for a long time after continue in motion: As the Sacred Poet plea$antly $ings,</P> <head><I>Qual l'alto Egeo,</I> &c.----------</head> <P>And that long continuing rough after a $torm, dependeth on <marg><I>Light bodies ea$ier to be moved than beavy, but le$s aut to con$erve the mo- tion.</I></marg> the gravity of the water: For, as I have el$ewhere $aid, light bo- dies are much ea$ier to be moved than the more grave, but yet are $o much the le$s apt to con$erve the motion imparted, when once the moving cau$e cea$eth. Whence it comes that the Aire, as being of it $elf very light and thin, is ea$ily mov'd by any very $mall force, yet it is withall very unable to hold on its motion, the Mover once cea$ing. Therefore, as to the Aire which envi- rons the Terre$trial Globe, I would fay, that by rea$on of its adherence, it is no le$$e carried about therewith then the Water; and e$pecially that part which is contained in its ve$$els; which <marg><I>Its more rational that the Air be commoved by the rugged $urface of the <*>ar h than b<*> the Cele$tial motion.</I></marg> ve$$els are the valleys enclo$ed with Mountains. And we may with much more rea$on affirm that this $ame part of the Air is carried round, and born forwards by the rugged parts of the Earth, than that the higher is whi<*>'d about by the motion of the Heavens, as ye <I>Peripateticks</I> maintain.</P> <P>What hath been hitherto $poken, $eems to me a $ufficient an- <marg><I>The revolution of the Earth con- f<*>med by a new argument taken from the Air.</I></marg> $wer to the allega ion of <I>Simputius</I>; yet neverthele$s with a new in$tance and $olution, founded upon an admirable experiment, I will $uperabundantly $atisfie him, and confirm to <I>Sagredus</I> the mobility of the Earth. I have told you that the Air, and in par- ticular that part of it which a$cendeth not above the tops of the highe$t Mountains, is carried round by the uneven parts of the Earths $urface: from whence it $hould $eem, that it mu$t of con- $equence come to pa$$e, that in ca$e the $uperficies of the Earth were not uneven, but $mooth and plain, no cau$e would remain for drawing the Air along with it, or at lea$t for revolving it with $o much uniformity. Now the $urface of this our Globe, is not all craggy and rugged, but there are exceeding great tracts very <foot>even,</foot> <p n=>401</p> even, to wit, the $urfaces of very va$t Seas, which being al$o far remote from the continuate ledges of Mountains which environ it, $eem to have no faculty of carrying the $uper-ambient Air along therewith: and not carrying it about, we may perceive what will of con$equence en$ue in tho$e places.</P> <P>SIMP. I was about to propo$e the very $ame difficulty, which I think is of great validity.</P> <P>SALV. You $ay very well <I>Simplicius,</I> for from the not finding in the Air that which of con$equence would follow, did this our Globe move round; you argue its immoveablene$$e. But in ca$e that this which you think ought of nece$$ary con$equence to be found, be indeed by experience proved to be $o; will you accept it for a $ufficient te$timony and an argument for the mobility of the $aid Globe?</P> <P>SIMP. In this ca$e it is not requi$ite to argue with me alone, for if it $hould $o fall out, and that I could not comprehend the cau$e thereof, yet haply it might be known by others.</P> <P>SALV. So that by playing with you, a man $hall never get, but be alwayes on the lo$ing hand; and therefore it would be better to give over: Neverthele$s, that we may not cheat our third man we will play on. We $aid even now, and with $ome addition we reitterate it, that the Ayr as if it were a thin and fluid body, and not $olidly conjoyned with the Earth, $eem'd not to be nece$$i- tated to obey its motion; unle$$e $o far as the craggine$s of the terre$trial $uperficies, tran$ports and carries with it a part there- of contigious thereunto; which doth not by any great $pace ex- ceed the greate$t altitude of Mountains: the which portion of Air ought to be $o much le$s repugnant to the terre$trial conver$ion, <marg><I>The vaporous parts of the earth, partake of its mo- tions.</I></marg> by how much it is repleat with vapours, fumes, and exhalations, matters all participating of terrene qualities, and con$equently apt of their own nature to the $ame motions. But where there are wanting the cau$es of motion, that is, where the $urface of the Globe hath great levels, and where there is le$s mixture of the terrene vapours, there the cau$e whereby the ambient Air is con- $trained to give entire obedience to the terre$trial conver$ion will cea$e in part; $o that in $uch places, whil$t the Earth revolveth to- wards the Ea$t, there will be continually a wind perceived which will beat upon us, blowing from the Ea$t towards the We$t: and $uch gales will be the more $en$ible, where the revolution of the Globe is mo$t $wift; which will be in places more remote from the Poles, and approaching to the greate$t Circle of the diurnal conver$ion. But now <I>de facto</I> experience much confi meth this Phylo$ophical argumentation; for in the $patious Seas, and in their parts mo$t remote from Land, and $ituate under the Torrid Zone, that is bounded by the Tropicks, where there are none of tho$e <foot>Eee $ame</foot> <p n=>402</p> <marg><I>Con$tant gales within the Tro- pieks blow towards the We$t.</I></marg> $ame terre$trial evaporations, we finde a perpetual gale move from the Ea$t with $o con$tant a bla$t, that $hips by favour there- of $ail pro$perou$ly to the <I>West-India's.</I> And from the $ame coa$ting along the <I>Mexican</I> $hore, they with the $ame felicity pa$s the <I>Pacifick</I> Ocean towards the <I>India's</I>; which to us are Ea$t, but <marg><I>The cour$e to the We$t</I>-India's <I>ea- $ie, the return dif- ficult.</I></marg> to them are We$t. Whereas on the contrary the Cour$e from thence towards the Ea$t is difficult and uncertain, and not to be made by the $ame Rhumb, but mu$t vere more to Land-ward, to recover other Winds, which we may call accidentary and tumul- tuary, produced from other Principles, as tho$e that inhabit the continent find by experience. Of which productions of Winds, the Cau$es are many and different, which $hall not at this time be <marg><I>Winds from Land make rough the Seas.</I></marg> mentioned. And the$e accidentary Winds are tho$e which blow indifferently from all parts of the Eatth, and make rough the Seas remote from the Equinoctial, and environed by the rugged Sur- face of the Earth; which is as much as to $ay environ'd with tho$e perturbations of Air, that confound that primary Gale. The which, in ca$e the$e accidental impediments were removed, would be continually felt, and e$pecially upon the Sea. Now $ee how the effect of the Water and Air $eem wonderfully to ac- cord with the Cele$tial ob$ervations, to confirm the mobility of our Terre$trial Globe.</P> <marg><I>Another ob$erva- tion taken from the Air in confirmati- on of the motion of the Earth.</I></marg> <P>SAGR. I al$o for a final clo$e will relate to you one particular, which as I believe is unknown unto you, and which likewi$e may $erve to confirm the $ame conclu$ion: You <I>Salviatus</I> alledged, That Accident which Sailers meet with between the Tropicks; I mean that perpetual Gale of Winde that beats upon them from the Ea$t, of which I have an account from tho$e that have many times made the Voyage: And moreover (which is very ob$er- vable) I under$tand that the Mariners do not call it a <I>Wind,</I> but <marg>Which Wind with our Engli$h Mariners is called the <I>Trade-wind.</I></marg> by another ^{*} name, which I do not now remember, taken haply from its $o fixed and con$tant Tenor; which when they have met with, they tie up their $hrouds and other cordage belonging to the Sails, and without any more need of touching them, though they be in a $leep, they can continue their cour$e. Now this con$tant Trade-wind was known to be $uch by its continual blowing with- out interruptions; for if it were interrupted by other Windes, it would not have been acknowledged for a $ingular Effect, and different from the re$t: from which I wlll infer, That it may be that al$o our Mediterranean Sea doth partake of the like accident; but it is not ob$erved, as being frequently altered by the conflu- ence of other windes. And this I $ay, not without good grounds, yea upon very probable conjectures whch came unto my know- ledge, from that which tendred it $elf to my notice on occa$ion of the voyage that I made into <I>Syria,</I> going Con$ul for this Nation <foot>to</foot> <p n=>403</p> to <I>Aleppo,</I> and this it is: That keeping a particular account and <marg><I>The voiages in the Mediterrane from Ea$t to We$t are made in $horter times than from We$t to Ea$t.</I></marg> memorial of the dayes of the departure and arrival of the Ships in the Ports of <I>Alexandria,</I> of <I>Alexandretta,</I> and this of <I>Venice</I>; in comparing $undry of them, which I did for my curio$ity, I found that in exactne$s of account the returns hither, that is the voiages from Ea$t to We$t along the Mediterrane, are made in le$s time then the contrary cour$es by 25. in the Hundred: So that we $ee that one with another, the Ea$tern windes are $tronger then the We$tern.</P> <P>SALV. I am very glad I know this particular, which doth not a little make for the confirmation of the Earths mobility. And although it may be alledged, That all the Water of the Mediter- rane runs perpetually towards the Straits-mouth, as being to di$imbogue into the Ocean, the waters of as many Rivers, as do di$charge them$elves into the $ame; I do not think that that cur- rent can be $o great, as to be able of it $elf alone to make $o no- table a difference: which is al$o manife$t by ob$erving that the water in the Pharo of <I>Sicily</I> runneth back again no le$s towards the Ea$t, than it runneth forwards towards the We$t.</P> <P>SAGR. I, that have not as <I>Simplicius,</I> an inclination to $ati$- fie any one be$ides my $elf, am $atisfied with what hath been $aid as to this fir$t particular: Therefore <I>Salviatus,</I> when you think it fit to proceed forward, I am prepared to hear you.</P> <P>SALV. I $hall do as you command me, but yet I would fain hear the opinion al$o of <I>Simplicius,</I> from who$e judgement I can argue how much I may promi$e to my $elf touching the$e di$- cour$es from the <I>Peripatetick</I> Schools, if ever they $hould come to their ears.</P> <P>SIMP. I de$ire not that my opinion $hould $erve or $tand for a mea$ure, whereby you $hould judge of others thoughts; for as I have often $aid, I am incon$iderable in the$e kinde of $tudies, and $uch things may come into the mindes of tho$e that are enter- ed into the deepe$t pa$$ages of Philo$ophy, as I could never think of; as having (according to the Proverb) $carce ki$t her Maid: yet neverthele$s, to give you my $udden thoughts, I $hall tell you, That of tho$e effects by you recounted, and particularly the la$t, there may in my judgement very $ufficient Rea$ons be given without the Earths mobility, by the mobility of the Heavens one- ly; never introducing any novelty more, than the inver$ion of that which you your $elf propo$e unto us. It hath been received <marg><I>It is demon$tra- ted inverting the argument, that the perpetual mo- tion of the Air from Ea$t to We$t cometh from the motion of Heaven?</I></marg> by the <I>Peripatetick</I> Schools, that the Element of Fire, and al$o a great part of the Aire is carried about according to the Diurnal conver$ion from Ea$t to We$t, by the contact of the Concave of the Lunar Orb, as by the Ve$$el their container. Now without going out of your track, I will that we determine the Quantity of <foot>Eee 2 the</foot> <p n=>404</p> <marg><I>It is demon$trated inverting the ar- gument, that the perpetual motion of the Air from Ea$t to We$t, cometh from the motion of Heaven.</I></marg> the Aire which partaketh of that motion to di$tend $o low as to the Tops of the highe$t Hills, and that likewi$e they would reach to the Earth, if tho$e Mountains did not impede them, which agreeth with what you $ay: For as you affirm, the Air, which is invironed by ledges of Mountains, to be carried about by the a$perity of the moveable Earth; we on the contrary $ay, That the whole Element of Air is carried about by the motion of Heaven, that part only excepted which lyeth below tho$e bodies, which is hindred by the a$perity of the immoveable Earth. And whereas you $aid, That in ca$e that a$perity $hould be removed, the Air would al$o cea$e to be whirld about; we may $ay, That the $aid a$perity being removed, the whole Aire would con- tinue its motion. Whereupon, becau$e the $urfaces of $pacious Seas are $mooth, and even; the Airs motion $hall continue upon tho$e, alwaies blowing from the Ea$t: And this is more $en$ibly perceived in Climates lying under the Line, and within the Tro- picks, where the motion of Heaven is $wifter; and like as that Cele$tial motion is able to bear before it all the Air that is at liberty, $o we may very rationally affirm that it contributeth the $ame motion to the Water moveable, as being fluid and not con- nected to the immobility of the Earth: And with $o much the <marg><I>The motion of the Water dependeth on the motion of Heaven.</I></marg> more confidence may we affirm the $ame, in that by your con- fe$$ion, that motion ought to be very $mall in re$ect of the efficient Cau$e; which begirting in a natural day the whole Terre$trial Globe, pa$$eth many hundreds of miles an hour, and e$pecially towards the Equinoctial; whereas in the currents of the open Sea, it moveth but very few miles an hour. And thus the voiages to- wards the We$t $hall come to be commodious and expeditious, not onely by rea$on of the perpetual Ea$tern Gale, but of the cour$e al$o of the Waters; from which cour$e al$o perhaps the Ebbing and Flowing may come, by rea$on of the different $citu- <marg><I>The flux and reo flux may d<*>d on the diurual mo- tion of Heaven.</I></marg> ation of the Terre$trial Shores: again$t which the Water coming to beat, may al$o return backwards with a contrary motion, like as experience $heweth us in the cour$e of Rivers; for according as the Water in the unevenne$s of the Banks, meeteth with $ome parts that $tand out, or make with their Meanders $ome Reach or Bay, here the Water turneth again, and is $een to retreat back a con$iderable $pace. Upon this I hold, That of tho$e effects from which you argue the Earths mobility, and alledge it as a cau$e of them, there may be a$$igned a cau$e $ufficiently valid, retaining the Earth $tedfa$t, and re$toring the mobility of Heaven.</P> <P>SALV. It cannot be denied, but that your di$cour$e is ingenious, & hath much of probability, I mean probability in appearance, but not in reality & exi$tence: It con$i$teth of two parts: In the fir$t it <foot>a$$ignes</foot> <p n=>405</p> a$$ignes a rea$on of the continual motion of the Ea$tern Winde, and al$o of a like motion in the Water. In the $econd, It would draw from the $ame Sour$e the cau$e of the Ebbing and Flowing. The fir$t part hath (as I have $aid) $ome appearance of probabi- lity, but yet extreamly le$s then that which we take from the Terre$trial motion. The $econd is not onely wholly improbable, but altogether impo$$ible and fal$e. And coming to the fir$t, <marg><I>A rea$on of the continual motion of the Air and Wa- ter may be given, making the Earth moveable, then by making it immove- able.</I></marg> whereas it is $aid that the Concave of the Moon carrieth about the element of Fire, and the whole Air, even to the tops of the higher Mountains. I an$wer fir$t, that it is dubious whether there be any element of Fire: But $uppo$e there be, it is much doubted of the Orbe of the Moon, as al$o of all the re$t; that is, Whether there be any $uch $olid bodies and va$t, or el$s, Whether beyond the Air there be extended a continuate expan$ion of a $ub$tance of much more tenuity and purity than our Air, up and down which the Planets go wandring, as now at la$t a good part of tho$e very Phylo$ophers begin to think: But be it in this or in <marg><I>Its improbable that the element of Fire $hould be carried round by the Con- cave of the Moon.</I></marg> that manner, there is no rea$on for which the Fire, by a $imple contract to a $uperficies, which you your $elf grant to be $mooth and ter$e, $hould be according to its whole depth carried round in a motion different from its natural inclination; as hath been de- fu$ely proved, and with $en$ible rea$ons demon$trated by^{+} <I>Il Sag-</I> <marg>+ A Treati$e of our Author formerly cited.</marg> <I>giatore</I>: Be$ides the other improbability of the $aid motions transfu$ing it $elf from the $ubtile$t Fire throughout the Air, much more den$e; and from that al$o again to the Water. But that a body of rugged and mountainous $urface, by revolving in it $elf, $hould carry with it the Air contiguous to it, and again$t which its promontaries beat, is not onely probable but nece$$ary, and experience thereof may be daily $een; though without $ee- ing it, I believe that there is no judgement that doubts thereof. As to the other part, $uppo$ing that the motion of Heaven did carry round the Air, and al$o the Water; yet would that motion for all that have nothing to do with the Ebbing and Flowing. For being that from one onely and uniform cau$e, there can fol- <marg><I>The Ebbing and Flowing cannot de- pend on the motion of Heaven.</I></marg> low but one $ole and uniform effect; that which $hould be di$co- vered in the Water, would be a continuate and uniform cour$e from Ea$t to We$t; and in that a Sea onely, which running com- pa$s environeth the whole Globe. But in determinate Seas, $uch as is the Mediterrane $hut up in the Ea$t, there could be no $uch motion. For if its Water might be driven by the cour$e of Heaven towards the We$t, it would have been dry many ages $ince: Be$ides that our Water runneth not onely towards the We$t, But returneth backwards towards the Ea$t, and that in or- dinal Periods: And whereas you $ay by the example of Rivers, that though the cour$e of the Sea were Originally that onely <foot>from</foot> <p n=>406</p> from Ea$t to We$t, yet neverthele$s the different Po$ition of the Shores may make part of the Water regurgitate, and return backwards: I grant it you, but it is nece$$ary that you take no- tice my <I>Simplicius,</I> that where the Water upon that account returneth backwards, it doth $o there perpetually; and where it runneth $traight forwards, it runneth there alwayes in the $ame manner; for $o the example of the Rivers $hewes you: But in the ca$e of the ebbing and flowing, you mu$t finde and give us $ome rea$on why it doth in the $elf $ame place run one while one way, and another while another; Effects that being contrary & irregular, can never be deduced from any uniform and con$tant Cau$e: And this Argument, that overthrows the Hypothe$is of the mo- tion contributed to the Sea from the Heavens diurnal motion, doth al$o confute that Po$ition of tho$e who would admit the $ole diurnal motion of the Earth, and believe that they are able with that alone to give a rea$on of the Flux and Reflux: Of which effect $ince it is irregular, the cau$e mu$t of nece$$ity be irregular and alterable.</P> <P>SIMP. I have nothing to reply, neither of my own, by rea$on of the weakne$s of my under$tanding; nor of that of others, for that the Opinion is $o new: But I could believe that if it were $pread among$t the Schools, there would not want Phylo$ophers able to oppo$e it.</P> <P>SAGR. Expect $uch an occa$ion; and we in the mean time if it $eem good to <I>Salviatus,</I> will proceed forward.</P> <P>SALV. All that which hath been $aid hitherto, pertaineth to the diurnal period of the ebbing and flowing; of which we have in the fir$t place demon$trated in general the primary and univer$al Cau$e, without which, no $uch effect would follow: Afterw ds pa$$ing to the particular Accidents, various, and in a certain $ort irregular, that are ob$erved therein: We have handled the $econ- dary and concommitant Cau$es upon which they depend. Now follow the two other Periods, Monethly, and Annual, which do not bring with them new and different Accidents, other than tho$e already con$idered in the diurnal Period; but they ope- rate on the $ame Accidents, by rendring them greater and le$$er in $everal parts of the Lunar Moneth, and in $everal times of the Solar Year; as if that the Moon and Sun did each conceive it $elf apart in operating and producing of tho$e Effects; a thing that totally cla$heth with my under$tanding, which $eeing how that this of Seas is a local and $en$ible motion, made in an im- men$e ma$s of Water, it cannot be brought to $ub$cribe to Lights, to temperate Heats, to predominacies by occult Quali- ties, and to $uch like vain Imaginations, that are $o far from be- ing, or being po$$ible to be Cau$es of the Tide; that on the con- <foot>trary</foot> <p n=>407</p> trary, the Tide is the cau$e of them, that is, of bringing them into the brains more apt for loquacity and o$tentation, than for the $peculation and di$covering of the more ab$tru$e $ecrets of Nature; which kind of people, before they can be brought to prononnce that wi$e, ingenious, and mode$t $entence, <I>I know it not,</I> $uffer to e$cape from their mouths and pens all manner of ex- travagancies. And the onely ob$erving, that the $ame Moon, and the $ame Sun operate not with their light with their motion, with great heat, or with temperate, on the le$$er reeeptaces of Water, but that to effect their flowing by heat, they mu$t be reduced to little le$$e than boiling, and in $hort, we not being able artificially to imitate any way the motions of the Tide, $ave only by the mo- tion of the Ve$$el, ought it not to $atisfie every one, that all the other things alledged, as cau$es of tho$e e$$ects, are vaine fancies, and altogether e$tranged from the Truth. I <marg><I>The al<*>ations in the effe<*>s argue alteration in the cau$e.</I></marg> $ay, therefore, that if it be true, that of one effect there is but one $ole primary cau$e, and that between the cau$e and effect, there is a firm and con$tant connection; it is nece$$ary that when- $oever there is $een a firm and con$tant alteration in the effect, there be a firm and con$tant alteration in the cau$e. And be- cau$e the alterations that happen in the ebbing and flowing in $everal parts of the Year and Moneths, have their periods firm and con$tant, it is nece$$ary to $ay, that a regular alteration in tho$e $ame times happeneth in the primary cau$e of the ebbings and flowings. And as for the alteration that in tho$e times happens <marg><I>The cau$es at large a$$igned of the Periods Mo- nethly and Annu- al of the ebbing and flowing.</I></marg> in the ebbings and flowings con$i$teth onely in their greatne$s; that is, in the Waters ri$ing and falling more or le$$e, and in running with greater or le$$e <I>impetus</I>; therefore it is nece$$ary, that that which is the primary cau$e of the ebbing and flowing, doth in tho$e $ame determinate times increa$e and dimini$h its force. But we have already concluded upon the inequality and irregularity of the motion of the Ve$$els containing the Water to be the primary cau$e of the ebbings and flowings. Therefore it is nece$$ary, that that irregularity, from time to time, corre- $pondently grow more irregular, that is, grow greater and le$$er. Now it is requi$ite, that we call to minde, that the irregularity, that is, the different velocity of the motions of the Ve$$els, to wit, of the parts of the Terre$trial Superficies, dependeth on their moving with a compound motion, re$ulting from the com- mixtion of the two motions, Annual and Diurnal, proper to the whole Terre$trial Globe; of which the Diurnal conver$ion, by one while adding to, and another while $ub$tracting from, the Annual motion, is that which produceth the irregularity in the compound motion; $o that, in the additions and $ub$tractions, that the Diurnal revolution maketh from the Annual motion, <foot>con-</foot> <p n=>408</p> con$i$teth the original cau$e of the irregular motion of the Ve$- $els, and con$equently of the Ebbing and Flowing: in$omuch <marg><I>The monethly and annual altera- tions of the tide can depend upon no- thing, $ave on the alteration of the additions & $ub- $tractions of the diurnal period from the annual.</I></marg> that if the$e additions and $ub$tractions $hould alwayes proceed in the $ame proportion, in re$pect of the Annual motion, the cau$e of the Ebbing and Flowing would indeed continue, but yet $o as that they would perpetually return in the $elf $ame man- ner: But we are to finde out the cau$e of making the $ame Eb- bings and Flowings in divers times greater and le$$er: There- fore we mu$t (if we will retain the identity of the cau$e) find the alteration in the$e <*>tions and $ub$tractions, that make them more & le$s potent, in producing tho$e effects which depend there- upon. But I $ee not how that potency and impotence can be intro- duced, unle$$e by making the $ame additions and $ub$tractions, one while greater, and another while le$$er; $o that the accelera- tion and the retardment of the compound motion, may be made, $ometimes in greater, and $ometimes in le$$er proportion.</P> <P>SAGR. I feel my $elf very gently led, as it were, by the hand, and though I finde no rubs in the way, yet neverthele$$e, like a blind man, I $ee not whether your Clue leadeth me, nor can I imagine where $uch a Journey will end.</P> <P>SALV. Though there be a great difference between my $low pac't Philo$ophy, and your more nimble Rea$on, yet neverthe- le$$e, in this particular which we are now upon, I do not much wonder, if the apprehen$ivene$$e of your wit be a little ob$cu- red by the dark and thick mi$t that hides the mark, at which we aime: and that which le$$eneth my admiration is, the remem- brance of the many hours, many dayes, yea more, many nights that I have con$umed in this contemplation, and of the many times that, de$pairing to bring it to a period, I have, for an in- couragement of my $elf, indeavoured to believe, by the exam- ple of the unfortunate <I>Orlando,</I> that that might not po$$ibly be true, which yet the te$timony of $o many credible men $et be- fore my eyes: wonder not, therefore, if this once, contrary to your cu$tome, you do not fore$ee what I intend: and if you will needs admire, I believe that the event, as far as I can judge un- expected, will make you cea$e your wonderment.</P> <P>SAGR. I thank God, that he did not permit that de$peration of yours to end in the <I>Exit</I> that is fabled of the mi$erable <I>Or- lando,</I> nor in that which haply is no le$$e fabulou$ly related of <I>Ari$totle,,</I> that $o neither my $elf nor others $hould be deprived of the di$covery of a thing, as ab$tru$e as it was de$irable: I be$eech you, therefore, to $atisfie my eager appetite as $oon as you can.</P> <P>SALV. I am ready to $erve you: We were upon an inquiry in what manner the additions and $ub$tractions of the Terre$tri- <foot>all</foot> <p n=>409</p> all conver$ion from the Annual motion, could be made, one while in a greater, and another while in a le$$er proportion; which diver$ity, and no other thing, could be a$$igned for the cau$e of the alterations, Monethly and Annual, that are $een in the greatne$$e of the Ebbings and Flowings. I will now con- $ider how this proportion of the additions and $ub$tractions of <marg><I>Three wayes of altering the pro- portion of the ad- ditions of the diur- nal Revolution to the annual motion.</I></marg> the Diurnal Revolution, and Annual motion may grow greater and le$$er three $everal wayes. One is by increa$ing and dimi- ni$hing the velocity of the Annual motion, retaining the additi- ons and $ub$tractions made by the Diurnal conver$ion in the $ame greatne$$e, becau$e the Annual motion being about three times greater, that is, more velocious than the Diurnal motion (con$idered likewi$e in the Grand Circle) if we increa$e it anew, the additions and $ub$tractions of the Diurnal motion will occa$ion le$$e alteration therein: but, on the other $ide, making it more $low, it will be altered in greater proportion, by that $ame diurnal motion, ju$t as the adding or $ub$tracting four degrees of velocity from one that moveth with twenty de- grees, altereth his cour$e le$$e, than tho$e very four degrees would do, added or $ub$tracted from one that $hould move onely with ten degrees. The $econd way would be, by making the additi- ons and $ub$tractions greater and le$$er, retaining the annual mo- tion in the $ame velocity; which is as ea$ie to be under$tood, as it is manife$t, that a velocity <I>v. gr.</I> of 20. degr. is more altered by the addition or $ub$traction of 10. deg. than by the addition or $ub$tra- ction of 4. The third way would be, in ca$e the$e two were joyned together, dimini$hing the annual motion, & increa$ing the diurnal additions and $ub$tractions. Hitherto, as you $ee, it was no hard matter <*>o attain, but yet it proved to me very hard to find by what means this might be effected in Nature. Yet in the end, <marg><I>That which <*> us is hard to be un- der$tood, i<*> with Nature ea$ie to be effected.</I></marg> I finde that $he doth admirably make u$e thereof, and in wayes almo$t incredible: I mean, admirable and incredible to us, but not to her, who worketh even tho$e very things, which, to our capacity, are of infinite wonder, with extraordinary facility and $implicity: and that which it is hard for us to under$tand, is ea- $ie for her to effect. Now to proceed, having $hewn that the proportion between the additions and $ub$tractions of the Diur- nal conver$ion and Annual motion may be made greater and le$- $er, two wayes, (and I $ay two, becau$e the third is comprized in the two fir$t) I adde, that Nature maketh u$e of them both: and farthermore, I $ubjoyn, that if $he did make u$e but of one alone, it would be nece$$ary to take away one of the two Perio- dical alterations. That of the Monethly Period would cea$e, if <marg><I>If the Diurnal motion $hould not alter, the annual Period would cea$e</I></marg> the annual motion $hould not alter. And in ca$e the additions and $ub$tractions of the diurnal revolution $hould continually <foot>Fff be</foot> <p n=>410</p> be equal, the alterations of the annual Period would fail.</P> <P>SAGR. It $eems then, that the Monethly alteration of eb- bings and flowings dependeth on the alteration of the annual motion of the Earth? And the annual alteration of tho$e eb- bings and flowings do, it $eems, depend on the additions and $ub$tractions of the diurnal conver$ion? And here now I finde my $elf wor$e puzzled than before, and more out of hope of being able to comprehend how this intricacy may be, which is more inextricable, in my judgment, than the Gordian knot. And I envy <I>Simplicius,</I> from who$e $ilence I argue that he doth ap- prehend the whole bu$ine$$e, and is acquit of that confu$ion which greatly puzzleth my brains.</P> <P>SIMP. I believe verily, <I>Sagredus,</I> that you are put to a a $tand; and I believe that I know al$o the cau$e of your con- fu$ion, which, if I mi$take not, ri$eth from your under$tanding part of tho$e particulars but even now alledged by <I>Salviatus,</I> and but a part. It is true likewi$e that I find my $elf free from the like confu$ion; but not for that cau$e as you think, to wit, be- cau$e I apprehend the whole, nay it happens upon the quite contrary account; namely, from my not comprehending any thing; and confu$ion is in the plurality of things, and not in nothing.</P> <P>SAGR. You $ee <I>Salviatus,</I> how a few checks given to <I>Simpli- cius</I> in the dayes preceding, have rendered him gentle, and brought him from the <I>capriol</I> to the <I>amble.</I> But I be$eech you without farther delay, put us both out of $u$pence.</P> <P>SALV. I will endeavour it to the utmo$t of my har$h way of expre$$ing my $elf, the obtu$ene$$e of which, the acutene$$e of your wit $hall $upply. The accidents of which we are to enquire the cau$es are two: The fir$t re$pecteth the varieties that happen in the ebbings and flowings in the Monethly Period; and the o- thr relateth to the Annual. We will fir$t $peak of the Moneth- ly, and then treat of the Annual; and it is convenient that we re$olve them all according to the Fundamentals and Hypothe$is already laid down, without introducing any novelty either in A- $tronomy, or in the Univer$e, in favour of the ebbings and flow- ings; therefore let us demon$trate that of all the $everal acci- dents in them ob$erved, the cau$es re$ide in the things already <marg><I>The true Hypo- the$is may di$patch its revolutions in a $horter time, in le$$er circles than in greater; the which is proved by two examples.</I></marg> known, and received for true and undoubted. I $ay therefore, that it is a truly natural, yea nece$$ary thing, that one and the $ame moveable made to move round by the $ame moving virtue in a longer time, do make its cour$e by a greater circle, rather than by a le$$er; and this is a truth received by all, and con- firmed by all experiments, of which we will produce a few. <marg><I>The fir$t ex- ample.</I></marg> In the wheel-clocks, and particularly in the great ones, to mo- <foot>derate</foot> <p n=>411</p> derate the time, the Artificers that make them accomodate a cer- tain voluble $taffe horozontally, and at each end of it they fa- $ten two Weights of Lead, and when the time goeth too $low, by the onely removing tho$e Leads a little nearer to the centre of the $taffe, they render its vibrations more frequent; and on the contrary to retard it, it is but drawing tho$e Weights more towards the ends; for $o the vibrations are made more $eldome, and con$equently the intervals of the hours are prolonged.</P> <P>Here the movent vertue is the $ame, namely the counterpoi$e, <marg><I>The $econd ex- ample.</I></marg> the moveables are tho$e $ame Weights of lead, and their vi- brations are more frequent when they are neerer to the centre, that is, when they move by le$$er circles. Hanging equal Weights at unequal cords, and being removed from their per- pendicularity, letting them go; we $hall $ee tho$e that are pen- dent at the $horter cords, to make their vibrations under $horter times, as tho$e that move by le$$er circles. Again, let $uch a kind of Weight be fa$tened to a cord, which cord let play upon a $taple fa$tened in the Seeling, and do you hold the other end of the cord in your hand, and having given the motion to the pendent Weight, whil$t it is making its vibrations, pull the end of the cord that you hold in your hand, $o that the Weight may ri$e higher and higher: In its ri$ing you $hall $ee the fre- quency of its vibrations encrea$e, in regard that they are made $ucce$$ively by le$$er and le$$er circies. And here I de$ire you to <marg><I>Two particular notable accidents in the</I> pendu<*> <I>and their vibrations.</I></marg> take notice of two particulars worthy to be ob$erved. One is that the vibrations of one of tho$e plummets are made with $uch a nece$$ity under $uch determinate times, that it is altogether impo$$ible to cau$e them to be made under other times, unle$$e it be by prolonging, or abreviating the cord; of which you may al$o at this very in$tant a$certain your $elves by experience, tying a $tone to a pack-threed, and holding the other end in your hand, trying whether you can ever by any artifice be able to $wing it this way and that way in other than one determinate time, unle$$e by lengthening or $hortening the $tring, which you will find to be ab$olutely impo$$ible. The other particular truly admirable is, that the $elf $ame <I>pendulum</I> makes its vibra- tions with one and the $ame frequency, or very little, and as it were in$en$ibly different, whether they be made by very great, or very $mall arches of the $elf-$ame circumference. I mean that whether we remove the <I>pendulum</I> from perpendicularity one, two, or three degrees onely, or whether we remove it 70. 80. nay to an entire quadrant, it being let go, will in the one ca$e and in the other make its vibrations with the $ame frequency, as well the former where it is to move by an arch of but four or $ix de- grees, as the $econd, where it is to pa$$e arches of 160. or more <foot>Fff 2 de-</foot> <p n=>412</p> degrees. Which may the better be $een, by hanging two weights at two $trings of equal length, and then removing them from per- pendicularity, one a little way, and the other very far; the which being $et at liberty, will go & return under the $ame times, the one by arches very $mall, & the other by very great ones, from whence followeth the conclu$ion of an admirable Problem; which is, <marg><I>Admirable Pro- blems of movea- bles de$cending by the Quadrant of a Circle, and of tho$e de$cending by all the cords of the whole Circle.</I></marg> That a Quadrant of a Circle being given (take a little diagram of the $ame, [in <I>Fig.</I> 3.]) as for in$tance: A B erect to the Hori- zon, $o as that it re$t upon the plain touching in the point B. and an Arch being made with a Hoop well plained and $moothed in the concave part, bending it according to the curvity of the Cir- cumference A D B. So that a Bullet very round and $mooth may freely run to and again within it (the rim of a Sieve is very proper for the experiment) I $ay, that the Bullet being put in any what ever place, neer or far from the lowe$t term B. As for in- $tance, putting it in the point C, or here in D, or in E; and then let go, it will in equal times, or in$en$ibly different arrive at the term B, departing from C, or from D, or from E, or from what- ever other place; an accident truly wonderfull. We may add another accident no le$s $trange than this, which is, That more- over by all the cords drawn from the point B to the points C, D, E; and to any other what$oever, taken not onely in the Qua- drant B A, but in all the whole circumference of the Circle the $aid moveable $hall de$cend in times ab$olutely equal; in$omuch that it $hall be no longer in de$cending by the whole Diameter erect perpendicularly upon the point B, then it $hall in de$cend- ing by B. C. although it do $ublend but one $ole degree, or a le$- $er Arch. Let us add the other wonder, which is, That the mo- tions of the falling bodies made by the Arches of the Quadrant A B; are made in $horter times than tho$e that are made by the cords of tho$e $ame Arches; $o that the $wifte$t motion, and made by a moveable in the $horte$t time, to arrive from the point A, to the term B, $hall be that which is made, not by the right line A, B, (although it be the $horte$t of all tho$e that can de drawn between the points A. B.) but by the circumference A D B. And any point being taken in the $aid Arch; as for example: The point D. and two co<*>ds drawn A D, and D. B. the moveable departing from the qoint A, $hall in a le$s time come to B, moving by the two cords A D and D B. than by the $ole cord A, B. But the $horte$t of all the times $hall be that of the fall by the Arch A D B. And the $elf $ame accidents are to be under$tood of all the other le$$er Arches taken from the lowermo$t term B. upwards.</P> <P>SAGR. No more, no more; for you $o confund and fill me with Wonders, and di$tract my thoughts $o many $everal wayes, <foot>that</foot> <p n=>413</p> that I fear I $hall have but a $mall part of it left free and di$in- gaged, to apply to the principal matter that is treated of, and which of it $elf is but even too ob$cure and intricate: So that I intreat you to vouch$afe me, having once di$patcht the bu$ine$s of the ebbings and flowings, to do this honour to my hou$e (and yours) $ome other dayes, and to di$cour$e upon the $o many other Problems that we have left in $u$pence; and which perhaps are no le$s curious and admirable, than this that hath been di$cu$$ed the$e dayes pa$t, and that now ought to draw to a con- clu$ion.</P> <P>SALV. I $hall be ready to $erve you, but we mu$t make more than one or two Se$$ions; if be$ides the other que$tions re$erved to be handled apart, we would di$cu$$e tho$e many that pertain to the local motion, as well of natural moveables, as of the reject- ed: an Argument largely treated of by our <I>Lyncean Accade- mick.</I> But turning to our fir$t purpo$e, where we were about to declare, That the bodies moving circularly by a movent virtue, which continually remaineth the $ame, the times of the circula- tions were prefixt and determined, and impo$$ible to be made longer or $horter, having given examples, and produced experi- ments thereof, $en$ible, and fea$ible, we may confirm the $ame truth by the experiences of the Cele$tial motions of the Planets; in which we $ee the $ame rule ob$erved; for tho$e that move by greater Circles, confirm longer times in pa$$ing them. A mo$t pertinent ob$ervation of this we have from the <I>Medicæan</I> Pla- nets, which in $hort times make their revolutions about <I>Jupiter</I>: In$omuch that it is not to be que$tioned, nay we may hold it for $ure and certain, that if for example, the Moon continuing to be moved by the $ame movent faculty, $hould retire by little and little in le$$er Circles, it would acquire a power of abreviating the times of its Periods, according to that <I>Pendulum,</I> of which in the cour$e of its vibrations, we by degrees $hortned the cord, that is contracted the Semidiameter of the circumferences by it pa$$ed. Know now that this that I have alledged an example of it in the Moon, is $een and verified e$$entially in fact. Let us call to mind, that it hath been already concluded by us, together with <I>Coperni-</I> <marg><I>The Earths an- nual motion by the Ecliptick, unequal by means of the Moons motion.</I></marg> <I>cus,</I> That it is not po$$ible to $eparate the Moon from the Earth, about which it without di$pute revolveth in a Moneth: Let us remember al$o that the Terre$trial Globe, accompanyed alwayes by the Moon, goeth along the circumference of the Grand Orb about the Sun in a year, in which time the Moon revolveth about the Earth almo$t thirteen times; from which revolution it follow- eth, that the $aid Moon $ometimes is found near the Sun; that is, when it is between the Sun and the Earth, and $ometimes much more remote, that is, when the Earth is $ituate between <foot>the</foot> <p n=>414</p> the Moon and Sun; neer, in a word, at the time of its conjun ction and change; remote, in its Full and Oppo$ition; and the greate$t vicinity differ the quantity of the Diameter of the Lu- nar Orb. Now if it be true that the virtue which moveth the Earth and Moon, about the Sun, be alwayes maintained in the $ame vigour; and if it be true that the $ame moveable moved by the $ame virtue, but in circles unequal, do in $horter times pa$$e like arches of le$$er circles, it mu$t needs be granted, that the Moon when it is at a le$$e di$tance from the Sun, that is in the time of conjunction, pa$$eth greater arches of the Grand Orb, than when it is at a greater di$tance, that is in its Oppp$ition and Full. And this Lunar inequality mu$t of nece$$ity be imparted to the Earth al$o; for if we $hall $uppo$e a right line produced from the centre of the Sun by the centre of the Terre$trial Globe, and prolonged as far as the Orb of the Moon, this $hall be the $emi- diameter of the Grand Orb, in which the Earth, in ca$e it were alone, would move uniformly, but if in the $ame $emidiameter we $hould place another body to be carried about, placing it one while between the Earth and Sun, and another while beyond the Earth, at a greater di$tance from the Sun, it is nece$$ary, that in this $econd ca$e the motion common to both, according to the circumference of the great Orb by means of the di$tance of the Moon, do prove a little $lower than in the other ca$e, when the Moon is between the Earth and Sun, that is at a le$$er di$tance. So that in this bu$ine$$e the very $ame happeneth that befals in the time of the clock; that lead which is placed one while farther $rom the centre, to make the vibrations of the $taffe or ballance le$$e frequent, and another while nearer, to make them thicker, repre$enting the Moon. Hence it may be manife$t, that the annual motion of the Earth in the Grand Orb, and under the Ecliptick, is not uniform, and that its ir- regularity proceedeth from the Moon, and hath its Monethly Periods and Returns. And becau$e it hath been concluded, that the Monethly and Annual Periodick alterations of the ebbings and flowings, cannot be deduced from any other cau$e than from the altered proportion between the annual motion and the additions and $ub$tractions of the diurnal conver$ion; and that tho$e alterations might be made two wayes, that is by altering the annual motion, keeping the quantity of the additions un- altered, or by changing of the bigne$$e of the$e, reteining the uniformity of annual motion. We have already found the fir$t of the$e, depending on the irregularity of the annual motion occa$ioned by the Moon, and which hath its Monethly Periods. It is therefore nece$$ary, that upon that account the ebbings and flowings have a Monethly Period in which they do grow <foot>greater</foot> <p n=>415</p> greater and le$$er. Now you $ee that the cau$e of the Monethly Period re$ideth in the annual motion; and withal you $ee how much the Moon is concerned in this bu$ine$s, and how it is there- with interrupted apart, without having any thing to do with either, with Seas or Waters.</P> <P>SAGR. If one that never had $een any kinde of Stairs or La- der, were $hewed a very high Tower, and asked if ever he hoped to climb to the top of it, I verily believe that he would an$wer he did not, not conceiving how one $hould come thither any way except by flying; but $hewing him a $tone of but a foot high, and asking him whether he thought he could get to the top of that, I am certain that he would an$wer he could; and farther, that he would not deny, but that it was not onely one, but ten, twenty, and an hundred times ea$ier to climb that: But now if he $hould be $hewed the Stairs, by means whereof, with the facility by him granted, it is po$$ible to get thither, whither he a little before had affirmed it was impo$$ible to a$cend, I do think that laughing at him$elf he would confe$s his dulne$s of apprehen$ion. Thus, <I>Salviatus,</I> have you $tep by $tep $o gently lead me, that, not without wonder, I finde that I am got with $mall pains to that height which I de$paired of arriving at. 'Tis true; that the Stair- ca$e having been dark, I did not perceive that I was got nearer to, or arrived at the top, till that coming into the open Air I di$- covered a great Sea, and $pacious Country: And as in a$cending one $tep, there is no labour; $o each of your propo$itions by it $elf $eemed to me $o plain, that thinking I heard but little or no- thing that was new unto me, I conceived that my benefit thereby had been little or none at all: Whereupon I was the more ama- zed at the unexpected <I>exit</I> of this di$cour$e, that hath guided me to the knowledge of a thing which I held impo$$ible to be de- mon$trated. One doubt onely remains, from which I de$ire to be freed, and this it is; Whether that if the motion of the Earth together with that of the Moon under the Zodiack are irregular motions, tho$e irregularities ought to have been ob$erved and ta- ken notice of by <I>A$tronomers,</I> which I do not know that they are: Therefore I pray you, who are better acquainted with the$e things than I, to free me from this doubt, and tell me how the ca$e $tands.</P> <P>SALV. You ask a rational que$tion, and an$wering to the Ob- <marg><I>Many things may remain as yet unob$erved in A- $tronomy.</I></marg> jection, I $ay; That although <I>A$tronomy</I> in the cour$es of many ages hath made a great progre$s in di$covering the con$titution and motions of the Cele$tial bodies, yet is it not hitherto arrived at that height, but that very many things remain undecided, and haply many others al$o undi$covered. It is to be $uppo$ed that the fir$t ob$ervers of Heaven knew no more but one motion common <foot>to</foot> <p n=>416</p> to all the Stars, as is this diurnal one: yet I believe that in few dayes they perceived that the Moon was incon$tant in keeping company with the other Stars; but yet withal, that many years pa$t, before that they di$tingui$hed all the Planets: And in par- ticular, I conceit that <I>Saturn</I> by its $lowne$s, and <I>Mercury</I> by rea- <marg>Saturn <I>for its $lowne$s, and</I> Mer- cury <I>for its rare- ne$s of appearing were among$t tho$e that were la$t ob- $erved.</I></marg> $on of its $eldom appearing, were the la$t that were ob$erved to be wand<*>ing and errant. It is to be thought that many more years run out before the $tations and retrogradations of the three $uperiour Planets were known, as al$o their approximations and rece$$ions from the Earth, nece$$ary occa$ions of introducing the Eccentrix and Epi<*>cles, things unknown even to <I>Ari$totle,</I> for that he makes no mention thereof. <I>Mercury,</I> and <I>Venus,</I> with their admirable apparitions; how long did they keep A$trono- mers in $u$pence, before that they could re$olve (not to $peak of any other of their qualities) upon their $ituation? In$omuch that the very order onely of the Mundane bodies, and the inte- gral $tructure of the parts of the Univer$e by us known, hath been doubted of untill the time of <I>Copernicus,</I> who hath at la$t given us notice of the true con$titution, and real $y$teme, according to which tho$e parts are di$po$ed; $o that at length we are certain that <I>Mercury, Venus,</I> and the other Planets do revolve about the Sun; and that the Moon revolveth about the Earth. But <marg><I>Particular $tru- ctures of the Orbs of the Planets not yet well re$olved.</I></marg> how each Planet governeth it $elf in its particular revolution, and how preci$ely the $tructure of its Orb is framed; which is that which is vulgarly called the <I>Theory</I> of the <I>Planets,</I> we cannot as yet undoubtedly re$olve. <I>Mars,</I> that hath $o much puzled our Modern A$tronomers, is a proof of this: And to the Moon her $elf there have been a$$igned $everal Theories, after that the $aid <I>Copernicus</I> had much altered it from that of <I>Ptolomy.</I> And to de$cend to our particular ca$e, that is to $ay, to the apparent mo- tion of the Sun and Moon; touching the former, there hath been ob$erved a certain great irregularity, whereby it pa$$eth the two <marg><I>The Sun pa$$- eth one half of the Zodiack nine days $ooner than the other.</I></marg> $emicircles of the Ecliptick, divided by the points of the Equi- noxes in very different times; in pa$$ing one of which, it $pend- eth about nine dayes more than in pa$$ing the other; a difference, as you $ee, very great and notable. But if in pa$$ing $mall arches, $uch for example as are the twelve Signs, he maintain a mo$t re- gular motion, or el$e proceed with paces, one while a little more $wift, and another more $low, as it is nece$$ary that it do, in ca$e the annual motion belong to the Sun onely in appearance, but in reality to the Earth in company with the Moon, it is what hath not hitherto been ob$erved, nor it may be, $ought. Touching <marg><I>The Moons mo- tion principally $ought in the ac- count of Eclip$es.</I></marg> the Moon in the next place, who$e re$titutions have been prin- cipally lookt into an account of the Eclip$es, for which it is $uf- ficient to have an exact knowledge of its motion about the Earth, <foot>it</foot> <p n=>417</p> it hath not been likewi$e with a perfect curio$ity inquired, what its cour$e is thorow the particular arches of the Zodiack. That therefore the Earth and Moon in running through the Zodiack, that is round the Grand Orb, do $omewhat accellerate at the Moons change, and retard at its full, ought not to be doubted; for that the $aid difference is not manife$t, which cometh to be unob$erved upon two accounts; Fir$t, Becau$e it hath not been lookt for. Secondly, Becau$e that its po$$ible it may not be very great. Nor is there any need that it $hould be great, for the pro- ducing the effect that we $ee in the alteration of the greatne$s of ebbings and flowings. For not onely tho$e alterations, but the <marg><I>Ebbings and flowings are petty things in com<*>ari- $on of the va$tne$s of Seas, and of the velocity of the mo- tion of the Terre- $trial Globe.</I></marg> Tides them$elves are but $mall matters in re$pect of the grandure of the $ubjects on which they work; albeit that to us, and to our littlene$s they $eem great. For the addition or $ubduction of one degree of velocity where there are naturally 700, or 1000, can be called no great alteration, either in that which conferreth it, or in that Which receiveth it: the Water of our Mediterrane carried about by the diurnal revolution, maketh about 700 miles an hour, (which is the motion common to the Earth and to it, and therefore not perceptible to us) & that which we $en$ibly di$cern to be made in the $treams or currents, is not at the rate of full one mile an hour, (I $peak of the main Seas, and not of the Straights) and this is that which altereth the fir$t, naturall, and grand mo- tion; and this motion is very great in re$pect of us, and of Ships: for a Ve$$el that in a $tanding Water by the help of Oares can make <I>v. g.</I> three miles an hour, in that $ame current will row twice as far with the $tream as again$t it: A notable difference in the motion of the Boat, though but very $mall in the motion of the Sea, which is altered but its $even hundredth part. The like I $ay of its ri$ing, and falling one, two, or three feet; and $carcely four or five in the utmo$t bounds of a $treight, two thou- $and, or more miles long, and where there are depths of hundreds of feet; this alteration is much le$s than if in one of the Boats that bring us fre$h Water, the $aid Water upon the arre$t of the Boat $hould ri$e at the Prow the thickne$s of a leaf. I conclude therefore that very $mall alterations in re$pect of the immen$e greatne$s, and extraordinary velocity of the Seas, is $ufficient to make therein great mutations in relation to our $mallne$s, and to our accidents.</P> <P>SAGR. I am fully $atisfied as to this particular; it remains to declare unto us how tho$e additions and $ub$tractions derived from the diurnal <I>Vertigo</I> are made one while greater, and ano- ther while le$$er; from which alterations you hinted that the an- nual period of the augmentations and diminutions of the eb- bings and flowings did depend.</P> <foot>Ggg SALV.</foot> <p n=>418</p> <P>SALV. I will u$e my utmo$t endeavours to render my $elf <marg><I>The cau$es of the inequality of the additions and $ub$tractions of the diurnal conver$ion from the annual motion.</I></marg> intelligible, but the difficulty of the accident it $elf, and the great attention of mind requi$ite for the comprehending of it, con$trains me to be ob$cure. The unequalities of the additions and $ub$tractions, that the diurnal motion maketh to or from the annual dependeth upon the inclination of the Axis of the di- urnal motion upon the plane of the Grand Orb, or, if you plea$e, of the Ecliptick; by means of which inclination the Equinoctial inter$ecteth the $aid Ecliptick, remaining inclined and oblique upon the $ame according to the $aid inclination of Axis. And the quantity of the additions importeth as much as the whole diame- ter of the $aid Equinoctial, the Earths centre being at the $ame time in the Sol$titial points; but being out of them it importeth le$$e and le$$e, according as the $aid centre $ucce$$ively approa- cheth to the points of the Equinoxes, where tho$e additions are le$$er than in any other places. This is the whole bu$ine$$e, but wrapt up in the ob$curity that you $ee.</P> <P>SAGR. Rather in that which I do no not $ee; for hitherto I comprehend nothing at all.</P> <P>SALV. I have already foretold it. Neverthele$$e we will try whether by drawing a Diagram thereof, we can give $ome $mall light to the $ame; though indeed it might better be $et forth by $olid bodies than by bare Schemes; yet we will help our $elves with Per$pective and fore-$hortning. Let us draw there- fore, as before, the circumference of the Grand Orb, [<I>as in Fig.</I> 4.] in which the point A is under$tood to be one of the Sol$titials, and the diameter A P the common Section of the Sol$titial Colure, and of the plane of the Grand Orb or Eclip- tick; and in that $ame point A let us $uppo$e the centre of the Terre$trial Globe to be placed, the Axis of which C A B, in- clined upon the Plane of the Grand Orb, falleth on the plane of the $aid Colure that pa$$eth thorow both the Axis of the Equino- ctial, and of the Ecliptick. And for to prevent confu$ion, let us only draw the Equinoctial circle, marking it with the$e chara- cters D G E F, the common $ection of which, with the plane of the grand Orb, let be the line D E, $o that half of the $aid E- quinoctial D F E will remain inclined below the plane of the Grand Orb, and the other half D G E elevated above. Let now the Revolution of the $aid Equinoctial be made, according to the order of the points D G E F, and the motion of the cen- tre from A towards E. And becau$e the centre of the Earth being in A, the Axis C B (which is erect upon the diameter of the Equinoctial D E) falleth, as hath been $aid, in the Sol$ti- tial Colure, the common Section of which and of the Grand Orb, is the diameter P A, the $aid line P A $hall <foot>be</foot> <p n=>419</p> be perpendicular to the $ame D E, by rea$on that the Colure is erect upon the grand Orb; and therefore the $aid D E, $hall be the Tangent of the grand Orb in the point A. So that in this Po$ition the motion of the Centre by the arch A E; that is, of one degree every day differeth very little; yea, is as if it were made by the Tangent D A E. And becau$e by means of the diurnal motion the point D, carried about by G, unto E, encrea$eth the motion of the Centre moved almo$t in the $ame line D E, as much as the whole diameter D E amounts unto; and on the other $ide dimini$heth as much, moving about the other $emicircle E F D. The additions and $ubductions in this place therefore, that is in the time of the $ol$tice, $hall be mea$ured by the whole diameter D E.</P> <P>Let us in the next place enquire, Whether they be of the $ame bigne$s in the times of the <I>E</I>quinoxes; and tran$porting the Centre of the Earth to the point I, di$tant a Quadrant of a Circle from the point A. Let us $uppo$e the $aid Equinoctial to be G E F D, its common $ection with the grand Orb D E, the Axis with the $ame inclination C B; but the Tangent of the grand Orb in the point I $hall be no longer D E, but another which $hall cut that at right Angles; and let it be this marked H I L, according to which the motion of the Centre I, $hall make its pro- gre$s, proceeding along the circumference of this grand Orb. Now in this $tate the Additions and Sub$tractions are no longer mea$ured by the diameter D E, as before was done; becau$e that diameter not di$tending it $elf according to the line of the annual motion H L, rather cutting it at right angles, tho$e terms D E, do neither add nor $ub$tract any thing; but the Additions and Sub$tractons are to be taken from that diameter that falleth in the plane that is errect upon the plane of the grand Orb, and that inter$ects it according to the line H L; which diameter in this ca$e $hall be this G F and the Adjective, if I may $o $ay, $hall be that made by the point G, about the $emicircle G E F, and the Ablative $hall be the re$t made by the other $emicircle F D G. Now this diameter, as not being in the $ame line H L of the annual motion, but rather cutting it, as we $ee in the point I, the term G being elevated above, and E depre$$ed below the plane of the grand Orb, doth not determine the Additions and Sub- $tractions according to its whole length, but the quantity of tho$e fir$t ought to be taken from the part of the line H L, that is in- tercepted between the perpendiculars drawn upon it from the terms G F; namely, the$e two G S, and F V: So that the mea- $ure of the additions is the line S V le$$er then G F, or then D E; which was the mea$ure of the additions in the Sol$tice A. And $o $ucce$$ively, according as the centre of the Earth $hall be con- <foot>Ggg 2 $tituted</foot> <p n=>420</p> $tituted in other points of the Quadrant A I, drawing the Tan- gents in the $aid points, and the perpndiculars upon the $ame fal- ling from the terms of the diameters of the Equinoctial drawn from the errect planes by the $aid Tangents to the plane of the grand Orb; the parts of the $aid Tangents (which $hall conti- nually be le$$er towards the Equinoctials, and greater towards the Sol$tices) $hall give us the quantities of the additions and $ub$tra- ctions. How much in the next place the lea$t additions differ from the greate$t, is ea$ie to be known, becau$e there is the $ame dif- ference betwixt them, as between the whole Axis or Diameter of the Sphere, and the part thereof that lyeth between the Polar- Circles; the which is le$s than the whole diameter by very near a twelfth part, $uppo$ing yet that we $peak of the additions and $ub$tractions made in the Equinoctial; but in the other Paral- lels they are le$$er, according as their diameters do dimini$h.</P> <P>This is all that I have to $ay upon this Argument, and all perhaps that can fall under the comprehen$ion of our knowledge, which, as you well know, may not entertain any conclu$ions, $ave onely tho$e that are firm and con$tant, $uch as are the three kinds of Pe- riods of the ebbings and flowings; for that they depend on cau$es that are invariable, $imple, and eternal. But becau$e that $e- condary and particular cau$es, able to make many alterations, in- termix with the$e that are the primary and univer$al; and the$e $econdary cau$es being part of them incon$tant, and not to be ob$erved; as for example, The alteration of Winds, and part (though terminate and fixed) unob$erved for their multiplicity, as are the lengths of the Straights, their various inclinations to- wards this or that part, the $o many and $o different depths of the Waters, who $hall be able, unle$s after very long ob$ervations, and very certain relations, to frame $o expeditious Hi$tories thereof, as that they may $erve for Hypoth e$es, and certain $uppo$itions to $uch as will by their combinations give adequate rea$ons of all the appearances, and as I may $ay, Anomalie, and particular irregula- rities that may be di$covered in the motions of the Waters? I will content my $elf with adverti$ing you, that the accidental cau$es are in nature, and are able to produce many alterations; for the more minute ob$ervations, I remit them to be made by tho$e that frequent $everal Seas: and onely by way of a conclu- $ion to this our conference, I will propo$e to be con$idered, how that the preci$e times of the fluxes and refluxes do not onely hap- pen to be altered by the length of Straights, and by the diffe- rence of depths; but I believe that a notable alteration may al$o proceed from the comparing together of $undry tarcts of Sea, different in greatne$s; and in po$ition, or, if you will, inclina- tion; which difference happeneth exactly here in the <I>Adriatick</I> <foot>Gulph,</foot> <p n=>421</p> Gulph, le$$e by far than the re$t of the Mediterrane, and placed in $o different an inclination, that whereas that hath its bounds that inclo$eth it on the Ea$tern part, as are the Coa$ts of <I>Syria,</I> this is $hut up in its more We$terly part: and becau$e the ebbings and flowings are much greater towards the extremities, yea, becau$e the Seas ri$ings and fallings are there onely greate$t, it may pro- bably happen that the times of Flood at <I>Venice</I> may be the time of low Water in the other Sea, which, as being much greater, and di$tended more directly from We$t to Ea$t, cometh in a certain $ort to have dominion over the <I>Adriatick:</I> and therefore it would be no wonder, in ca$e the effects depending on the pri- mary cau$es, $hould not hold true in the times that they ought, and that corre$pond to the periods in the <I>Adriatick,</I> as it doth in the re$t of the Mediterrane. But the$e Particularities require long Ob$ervations, which I neither have made as yet, nor $hall I ever be able to make the $ame for the future.</P> <P>SAGR. You have, in my opinion, done enough in opening us the way to $o lofty a $peculation, of which, if you had given us no more than that fir$t general Propo$ition that $eemeth to me to admit of no reply, where you declare very rationally, that the Ve$$els containing the Sea-waters continuing $tedfa$t, it would be impo$$ible, according to the common cour$e of Nature, that tho$e motions $hould follow in them which we $ee do follow; and that, on the other $ide, granting the motions a$cribed, for o- ther re$pects, by <I>Copernicus</I> to the Terre$trial Globe, the$e $ame alterations ought to en$ue in the Seas, if I $ay you had told us no more, this alone in my judgment, $o far exceeds the vanities in- troduced by $o many others, that my meer looking on them makes me nau$eate them, and I very much admire, that among men of $ublime wit, of which neverthele$s there are not a few, not one hath ever con$idered the incompatibility that is between the reciprocal motion of the Water contained, and the immobi- lity of the Ve$$el containing, which contradiction $eemeth to me now $o manife$t.</P> <P>SALV. It is more to be admired, that it having come into the <marg><I>One $ingle moti- on of the terre$tri- al Globe $ufficeth not to produce the Ebbing & Flowing</I></marg> thoughts of $ome to refer the cau$e of the Tide to the motion of the Earth, therein $hewing a more than common apprehen$ion, they $hould, in afterwards driving home the motion clo$e with no $ide; and all, becau$e they did not $ee that one $imple and uniform motion, as <I>v. gr.</I> the $ole diurnal motion of the Terre- $trial Globe, doth not $uffice, but that there is required an une- ven motion, one while accelerated, and another while retarded: for when the motion of the Ve$$els are uniforme, the waters contained will habituate them$elves thereto, without ever ma- king any alteration. To $ay al$o (as it is related of an ancient <foot>Mathe-</foot> <p n=>422</p> <marg><I>The opinion of</I> Seleucus <I>the Ma- thematician cen$u- red.</I></marg> Mathematician) that the motion of the Earth meeting with the motion of the Lunar Orb, the concurrence of them occa$ioneth the Ebbing and Flowing, is an ab$olute vanity, not onely be- cau$e it is not expre$t, nor $een how it $hould $o happen, but the fal$ity is obvious, for that the Revolution of the Earth is not con- trary to the motion of the Moon, but is towards the $ame way. So that all that hath been hitherto $aid, and imagined by others, is, in my judgment, altogether invalid. But among$t all the famous men that have philo$ophated upon this admirable effect <marg>Kepler <I>is with ve$pect blamed.</I></marg> of Nature, I more wonder at <I>Kepler</I> than any of the re$t, who being of a free and piercing wit, and having the motion a$cri- bed to the Earth, before him, hath for all that given his ear and a$$ent to the Moons predominancy over the Water, and to oc- cult properties, and $uch like trifles.</P> <P>SAGR. I am of opinion, that to the$e more $paculative per- $ons the $ame happened, that at pre$ent befalls me, namely, the not under$tanding the intricate commixtion of the three Periods Annual, Monethly, and Diurnal; And how their cau$es $hould $eem to depend on the Sun, and on the Moon, without the Suns or Moons having any thing to do with the Water; a bu$ine$$e, for the full under$tanding of which I $tand in need of a little longer time to con$ider thereof, which the novelty and difficulty of it hath hitherto hindred me from doing: but I de$pair not, but that when I return in my $olitude and $ilence to ruminate that which remaineth in my fancy, not very well dige$ted, I $hall make it my own. We have now, from the$e four dayes Di$- cour$e, great atte$tations, in favour of the <I>Copernican</I> Sy$teme, among$t which the$e three taken: the fir$t, from the Stations and Retrogradations of the Planets, and from their approaches, and rece$$ions from the Earth; the $econd, from the Suns revolving in it $elf, and from what is ob$erved in its $pots; the third, from the Ebbing and Flowing of the Sea do $hew very rational and concluding.</P> <P>SALV. To which al$o haply, in $hort, one might adde a fourth, and peradventure a fifth; a fourth, I $ay, taken from the fixed $tars, $eeing that in them, upon exact ob$ervations, tho$e minute mutations appear, that <I>Copernicus</I> thought to have been in$en$ible. There $tarts up, at this in$tant, a fifth novelty, from which one may argue mobility in the Terre$trial Globe, by <marg>Sig. Cæ$are Mar- $ilius <I>ob$erveth the Meridian to be moveable.</I></marg> means of that which the mo$t Illu$trious <I>Signore Cæ$are,</I> of the noble Family of the <I>Mar$ili<*></I> of <I>Bologna,</I> and a <I>Lyncean</I> Aca- demick, di$covereth with much ingenuity, who in a very learned Tract of his, $heweth very particularly how that he had ob$erved a continual mutation, though very $low in the Meridian line, of which Treati$e, at length, with amazement, peru$ed by me, <foot>I</foot> <p n=>423</p> I hope he will communicate Copies to all tho$e that are Students of Natures Wonders.</P> <P>SAGR. This is not the fir$t time that I have heard $peak of the exqui$ite Learning of this Gentleman, and of his $hewing him$elf a zealous Patron of all the Learned, and if this, or any other of his Works $hall come to appear in publique, we may be aforehand a$$ured, that they will be received, as things of great value.</P> <P>SALV. Now becau$e it is time to put an end to our Di$cour- $es, it remaineth, that I intreat you, that if, at more lea$ure go- ing over the things again that have been alledged you meet with any doubts, or $cruples not well re$olved, you will excu$e my over$ight, as well for the novelty of the Notion, as for the weakne$$e of my wit, as al$o for the grandure of the Subject, as al$o finally, becau$e I do not, nor have pretended to that a$- $ent from others, which I my $elf do not give to this conceit, which I could very ea$ily grant to be a <I>Chymæra,</I> and a meer paradox; and you <I>Sagredus,</I> although in the Di$cour$es pa$t you have many times, with great applau$e, declared, that you were plea$ed with $ome of my conjectures, yet do I believe, that that was in part more occa$ioned by the novelty than by the cer- tainty of them, but much more by your courte$ie, which did think and de$ire, by its a$$ent, to procure me that content which we naturally u$e to take in the approbation and applau$e of our own matters: and as your civility hath obliged me to you; $o am I al$o plea$ed with the ingenuity of <I>Simplicius.</I> Nay, his con$tancy in maintaining the Doctrine of his Ma$ter, with $o much $trength & undauntedne$s, hath made me much to love him. And as I am to give you thanks, <I>Sagredus,</I> for your courteous a$- fection; $o of <I>Simplicius,</I> I ask pardon, if I have $ometimes moved him with my too bold and re$olute $peaking: and let him be a$$ured that I have not done the $ame out of any inducement of $ini$ter affection, but onely to give him occa$ion to $et before us more lofty fancies that might make me the more knowing.</P> <P>SIMP. There is no rea$on why you $hould make all the$e ex- cu$es, that are needle$$e, and e$pecially to me, that being accu- $tomed to be at Conferences and publique Di$putes, have an hundred times $een the Di$putants not onely to grow hot and an- gry at one another, but likewi$e to break forth into injurious words, and $ometimes to come very neer to blows. As for the pa$t Di$cour$es, and particulatly in this la$t, of the rea$on of the Ebbing and Flowing of the Sea, I do not, to $peak the truth, very well apprehend the $ame, but by that $light <I>Idea,</I> what e- ver it be, that I have formed thereof to my $elf, I confe$$e that your conceit $eemeth to me far more ingenuous than any of all <foot>tho$e</foot> <p n=>424</p> tho$e that I ever heard be$ides, but yet neverthele$$e I e$teem it not true and concluding: but keeping alwayes before the eyes of my mind a $olid Doctrine that I have learn't from a mo$t learned and ingenuous per$on, and with which it is nece$$ary to $it down; I know that both you being asked, Whether God, by his infinite Power and Wi$dome might confer upon the Element of Water the reciprocal motion which we ob$erve in the $ame in any other way, than by making the containing Ve$$el to move; I know, I $ay, that you will an$wer, that he might, and knew how to have done the $ame many wayes, and tho$e unimaginable to our $hallow under$tanding: upon which I forthwith conclude, that this being granted, it would be an extravagant boldne$$e for any one to goe about to limit and confine the Divine Power and Wi$dome to $ome one particular conjecture of his own.</P> <P>SALV. This of yours is admirable, and truly Angelical Do- ctrine, to which very exactly that other accords, in like manner divine, which whil$t it giveth us leave to di$pute, touching the con$titution of the World, add<*>th withall (perhaps to the end, that the exerci$e of the minds of men might neither be di$cou- raged, nor made bold) that we cannot find out the works made by his hands. Let therefore the Di$qui$ition permitted and or- dain'd us by God, a$$i$t us in the knowing, and $o much more admiring his greatne$$e, by how much le$$e we finde our $elves too dull to penetrate the profound Aby$$es of his infinite Wi$- dome.</P> <P>SAGR. And this may $erve for a final clo$e of our four dayes Di$putations, after which, if it $eem good to <I>Salviatus,</I> to take $ome time to re$t him$elf, our curio$ity mu$t, of nece$$ity, grant him the $ame, yet upon condition, that when it is le$$e incommo- dious for him, he will return and $atisfie my de$ire in particular concerning the Problemes that remain to be di$cu$t, and that I have $et down to be propounded at one or two other Conferen- ces, according to our agreement: and above all, I $hall very impatiently wait to hear the Elements of the new Science of our <I>Academick</I> about the natural and violent local Motions. And in the mean time, we may, according to our cu$tome, $pend an hour in taking the Air in the <I>Gondola</I> that waiteth for us.</P> <head><I>FINIS.</I></head> <pb> <fig> <cap><I>Place this Plate at the end of the fourth</I></cap> <cap>Dialogue</cap> <pb> <head>THE Ancient and Modern DOCTRINE OF Holy Fathers, AND Iudicious Divines,</head> <head>CONCERNING</head> <head>The ra$h citation of the Te$timony of SACRED SCRIPTURE, in Conclu$ions meerly Natural, and that may be proved by Sen$ible Fxperiments, and Nece$$ary Demon$trations.</head> <head>Written, $ome years $ince, to Gratifie The mo$t SERENE CHRISTINA LOTHARINGA, <I>Arch- Dutche$s</I> of <I>TVSCANR</I>;</head> <head>By GALILÆO GALILÆI, A Gentleman of <I>Florence,</I> and Chief Philo$opher and Mathematician to His mo$t Serene Highne$s the Grand <I>DVKE.</I></head> <head><I>And now rendred into Engli$h from the Italian,</I> BY THOMAS SALUSBURY.</head> <P><I>Naturam Rerum invenire, difficile; & ubi inveneris, indicare in vulgus, nefas.</I> Plato.</P> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, 1661, Hhh</head> <p n=>427</p> <head>TO Her mo$t Serene HIGHNES THE Gran Duche$s Mother.</head> <P>Some years $ince, as Your mo$t Serene Highne$s well knoweth, I did di$cover many particulars in Hea- ven that had been un$een and unheard of untill this our Age; which, as well for their Novelty, as for certain con$equences which depend upon them, cla$hing with $ome Phy$ical Propo$itions commonly recei- ved by the Schools, did $tir up again$t me no $mall number of $uch as profe$$ed the vulgar Philo$ophy in the Univer$ities; as if I had with my own hand newly placed the$e things in Heaven to ob$cure and di$turb Nature and the Sciences: who forgetting that the multitude of Truths contribute, and concur to the inve- $tigation, augmentation, and e$tabli$hment of the Arts, and not to their diminution, and de$truction; and at the $ame time $hewing them$elves more affectionate to their own Opinions, than to Truth, went about to deny, and to di$prove tho$e Novelties; of which their very $en$e, had they but plea$ed to have inten$ly be- held them, would have rendered them thorowly a$$ured. And to this purpo$e they alledged $undry things, and publi$hed cer- tain Papers fraughted with vain di$cour$es; and which was a more gro$s errour, interwoven with the atte$tations of the Sacred Scriptures, taken from places by them not rightly under$tood, and which did not any thing concern the point for which they were produced Into which errour perhaps they would not have run, if they had but been adverti$ed of a mo$t profitable Document which S. <I>Augu$tine</I> giveth us, concerning our pro- ceeding warily, in making po$itive determinations in points that <foot>Hhh 2 are</foot> <p n=>428</p> are ob$cure and hard to be under$tood by the meer help of ratiocination; where treating (as we) of a certain natural conclu- $ion concerning Cele$tial Bodies, he thus writes: <I>(a) But now</I> <marg><I>(a) Nunc au- tem, $ervatâ $em- per moderatione piæ gravitatis, nihil credere de re ob- $curâ temerè de- bemus, ne fortè, quod po$tea veritas patefecerit, quam- vis Libris Sanct is, $ive Te$tamenti Veteris, $ive No- vi, nisllo modo <*>$$ po$$it adver$um, tamen propter a- morem no$tr<*> erro- ris, oderimus.</I></marg> <I>having evermore a re$pect to the moderation of pious Gravity, we ought to believe nothing unadvi$edly in a doubtful point; le$t we conceive a prejudice again$t that, in favour to our Errour, which Truth hereafter may di$cover to be no wi$e contrary to the Sacred Books either of the Old, or New Te$tament.</I></P> <P>It hath $ince come to pa$s, that Time hath by degrees di$co- vered to every one the truths before by me indicated: and to- gether with the truth of the fact, a di$covery hath been made of the difference of humours between tho$e who $imply and with- out pa$$ion did refu$e to admit $uch like <I>Phænomena</I> for true, and tho$e who to their incredulity had added $ome di$compo$ed af- fection: For as tho$e who were better grounded in the Science of <marg>Lib_{+} 2. Gene$i ad Literam in fine.</marg> A$tronomy, and Natural Philo$ophy, became $atisfied upon my fir$t ntimation of the news; $o all tho$e who $tood not in the Negative, or in doubt for any other rea$on, but becau$e it was an unlookt-for-Novelty, and becau$e they had not an occa$ion of $eeing a $ensible experiment thereof, did by degrees come to $a- risfie them$elves: But tho$e, who be$ides the love they bore to their fir$t Errour, have I know not what imaginary intere$s to render them di$affected; not $o much towards the things, as to- wards the Author of them, not being able any longer to deny them, conceal them$elves under an ob$tinate $ilence; and being exa$perated more than ever by that whereby tho$e others were $atisfied and convinced, they divert their thoughts to other pro- jects, and $eek to prejudice me $ome other wayes: of whom I prore$s that I would make no more account than I have done of tho$e who heretofore have contradicted me (at whom I alwaies laugh, as being a$$ured of the i$$ue that the bu$ine$s is to have) but that I $ee that tho$e new Calumnies and Per$ecutions do not determine in our greater or le$ier Learning (in which I will $carce pretend to any thing) but extend $o far as to attempt to a$per$e me with Crimes which ought to be, and are more abhorred by me than Death it $elf: Nor ought I to content my $elf that they are known to be unju$t by tho$e onely who know me and them, but by all men what$oever. They per$i$ting therefore in their fir$t Re$olution, Of ruining me and what$oever is mine, by all imaginable waies; and knowing how that I in my Studies of A$tronomy and Philo$ophy hold, as to the Worlds Sy$teme, That the Sun, without changing place, is $ituate in the Centre of the Conver$ion of the Cele$tial Orbes; and that the Earth, convertible about its own Axis, moveth it $elf about the Sun: And moreover under$tanding, that I proceed to maintain this Po- <foot>$ition</foot> <p n=>429</p> $ition, not onely by refuting the Rea$ons of <I>Ptolomy</I> and <I>Ari$to- tle,</I> but by producing many on the contrary; and in particular, $ome Phy$ical pertaining to Natural Effects, the cau$es of which perhaps can be by no other way a$$igned; and others A$trono- mical depending upon many circum$tances and encounters of new Di$coveries in Heaven, which manife$tly confute the Ptolo- maick Sy$teme, and admirably agree with and confirm this other Hypothe$is: and po$$ibly being a$hamed to $ee the known truth of other Po$itions by me a$$erted, different from tho$e that have been commonly received; and therefore di$tru$ting their de- fence $o long as they $hould continue in the Field of Philo$o- phy: for the$e re$pects, I $ay, they have re$olved to try whe- ther they could make a Shield for the fallacies of their Argu- ments of the Mantle of a feigned Religion, and of the Autho- rity of the Sacred Scriptures, applyed by them with little judg- ment to the confutation of $uch Rea$ons of mine as they had neither under$tood, nor $o much as heard.</P> <P>And fir$t, they have indeavoured, as much as in them lay, to divulge an opiniou thorow the Univer$e, that tho$e Propo$itions are contrary to the Holy Letters, and con$equently Damnable and Heretical: And thereupon perceiving, that for the mo$t part, the inclination of Mans Nature is more prone to imbrace tho$e enterprizes, whereby his Neighbour may, although un- ju$tly, be oppre$$ed, than tho$e from whence he may receive ju$t incouragement; it was no hard matter to find tho$e Com- plices, who for $uch (that is, for Damnable and Heretical) did from their Pulpits with unwonted confidence preach it, with but an unmerciful and le$s con$iderate injury, not only to this Do- ctrine, and to its followers, but to all Mathematicks and Ma- thematicians together. Hereupon a$$uming greater confidence, and vainly hoping that that Seed which fir$t took root in their un- $ound mindes, might $pread its branches, and a$cend towards Heaven, they went $cattering rumours up and down among the People, That it would, ere long be condemned by Supreme Au- thority: and knowing that $uch a <I>Cen$ure</I> would $upplant not onely the$e two Conclu$ions of the Worlds Sy$teme, but would make all other A$tronomical and Phy$ical Ob$ervations that have corre$pondence and nece$$ary connection therewith to become damnable, to facilitate the bu$ine$s they $eek all they can to make this opinion (at lea$t among the vulgar) to $eem new, and peculiar to my $elf, not owning to know that <I>Nicholas Coper- nicus</I> was its Authour, or rather Re$torer and Confirmer: a per- $on who was not only a Catholick, but a Prie$t, Canonick, and $o e$teemed, that there being a Di$pute in the <I>Lateran Council,</I> under <I>Leo</I> X. touching the correction of the Eccle$ia$tick Ca- <foot>lender</foot> <p n=>430</p> lendar, he was $ent for to <I>Rome</I> from the remote$t parts of <I>Germany,</I> for to a$$i$t in this Reformation, which for that time was left imperfect, onely becau$e as then the true mea$ure of the Year and Lunar Moneth was not exactly known: whereupon it was given him in charge by the Bi$hop of <I>Sempronia,</I> at that time Super-intendent in that Affair, to $earch with reiterated $tudies and pains for greater light and certainty, touching tho$e Cœle$tial Motions. Upon which, with a Labour truly <I>Atlantick</I> and with his admirable Wit, $etting him$elf again to that Study, he made $uch a progre$s in the$e Sciences, and reduced the knowledge of the Cœle$tial Motions to $uch exactne$$e, that he gained the title of an Excellent <I>A$tronomer.</I> And, according unto his Doctrine, not only the Calendar hath been $ince regu- lated, but the Tables of all the Motions of the Planets have al- $o been calculated: and having reduced the $aid Doctrine into $ix Books, he publi$hed them to the World at the in$tance of the Cardinal of <I>Capua,</I> and of the Bi$hop of <I>Culma.</I> And in regard that he had re-a$$umed this $o laborious an enterprize by the order of The Pope; he dedicated his Book <I>De Revolutioni- bus Cœle$tibus</I> to His Succe$$our, namely <I>Paul</I> III. which, being then al$o Printed, hath been received by The Holy Church, and read and $tudied by all the World, without any the lea$t um- brage of $cruple that hath ever been conceived at his Doctrine; The which, whil$t it is now proved by manife$t Experiments and nece$$ary Demon$trations to have been well grounded, there want not per$ons that, though they never $aw that $ame Book in- tercept the reward of tho$e many Labours to its Authour, by cau$ing him to be cen$ured and pronounced an Heretick; and this, only to $atisfie a particular di$plea$ure conceived, without any cau$e, again$t another man, that hath no other intere$t in <I>Copernicus,</I> but only as he is an approver of his Doctrine.</P> <P>Now in regard of the$e fal$e a$per$ions, which they $o unju$tly $eek to throw upon me, I have thought it nece$$ary for my ju$ti- fication before the World (of who$e judgment in matters of Religion and Reputation I ought to make great e$teem) to di$cour$e concerning tho$e Particulars, which the$e men produce to $candalize and $ubvert this Opinion, and in a word, to con- demn it, not only as fal$e, but al$o as Heretical; continually making an Hipocritical Zeal for Religion their Shield; going a- bout moreover to intere$t the Sacred Scriptures in the Di$pute, and to make them in a certain $en$e Mini$ters of their deceiptful purpo$es: and farthermore de$iring, if I mi$take not, contrary to the intention of them, and of the Holy Fathers to extend (that I may not $ay abu$e) their Authority, $o as that even in Conclu$ions meerly Natural, and not <I>de Fide,</I> they would have us altogether <foot>leave</foot> <p n=>431</p> leave Sen$e and Demon$trative Rea$ons, for $ome place of Scri- pture which $ometimes under the apparent words may contain a different $en$e. Now I hope to $hew with how much greater Piety and Religious Zeal I proceed, than they do, in that I propo$e not, that the Book of <I>Copernicus</I> is not to be condemn- ed, but that it is not to be condemned, as they would have it; without under$tanding it, hearing it, or $o much as $eeing it; and e$pecially he being an Author that never treateth of matters of Religion or Faith; nor by Rea$ons any way depending on the Authority of Sacred Scripoures whereupon he may have erroni- ou$ly interpreted them; but alwaies in$i$ts upon Natural Conclu- $ions belonging to the Cele$tial Motions, handled with A$trono- mical and Geometrical Demon$trations. Not that he had not a <marg><I>(c) Si fort a$$eerunt Matæologi, qui cum omnium Ma- thematicum igna- ri $int, tamen de tis judicium a$$u- munt, propter ali- quem locum Scri- ptur æ, malè ad $u- um propo$itum, de- tortum, au$i fue- rint hoc meum in- $titutum reprehen- dere ac in$ectari, illos nihil moror, adeò ut etiam illo- rum judicium, tan- guam temera ium contemnam. Non enim ob$curum e$t, Lact antium, cele- lebrem alioqui Scriptorem, $ed Mathematicum parvum, admodum pueriliter de forma Terræ loqui, cùm deridet eos, qui Terram, Globi for- mam habere prodi- derunt. Itaque non debet mirum vide- ri $tudio$is, $i qui tales, nos ettam ri- debunt. Mathema- ta Mathematicis $cribuntur; quibus & hi no$tri labo- res, ($i me non fal- lit opinio) vide- buntur etiam Rei- publicæ Eccle$ia- $ticæ conducere a- liquid, cujus Prin- cipatum Tua San- ctitas nunc teness.</I></marg> re$pect to the places of the Sacred Leaves, but becau$e he knew very well that his $aid Doctrine being demon$trated, it could not contradict the Scriptures, rightly, and according to their true meaning under$tood. And therefore in the end of his Epi$tle Dedicatory, $peaking to The Pope, he $aith thus: <I>(b) If there $hould chance to be any Matæologi$ts, who though ignorant in all the Mathematicks, yet pretending a skill in tho$e Learnings, $hould dare, upon the authority of $ome place of Scripture wre$ted to their purpo$e, to condemn and cen$ure this my Hypothe$is, I value them not, but $hall $light their incon$iderate Judgement. For it is not unknown, that</I> Lactantius (<I>otherwi$e a Famous Author, though mean Mathematician) writeth very childi$hly touching the Form of the Earth, when he $coffs at tho$e who affirm the Earth to be in Form of a Globe. So that it ought not to $eem $trange to the Ingenious, if any $uch $hould likewi$e now deride us. The Ma- thematicks are written for Mathematitians, to whom (if I deceive not my $elf) the$e Labours of mine $hall $eem to add $omething, as al$o to the Common-weale of the Church, who$e Government is now in the hands of Your Holine$s.</I></P> <P>And of this kinde do the$e appear to be who indeavour to per$wade that <I>Copernicus</I> may be condemned before his Book is read; and to make the World believe that it is not onely lawfull but commendable $o to do, produce certain Authorities of the Scripture, of Divines, and of Councils; which as they are by me had in reverence, and held of Supream Authority, in$omuch that I $hould e$teem it high temerity for any one to contradict them whil$t they are u$ed according to the In $titutes of Holy Church, $o I believe that it is no errour to $peak, $o long as one hath rea- $on to $u$pect that a per$on hath a de$ire, for $ome concern of his own, to produce and alledge them, to purpo$es different from tho$e that are in the mo$t Sacred intention of The Holy Church. Therefore I not onely prote$t (and my $incerity $hall manife$t it <foot>$elf)</foot> <p n=>432</p> $elf) that I intend to $ubmit my $elf freely to renounce tho$e et- rors, into which, through ignorance, I may run in this Di$cour$e of matters pertaining to Religion; but I farther declare, that I de$ire not in the$e matters to engage di$pute with any one, al- though it $hould be in points that are di$putable: for my end endeth onely to this, That if in the$e con$iderations, be$ides my own profe$$ion, among$t the errours that may be in them, there be any thing apt to give others an hint of $ome Notion beneficial to the Holy Church, touching the determining about the <I>Coper- nican</I> Sy$teme, it may be taken and improved as $hall $eem be$t to my Superiours: If not, let my Book be torn and burnt; for that I do neither intend, nor pretend to gain to my $elf any fruit from my writings, that is not Pious and Catholick. And more- over, although that many of the things that I ob$erve have been $poken in my own hearing, yet I $hall freely admit and grant to tho$e that $pake them, that they never $aid them, if $o they plea$e, but confe$s that I might have been mi$taken: And therefore what I $ay, let it be $uppo$ed to be $poken not by them, but by tho$e which were of this opinion.</P> <P>The motive therefore that they produce to condemn the Opi- nion of the Mobility of the Earth, and Stability of the Sun, is, that reading in the Sacred Leaves, in many places, that the Sun mo- veth, that the Earth $tandeth $till; and the Scripture not being capable of lying, or erring, it followeth upon nece$$ary con$e- quence, that the Po$ition of tho$e is Erronious and Heretical, who maintain that the Sun of it $elf is immoveable, and the Earth moveable.</P> <P>Touching this Rea$on I think it fit in the fir$t place, to con- $ider, That it is both piou$ly $poken, and prudently affirmed, That the Sacred Scripture can never lye, when ever its true meaning is under$tood: Which I believe none will deny to be many times very ab$truce, and very different from that which the bare $ound of the words $ignifieth. Whence it cometh to pa$s, that if ever any one $hould con$tantly confine him$elf to the naked Gram- matical Sence, he might, erring him$elf, make not only Contra- dictions and Propo$itions remote from Truth to appear in the Scriptures, but al$o gro$s Here$ies and Bla$phemies: For that we $hould be forced to a$$ign to God feet, and hands, and eyes, yea more corporal and humane affections, as of Anger, of Repen- tance, of Hatred, nay, and $ometimes the Forgetting of things pa$t, and Ignorance of tho$e to come: Which Propo$itions, like as ($o the Holy Gho$t affirmeth) they were in that manner pro- nounced by the Sacred Scriptures, that they might be accommo- dated to the Capacity of the Vulgar, who are very rude and un- learned; $o likewi$e, for the $akes of tho$e that de$erve to be di- <foot>$tingui$hed</foot> <p n=>433</p> $tingui$hed from the Vulgar, it is nece$$ary that grave and skilful Expo$itors produce the true $en$es of them, and $hew the parti- cular Rea$ons why they are dictated under $uch and $uch words. And this is a Doctrine $o true and common among$t Divines, that it would be $uperfluous to produce any atte$tation thereof.</P> <P>Hence methinks I may with much more rea$on conclude, that the $ame holy Writ, when ever it hath had occa$ion to pronounce any natural Conclu$ion, and e$pecially, any of tho$e which are more ab$truce, and difficult to be under$tood, hath not failed to ob$erve this Rule, that $o it might not cau$e confu$ion in the mindes of tho$e very people, and render them the more contu- macious again$t the Doctrines that were more $ublimely my$teri- ous: For (like as we have $aid, and as it plainly appeareth) out of the $ole re$pect of conde$cending to Popular Capacity, the Scripture hath not $crupled to $hadow over mo$t principal and fundamental Truths, attributing, even to God him$elf, qualities extreamly remote from, and contrary unto his E$$ence. Who would po$itively affirm that the Scripture, laying a$ide that re- $pect, in $peaking but occa$ionally of the Earth, of the Water, of the Sun, or of any other Creature, hath cho$en to confine it $elf, with all rigour, within the bare and narrow literal $en$e of the words? And e$pecially, in mentioning of tho$e Crea- tures, things not at all concerning the primary In$titution of the $ame Sacred Volume, to wit, the Service of God, and the $alvation of Souls, and in things infinitely beyond the appre- hen$ion of the Vulgar?</P> <P>This therefore being granted, methinks that in the Di$cu$$ion of Natural Problemes, we ought not to begin at the authority of places of Scripture; but at Sen$ible Experiments and Ne- ce$$ary Demon$trations: For, from the Divine Word, the Sacred Scripture and Nature did both alike proceed; the fir$t, as the Holy Gho$ts In$piration; the $econd, as the mo$t ob$er- vant Executrix of Gods Commands: And moreover it being convenient in the Scriptures (by way of conde$cen$ion to the under$tanding of all men) to $peak many things different, in appearance; and $o far as concernes the naked $igni$ication of the words, from ab$olute truth: But on the contrary, Nature being inexorable and immutable, and never pa$$ing the bounds of the Laws a$$igned her, as one that nothing careth whether her ab$tru$e rea$ons and methods of operating be, or be not ex- po$ed to the Capacity of Men; I conceive that that, concer- ning Natural Effects, which either Sen$ible Experience $ets be- fore our eyes, or Nece$$ary Demon$trations do prove unto us, ought not, upon any account, to be called into que$tion, much <foot>Iii le$s</foot> <p n=>434</p> le$s condemned upon the te$timony of Texts of Scripture, which may, under their words, couch Sen$es $eemingly contrary there- to; In regard that every Expre$$ion of Scripture is not tied to $o $trict conditions, as every Effect of Nature: Nor doth God le$s admirably di$cover him$elf unto us in Nature's Actions, than in the Scriptures Sacred Dictions. Which peradventure <I>Tertul-</I> <marg><I>Nos definimus, Deum, primò N.- tura cogno$cen- dum; Deinde, Do- ctrina recogno$cen- dum: Natura ex operibus; Doctri- na ex pr ædicatio- nibus.</I></marg> <I>lian</I> intended to expre$s in tho$e words<I>: (c) We conclude, God is known; fir$t, by Nature, and then again more particularly known by Doctrine: by Nature, in his Works; by Doctrine, in his Word preached.</I></P> <P>But I will not hence affirm, but that we ought to have an ex- traordinary e$teem for the Places of Sacred Scripture, nay, being <marg>Tertul. adver. Marcion. lib. 1. cap. 18.</marg> come to a certainty in any Natural Conclu$ions, we ought to make u$e of them, as mo$t appo$ite helps to the true Expo- $ition of the $ame Scriptures, and to the inve$tigation of tho$e Sen$es which are nece$$arily conteined in them, as mo$t true, and concordant with the Truths demon$trated.</P> <P>This maketh me to $uppo$e, that the Authority of the Sacred Volumes was intended principally to per$wade men to the be- lief of tho$e Articles and Propo$itions, which, by rea$on they $urpa$s all humane di$cour$e, could not by any other Science, or by any other means be made credible, than by the Mouth of the Holy Spirit it $elf. Be$ides that, even in tho$e Propo$itions, which are not <I>de Fide,</I> the Authority of the $ame Sacred Leaves ought to be preferred to the Authority of all Humane Sciences that are not written in a Demon$trative Method, but either with bare Narrations, or el$e with probable Rea$ons; and this I hold to be $o far convenient and nece$$ary, by how far the $aid Di- vine Wi$dome $urpa$$eth all humane Judgment and Conjecture. But that that $elf $ame God who hath indued us with Sen$es, Di$cour$e, and Under$tanding hath intended, laying a$ide the u$e of the$e, to give the knowledg of tho$e things by other means, which we may attain by the$e, $o as that even in tho$e Natural Conclu$ions, which either by Sen$ible Experiments or Nece$$ary Demon$trations are $et before our eyes, or our Under$tanding, we ought to deny Sen$e and Rea$on, I do not conceive that I am bound to believe it; and e$pecially in tho$e Sciences, of which but a $mall part, and that divided into Conclu$ions is to be found in the Scripture: Such as, for in$tance, is that of <I>A$tro- nomy,</I> of which there is $o $mall a part in Holy Writ, that it doth not $o much as name any of the Planets, except the Sun and the Moon, and once or twice onely <I>Venus</I> under the name of <I>Luci- fer.</I> For if the Holy Writers had had any intention to per$wade People to believe the Di$po$itions and Motions of the Cœle$tial Bodies; and that con$equently we are $till to derive that know- <foot>ledge</foot> <p n=>435</p> ledge from the Sacred Books they would not, in my opinion, have $poken $o little thereof, that it is as much as nothing, in compa- ri$on of the infinite admirable Conclu$ions, which in that Sci- ence are comprized and demon$trated Nay, that the Authours of the Holy Volumes did not only not pretend to teach us the Con$titutions and Motions of the Heavens and Stars, their Fi- gures, Magnitudes, and Di$tances, but that intentionally (al- beit that all the$e things were very well known unto them) they <marg><I>(c) Queri etia<*> $olet, quæ forma & figura Cæli cre- denda $it $ecun- dum Scripturas no$tras: Multi e- nim multum di$- put ant de iis rebus, quas majori pru- dentia no$tri Auto- res omi$erunt, ad beatam vitam non profutur as di$cen- libus, & occupan- tes (quod prius e$t) multum prolixa, & rebus $alubri- bus impendenda temporum $patia. Quid enim ad me pertinet, utrum Cælum, $icut Sphæ- ra, undique conclu- dat Terram, in media. Mundi mo- le libratam; an eam ex una par- te de$uper, ve- lut di$cus, ope- riat? Sed quia de Fide agitur S cripiurærum, propter illam cau$am, quam non $emel commemoravimus, Ne $cilicet qui$quam eloquia divina non intelligens, cum de his rebus tale aliquid vel invenerit in Libris No$tris, vel ex illis audiverit, quod perceptis a$$ertionibus adver $ari videatur, nullo modo eis, cetera utilia monentibus, vel narrantibus, vel pranuntiantibus, credat: Breviter di$cendum e$t, de figura Cæli, hoc $ci$$e Autores no$tros, quod verit as ha- bet: Sed Spiritum Dei, qui per ip$os loquebstur, nolui$$e i$ta docere homines, nulli ad $alutem profutura.</I> D. Augu$t. Lib. 2. De Gen. ad literam, Cap. 9. Idem etiam legitur apud <I>Petrum Lombardum</I> Magi$trum Sententiarum.</marg> forbore to $peak of them, is the opinion of the Mo$t Holy & Mo$t Learned Fathers: and in S. <I>Augu$tine</I> we read the following words. <I>(c) It is likewi$e commonly asked, of what Form and Figure we may believe Heaven to be, according to the Scriptures: For many contend much about tho$e matters, which the greater pru- dence of our Authors hath forborn to $peak of, as nothing further- ing their Learners in relation to able$$ed life; and, (which is the chiefe$t thing) taking up much of that time which $hould be $pent in holy exerci$es. For what is it to me whether Heaven, as a Sphere, doth on all $ides environ the Earth, a Ma$s ballanced in the middle of the World; or whether like a Di$h it doth onely cover or overca$t the $ame? But becau$e belief of Scripture is urged for that cau$e, which we have oft mentioned, that is, That none through ignorance of Divine Phra$es, when they $hall find any thing of this nature in, or hear any thing cited out of our Bibles which may $eem to oppo$e manife$t Conclu$ions, $hould be induced to $u$pect their truth, when they admoni$h, relate, & deliver more profitable matters Briefly be it $poken, touching the Figure of Heaven, that our Au- thors knew the truth: But the H. Spirit would not, that men $hould learn what is profitable to none for $alvation.</I></P> <P>And the $ame intentional $ilence of the$e $acred Penmen in determining what is to be believed of the$e accidents of the Ce- le$tial Bodies, is again hinted to us by the $ame Father in the en- $uing 10. Chapter upon the Que$tion, Whether we are to believe that Heaven moveth, or $tandeth $till, in the$e words: <I>(d) There</I> <marg><I>(d) De Motu etiam Cæli, non- nulli fratres quæ- $tionem movent, u- trum $tet, an mo- veatur; quia $i mo- vetur, inquiunt, quomodo Firma- mentum e$t? Si autem $tat, quomo- do Sydera quæ i<*> ip$o fixa credun- tur, ab Oriente in Occidentem circ<*> eunt, Septentrio- nalibus breviores gyros juxta cardi- nem perag entibus; ut Cælum, $i est a- lius nobis occultus cardo, ex alio ver- tice, $icut Sphæra; $i autem nullus a- lius cardo e$t, vel uti di$cus rotari videatur? Quibus re$pondeo, Multum $ubtilibus & labo- rio$is rationibus i$ta perquiri, ut ve- re percipiatur, u- trum ita, an non ita $it, quibus ine- undis atque tra- ctandis, nec mihi jam tempus e$t, nec illis e$$e debet, quos ad $alutem $uam, è Sanctæ Eccle$iæ nece$$aria utilitate cupimus informa- ri:</I></marg> <I>are $ome of the Brethren that $tart a que$tion concerning the motion of Heaven, Whether it be fixed, or moved: For if it be moved ($ay they) how is it a Firmament? If it $tand $till, how do the$e Stars which are held to be fixed go round from Ea$t to We$t, the more Norchern performing $horter Circuits near the Pole; $o that Heaven, if there be another Pole, to us unknown, may $eem to re- volve upon $ome other Axis; but if there be not another Pole, it may be thought to move as a Di$cus? To whom I reply, That</I> <foot>Iii 2 the$e</foot> <p n=>436</p> <I>the$e points require many $ubtil and profound Rea$ons, for the making out whether they be really $o, or no; the undertakeing and di$eu$$ing of which is neither con$i$tent with my lea$ure, nor their duty, vvhom I de$ire to in$truct in the nece$$ary matters more di- rectly conducing to their $alvation, and to the benefit of The Holy Church.</I></P> <P>From which (that we may come nearer to our particular ca$e) it nece$$arily followeth, that the Holy Gho$t not having intend- ed to teach us, whether Heaven moveth or $tandeth $till; nor whether its Figure be in Form of a Sphere, or of a Di$cus, or di- $tended <I>in Planum</I>: Nor whether the Earth be contained in the Centre of it, or on one $ide; he hath much le$s had an intention to a$$ure us of other Conclu$ions of the $ame kinde, and in $uch a manner, connected to the$e already named, that without the dedermination of them, one can neither affirm one or the other part; which are, The determining of the Motion and Re$t of the $aid Earth, and of the Sun. And if the $ame Holy Spirit hath purpo$ely pretermitted to teach us tho$e Propo$itions, as nothing concerning his intention, that is, our $alvation; how can it be af- firmed, that the holding of one part rather than the other, $hould be $o nece$$ary, as that it is <I>de Fide,</I> and the other erronious? Can an Opinion be Heretical, and yet nothing concerning the $alvation of $ouls? Or can it be $aid that the Holy Gho$t purpo- $ed not to teach us a thing that concerned our $alvation? I might <marg>* Card. Baronius.</marg> here in$ert the Opinion of an Eccle$ia$tical ^{*} Per$on, rai$ed to the <marg><I>Spiritu $ancti mentem fui$$e, nos docere, quomodo ad Cælum eatur: non autem, quomodo Cælum gradiatur.</I> Cardinal. Bar.</marg> degree of <I>Eminenti$$imo,</I> to wit, <I>That the intention of the Holy Gho$t, is to teach us how we $hall go to Heaven, and not how Hea- ven goeth.</I></P> <P>But let us return to con$ider how much nece$$ary Demon$tra- tions, and $en$ible Experiments ought to be e$teemed in Natural Conclu$ions; and of what Authority Holy and Learned Divines have accounted them, from whom among$t an hundred other atte- <marg><I>(e) Illud etiam diligenter caven- dum, & emnino fugiendum e$t, ne in tractanda</I> Mo- $is <I>Dectrina, quic- quam affirmate & a$$everanter $en- tiamus & dica- mus, quod repug- net manife$tis ex- perimentis & rationibus Philo$ophiæ, vel aliarum Di$ciplinarum. Namque cum Verum omne $emper cum Vero congruat, non pote$t Verit as Sacrarum Litterarum, Veris Rationibus & Experimentis Humanarum Doctrina- rum e$$e contraria.</I> Perk. in Gen. circa Principium.</marg> $tations, we have the$e that follow: <I>(e) We must al$o carefully heed and altogether avoid in handling the Doctrine of</I> Mo$es, <I>to avouch or $peak any thing affirmatively and confidently which contradicteth the manife$t Experiments and Rea$ons of Philo$o- phy, or other Sciences. For $ince all Truth is agreeable to Truth, the Truth of Holy Writ cannot be contrary to the $olid Rea$ons and Experiments of Humane Learning.</I></P> <marg><I>(f) Si manife- $tæ certæque Rati- oni, velut $ancta- rum Litterarum objicitur autori- ritas, non intelli- git, qui hoc facit; & non Scripturæ $en$um (ad quem penetrare non po- tuit) $ed $uum po- tius objicit verita- ti: nec id quod in sa, $ed quod in $e- ip$o velue pro c<*> invenit, opponit.</I></marg> <P>And in St. <I>Augu$tine</I> we read: <I>(f) If any one $hall object the Authority of Sacred Writ, again$t clear and manife$t Rea$on, he that doth $o, knows not what he undertakes: For he objects</I> <foot><I>again$t</I></foot> <p n=>437</p> <I>again$t the Truth, not the $en$e of the Scripture (which is be- yond his comprehen$ion) but rather his own; not what is in it, but what, finding it in him$elf, he fancyed to be in it.</I></P> <P>This granted, and it being true, (as hath been $aid) that two Truths cannot be contrary to each other, it is the office of a Judicious Expo$itor to $tudy to finde the true Sen$es of Sacred Texts, which undoubtedly $hall accord with tho$e Natural Con- clu$ions, of which manife$t Sen$e and Nece$$ary Demon$trations <marg>Epi$t. 7. ad Mar- cellinum.</marg> had before made us $ure and certain. Yea, in regard that the Scriptures (as hath been $aid) for the Rea$ons alledged, admit in many places Expo$itions far from the Sen$e of the words; and moreover, we not being able to affirm, that all Interpreters $peak by Divine In$piration; For (if it were $o) then there would be no difference between them about the Sen$es of the $ame places; I $hould think that it would be an act of great pru- dence to make it unlawful for any one to u$urp Texts of Scri- pture, and as it were to force them to maintain this or that Natu- rall Conclu$ion for truth, of which Sence, & Demon$trative, and nece$$ary Rea$ons may one time or other a$$ure us the contrary. For who will pre$cribe bounds to the Wits of men? Who will a$$ert that all that is $en$ible and knowable in the World is al- ready di$covered and known? Will not they that in other points di$agree with us, confe$s this (and it is a great truth) that <I>Ea quæ $cimus, $int minima pars eorum quæ ignoramus</I>? That tho$e Truths which we know, are very few, in compari$on of tho$e which we know not? Nay more, if we have it from the Mouth <marg>Eccle$ia$t. cap. 3.</marg> of the Holy Gho$t, that <I>Deus tradidit Mundum di$putationi eorum, ut non inveniat homo opus, quod operatus e$t Deus ab initio ad finem:</I> One ought not, as I conceive, to $top the way to free Philo$ophating, touching the things of the World, and of Nature, as if that they were already certainly found, and all ma- nife$t: nor ought it to be counted ra$hne$s, if one do not fit down $atisfied with the opinions now become as it were com- mune; nor ought any per$ons to be di$plea$ed, if others do not hold, in natural Di$putes to that opinion which be$t plea$eth them; and e$pecially touching Problems that have, for thou$ands of years, been controverted among$t the greate$t Philo$ophers, as is the Stability of the Sun, and Mobility of the Earth, an opinion held by <I>Pythagoras,</I> and by his whole Sect; by <I>Heraclides Pon- ticus,</I> who was of the $ame opininion; by <I>Phylolaus,</I> the Ma$ter of <I>Plato</I>; and by <I>Plato</I> him$elf, as <I>Ari$totle</I> relateth, and of which <I>Plutarch</I> writeth in the life of <I>Numa,</I> that the $aid <I>Plato,</I> when he was grown old, $aid, It is a mo$t ab$urd thing to think otherwi$e: The $ame was believed by <I>Ari$tarchus Samius,</I> as we have it in <I>Archimedes</I>; and probably by <I>Archimedes</I> him- <foot>$elf;</foot> <p n=>438</p> $elf; by <I>Nicetas</I> the Philo$opher, upon the te$timony of <I>Scicero,</I> and by many others. And this opinion hath, finally, been am- plified, and with many Ob$ervations and Demon$trations con- firmed by <I>Nicholaus Copernicus.</I> And <I>Seneca,</I> a mo$t eminent Philo$opher, in his Book <I>De Cometis,</I> advertizeth us that we ought, with great diligence, $eek for an a$$ured knowledge, whether it be Heaven, or the Earth, in which the Diurnal Con- ver$ion re$ides.</P> <P>And for this cau$e, it would probably be prudent and pro$i- table coun$el, if be$ides the Articles which concern our Salvati- on, and the e$tabli$hment of our Faith (again$t the $tability of which there is no fear that any valid and $olid Doctrine can e- ver ri$e up) men would not aggregate and heap up more, with- out nece$$ity: And if it be $o, it would certainly be a prepo$te- rous thing to introduce $uch Articles at the reque$t of per$ons who, be$ides that we know not that they $peak by in$piration of Divine Grace, we plainly $ee that there might be wi$hed in them the under$tanding which would be nece$$ary fir$t to enable them to comprehend, and then to di$cu$s the Demon$trations wherewith the $ubtiler Sciences proceed in confirming $uch like Conclu$ions. Nay, more I $hould $ay, (were it lawful to $peak my judgment freely on this Argument) that it would haply more $uit with the <I>Decorum</I> and Maje$ty of tho$e Sacred Vo- lumes, if care were taken that every $hallow and vulgar Writer might not authorize his Books (which are not $eldome grounded upon fooli$h fancies) by in$erting into them Places of Holy Scri- pture, interpreted, or rather di$torted to Sen$es as remote from the right meaning of the $aid Scripture, as they are neer to deri- ri$ion, who not without o$tentation flouri$h out their Writings therewith. Examples of $uch like abu$es there might many be produced, but for this time I will confine my $elf to two, not much be$ides the$e matters of <I>A$tronomy:</I> One of which, is that of tho$e Pamphlets which were publi$hed again$t the <I>Medicean</I> Planets, of which I had the fortune to make the di$covery; a- gain$t the exi$tence of which there were brought many places of Sacred Sctipture: Now, that all the World $eeth them to be Planets, I would gladly hear with what new interpretations tho$e very Antagoni$ts do expound the Scripture, and excu$e their own $implicity. The other example is of him who but very lately hath Printed again$t <I>A$tronomers</I> and <I>Philo$ophers,</I> that the Moon doth not receive its light from the Sun, but is of its own nature re$plendent: which imagination he in the clo$e confirm- eth, or, to $ay better, per$wadeth him$elf that he confirmeth by $undry Texts of Scripture, which he thinks cannot be reconciled unle$$e his opinion $hould be true and nece$$ary. Neverthele$$e, <foot>the</foot> <p n=>439</p> the Moon of it $elf is Tenebro$e, and yet it is no le$$e lucid than the Splendor of the Sun.</P> <P>Hence it is manife$t, that the$e kinde of Authors, in regard they did not dive into the true Sence of the Scriptures, would (in ca$e their authority were of any great moment) have impo$ed a nece$- $ity upon others to believe $uch Conclu$ions for true as were re- pugnant to manife$t Rea$on, and to Sen$e. Which abu$e <I>Deus avertat,</I> that it do not gain Countenance and Authority; for if it $hould, it would in a $hort time be nece$$ary to pro$cribe and in- hibit all the Contemplative Sciences. For being that by nature the number of $uch as are very unapt to under$tand perfectly both the Sacred Scriptures, and the other Sciences is much great- er than that of the skilfull and intelligene; tho$e of the fir$t $ort $uperficially running over the Scriptures, would arrogate to them- $elves an Authority of decreeing upon all the Que$tions in Na- ture, by vertue of $ome Word by them mi$onder$tood, and pro- duced by the Sacred Pen-men to another purpo$e: Nor would the $mall number of the Intelligent be able to repre$s the furious Torrent of tho$e men, who would finde $o many the more fol- lowers, in that the gaining the reputation of Wi$e men without pains or Study, is far more grateful to humane Nature, than the con$uming our $elves with re$tle$s contemplations about the mo$t painfull Arts. Therefore we ought to return infinite thanks to Almighty God, who of his Goodne$s freeth us from this fear, in that he depriveth $uch kinde of per$ons of all Authority and, re- po$eth the Con$ulting, Re$olving, and Decreeing upon $o im- portant Determinations in the extraordinary Wi$dom and Can- dor of mo$t Sacred Fathers; and in the Supream Authority of tho$e, who being guided by his Holy Spirit, cannot but determin Holily: So ordering things, that of the levity of tho$e other men, there is no account made. This kinde of men are tho$e, as I be- lieve, again$t whom, not without Rea$on, Grave, and Holy Wri- ters do $o much inveigh; and of whom in particular S. <I>Hierom</I> <marg><I>(g) Hanc (Sci- licer Sacram Scri- pturam) garrula arus, hanc deli- <*>w $en x hanc So- phi$ta verbo$us, h<*>ver$i præ- $<*>munt, lacerant, docent, anteguans di$cant. Alij, addacto $upercilio, grandia verba trutinantes, inter mulierculas, de Sacris Litteris Philo$ophantur. Alij di$cunt, prob pudor! à fæminis, quod viros docent, & ne parum hoc $it, quadam faci- litate verborum, imo audaciâ, edi$- $erunt aliis, quod ip$i non intelli- gunt. Taceo de mei $imilibus, qui $i fortè ad Scriptu- ras Sanctas, po$t $eculares litteras venerint, & $er- mone compo$ito, aurem populi mul- $erint; quicquid dixerint, hoc le- gem Dei putant: nec $cire dignan- tur, quid Prophe- tæ, quid Apo$toli $en$erint, $ed ad $en$um $uum, in- congrua aptant te- $timonia: Qua$i grande $it, & non vitioci$$imum do- cendi genus, de- pravare $ententi- as, & ad volun- tatem $uam Scri- pturamtrahere re- pugnantem.</I> Je- ron. Epi$t. ad <I>Paul.</I> 103.</marg> writeth: <I>(g) This</I> (Scilicet <I>the Sacred Scripture) the talking old woman, the doting old man, the talkative Sophi$ter, all venture upon, lacerate, teach, and that before they have learnt it. Others induced by Pride, diving into hard words, Philo$ophate among$t Women, touching the Holy Scriptures. Others (Oh $hame- ful!) Learn of Women what they teach to Men; and, as if this were nothiug, in a certain facility of words, I may $ay of confi- dence, expound to others what they under$tand not them$elves. I forbear to $peak of tho$e of my own Profe$$ion, who, if after Hu- mane Learning they chance to attain to the Holy Scriptures, and tickle the ears of the people with affected and Studied expre$$ions, they affirm that all they $ay, is to be entertained as the Law of God</I>; <foot><I>and</I></foot> <p n=>440</p> <I>and not $tooping to learn what the Prophets and Apo$tles held, they force incongruous te$timonies to their own Sen$e: As if it were the genuine, and not corrupt way of teaching to deprave Sen- tences, and Wre$t the Scripture according to their own $ingular and contradictory humour.</I></P> <P>I will not rank among the$e $ame $ecular Writers any <I>Theo- logi$ts,</I> whom I repute to be men of profound Learning, and $o- ber Manners, and therefore hold them in great e$teem and vene- ration: Yet I cannot deny but that I have a certain $cruple in my mind, and con$equently am de$irous to have it removed, whil$t I hear that they pretend to a power of con$training others by Authority of the Scriptures to follow that opinion in Natu- ral Di$putations, which they think mo$t agreeth with the Texts of that: Holding withall, that they are not bound to an$wer the Rea$ons and Experiments on the contrary: In Explication and Confirmation of which their judgement they $ay, That <I>The- ologie</I> being the Queen of all the Sciences, $he ought not upon any account to $toop to accomodate her $elf to the Po$itions of the re$t, le$s worthy, and inferior to her: But that they ought to refer them$elves to her (as to their Supream Empere$s) and change and alter their Conclu$ions, according to <I>Theological</I> Statutes and Decrees. And they further add, That if in the inferior Science there $hould be any Conclu$ion certain by ver- tue of Demon$trations or experiments, to which there is found in Scripture another Conclu$ion repugnant; the very Profe$$ors of that Science ought of them$elves to re$olve their Demon$trati- ons, and di$cover the falacies of their own Experiments, without repairing to Theologers and Textuaries, it not $uiting (as hath been $aid) with the dignity of <I>Theologie</I> to $toop to the inve$tiga- tion of the falacies of the inferior Sciences: But it $ufficeth her, to determine the truth of the Conclu$ion with her ab$olute Au- thority, and by her infallibility. And then the Natural Conclu- $ions in which they $ay that we ought to bide by the meer Au- thority of the Scripture, without glo$$ing, or expounding it to Sen$es different from the Words, they affirm to be Tho$e of which the Scripture $peaketh alwaies in the $ame manner; and the Holy Fathers all receive, and expound to the $ame Sen$e.</P> <P>Now as to the$e Determinations, I have had occa$ion to con$i- der $ome particulars (which I will purpo$e) for that I was made cautious thereof, by tho$e who under$tand more than I in the$e bu$ine$$es, and to who$e judgments I alwaies $ubmit my $elf. And fir$t I could $ay, that there might po$$ibly a certain kinde of equivocation interpo$e, in that they do not di$tingui$h the prehe- minences whereby Sacred <I>Theologie</I> meriteth the Title of Queen. <foot>For</foot> <p n=>441</p> For it might be called $o, either becau$e that that which is taught by all the other Sciences, is found to be comprized and demon$tra- ted in it, but with more excellent means, and with more $ublime Learning; in like manner, as for example; The Rules of mea$uring of Land, & of Accountant$hip are much more excellently contain- ed in the Arithmatick and Geometry of <I>Euclid,</I> than in the Practi- $es of Surveyours and Accomptants: Or becau$e the Subject about which <I>Theologie</I> is conver$ant, excelleth in Dignity all the other Subjects, that are the Matters of other Sciences: As al$o becau$e its Documents are divulged by nobler waies. That the Title and Authority of Queen belongeth to <I>Theologie</I> in the fir$t Sen$e, I think that no Theologers will affirm, that have but any in-$ight into the other Sciences; of which there are none (as I be- lieve) that will $ay that Geometry, A$tronomy Mu$ick, and Me- dicine are much more excellently and exactly contained in the Sacred Volumes, than in the Books of <I>Archimedes,</I> in <I>Ptolomy,</I> in <I>Boetius,</I> and in <I>Galen.</I> Therefore it is probable that the Regal Preheminence is given her upon the $econd account, namely, By rea$on of the Subject, and the admirable communicating of the Divine Revelations in tho$e Conclu$ions which by other means could not be conceived by men, and which chiefly concern the acqui$t of eternal Beatitude. Now if <I>Theologie</I> being conver- $ant about the loftie$t Divine Contemplation, and re$iding for Dignity in the Regal Throne of the Sciences, (whereby $he be- cometh of highe$t Authority) de$cendeth not to the more mean and humble Speculations of the inferior Sciences: Nay; (as hath been declared above) hath no regard to them, as not concerning Bearitude; the Profe$$ors thereof ought not to arrogate to them- $elves the Authority to determin of Controver$ies in tho$e Pro- fe$$ions which have been neither practi$ed nor $tudied by them. For this would be as if an Ab$olute Prince, knowing that he might freely command, and cau$e him$elf to be obeyed, $hould (being neither Phi$itian nor Architect) undertake to admini$ter Medicines, and erect Buildings after his own fa$hion, to the great endangering af the lives of the poor Patients, and to the manife$t de$truction of the Edifices.</P> <P>Again, to command the very Profe$$ors of <I>A$tronomy,</I> that they of them$elves $ee to the confuting of their own Ob$erva- tions and Demon$trations, as tho$e that can be no other but Falacies and Sophi$mes, is to enjoyn a thing beyond all po$$ibi- lity of doing: For it is not onely to command them that they do not $ee that which they $ee, and that they do not under$tand that which they under$tand; but that in $eeking, they finde the contrary of that which they happen to meet with. Therefore be- fore that this is to be done, it would be nece$$ary that they were <foot>Kkk $hewed</foot> <p n=>442</p> $hewed the way how to make the Powers of the Soul to command one another, and the inferior the Superior; $o that the imaginati- on and will might, and $hould believe contrary to what the Intel- lect under$tands: I $till mean in Propo$itions purely Natural, and which are not <I>de Fide,</I> and not in the Supernatural, which are <I>de Fide.</I></P> <P>I would entreat the$e Wi$e and Prudent Fathers, that they would withal diligence con$ider the difference that is between Opinable and Demon$trative Doctrines: To the end, that well weighing in their minds with what force Nece$$ary Illations ob- lige, they might the better a$certain them$elves, that it is not in the Power of the Profe$$ors of Demon$trative Sciences to change their Opinions at plea$ure, and apply them$elves one while to one $ide, and another while to another; and that there is a great difference between commanding a Methametitian or a Philo$o- pher, and the di$po$ing of a Lawyer or a Merchant; and that the demon$trated Conclu$ions touching the things of Nature and of the Heavens cannot be changed with the $ame facility, as the Opinions are touching what is lawful or not in a Contract, Bar- gain, or Bill of Exchange. This difference was well under$tood by the Learned and Holy Fathers, as their having been at great pains to confute many Arguments, or to $ay better, many Phi- <marg><I>(g) Hoc indu- bitanter tenendum e$t, ut quicquid Sapientes hujus Mundi, de Natu- ra rerum veraci- ter demon$trare potuerint, o$tenda- mus, no$tris libris non e$$e contrari- um: quicquid au- tem illi, in $uis vo- lumintbus, contra- rium Sacris Lit- teris docent, $ine ulla dubitatione credamus, id fal$i$- $imum e$$e, & quo- quo modo po$$u- mus, etiam o$ten- damus; atque it a teneamus Fidem Domini no$tri, in qua$unt ab$conditi omnes the$auri Sapientiæ, ut ne- que fal$æ Philo$o- phiæ loquacitate $educamur, neque $imulata Religio- nis $uper$titione terreamur.</I></marg> lo$ophical Fallacies, doth prove unto us; and as may expre$ly be read in $ome of them, and particularly we have in S. <I>Augu$tine</I> the following words: <I>(g) This is to be held for an undoubt- ed Truth, That we may be confident, that whatever the Sages of this World have demon$trated touching Natural Points, is no waies contrary to our Bibles: And in ca$e they teach any thing in their Books that is contrary to the Holy Scriptures, we may without any $cruple conclude it to be mo$t fal$e; And aceording to our ability let us make the $ame appear: And let us $o keep the Faith of our Lord, in whom are hidden all the Trea$ures of Wi$dom; that we be neither $educed with the Loquacity of fal$e Philo$ophy, nor $cared by the $uper$tition of a counterfeit Religion.</I></P> <P>From which words, I conceive that I may collect this Do- ctrine, namely, That in the Books of the Wi$e of this World, there are contained $ome Natural truths that are $olidly demon- $trated, and others again that are barely taught; and that as to the fir$t $ort, it is the Office of wi$e Divines to $hew that they are not contrary to the Sacred Scriptures; As to the re$t, taught, but not nece$$arily demon$trated, if they $hall contain any thing contrary to the Sacred Leaves, it ought to be held undoubtedly fal$e, and $uch it ought by all po$$ible waies to be demon- $trated.</P> <marg>Gen. ad Litteram. <I>lib</I> I. Cap. 25.</marg> <P>If therefore Natural Conclu$ions veritably demon$trated, are <foot>not</foot> <p n=>443</p> not to be po$tpo$ed to the Places of Scripture, but that it ought to be $hewn how tho$e Places do not interfer with the $aid Con- clu$ions; then its nece$$ary before a Phy$ical Propo$ition be condemned, to $hew that it is not nece$$arily demon$trated; and this is to be done not by them who hold it to be true, but by tho$e who judge it to be fal$e. And this $eemeth very rea$onable, and agreeable to Nature; that is to $ay, that they may much more ea$ily find the fallacies in a Di$cour$e, who believe it to be fal$e, than tho$e who account it true and concludent. Nay, in this particular it will come to pa$$e, that the followers of this o- pinion, the more that they $hall turn over Books, examine the Arguments, repeat the Ob$ervations, and compare the Experi- ments, the more $hall they be confirmed in this belief. And your Highne$s knoweth what happened to the late Mathematick Pro- fe$$or in the Univer$ity of <I>Pi$a,</I> Who betook him$elf in his old age to look into the Doctrine of <I>Copernicus,</I> with hope that he might be able $olidly to confute it (for that he held it $o far to be fal$e, as that he had never $tudied it) but it was his fortune, that as $oon as he had under$tood the grounds, proceedings, and demon$trations of <I>Copernicus,</I> he found him$elf to be per$waded, and of an oppo$er became his mo$t confident Defender. I might al$o nominate other ^{*} Mathematicians, who being moved <marg>* P. Clavius the Je$uite.</marg> by my la$t Di$coveries, have confe$$ed it nece$sary to change the formerly received Con$titution of the World, it not being able by any means to $ub$i$t any longer.</P> <P>If for the bani$hing this Opinion and Hypothe$is out of the World, it were enough to $top the mouth of one alone, as it may be they per$wade them$elves who mea$uring others judge- ments by their own, think it impo$$ible that this Doctrine $hould be able to $ub$i$t and finde any followers, this would be very ea- $ie to be done, but the bu$ine$s $tandeth otherwi$e: For to execute $uch a determination, it would be nece$$ary to prohibite not onely the Book of <I>Copernicus,</I> and the Writings of the o- ther Authors that follow the $ame opinion, but to interdict the whole Science of <I>A$tronomy</I>; and which is more, to forbid men looking towards Heaven, that $o they might not $ee <I>Mars</I> and <I>Venus</I> at one time neer to the Earth, and at another farther off, with $uch a difference that the latter is found to be fourty times, and the former $ixty times bigger in $urface at one time than at another; and to the end, that the $ame <I>Venus</I> might not be di$covered to be one while round, and another while forked, with mo$t $ubtil hornes: and many other $en$ible Ob$ervations which can never by any means be reconciled to the <I>Ptolomaick</I> Sy$teme, but are unan$werable Arguments for the <I>Copernican.</I></P> <P>But the prohibiting of <I>Copernicus</I> his Book, now that by many <foot>Kkk 2 new</foot> <p n=>444</p> new Ob$ervations, and by the application of many of the Lear- ned to the reading of him, his Hypothe$is and Doctrine doth every day appear to be more true, having admitted and tolerated it for $o many years, whil$t he was le$$e followed, $tudied, and confirmed, would $eem, in my judgment, an affront to Truth, and a $eeking the more to ob$cure and $uppre$$e her, the more $he $heweth her $elf clear and per$picuous.</P> <P>The aboli$hing and cen$uring, not of the whole Book, but onely $o much of it as concerns this particular opinion of the <I>Earths Mobility,</I> would, if I mi$take not, be a greater detriment to $ouls, it being an occa$ion of great $candal, to $ee a Po$ition proved, and to $ee it afterwards made an Here$ie to believe it.</P> <P>The prohibiting of the whole Science, what other would it be but an open contempt of an hundred Texts of the Holy Scri- ptures, which teach us, That the Glory, and the Greatne$$e of Almighty God is admirably di$cerned in all his Works, and di- vinely read in the Open Book of Heaven? Nor let any one think that the Lecture of the lofty conceits that are written in tho$e Leaves fini$h in only beholding the Splendour of the Sun, and of the Stars, and their ri$ing and $etting, (which is the term to which the eyes of bruits and of the vulgar reach) but there are couched in them my$teries $o profound, and conceipts $o $ub- lime, that the vigils, labours, and $tudies of an hundred and an hundred acute Wits, have not yet been able thorowly to dive into them after the continual di$qui$ition of $ome thou$ands of years. But let the Unlearned believe, that like as that which their eyes di$cern in beholding the a$pect of a humane body, is very little in compari$on of the $tupendious Artifices, which an exqui$ite and curious Anatomi$t or Philo$opher finds in the $ame when he is $earching for the u$e of $o many Mu$cles, Tendons, Nerves, and Bones; and examining the Offices of the Heart, and of the other principal Members, $eeking the $eat of the vi- tal Faculties, noting and ob$erving the admirable $tructures of the In$truments of the Sen$es, and, without ever making an end of $atisfying his curio$ity and wonder, contemplating the Re- ceptacles of the Imagination, of the Memory, and of the Un- der$tanding; So that which repre$ents it $elf to the meer $ight, is as nothing in compari$on and proportion to the $trange Won- ders, that by help of long and accurate Ob$ervations the Wit of Learned Men di$covereth in Heaven. And this is the $ub- $tance of what I had to con$ider touching this particular.</P> <P>In the next place, as to tho$e that adde, That tho$e Natural Propo$itions of which the Scripture $till $peaks in one con$tant teno<*>, and which the Fathers all unanimou$ly receive in the $ame $en$e, ought to be accepted according to the naked and <foot>literal</foot> <p n=>445</p> literal $en$e of the Words, without glo$$es and interpretations; and received and held for mo$t certain and true; and that con- $equently the Mobility of the Sun, and Stability of the Earth, as being $uch, are <I>de Fide</I> to be held for true, and the contrary opinion to be deemed Heretical: I $hall propo$e to con$idera- tion, in the fir$t place, That of Natural Propo$itions, $ome there are, of which all humane Science and Di$cour$e can furni$h us only with $ome plau$ible opinion, and probable conjecture ra- ther than with any certain and demon$trative knowledge; as for example, whether the Stars be animated: Others there are, of which we have, or may confidently believe that we may have, by Experiments, long Ob$ervations, and Nece$$ary Demon$tra- tions an undubitable a$$urance; as for in$tance, whether the Earth and Heavens move, or not; whether the Heavens are Spherical, or otherwi$e. As to the fir$t $ort, I doubt not in the lea$t, that if humane Ratiocinations cannot reach them, and that con$equently there is no Science to be had of them, but on- ly an Opinion or Belief, we ought fully and ab$olutely to com- ply with the meer Verbal Sen$e of the Scripture: But as to the other Po$itions, I $hould think (as hath been $aid above) That we are fir$t to a$certain our $elves of the fact it $elf, which will a$$i$t us in finding out the true $en$es of the Scriptures; which $hall mo$t certainly be found to accord with the fact demon$tra- ted, for two truths can never contradict each other. And this I take to be a Doctrine orthodox and undoubted, for that I $inde it written in Saint <I>Augu$tine,</I> who $peaking to our point of the Figure of Heaven, and what it is to be believed to be, in regard that which A$tronomers affirm concerning it $eemeth to be, contrary to the Scripture, (they holding it to be rotund, and the Scripture calling it as it were a ^{*} Curtain, determi- <marg>* <I>Pelle,</I> a Skin in the Original, out in our Bibles a Curtain.</marg> neth that we are not at all to regard that the Scripture contra- dicts A$tronomers; but to believe its Authority, if that which they $ay $hall be fal$e, and founded, only on the conjectures of humane infirmity: but if that which which they affirm be pro- ved by indubitable Rea$ons, this Holy Father doth not $ay, that the A$tronomers are to be enjoyned, that they them$elves re$olving and renouncing their Demon$trations do declare their Conclu$ion to be fal$e, but $aith, that it ought to be de- mon$trated, That what is $aid in Scripture of a Curtain is not contrary to their true Demon$trations. The$e are his words: <marg>(h) <I>Sed ait ali- quis, quomodo non e$t coutrarium iis, qui figur am Sphæ- ræ Cœlo tribunt, quod $criptum e$t en Libris No$tris,</I> Qui extendit Cœ- lum, $icut pellem? <I>Stt $ane contrari- um, $i fal$um e$t, quod illi dicunt: hoc enim verum e$t, quod Divina dicit authoritas, potius quans illud, quod h<*>mana in- firmitas conjicit. Sed $i forte illud talibus illi docu- mentis probare po- tuerint, at dubi- tari inde non debe- at; demon$trandum e$t, hoc quod apud nos e$t de Pelle di- ctum, veris illis rationibus non e$$e contrarium.</I></marg> <I>(h) But $ome object; How doth it appear, that the $aying in our Bibles,</I> Who $tretcheth out the Heaven as a Curtain, <I>maketh not again$t tho$e who maintain the Heavens to be in figure of a Sphere? Let it be $o, if that be fal$e which they affirme: For that is truth which is $poke by Divine Authority, rather than</I> <foot><I>that</I></foot> <p n=>446</p> <I>that which proceeds from Humane In$irmity. But if peradven- ture they $hould be able to prove their Po$ition by $uch Experiments as puts it out of que$tion, it is to be proved, that what is $aid in Scripture concerning a Curtain, doth in no wi$e contradict their manife$t Rea$ons.</I></P> <P>He proceedeth afterwards to admoni$h us that we ought to be no le$s careful and ob$ervant in reconciling a Text of Scripture with a demon$trated Natural Propo$ition, than with another Text of Scripture which $hould $ound to a contrary Sen$e. Nay methinks that the circum$pection of this Saint is worthy to be ad- mired and imitated, who even in ob$cure Conclu$ions, and of which we may a$$ure our $elves that we can have no knowledge or Science by humane demon$tration, is very re$erved in deter- mining what is to be believed, as we $ee by that which he wri- teth in the end of his $econd Book, <I>de Gene$i ad Litteram,</I> $peak- ing, whether the Stars are to be believed animate: <I>(i) Which</I> <marg><I>(i) Quod licet in pra$enti facile non po$$it comprehendi; arbitror tamen, in proce$$is tract an- dærum Scriptura- rum, opportuntora loca po$$e occurre- re, ubinobis de hac re, $ecundum San- ctæ auctoritatis Litteras, et$i non o$tendere certum aliquid, tamen cre- dere licebit. Nunc autem, $ervat â $emper moderatio- ne piæ gravitatis, nihil credere dere ob$cura temere debemus; ne fortè, quoà po$tea verit as patefecerit, quam- vis Libris San- ctis, $ive Te$ta- menti veteris, $ive, novi nullo modo e$- $e po$$it æever$um, tamen propter a- morem no$tri er- roris, oderimus.</I></marg> <I>particular, although (at pre$ent) it cannot ea$ily be comprehended, yet I $uppo$e in our farther Progre$s of bandling the Scriptures, we may meet with $ome more pertinent places, upon which it will be permitted us (if not to determin any thing for certain, yet) to $ugge$t $omewhat concerning this matter, according to the dictates of Sacred Authority. But now, the moderation of pious gravity being alwaies ob$erved, we ought to receive nothing ra$hly in a doubtful point, lea$t perhaps we reject that out of re$pect to our Errour, which hereafter Truth may di$cover, to be in no wi$e repugnant to the Sacred Volumes of the Old and New Te- $tament.</I></P> <P>By this and other places (if I deceive not my $elf) the intent of the Holy Fathers appeareth to be, That in Natural que$tions, and which are not <I>de Fide,</I> it is fir$t to be con$idered, whether they be indubitably demon$trated, or by $en$ible Experiments known; or whether $uch a knowledge and demon$tration is to be had; which having obtained, and it being the gift of God, it ought to be applyed to find out the true Sences of the Sacred Pa- ges in tho$e places, which in appearance might $eem to $peak to a contrary meaning: Which will unque$tionably be pierced into by Prudent Divines, together with the occa$ions that moved the <marg>Id. D Aug. in Gen. <I>ad Lute- ram,</I> lib. 1. <I>in fine.</I></marg> Holy Gho$t, (for our exerci$e, or for $ome other rea$on to me un- known) to veil it $elf $ometimes under words of different $igni- fications.</P> <P>As to the other point, Of our regarding the Primary Scope of tho$e Sacred Volumes, I cannot think that their having $poken alwaies in the $ame tenour, doth any thing at all di$turb this Rule. For if it hath been the Scope of the Scripture by way of conde$cention to the capacity of the Vulgar at any time, to ex- <foot>pre$s</foot> <p n=>447</p> pre$s a Propo$ition in words, that bear a $en$e different from the E$$ence of the $aid Propo$ition; why might it not have ob$erved the $ame, and for the $ame re$pect, as often as it had occa$ion to $peak of the $ame thing? Nay I conceive, that to have done otherwi$e, would but have encrea$ed the confu$ion, and dimi- ni$hed the credit that the$e Sacred Records ought to have a- mong$t the Common People.</P> <P>Again, that touching the Re$t and Motion of the Sun and Earth, it was nece$$ary, for accommodation. to Popular Capa- city, to a$$ert that which the Litteral $en$e of the Scripture im- porteth, experience plainly proveth: For that even to our dayes people far le$s rude, do continue in the $ame Opinion upon Rea- $ons, that if they were well weighed and examined, would be found to be extream trivial, and upon Experiments, either whol- ly fal$e, or altogether be$ides the purpo$e. Nor is it worth while to go about to remove them from it, they being incapable of the contrary Rea$ons that depend upon too exqui$ite Ob$er- vations, and too $ubtil Demon$trations, grounded upon Ab$tra- ctions, which, for the comprehending of them, require too $trong an Imagination. Whereupon, although that the Stability of Heaveu, and Motion of the Earth $hould be more than certain and demon$trated to the Wi$e; yet neverthele$s it would be nece$$ary, for the con$ervation of credit among$t the Vulgar, to affirm the contrary: For that of a thou$and ordinary men, that come to be que$tioned concerning the$e particulars, its probab e that there will not be found $o much as one that will not an- $wer that he thinketh, and $o certainly he doth, that the Sun moveth, and the Earth $tandeth $till. But yet none ought to take this common Popular A$$ent to be any Argument of the truth of that which is affirmed: For if we $hould examine the$e very men touching the grounds and motives by which they are induced to believe in that manner; and on the other $ide $hould hear what Experiments and Demon$trationslper$wade tho$e few others to believe the contrary, we $hould finde the$e latter to be moved by mo$t $olid Rea$ons, and the former by $imple appearances, and vain and ridiculous occurrences. That therefore it was nece$$ary to a$$ign Motion to the Sun, and Re$t to the earth, le$t the $hallow capacity of the Vulgar $hould be confounded, amu$ed, and rendred ob$tinate and contumacious, in giving credit to the principal Articles, and which are ab$olute- ly <I>de fide,</I> it is $ufficiently obvious. And if it was nece$$ary $o to do, it is not at all to be wondred at, that it was with extraor- dinary Wi$dom $o done, in the Divine Scriptures.</P> <P>But I will alledge further, That not onely a re$pect to the Incapacity of the Vulgar, but the current Opinion of tho$e times <foot>made</foot> <p n=>448</p> made the Sacred Writers, in the points that were not nece$$ary to $alvation, to accommodate them$elves more to the received u$e, than to the true E$$ence of things: Of which S. <I>Hierom</I> treating, writeth: <I>(k) As if many things were not $poken in</I> <marg>(k) <I>Qua$i non multa in Scriptu- ris Sanctis dican- tur juxta opinio- nem illius tempor is quo ge$t a referant, & non juxta quod rei veritas contine- bat.</I> D. Hiero. in c. 28. Jerem.</marg> <I>the Holy Scriptures according to the judgement of tho$e times in which they were acted, and not according to that which truth contained.</I> And el$ewhere, the $ame Saint: <I>(l) It is the cu- $tome for the Pen-men of Scripture, to deliver their Judgments in many things, according to the common received opinion that their times had of them.</I> And ^{*} S. <I>Thomas Aquinas</I> in <I>Job</I> upon tho$e words, <I>Qui extendit Aquilonem $uper vacuum, & appendit</I> <marg>(l) <I>Con$uctudi- nis Scripturarum e$t, ut opinionem multarum rerum $ic narret Hi$tori- cus, quomodo eo tempore ab omni- bus credebatur.</I> In cap. 13. Matth.</marg> <I>Terram $uper nihilum</I>: Noteth that the Scripture calleth that $pace <I>Vacuum</I> and <I>Nihilum,</I> which imbraceth and invironeth the Earth, and which we know, not to be empty, bat filled with Air; Neverthele$$e, $aith he, The Scripture to comply with the appre- hen$ion of the Vulgar, who think that in that $ame $pace there is nothing, calleth it <I>Vacuum</I> and <I>Nihilum.</I> Here the words of <marg>* D. Thomas, in cap. 26. Job. v. 7.</marg> S. <I>Thomas, Quod de $uperiori Hæmi$phærio Cœli nibil nobis ap- paret, ni$i $patium aëre plenum, quod vulgares homines reputant Vacnum; loquitur enim $ecundum exi$timationem vulgarium ho- minum, prout e$t mos in Sacra Scriptura.</I> Now from this Place I think one may very Logically argue, That the Sacred Scripture for the $ame re$pect had much more rea$on to phra$e the Sun mo- veable, and the Earth immoveable. For if we $hould try the ca- pacity of the Common People, we $hould find them much more unapt to be per$waded of the $tability of the Sun, and Motion of the Earth, than that the $pace that environeth it is full of Air. Therefore if the $acred Authors, in this point, which had not $o much difficulty to be beat into the capacity of the Vulgar, have notwith$tanding forborn to attempt per$wading them unto it, it mu$t needs $eem very rea$onable that in other Propo$itions much more ab$tru$e they have ob$erved the $ame $tile. Nay <I>Copernicus</I> him$elf, knowing what power an antiquated cu$tome and way of conceiving things become familiar to us from our infancy hath in our Fancy, that he might not increa$e confu$ion and dif- ficulty in our apprehen$ions, after he had fir$t demon$trated, That the Motions which appear to us to belong to the Sun, or to the Firmament, are really in the Earth; in proceeding after- wards to reduce rhem into Tables, and to apply them to u$e, he calleth them the Motions of the Sun, and of the Heaven that is above the Planets; expre$ly terming them the Ri$ing and Set- ting of the Sun and Stars; and mutations in the obliquity of the Zodiack, and variations in the points of the Equinoxes, the Middle Motion, <I>Anomalia, Pro$thaphære$is</I> of the Sun; and $uch other things; which do in reality belong to the Earth: But be- <foot>cau$e</foot> <p n=>449</p> cau$e being joyned to it, and con$equently having a $hare in eve- ry of its motions, we cannot immediately di$cern them in her, but are forced to refer them to the Cele$tial Bodies in which they appear; therefore we call them as if they were made there, where they $eem to us to be made. Whence it is to be noted how ne- ne$$ary it is to accommodate our di$cour$e to our old and accu- $tomed manner of under$tanding.</P> <P>That, in the next place, the common con$ent of Fathers, in re- ceiving a Natural Propo$ition of Scripture, all in the $ame $en$e ought to Authorize it $o far, as to make it become a matter of Faith to believe it to be ^{*} $o, I $hould think that it ought at mo$t <marg>* Namely, ac- cording to the Lit- teral Sen$e.</marg> to be under$tood of tho$e Conclu$ions onely, which have beenby the $aid Fathers di$cu$$ed, and $ifted with all po$$ible diligence, and debated on the one $ide, and on the other, and all things in the end concurring to di$prove the one, and prove the other. But the Mobility of the Earth, and Stability of the Sun, are not of this kinde; For, that the $aid Opinion was in tho$e times total- ly buried, and never brought among$t the Que$tions of the Schools, and not con$idered, much le$s followed by any one: So that it is to be believed that it never $o much as entered into the thought of the Fathers to di$pute it, the Places of Scripture, their own Opinion, and the a$$ent of men having all concurred in the $ame judgement, without the contradiction of any one, $o far as we can finde.</P> <P>Be$ides, it is not enough to $ay that the Fathers all admit the $tability of the Earth, &c. Therefore to believe it is a matter of Faith: But its nece$$ary to prove that they have condemned the contrary Opinion: For I may affirm and bide by this, That their not having occa$ion to make $atisfaction upon the $ame, and to di$cu$s it, hath made them to omit and admit it, onely as cur- rent, but not as re$olved and proved And I think I have very good Rea$on for what I $ay; For either the Fathers did make reflection upon this Conclu$ion as controverted, or not: If not, then they could determin nothing concerning it no not in their private thoughts; and their incogitance doth not oblige us to receive tho$e Precepts which they have not, $o much as in their intentions enjoyned. But if they did reflect and con$ider there- on, they would long $ince have condemned it, if they had judged it erroneous; which we do not find that they have done. Nay, after that $ome Divines have began to con$ider it, we find that they have not deem'd it erroneous; as we read in the Commentaries of <I>Didacus a Stunica</I> upon <I>Job,</I> in <I>Cap. 9, v. 6.</I> on the words, <I>Qui com- movet Terram de loco $uo,</I> &c. Where he at large di$cour$eth upon the <I>Copernican</I> Hypothe$is, and concludeth, <I>That the Mobility of the Earth, is not contrary to Scripture.</I></P> <P>Withal, I may ju$tly que$tion the truth of that determination, namely, That the Church enjoyneth us to hold $uch like Natural <foot>Lll Con-</foot> <p n=>450</p> Conclu$ions as matters of Faith, onely becau$e they bear the $tamp of an unanimous Interpretation of all the Fathers: And I do $uppo$e that it may po$$ibly be, that tho$e who hold in this manner, might po$$ibly have gone about in favour of their own Opinion, to have amplified the Decretal of the Councils; which I cannot finde in this ca$e to prohibit any other, $ave onely, <I>Per- verting to Sen$es contrary to that of Holy Church, or of the concurrent con$ent of Fathers, tho$e places, and tho$e onely that do pertain either to Faith or Manners, or concern our edification in the Doctrine of Chri$tianity: And thus $peaks the Council of Trent. Se$$.</I> 4. But the Mobility or Stability of the Earth, or <marg><I>Concil. Trid. Se$$.</I> 4.</marg> of the Sun, are not matters of Faith, nor contrary to Manners, nor is there any one, that for the $tabli$hing of this Opinion, will pervert places of Scripture in oppo$ition to the Holy Church, or to the Fathers: Nay, Tho$e who have writ of this Doctrine, did never make u$e of Texts of Scripture; that they might leave it $till in the brea$ts of Grave and Prudent Divines to interpret the $aid Places, according to their true meaning.</P> <P>And how far the Decrees of Councills do comply with the Ho- ly Fathers in the$e particulars, may be $ufficiently manife$t, in that they are $o far from enjoyning to receive $uch like Natural Conclu$ions for matters of Faith, or from cen$uring the contrary Opinions as erronious; that rather re$pecting the Primitive and primary intention of the Holy Church, they do adjudge it un- profitable to be bu$ied in examining the truth thereof. Let your Highne$s be plea$ed to hear once again what S. <I>Augu$tine</I> an$wers to to tho$e Brethren who put the Que$tion, Whether it <marg>(*) <I>His re- $pondeo, multum $ubüliter, & labo- rio$is ratiombus, i$ta perqui<*>, ut vere percipiatur, ntrum ita, an non ita $it: quibus in- eundis atque tra- ctandis, nec mihi jam tempus e$t, nec illis e$$e debet, quos ad $alutem $uam, Sanctæ Ec- cle$iæ nece$$ariam utilitatem cupi mus informari.</I></marg> be true that Heaven moveth, or $tandeth $till? (*) <I>To the$e I an$wer, That Points of this nature require a curious and pro- found examination, that it may truly appear whether they be true or fal$e; a work incon$i$tent with my lea$ure to under- take or go thorow with, nor is it any way nece$$ary for tho$e, whom we de$ire to inform of the things that more nearly concern their own $alvation and The Churches Be- nefit.</I></P> <P>But yet although in Natural Propo$itions we were to take the re$olution of condemning or admitting them from Texts of Scri- pture unanimou$ly expounded in the $ame Sen$e by all the Fa- thers, yet do I not $ee how this Rule can hold in our Ca$e; for that upon the $ame Places we read $everal Expo$itions in the Fathers; <marg><I>(m) Non Solem, $ed Primum Mobile immotum con$ti- ti$$e</I>: Dioni$. Areop.</marg> <I>(m) Diony$ius Areopagita</I> $aying, <I>That the Primum Mobile, and not the Sun $tand $till.</I> Saint <I>Augu$tine</I> is of the $ame Opinion; <I>(n) All the Cele$tial Bodies were immoveable.</I> And with them <marg><I>(n) Omnia cor- pora Cæle$tia, im- mota $ub$titi$$e</I>:</marg> concurreth <I>Abulen$is.</I> But which is more, among$t the Jewi$h Authors (whom <I>Jo$ephus</I> applauds) $ome have held, <I>(o) That</I> <foot><I>the</I></foot> <p n=>451</p> <I>The Sun did not really $tand $till, but $eemed $o to do, during the</I> <marg><I>(o) Solem re- vera non $ub$titi$- $e immorum, $ed pro brevi tempore, intra quod I$ræeli- tæ, ho$tes $uos fu- derunt, id ita vi- $um e$$e.</I></marg> <I>$hort time in which I$rael gave the overthrow to their Enemies.</I> So for the Miracle in the time of <I>Hezekiah, Paulus Burgen$is</I> is of opinion that it was not wrought on the Sun, but on the Diall. But that, in $hort, it is nece$$ary to Glo$$e and Interpret the words of the Text in <I>Jo$hua,</I> when ever the Worlds Sy$teme <marg>I$a<*> Cap. 38.</marg> is in di$pute, I $hall $hew anon. Now finally, granting to the$e Gentlemen more than they demand, to wit, That we are whol- ly to acquie$ce in the judgment of Judicious Divines, and that in regard that $uch a particular Di$qui$ition is not found to have been made by the Ancient Fathers, it may be undertaken by the Sages of our Age, who having fir$t heard the Experiments, Ob$ervations, Rea$ons, and Demon$trations of Philolophers and Aftronomers, on the one $ide, and on the other ($eeing that the Controver$ie is about Natural Problems, and Nece$$ary <I>Dilem- ma's,</I> and which cannot po$$ibly be otherwi$e than in one of the two manners in controver$ie) they may with competent cer- tainty determine what Divine In$pirations $hall dictate to them. But that without minutely examining and di$cu$$ing all the Rea- $ons on both $ides; and without ever comming to any certainty of the truth of the Ca$e, $nch a Re$olution $hould be taken, Is not to be hoped from tho$e who do not $tick to hazzard the Ma- je$ty and Dignity of the Sacred Scripture, in defending the re- putation of their vain Fancies; Nor to be feared from tho$e who make it their whole bu$ine$$e, to examine with all in- ten$ne$s, what the Grounds of this Doctrine are; and that only in an Holy Zeal for Truth, the Sacred Scriptures, and for the Maje$ty, Dignity, and Authority, in which every Chri$tian $hould indeavour to have them maintained. Which Dignity, who $eeth not that it is with greater Zeal de$ired and procured by tho$e who, ab$olutely $ubmitting them$elves to the Holy Church, de$ire, not that this, or that opinion may be prohibi- ted, but onely that $uch things may be propo$ed to con$idera- tion, as may the more a$certain her in the $afe$t choice, than by tho$e who being blinded by their particular Intere$t, or $timula- ted by malitious $ugge$tions, preach that $he $hould, without more ado, thunder out Cur$es, for that $he had power $o to do: Not con$idering that all that may be done is not alwayes conve- nient to be done. The Holy Fathers of old were not of this opinion, but rather knowing of how great prejudice, and how much again$t the primary intent of the Catholick Church, it would be to go about from Texts of Scripture to decide Natu- ral Conclu$ions, touching which, either Experiments or nece$$ary Demon$trations, might in time to come evince the contrary, of that which the naked $en$e of the Words $oundeth, they have <foot>Lll 2 not</foot> <p n=>452</p> not only proceeded with great circum$pection, but have left the following Precepts for the in$truction of others. <I>(p) In points</I> <marg>(p) <I>In rebus ob- $ouris, atque a no- $tris oculis remi- ti$$imis, $iqua inde $cripta etiam divi- næ legerimus, quæ po$$int $alva fide, qua imbuimur, a- liis atque altis pa- rere $entextiis, in nullam earum nos præcipiti affirma- tione ita projici- amu<*>, ut $i forte ailigentiùs di$cu$- $a veritas eã recte labefact averit, corruamus: non pro $ententia Divinarum Scripturarum, $ed pro no$tra ita dimicantes, ut eam velimus Scripturarum e$$e, quæ no$tra e$t, cum potius eam quæ Scripturarum e$t, no$tram e$$e velle debeamus,</I> Divus Augu$tin. in Gen. ad Litteram, lib. 2. c. 18. & $eq.</marg> <I>ob$cure and remote from our Sight, if we come to read any thing out of Sacred Writ, that, with a</I> Salvo <I>to the Faith that we have imbued, may corre$pond with $everal con$tructions, let us not $o farre throw our $elves upon any of them with a precipitous ob- $tinacy, as that if, perhaps the Truth being more diligently $earch't into, it $hould ju$tly fall to the ground, we might fall together with it: and $o $hew that we contend not for the $en$e of Divine Scriptures, but our own, in that we would have that which is our own to be the $en$e of Scriptures, when as we $hould ra- ther de$ire the Scriptures meaning to be ours.</I></P> <P>He goeth on, and a little after teacheth us, that no Propo$i- tion can be again$t the Faith, unle$$e fir$t it be demon$trated <marg>(q) <I>Tam diu non e$t extra fidem, do- nec Veritate cer- ti$$ima refellatur. Quod $i fæctum fuerit, non hoc ha- bebut Divina Scri- ptura, $ed hoc $en- $er at humana Ig- norantia.</I> Ibid.</marg> fal$e; $aying, <I>(q) Tis not all the while contrary to Faith, until it be di$proved by mo$t certain Truth, which if it $hould $o be, the Holy Scripture affirm'd it not, but Humane Ignorance $uppo$ed it.</I> Whereby we $ee that the $en$es which we impo$e on Texts of Scripture, would be fal$e, when ever they $hould di$agree with Truths demon$trated. And therefore we ought, by help of de- mon$trated Truth, to $eek the undoubted $en$e of Scripture: and not according to the $ound of the words, that may $eem true to our weakne$$e, to go about, as it were, to force Na- ture, and to deny Experiments and Nece$$ary Demon$tra- tions.</P> <P>Let Your Highne$$e be plea$ed to ob$erve farther, with how great circum$pection this Holy Man proceedeth, before he af- firmeth any Interpretation of Scripture to be $ure, and in $uch wi$e certain, as that it need not fear the encounter of any diffi- culty that may procure it di$turbance, for not contenting him$elf that $ome $en$e of Scripture agreeth with $ome Demon- <marg>(r) <I>Si autem hoc verum e$$e ve- ra ratio demon- $traverit, adhuc incertum erit, u- trum hoc in illis verbis Sanctorum Librorum, Scrip- tor $entiri volue- rit, an aliquid a- liud non minus ve- rum. Quod $i cætera contextio $ermonis non hoc eum volui$$e probaverit, non ideo fal$um erit aliud, quod ip$e intelligi voluit, $ed & verum, & quod utilius cogno$catur.</I></marg> $tration, he $ubjoynes. <I>(r) But if right Rea$on $hall demon- $trate this to be true, yet is it que$tionable whether in the$e words of Sacred Scripture the Pen-man would have this to be under- $tood, or $omewhat el$e, no le$$e true. And in ca$e the Context of his Words $hall prove that he intended not this, yet will not that which he would have to be under$tood be therefore fal$e, but mo$t true, aad that which is more profitable to be known.</I></P> <P>But that which increa$eth our wonder concerning the cir- <foot>cum$pection,</foot> <p n=>453</p> cum$pection, wherewith this Pious Authour proceedeth, is, <marg><I>($) Si autem con- textio Scripturæ, hoc volui$$e intel- ligi Scriptorem, non repugnaverit, adhuc re$tabit quærere, utrum & aliud non potuerit.</I></marg> that not tru$ting to his ob$erving, that both Demon$trative Rea$ons, and the $en$e that the words of Scripture and the re$t of the Context both precedent and $ub$equent, do con$pire to prove the $ame thing, he addeth the following words.</P> <P><I>($) But if the Context do not hold forth any thing that may</I> <marg><I>(t) Quod $i & aliud potui$$e inve- nerimus, incertum erit; quidnam eo- rum ille voluerit: aut utrumque vo- lui$$e non inconve- nienter creditur, $i utriu$que $en<*>entiæ certa circum$t an- tia $ufragatur.</I></marg> <I>di$prove this to be the Authors Sen$é, it yet remains to enquire, Whether the other may not be intended al$o.</I> And not yet re$olving to accept of one Sen$e, or reject another, but thinking that he could never u$e $ufficient caution, he proceedeth: <I>(t) But if $o be we finde that the other may be al$o meant, it will be doubted which of them he would have to $tand; or which in probability he may be thought to aim at, if the true circum$tances on both $ides be weighed.</I> And la$tly, intending to render a Rea$on of this his <marg><I>(u) Plerumque enim accidit, at a- liquid de Terra, d<*> Celo, de ceter is hu- jus mundi elemen- tis, de motu, con- ver$ione, vel ctiam magnitudine & intervallis Syde- rum, de certis de- fectibus Solis, & Lunæ, de eircuiti- bus annorum & temporum; de Na- turis animalium, fruticum, lapidum, atque buju$modi ceter is, etiam non Chri$tianus ita no- verit, ut cir<*>$$ima ratione vel experi- entiâ teneat. Tur- pe autem e$t nimis & pernicio$um, ae maxime caven- dum, at Chri$tia- num de his rebus qua$i $ecundum Chri$tianaslitter as loquentem, ita de- lirare quilibet in- fiàelis audiat, ut, quemadmodum di- citur, toto Cælo er- rærecon$piciens, ri- $ũtenere vix po$$it: & non tam mole- $tum e$t, quod er- rans homo deride- retur, $ed quod au- ctores no$tri, ab tis qui foris $unt, ta- lia $en$i$$e credun- tur, & cum magno exitio eorum, de quorum $alute $atagimus, tanquam indocti reprehenduntur atque re$puuntur. Cum enim quemquam de numero Chri$tiano um eainre, quam ip $i optime nor<*>int, deprehenderint, & venam $enten- tiam $uam de no$tris libris a$$erent; quo pacts illis Libris credituri $unt, de Re$urrectione Mortuorum, & de $p<*> vit æ eternæ, Regnoque Celorum; quando de his rebus quas jam experiri, vel indubitatis rationibus percipere potueru<*> fallaciter putaverint e$$e con$criptos.</I></marg> Rule, by $hewing us to what perils tho$e men expo$e the Scri- ptures, and the Church; who, more re$pecting the $upport of their own errours, than the Scriptures Dignity, would $tretch its Authority beyond the Bounds which it pre$cribeth to it $elf, he $ubjoyns the en$uing words, which of them$elves alone might $uffice to repre$s and moderate the exce$$ive liberty, which $ome think that they may a$$ume to them$elves: <I>(u) For it many times falls out, that a Chri$tian may not $o fully under$tand a Point concerning the Earth, lieaven, and the re$t of this Worlds Elements; the Motion, Conver$ion, Magnitude, and Di$tances of the Stars, the certain defects of the Sun and Moon, the Revoluti- ons of Years and Times, the Nature of Animals, Fruits, Stones, and other things of like nature, as to defend the $ame by right Rea$on, or make it out by Experiments. But its too great an ab- $urdity, yea mo$t pernicious, and chiefly to be avoided, to let an Infidel finde a Chri$tian $o $tupid, that he $hould argue the$e mat- ters; as if they were according to Chri$tian Doctrine; and make him (as the Proverb $aith) $carce able to contain his laughter, $ee- ing him $o far from the Mark Nor is the matter $o much that one in an errour $hould be laught at, but that our Authors $hould be thought by them that are without, to be of the $ame Opinion, and to the great prejudice of tho$e, who$e $alvation we wait for, $en$urcd and rejected as unlearned. For when they $hal confute any one of the Chri$tians in that matter, which they them$elvs thorowly under- $tand, and $hall thereupon expre$s their light e$teem of our Books; how $hall the$e Volumes be believed touching the Re$urrection of the Dead, the Hope of eternal Life, and the Kingdom of Heaven; when, as to the$e Points which admit of pre$ent Demon$tration, or undoubted Rea$ons, they conceive them to be fal$ly written.</I></P> <foot>And</foot> <p n=>454</p> <P>And how much the truly Wi$e and Prudent Fathers are di$- plea$ed with the$e men, who in defence of Propo$itions which they do not under$tand, do apply, and in a certain $en$e pawn Texts of Scripture, and afterwards go on to encrea$e their fir$t Errour, by producing other places le$s under$tood than the for- mer. The $ame Saint declareth in the expre$$ions following: <marg><I>(y) Quid enim mole$t<*>æ, tri$tiæque ingerant prudenti- bus fratribus, te- nerarij præ$umpto- res, $atis dici non pote$t, cum, $i quando de fal$a & prava opinione $ua reprehendi & con- vinci cæperint, ab iis qui no$trorum librorum auctori- tate, & aperli$$ima falfitate dixerunt, eo$dnm libros San- ctos, unde id pro- bent, proferre co- nantur; vel etiam memoriter, quæ ad te$timonium vale- re arbitrantur, multa inde verba pronunciant, non intelligentes, neque quæ loquuntur, ne- que de quibus af- firmant.</I></marg> <I>(x) What trouble and $orrow weak undertakers bring upon their knowing Brethren, is not to be expre$$ed; $ince when they begin to be told and convinced of their fal$e and un$ound Opinion, by tho$e who have no re$pect for the Authority of our Scriptures, in defence of what through a fond Temerity, and mo$t manife$t fal- $ity, they have urged; they fall to citing the $aid Sacred Books for proof of it, or el$e repeat many words by heart out of them, which they conceive to make for their purpo$e; not knowing either what they $ay, or whereof they affirm.</I></P> <P>In the number of the$e we may, as I conceive, account tho$e, who, being either unwilling or unable to under$tand the De- mon$trations and Experiments, wherewith the Author and fol- lowers of this Opinion do confirm it, run upon all occa$ions to the Scriptures, not con$idering that the more they cite them, and the more they per$i$t in affirming that they are very clear, and do admit no other $en$es, $ave tho$e which they force upon them, the greater injury they do to the Dignity of them (if we allowed that their judgments were of any great Authority) in ca$e that the Truth coming to be manife$tly known to the con- trary, $hould occa$ion any confu$ion, at lea$t to tho$e who are $eparated from the Holy Church; of whom yet $he is very $olici- tous, and like a tender Mother, de$irous to recover them again into her Lap. Your Highne$s therefore may $ee how præpo$terou$- ly tho$e Per$ons proceed, who in Natural Di$putations do range Texts of Scripture in the Front for their Arguments; and $uch Texts too many times, as are but $uperficially under$tood by them.</P> <P>But if the$e men do verily think, & ab$olutely believe that they have the true $ence of Such a particular place of Scripture, it mu$t needs follow of con$equence, that they do likewi$e hold for certain, that they have found the ab$olute truth of that Natural Conclu$i- on, which they intend to di$pute<*> And that withall, they do know that they have a great advantage of their Adver$ary, who$e Lot it is to defend the part that is fal$e; in regard that he who maintain- eth the Truth, may have many $en$ible experiments, and many ne- ce$$ary Demon$trations on his $ide; whereas his Antagoni$t can make u$e of no other than deceitful appearances, <I>Paralogi$ms</I> and <I>Sophi$ms.</I> Now if they keeping within natural bounds, & produ- cing no other Weapons but tho$e of Philo$ophy, pretend however, to have $o much advantage of their Enemy; why do they after- <foot>wards</foot> <p n=>455</p> wards in coming to engage, pre$ently betake them$elves to a Wea- pon inevitable & dreadful to terrifie their Opponent with the $ole beholding of it? But if I may $peak the truth, I believe that they are the fir$t that are affrighted, and that perceiving them$elves unable to bear up again$t the a$$aults of their Adver$ary, go about to find out ways how to keep them far enough off, forbidding unto them the u$e of the Rea$on which the Divine Bounty had vouch$afed them, & abu$ing the mo$t equitable Authority of $acred Scripture, which rightly under$tood and applyed, can never, according to the common Maxime of Divines, oppo$e the Manife$t Experi- ments, or Nece$$ary Demon$trations. But the$e mens running to the Scriptures for a Cloak to their inability to comprehend, not to $ay re$olve the Rea$ons alledged again$t them, ought (if I be not mi$taken) to $tand them in no $tead: the Opinion which they oppo$e having never as yet been condemned by Holy Church. So that if they would proceed with Candor, they $hould either by $ilence confe$s them$elves unable to handle $uch like points, or fir$t con$ider that it is not in the power of them or others, but onely in that of the Pope, and of Sacred Councils to <marg>If this pa$$age $eem har$h, the Reader mu$t re- member that I do but Tran$late.</marg> cen$ure a Po$ition to be Erroneous: But that it is left to their freedome to di$pute concerning its fal$ity. And thereupon, knowing that it is impo$$ible that a Propo$ition $hould at the $ame time be True and Heretical; they ought, I $ay, to imploy them$elves in that work which is mo$t poper to them, namely, in demon$trating the fal$ity thereof: whereby they may $ee how needle$$e the prohibiting of it is, its fal$hood being once di$covered, for that none would follow it: or the Prohibition would be $afe, and without all danger of Scandal. Therefore fir$t let the$e men apply them$elves to examine the Arguments of <I>Copernicus</I> and others; and leave the condemning of them for Erroneous and Heretical to whom it belongeth: But yet let them not hope ever to finde $uch ra$h and precipitous Determina- tions in the Wary and Holy Fathers, or in the ab$olute Wi$- dome of him that cannot erre, as tho$e into which they $uffer them$elves to be hurried by $ome particular Affection or Inte- re$t of their own. In the$e and $uch other Po$itions, which are not directly <I>de Fide,</I> certainly no man doubts but His Holine$s hath alwayes an ab$olute power of Admitting or Condemn- ing them, but it is not in the power of any Creature to make them to be true or fal$e, otherwi$e than of their own nature, and <I>de facto</I> they are.</P> <P>Therefore it is in my judgment more di$cretion to a$$ure us fir$t of the nece$$ary and immutable Truth of the Fact, (over which none hath power) than without that certainty by condem- ning one part to deprive ones $elf of that authority of freedome <foot>to</foot> <p n=>456</p> to elect, making tho$e Determinations to become nece$$ary, which at pre$ent are indifferent and arbitrary, and re$t in the will of Supreme Authority. And in a word, if it be not po$- $ible that a Conclu$ion $hould be declared Heretical, whil$t we are not certain, but that it may be true, their pains are in vain who pretend to condemn the Mobility of the Earth and Stabili- ty of the Sun, unle$$e they have fir$t demon$trated it to be im- po$$ible and fal$e.</P> <P>It remaineth now, that we con$ider whether it be true, that the Place in <I>Jo$huab</I> may be taken without altering the pure $ig- nification of the words: and how it can be that the Sun, obey- ing the command of <I>Jo$huah,</I> which was, <I>That it $hould $tand $till,</I> the day might thereupon be much lengthened. Which bu- $ine$$e, if the Cele$tial Motions be taken according to the <I>Ptolo- maick</I> Sy$teme, can never any wayes happen, for that the Sun moving thorow the Ecliptick, according to the order of the Signes, which is from Ea$t to We$t (which is that which maketh Day and Night) it is a thing manife$t, that the Sun cea$ing its true and proper Motion, the day would become $horter and not longer; and that on the contrary, the way to lengthen it would be to ha$ten and velocitate the Suns motion; in$omuch that to cau$e the Sun to $tay above the Horizon for $ome time, in one and the $ame place, without declining towards the We$t, it would be nece$$ary to accelerate its motion in $uch a manner as that it might $eem equal to that of the <I>Primum Mobile,</I> which would be an accelerating it about three hundred and $ixty times more than ordinary. If therefore <I>Jo$huah</I> had had an intention that his words $hould be taken in their pure and proper $ignification, he would have bid the Sun to have accelerated its Motion $o, that the Rapture of the <I>Primum Mobile</I> might not carry it to the We$t: but becau$e his words were heard by people which hap- ly knew no other Cele$tial Motion, $ave th<*> grand and common one, from Ea$t to We$t, $tooping to their Capacity, and having no intention to teach them the Con$titution of the Spheres, but only that they $hould perceive the greatne$s of the Miracle wrought, in the lengthening of the Day, he $poke according to their apprehen$ion. Po$$ibly this Con$ideration moved <I>Diony- $ius Areopagita</I> to $ay that in this Miracle the <I>Primum Mobile</I> $tood $till, and this $topping, all the Cele$tial Spheres did of con$equence $tay: of which opinion is S. <I>Augu$tine</I> him$elf, and <I>Abulen$is</I> at large confirmeth it. Yea, that <I>Jo$hua's</I> intention was, that the whole Sy$teme of the Cele$tial Spheres $hould $tand $till, is collected from the command he gave at the $ame time to the Moon, although that it had nothing to do in the lengthening of the day; and under the injunction laid upon the <foot>Moon,</foot> <p n=>457</p> Moon, we are to under$tand the Orbes of all the other Planets, pa$$ed over in $ilence here, as al$o in all other places of the Sacred Scriptures; the intention of which, was not to reach us the A$tro- nomical Sciences. I $uppo$e therefore, (if I be not deceived) that it is very plain, that if we allow the <I>Ptolemaick</I> Sy$teme, we mu$t of nece$$ity interpret the words to $ome $en$e different from their $trict $ignification. Which Interpretation (being admo- ni$hed by the mo$t u$efull precepts of S. <I>Augu$tine)</I> I will not affirm to be of nece$$ity this above-mentioned, $ince that $ome other man may haply think of $ome other more proper, and more agreeable Sen$e.</P> <P>But now, if this $ame pa$$age may be under$tood in the <I>Coper- nican</I> Sy$teme, to agree better with what we read in <I>Jo$huah,</I> with the help of another Ob$ervation by me newly $hewen in the Body of the Sun; I will propound it to con$ideration, $peak- ing alwaies with tho$e $afe Re$erves; That I am not $o affectio- nate to my own inventions, as to prefer them before tho$e of other men, and to believe that better and more agreeable to the intention of the Sacred Volumes cannot be produced.</P> <P>Suppo$ing therefore in the fir$t place, that in the Miracle of <I>Jo$huah,</I> the whole Sy$teme of the Cele$tial Revolutions $tood $till, according to the judgment of the afore-named Authors: And this is the rather to be admitted, to the end, that by the $taying of one alone, all the Con$titutions might not be con- founded, and a great di$order needle$ly introduced in the whole cour$e of Nature: I come in the $econd place to con$ider how the Solar Body, although $table in one con$tant place, doth neverthe- le$s revolve in it $elf, making an entire Conver$ion in the $pace of a Month, or thereabouts; as I conceive I have $olidly demon- $trated in my Letters <I>Delle Machie Solari</I>: Which motion we $en$ibly $ee to be in the upper part of its Globe, inclined to- wards the South; and thence towards the lower part, to encline towards the North, ju$t in the $ame manner as all the other Orbs of the Planets do. Thirdly, If we re$pect the Nobility of the Sun, and his being the Fountain of Light, by which, (as I nece$- $arily demon$trate) not onely the Moon and Earth, but all the other Planets (all in the $ame manner dark of them$elves) become illuminated; I conceive that it will be no unlogicall Illation to $ay, That it, as the Grand Mini$ter of Nature, and in a certain $en$e the Soul and Heart of the World, infu$eth into the other Bodies which environ it; not onely Light, but Motion al$o; by revol- ving ^{*} in it $elf: So that in the $ame manner that the motion of <marg>* <I>i. i.</I> On its own Axis.</marg> the Heart of an Animal cea$ing, all the other motions of its Members would cea$e; $o, the Conver$ion of the Sun cea$ing, the Conver$ions of all the Planets would $tand $till. And though <foot>Mmm I</foot> <p n=>458</p> I could produce the te$timonies of many grave Writers to prove the admirable power and influence of the Sun, I will content my $elf with one $ole place of Holy <I>Dioni$ius Areopagita</I> in his Book <marg>(*) <I>Lux ejus colli- git, convertitque ad $e omnia, quæ vi- dentur, quæ mo- ventur, quæ illu- $trantur, quæ ca- le$cunt, & uno no- mine ea, quæ ab e- jus $plendore cen- tinentur. Itaque Sol <G>Hli<34></G> dicitur, quod omnia con- greger, colligatque di$per$a.</I></marg> <I>de Divinis Nominibus</I>; who thus writes of the Sun: ^{(*)} <I>His Light gathereth and converts all things to him$elf, which are $een, moved, illu$trated, wax hot, and (in a word) tho$e things which are pre$erved by his $plendor: Wherefore the Sun is called</I> <G>Hli<*>,</G> <I>for that he collecteth and gathereth together all things di$per$ed.</I> And a little after of the Sun again he adds; ^{(*)} <I>If this Sun which wo $ee, as touching the E$$ences and Qualities of tho$e things which fall within our Sen$e, being very many and different; yet if he who is one, and equally be$towes his Light, doth renew, nouri$h, defend, perfect, divide, conjoyn, cheri$h, make fruitfull,</I> <marg>(*) <I>Si enim Sol hic quem vi- domus, eorum quæ $ub $en$um ca- dunt, e$$entias & qualitates, quæ que muliæ $int ac di$- $imiles, tam<*>n ip$e qui unus e$t, æqua- literque lumen fundit, renovat, a- lit, tuetur, perficit, dividit, conjungit, fovet, fæcunda red- dit, auget, mutat, firmat, edit, movet, vitaliaq; facit om- nia: & unaquæq; res hujus univer- $itatis, pro cæptu $uo, unius atque e- ju$dem Solis e$t particeps, cau$æ$- que multorum, quæ participant, in $e æquabiliter an- ticipatas habet, certe majori ratio- ne,</I> &c.</marg> <I>encrea$e, change, fix, produce, move, and fa$hion all living crea- tures: And every thing in this Vniver$e at his Plea$ure, is par- taker of one and the $ame Sun; and the cau$es of many things which participate of him, are equally auticipated in him: Certain- ly by greater rea$on</I>; &c. The Sun therefore being the Foun- tain of Light and, Principle of Motion, God intending, that at the Command of <I>Jo$hua,</I> all the Worlds Sy$teme, $hould con- tinue many hours in the $ame $tate, it $ufficeth to make the Sun $tand $till, upon who$e $tay (all the other Conver$ions cea$ing) the Earth, the Moon, the Sun did abide in the $ame Con$titution as before, as likewi$e all the other Planets: Nor in all that time did the Day decline towards Night, but it was miraculou$ly pro- longed: And in this manner, upon the $tanding $till of the Sun, without altering, or in the lea$t di$turbing the other A$pects and mutual Po$itions of the Stars, the Day might be lengthned on Earth; which exactly agreeth with the Litteral $en$e of the Sacred Text.</P> <P>But that of which, if I be not mi$taken, we are to make no $mall account, is, That by help of this <I>Copernican</I> Hypothe$is, we have the Litteral, apert, and Natural Sen$e of another parti- cular that we read of in the $ame Miracle; which is, That the Sun $tood $till <I>in Medio Cæli</I>: Upon which pa$$age grave Divines rai$e many que$tions, in regard it $eemeth very probable, That when <I>Jo$huah</I> de$ired the lengthning of the Day, the Sun was near $etting, and not in the Meridian; for if it had been in the Meridian, it being then about the Summer <I>Sol$tice,</I> and con- $equently the dayes being at the longe$t, it doth not $eem likely that it was nece$$ary to pray for the lengthning of the day, to pro$ecute Victory in a Battail, the $pace of $even hours and more, which remained to Night, being $ufficient for that purpo$e. Upon which Grave Divines have been induced to think that the Sun was near $etting: And $o the words them$elves $eem to <foot>$ound</foot> <p n=>459</p> $ound, $aying, <I>Ne movearis Sol, ne movearis.</I> For if it had been in the Meridian, either it had been needle$s to have asked a Miracle, or it would have been $ufficient to have onely praid for $ome retardment. Of this opinion is <I>Cajetan,</I> to which $ub- $cribeth <I>Magaglianes,</I> confirming it by $aying, that <I>Jo$hua</I> had that very day done $o many other things before his commanding the Sun, as were not po$$ibly to be di$patch't in half a day. Whereupon they are forced to read the Words <I>in Medio Cœli</I> (to confe$s the truth) with a little har$hne$s, $aying that they import no more than this: <I>That the Sun $tood $till, being in our Hemi$phere, that is, above the Horizon.</I> But (if I do not erre) <marg><I>Solem $teti$$e, dum adhuc in He- mi$pharto no$tro, $upra $cilicet Ho- rizontem exi$teret.</I> Cajetan <I>in loce.</I></marg> we $hall avoid that and all other har$h expo$itions, if according to the <I>Copernican</I> Sy$teme we place the Sun in the mid$t, that is, in the Centre of the Cœle$tial Orbes, and of the Planetary Conver$ions, as it is mo$t requi$ite to do. For $uppo$ing any hour of the day (either Noon, or any other, as you $hall plea$e neerer to the Evening) the Day was lengthened, and all the Cœle$tial Revolutions $tayed by the Suns $tanding $till, <I>In the mid$t,</I> that is, <I>in the Centre of Heaven,</I> where it re$ides: A Sen$e $o much the more accomodate to the Letter (be$ides what hath been $aid already) in that, if the Text had de$ired to have affirmed the Suns Re$t to have been cau$ed at Noon-day, the proper expre$$ion of it had been to $ay, <I>It $tood $till at Noon-day,</I> or <I>in the Meridian Circle,</I> and not <I>in the mid$t of Heaven</I>: In regard that the true and only <I>Middle</I> of a Spherical Body (as is Heaven) is the Centre.</P> <P>Again, as to other places of Scripture, which $eem contrary to this po$ition, I do not doubt but that if it were acknowledged for True and Demon$trated tho$e very Divines who $o long as they repute it fal$e, hold tho$e places incapable of Expo$itions that agree with it would finde $uch Interpretations for them, as $hould very well $uit therewith; and e$pecially if to the know- ledge of Divine Learning they would but adde $ome knowledge of the A$tronomical Sciences: And as at pre$ent, whil$t they deem it fal$e they think they meet in Scripture only with $uch places as make again$t it, if they $hall but once have entertained another conceipt thereof, they would meet peradventure as many others that accord with it, and haply would judge, that the Holy Church doth very appo$itly teach, That God placed the Sun in the Centre of Heaven, and that thereupon by revolving it in it $elf, after the manner of a Wheel, He contributed the ordinary Cour$es to the Moon and other Erratick Stars, whil$t that $he Sings,</P> <P><I>Cœli Deus $ancti$$ime,</I></P> <P><I>Qui lucidum Centrum Poli,</I></P> <foot>Mmm 2 <I>Candore</I></foot> <p n=>460</p> <P><I>Candore ping is igneo,</I></P> <P><I>Augens decoro lumine,</I></P> <P><I>Quarto die, qui flammeam</I></P> <P><I>Solis rotam con$tituens</I></P> <P><I>Lunœ mini$tras ordinem,</I></P> <P><I>Vago$que cur$us Syderum.</I></P> <P>They might $ay, that the Name of <I>Firmament</I> very well a- greeth, <I>ad literam,</I> to the Starry Sphere, and to all that which is above the Planetary Conver$ions; which according to this Hy- pothe$is is altogether <I>firme</I> and immoveable. <I>Ad litteram</I> (the Earth moving circularly) they might under$tand its <I>Poles,</I> where it's $aid, <I>Nec dum Terram fecerat, & flumina, &</I> Cardi- nes <I>Orbis Terrœ,</I> Which <I>Cardines</I> or ^{*} <I>liinges</I> $eem to be a$cribed <marg>* Or Poles.</marg> to the Earth in vain, if it be not to turn upon them.</P> <head><I>FINIS.</I></head> <fig> <p n=>461</p> <head>AN ABSTRACT OF THE Learned Treati$e OF JOHANNIS KEPL<I>E</I>RUS, The Emperours <I>Mathematician</I>: ENTITULED <I>His Introduction upon</I> MARS:</head> <P>It mu$t be confe$$ed, that there are very many who are devoted to Holine$$e, that di$$ent from the Judgment of <I>Co- pernicus,</I> fearing to give the Lye to the Holy Gho$t $peaking in the Scriptures, if they $hould $ay, that the Earth mo- veth, and the Sun $tands $till. But let $uch con$ider, that $ince we judge of ve- ry many, and tho$e the mo$t principal things by the Sen$e of Seeing, it is impo$$ible that we $hould ali- enate our Speech from this Sen$e of our Eyes. Therefore many things daily occur, of which we $peak according to the Sen$e of Sight, when as we certainly know that the things them$elves are otherwi$e. An Example whereof we have in that Ver$e of <I>Virgil</I>;</P> <head><I>Provehimur portu, Terrœque urbe$que recedunt.</I></head> <P>So when we come forth of the narrow $traight of $ome Val- ley, we $ay that a large Field di$covereth it $elf. So Chri$t to <I>Peter, Duc in altum</I>; [Lanch forth into the Deep, or on high,] as if the Sea were higher than its Shores; For $o it $eemeth to the Eye, but the Opticks $hew the cau$e of this fallacy. Yet Chri$t u$eth the mo$t received Speech, although it proceed from this delu$ion of the Eyes. Thus we conceive of the Ri$ing and <foot>Setting</foot> <p n=>462</p> Setting of the Stars, that is to $ay, of their A$cen$ion and De$- cen$ion; when at the $ame time that we affirm the Sun ri$eth, o- thers $ay, that it goeth down. See my <I>Optices A$tronomiœ, cap.</I> 10. <I>fol.</I> 327 So in like manner, the <I>Ptolomaicks</I> affirm, that the Planets <I>$tand $till,</I> when for $ome dayes together they $eem to be fixed, although they believe them at that very time to be moved in a direct line, either downwards to, or upwards from the Earth. Thus the Writers of all Nations u$e the word <I>Sol$titi- um,</I> and yet they deny that the Sun doth really $tand $till. Like- wi$e there will never any man be $o devoted to <I>Copernicus,</I> but he will $ay, the Sun entereth into <I>Cancer</I> and <I>Leo,</I> although he granteth that the Earth enters <I>Capricorn</I> or <I>Aquarius</I>: And $o in other ca$es of the like nature. But now the Sacred Scriptures, $peaking to men of vulgar matters (in which they were not in- tended to in$truct men) after the manner of men, that $o they might be under$tood by men, do u$e $uch Expre$$ions as are granted by all, thereby to in$inuate other things more My$terious and Divine. What wonder is it then, if the Scripture $peaks according to mans apprehen$ion, at $uch time when the Truth of things doth di$$ent from the Conception that all men, whe- ther Learned or Unlearned have of them? Who knows not that it is a Poetical allu$ion, <I>P$al.</I> 19. where, whil$t under the $i- militude of the Sun, the Cour$e of the Go$pel, as al$o the Pere- grination of our Lord Chri$t in this World, undertaken for our $akes, is de$cribed, <I>The Sun</I> is $aid <I>to come forth of his Taberna- cle</I> of the Horizon, <I>as a Bridegroom out of his Chamber, re- joycing as a Giant to run a Race</I>? Which <I>Virgil</I> thus imitates;</P> <head><I>Tithono croceum linquens Auror a cubile</I>:</head> <P>For the fir$t Poets were among$t the Jews. The P$almi$t knew that the Sun went not forth of the Horizon, as out of its Tabernacle, & yet it $eemeth to the Eye $o to do: Nor did he believe, that the Sun moved, for that it appeared to his $ight $o to do. And yet he $aith both, for that both were $o to his $eeming. Neither is it to be adjudged fal$e in either Sen$e: for the perception of the Eyes hath its verity, fit for the more $ecret purpo$e of the P$al- mi$t in $hadowing forth the current pa$$age o$ the Go$pel, as al$o the Peregrination of the Son of God. <I>Jo$hua</I> likewi$e mentioneth the Vallies on or in, which the Sun and Moon mo- ved, for that they appeared to him at <I>Jordan</I> $o to do: And yet both the$e Pen-men may obtain their ends. <I>David,</I> (and with him <I>Syracides</I>) the magnificence of God being made known, which cau$ed the$e things to be in this manner repre$ented to $ight, or otherwi$e, the my$tical meaning, by means of the$e Vi$ibles being di$cerned: And <I>Jo$hua,</I> in that the Sun, as to his <foot>Sen$e</foot> <p n=>463</p> Sen$e of Seeing, $taid a whole day in the mid$t of Heaven, where- as at the $ame time to others it lay hid under the Earth. But in- cogitant per$ons onely look upon the contrariety of the words, <I>The Sun $tood $till,</I> that is, <I>The Earth $tood $till</I>; not con$idering that this contradiction is confined within the limits of the Op- ticks and A$tronomy: For which cau$e it is not outwardly ex- po$ed to the notice and u$e of men: Nor will they under$tand that the onely thing <I>Jo$huah</I> prayed for, was that the Mountains might not intercept the Sun from him; which reque$t he expre$- $ed in words, that $uited with his Ocular Sen$e: Be$ides it had been very un$ea$onable at that time to think of A$tronomy, or the Errours in Sight; for if any one $hould have told him that the Sun could not really move upon the Valley of <I>Ajalon,</I>, but onely in relation to Sen$e, would not <I>Jo$huah</I> have replyed, that his de$ire was that the day might be prolonged, $o it were by any means what$oever? In like manner would he have an$wered if any one had $tarted a que$tion about the Suns Mobility, and the Earths Motion. But God ea$ily under$tood by <I>Jo$huahs</I> words what he asked for, and by arre$ting the Earths Motion, made the Sun in his apprehen$ion $eem to $tand $till. For the $umm of <I>Jo$huahs</I> Prayer amounts to no more but this, that it might thus appear to him, let it in the mean time <I>be what it would</I> of it $elf. For that its $o $eeming, was not in vain and ridiculous, but accompanied with the de$ired effect. But read the tenth <I>Chap.</I> of my <I>Book,</I> that treats of <I>the Optick part of A- $tronomy,</I> where thou $halt finde the Rea$ons why the Sun doth in this manner $eem to all mens thinking to be moved, and not the Earth; as namely, becau$e the Sun appeareth $mall; and the Earth bigg. Again, the Motion of the Sun is not di$cerned by the eye, by rea$on of his $eeming tardity, but by ratiocina- tion onely; in that after $ome time it varieth not its proximity to $uch and $uch Mountains. Therefore it is impo$$ible that Rea- $on, unle$s it be fir$t in$tructed, $hould frame to it $elf any other apprehen$ion, than that the Earth with Heavens Arch placed over it, is as it were a great Hou$e, in which, being immoveable, the Sun like a Bird flying in the Air, pa$$eth in $o $mall a Species out of one Climate into another. Which imagination of all Man-kinde being thus, gave the fir$t line in the Sacred Leaves: ^{*} <I>In the beginning</I> ($aith <I>Mo$es) God created the Heaven and the <marg>* Gen. <I>Chv. 1. v.</I> 1.</marg> Earth</I>; for that the$e two are mo$t obvious to the eye. As if <I>Mo$es</I> $hould have $aid thus to Man; This whole Mundane Fa- brick which thou $ee$t, lucid above, and dark, and of a va$t ex- tent beneath, wherein thou ha$t thy being, and with which thou art covered, was created by God.</P> <P>In another place Man is que$tioned; <I>Whether he can finde out</I> <foot><I>the</I></foot> <p n=>464</p> <I>the height of Heaven above, or depth of the Earth beneath</I>: for that each of them appeareth to men of ordinary capacity, to have equally an infinite extent. And yet no man that is in his right mind will by the$e words circum$cribe and bound the diligence of A$tronomers, whether in demon$trating the mo$t contemptible Minuity of the Earth, in compari$on of Heaven, or in $earching out A$tronomical <I>Di$tances</I>: Since tho$e words $peak not of the Rational, but real Dimention; which to a Humane Body, whil$t confin'd to the Earth, and breathing in the open Air, is al- together impo$$ible. Read the whole 38. Chapter of <I>Job,</I> and compare it with tho$e Points which are di$puted in A$tronomy, <marg>* P$al. 24. 2.</marg> and Phy$iologie. If any one do alledge from <I>P$al.</I> 24. That ^{*} <I>The Earth is founded upon the Seas,</I> to the end that he may thence infer $ome new Principle in Philo$ophy, ab$urd to hear; as, That the Earth doth float upon the Waters; may it not truly be told him, That he ought not to meddle with the Holy Spirit, nor to bring him with contempt into the School of Phy$iologie. For the P$almi$t in that place means nothing el$e but that which men fore-know, and daily $ee by experience; namely, That the Earth (being lifted up after the $eparation of the Wa- ters) doth $wim between the Grand Oceans, and float about the Sea. Nor is it $trange that the expre$$ion $hould be the $ame where the <I>I$raelites</I> $ing, ^{*} <I>That they $ate on the River of Baby- <marg>P$al. 137. 1.</marg> lon</I>; that is, <I>by</I> the River $ide. or on the Banks of <I>Euphrates</I> and <I>Tygris.</I></P> <P>If any one receive this Reading without $cruple, why not the other; that $o in tho$e $ame Texts which are wont to be alledged again$t the Motion of the Earth, we may in like manner turn our eyes from Natural Philo$ophy, to the $cope and intent of Scri- pture. <I>One Generation pa$$eth away,</I> ($aith <I>Eccle$ia$tes) and a- <marg>* Chap. 1. v. 4, to 9.</marg> nother Generation cometh: But the Earth abideth for ever.</I> ^{*} As if <I>Solomon</I> did here di$pute with A$tronomers, and not rather put men in minde of their Mutability; when as the Earth, Mankindes habitation, doth alwaies remain the $ame: The Suns Motion doth continually return into what it was at fir$t: The Wind is acted in a Circle, and returns in the $ame manner: The Rivers flow from their Fountains into the Sea, and return again from thence unto their Fountains: To conclude, The Men of this Age dying, others are born in their room; the Fable of Life is ever the $ame; there is nothing new under the Sun. Here is no reference to any Phy$ical Opinion. <G><*>on<*>esi\a</G> is Moral of a thing in it $elf manife$t, and $een by the eyes of all, but little regarded: Tis that therefore which <I>Solomon</I> doth inculcate. For who knows not that the Earth is alwaies the $ame? Who $ees not that the Sun dothari$e from the Ea$t; That the Rivers continually run into <foot>the</foot> <p n=>465</p> the Sea; That the vici$$itudes of the Windes return into their primitive State; That $ome men $ucceed others? But who con- $idereth that the $elf-$ame <I>Scene</I> of Life is ever acting, by diffe- rent per$ons; and that nothing is <I>new</I> in humane affairs? There- fore <I>Solomon</I> in$tancing in tho$e things which all men $ee, doth put men in minde of that which many thorowly know, but too $lightly con$ider.</P> <P>But the 104. <I>P$alm</I> is thought by $ome to contain a Di$cour$e altogether Phy$ical, in regard it onely concerns Natural Philo$o- phy. Now God is there $aid, <I>To have laid the Foundations of <marg>P$al. 104. v. 5.</marg> the Earth, that it $hould not be removed for ever.</I> But here al- $o the P$almi$t is far from the Speculation of Phy$ical Cau$es: For he doth wholly acquie$ce in the Greatne$$e of God, who did all the$e things, and $ings an Hymne to God the Maker of them, in which he runneth over the World in order, as it appeared to his eyes. And if you well con$ider this P$alme, it is a Paraphra$e upon the $ix dayes work of the Crea- tion: For as in it the three fir$t dayes were $pent in the Separa- tion of Regions; the fir$t of Light from the exteriour Dark- ne$s; the $econd, of the Waters from the Waters, by the inter- po$ition of the Firm ament; the third, of the Sea from Land; when al$o the Earth was cloathed with Herbage and Plants: And the three la$t dayes were $pent in the filling the Re- gions thus di$tingui$hed; the fourth, of Heaven; the fifth, of the Seas and Aire; the fixth, of the Earth: So here in this P$alme there are $o many di$tinct parts pro- portionable to the Analogy of the $ix dayes Works. For in <I>Ver$e</I> 2. he cloaths and covereth the Creator with Light (the fir$t of Creatures, and work of the fir$t day) as with a Garment. The $econd part beginneth at <I>Ver$e</I> 3. and treats of the Waters above the Heavens, the extent of Heaven and of Me- teors (which the P$almi$t $eemeth to intend by the Waters a- bove) as namely of Clouds, Winds, Whirl-winds, Lightnings. The third part begins at <I>Ver$e</I> 6. and doth celebrate the Earth as the foundation of all tho$e things which he here con$idereth. For he referreth all things to the Earth, and to tho$e Animals which inhabit it, for that in the judgment of Sight the two prin- cipal parts of the World are Heaven and Earth. He therefore here ob$erveth that the Earth after $o many Ages hath not falte- red, tired, or decayed; when as notwith$tanding no man hath yet di$covered upon what it is founded. He goeth not about to teach men what they do not know, but putteth them in minde of what they neglect, to wit, the Greatne$$e and Power of God in creating $o huge a Ma$s $o firm and $tedfa$t. If an A$trono- mer $hould teach that the Earth is placed among the Planets, he <foot>Nnn over-</foot> <p n=>466</p> overthroweth not what the P$almi$t here $aith, nor doth he con- tradict Common Experience; for it is true notwith$tanding, that the Earth, the Structure of God its Architect, doth not de- cay (as our Buildings are wont to do) by age, or con$ume by wormes, nor $way and leane to this or that $ide; that the Seats and Ne$ts of Living Creatures are not mole$ted; that the Mountains and Shores $tand immoveable again$t the violence of the Winds and Waves, as they were at the beginning. But the P$almi$t addeth a mo$t Elegant Hypothe$is of the Separation of the Waters from the Continent or Main-land, and adorns it with the production of Fountains, and the benefits that Springs and Rocks exhibit to Birds and Bea$ts. Nor doth he omit the apparelling the Earths Surface, mentioned by <I>Mo$es</I> among$t the works of the third Day, but more $ublimely de$cribeth it in his Ca$e in expre$$ions infu$ed from Divine In$piration; and flouri- $heth out the commemoration of the many commodities which redound from that Exornation for the Nouri$hment and Com- <marg>* Shelter.</marg> fort of Man, and ^{*} Covert of Bea$ts. The fourth part begins at <I>Ver$e</I> 20. celebrating the fourth dayes work, <I>viz.</I> The Sun and Moon, but chiefly the commodiou$ne$$e of tho$e things, which in their Sea$ons befall to all Living Creatures and to Man; this being the $ubject matter of his Di$cour$e: So that it plain- ly appeareth he acted not the part of an A$tronomer. For if he had, he would not then have omitted to mention the five Planets, than who$e moiton nothing is more admirable, nothing more ex- cellent, nothing that can more evidently $et forth the Wi$dome of the Creator among$t the Learned. The fifth part begins, <I>Ver$e</I> 25. with the fifth Dayes work. And it $tores the Seas with Fi$hes, and covers them with Ships. The $ixth part is more ob- $curely hinted at, <I>Ver$e</I> 28. and alludeth to the Land-Creatures that were created the $ixth day. And la$tly, he declareth the goodne$$e of God in general, who daily createth and pre$erveth all things? So that whatever he $aid of the World is in relation to Living Creatures; He $peaks of nothing but what is granted on all hands; for that it was his intent to extol things known, and not to dive into hidden matters, but to invite men to con- template the Benefits that redouud unto them from the works of each of the$e dayes.</P> <P>And I do al$o be$eech my Reader, not forgetting the Divine Goodne$$e conferred on Mankind; the con$ideration of which the P$almi$t doth chiefly urge, that when he returneth from the Temple, and enters into the School of <I>A$tronomy,</I> he would with me prai$e and admire the Wi$dome and Greatne$$e of the Creator, which I di$cover to him by a more narrow explication of the Worlds Form, the Di$qui$ition of Cau$es, and Detection <foot>of</foot> <p n=>467</p> of the Errours of Sight: And $o he will not onely extoll the Bounty of God in the pre$ervation of Living Creatures of all kindes, and e$tabli$hment of the Earth; but even in its Motion al$o, which is $o $trange, $o admirable, he will acknowledge the Wi$dome of the Creator. But he who is $o $tupid as not to comprehend the Science of <I>A$tronomy,</I> or $o weak and $crupu- lous as to think it an offence of Piety to adhere to <I>Copernicus,</I> him I advi$e, that leaving the Study of <I>A$tronomy,</I> and cen$uring the opinions of Philo$ophers at plea$ure, he betake him$elf to his own concerns, and that de$i$ting from further pur$uit of the$e intricate Studies, he keep at home and manure his own Ground; and with tho$e Eyes wherewith alone he $eeth, being eleva- ted towards this to be admired Heaven, let him pour forth his whole heart in thanks and prai$es to God the Creator; and a$- $ure him$elf that he $hall therein perform as much Wor$hip to God, as the <I>A$tronomer,</I> on whom God hath be$towed this Gift, that though he $eeth more clearly with the Eye of his Under- $tanding; yet whatever he hath attained to, he is both able and willing to extoll his God above it.</P> <P>And thus much concerning the Authority of Sacred Scripture. Now as touching the opinions of the Saints about the$e Natural Points. I an$wer in one word, That in Theology the weight of Authority, but in Philo$ophy the weight of Rea$on is to be con- $idered. Therefore Sacred was <I>Lactantius,</I> who denyed the Earths rotundity; Sacred was <I>Augu$tine,</I> who granted the Earth to be round, but denyed the <I>Antipodes</I>; Sacred is the ^{*}Liturgy of <marg>* Officium</marg> our Moderns, who admit the $mallne$$e of the Earth, but deny its Motion: But to me more $acred than all the$e is Truth, who with re$pect to the Doctors of the Church, do demon$trate from Philo$ophy that the Earth is both round, circumhabited by <I>Antipodes,</I> of a mo$t contemptible $malne$$e, and in a word, that it is ranked among$t the Planets.</P> <foot>Nnn 2 AN</foot> <p n=>468</p> <head>AN ABSTRACT OF Some pa$$ages in the Commentaries of Didacus à Stunica, OF SALAMANCA Upon <I>JOB:</I></head> <head>The Toledo Edition, Printed by <I>JOHN RODERICK, Anno</I> 1584, in <I>Quarto,</I> Pag. 205. & <I>$eqq.</I> on the$e Words, Chap. 9. Ver$e 6.</head> <head><I>Who $haketh the Earth out of her place, and the Pil- lars thereof Tremble.</I></head> <P>The Sacred Pen-man here $ets down another ef- fect whereby God $heweth his Ahnighty Po- wer, joyned with infinite Wi$dom. Which place, though it mu$t be confe$$ed very diffi- cult to under$tand, might be greatly cleared by the Opinion of the <I>Pythagorians,</I> who hold the Earth to be moved of its own Na- ture, and that the Motion of the Stars can no other way be a$cer- tained, they being $o extreamly different in tardity and velocity. Of which judgement was <I>Philolaus,</I> and <I>Heraclides Ponticus,</I> as <I>Plutarch</I> relateth in his Book <I>De Placitis Philo$ophorum</I>: Who were followed by <I>Numa Pompilius,</I> and, which I more regard, The Divine <I>Plato</I> in his old age; in$omuch that he affirmed that it was mo$t ab$urd to think otherwi$e, as the $ame <I>Plutarch</I> tells us in his ^{*} <I>Numa.</I> And <I>Hypocrates</I> in his Book <I>De Flatibus,</I> <marg>* <I>In vita ejus.</I></marg> calleth the Air <G>thsghso)xh)ma,</G> <I>i. e.</I> The Earths Chariot. But in this <foot>our</foot> <p n=>469</p> our Age, <I>Copernicus</I> doth demon$trate the cour$es of the Pla- nets to be according to this Opinion. Nor is it to be doubted but that the Planets Places may be more exactly and certainly a$$igned by his Doctrine, than by <I>Ptolomies</I> Great Almoge$t of Sy$teme, or the Opinions of any others. For its manife$t, that <I>Ptolomy</I> could never de$cribe either the Motion of the Equi- noxes, or a$$ign the certain and po$itive beginning of the Year <*> the which he ingeniou$ly confe$$eth in <I>Lih.</I> 3. <I>De Almage$t. Mag- num. Ch.</I> 2. and which he leaveth to be di$covered in after times by tho$e A$tronomers, who coming into the World much later than he, might be able to invent $ome way to make more accurate ob$ervations. And although the ^{*} <I>Alphon$ines & Thebith Ben Core</I> <marg>* Followers of that Learned Kings Hypothe- $is.</marg> have attempted to explain them; yet it appeareth that they have done as much as nothing. For the Po$itions of the <I>Alphon$ines</I> di$agree among$t them$elves, as <I>Ricius</I> proveth. And although the Rea$on of <I>Thebith</I> be more acute, and that thereby he de- termined the certain beginning of the year, (being that which <I>Ptolomy</I> $ought for) yet it is now clear, that the Progre$$ions of the Equinoxes are much longer than he conceived they could be. Moreover, the Sun is found to be much nearer to us than it was <marg>* That is 5000 miles; eight of the$e making an <I>Italian,</I> or <I>Engli$h</I> mile of a 1000. paces, every paco containing 5. Feet.</marg> held to be in times pa$t, by above fourty thou$and ^{*} <I>Stadia,</I> or furlongs. The Cau$e and Rea$on of who$e Motion, neither <I>Ptolomy</I> nor any other A$trologers could ever comprehend: And yet the Rea$ons of the$e things are mo$t plainly explained and demon$trated by <I>Copernicus</I> from the Motion of the Earth, with which he $heweth that all the other <I>Phœnomena</I> of the Univer$e do more aptly accord. Which opinion of his is not in the lea$t contradicted by what <I>Solomon</I> $aith in ^{*} <I>Eccle$ia$tes: But the <marg>* Chap. 1. v. 4.</marg> Earth abideth for ever.</I> For that Text $ignifieth no more but this, That although the $ucce$$ion of Ages, and generations of Men on Earth, be various; yet the Earth it $elf is $till one and the $ame, and continueth without any $en$ible alteration; For the words run thus: <I>One Generation pa$$eth away, and another Generation cometh; but the Earth abideth for ever.</I> So that it hath no coherence with its Context, (as Philo$ophers $hew) if it <marg>The Motion of the Earth, not a- gain$t Scripture.</marg> be expounded to $peak of the Earths immobility. And al- though in this Chapter <I>Eccle$ia$tes,</I> and in many others, Holy Writ a$cribes Motion to the Sun, which <I>Copernicus</I> will have to $tand fixed in the Centre of the Univer$e; yet it makes nothing again$t his Po$ition. For the Motion that belongs to the Earth, is by way of $peech a$$igned to the Sun, even by <I>Copernicus</I> him- $elf, and tho$e who are his followers, $o that the Revolution of the Earth is often by them phra$ed, The Revolution of the Sun. To conclude, No place can be produced out of Holy Scripture, which $o clearly $peaks the Earths Immobility, as this doth its <foot>Mo-</foot> <p n=>470</p> Mobility. Therefore this Text, of which we have $poken, is ea- $ily reconciled to this Opinion. And to $et forth the Wonder- ful power and Wi$dome of God, who can indue and actuate the Frame of the Whole Earth (it being of a mon$trous weight by Nature) with Motion, this our Divine pen-man addeth; <I>And the pillars thereof tremble:</I> As if he would teach us, from the Doctrine laid down, that it is moved from its Foundations.</P> <fig> <pb> <head>AN EPISTLE Of the Reverend Father <I>PAOLO ANTONIO FOSCARINI,</I> A CARMELITE; Concerning The <I>PYTHAGORIAN</I> and <I>COPERNICAN</I> Opinion OF The Mobility of the <I>EARTH,</I> AND Stability of the <I>SVN;</I> AND Of the New Sy$teme or Con$titution OF THE WORLD.</head> <head>IN WHICH, The Authorities of <I>SACRED SCRIPTVRE,</I> and <I>ASSERTIONS</I> of <I>DIVINES,</I> commonly alledged again$t this Opinion, are Reconciled.</head> <head>WRITTEN To the mo$t Reverend FATHER, SEBASTIANO FANTONI, General of the Order of CARMELITES.</head> <head><I>Engli$hed from the Original,</I> BY <I>THOMAS SALVSBVRIE.</I></head> <head><I>So quis indiget $apientia, po$tulet à Deo.</I> Jacobi 1. ver$u. 5.</head> <head><I>Optavi, & datus e$t mihi $en$us.</I> Sapientiæ 7. ver$u. 7.</head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, MDCLXI.</head> <p n=>473</p> <head>To the Mo$t Reverend Father SEBASTIANO FANTONI, <I>General of the Order of</I> CARMELITES.</head> <P>In obedience to the command of the No- ble <I>Signore Vincenzo Carraffa,</I> a Neapo- litan, and Knight of S. <I>John of Jeru- $alem,</I> (a per$on, to $peak the truth, of $o great Merit, that in him Nobility of Birth, Affability of Manners, Univer$al knowledge of Arts and things, Piety and Vertue do all contend for prehemi- nence) I re$olved with my $elf to un- dertake the Defence of the Writings of the New, or rather Re- newed, and from the Du$t of Oblivion (in which it hath long lain hid) lately Revived Opinion, <I>Of the Mobility of the Earth, and Stability of the Sun,</I> in times pa$t found out fir$t by <I>Pytha- goras,</I> and at la$t reduced into Practice by <I>Copernicus</I>; who like- wi$e hath deduced the Po$ition of the Sy$teme and Con$titution of the World and its parts from that Hypothe$is: on which Subject I have formerly writ to You, Mo$t Reverend Sir: But in regard I am bound for <I>Rome</I> to preach there by your Com- mand; and $ince this Speculation may $eem more proper for a- nother Treati$e, to wit, a Volume of <I>Co$mography,</I> which I am in hand with, and which I am daily bu$ie about, that it may come forth in company with my <I>Compendium of the Liberal Arts,</I> which I have already fini$hed, rather than now to di$cu$s it by it $elf, I thought to forbear, imparting what I have done for the pre$ent; Yet I was de$irous to give, in the mean time, a brief ac- count of this my Determination, and to $hew You, Mo$t Reve- rend Father, (to whom I owe all my indeavours, and my very $elf) the Foundations on which this Opinion may be grounded, lea$t, whil$t otherwi$e it is favoured with much probability, it be found in reality to be extreamly repugnant (as at fir$t $ight it <foot>Ooo $eems)</foot> <p n=>474</p> $eems) not onely to Phy$ical Rea$ons, and Common Principles received on all hands (which cannot do $o much ha<*>m) but al$o (which would be of far wor$e con$equence) to many Authori- ties of $acred Scripture: Upon which account many at their fir$t looking into it, explode it as the mo$t fond Paradox and Mon$trous <I>Capriccio</I> that ever was heard of. Which thing pro- ceeds only from an antiquated and long confirmed Cu$tome, which hath $o hardened men in, and habituated them to Vul- gar, Plau$ible, and for that cau$e by all men (a$well learned as unlearned) Approved Opinions, that they cannot be removed one $tep from them: So great is the force of Cu$tome (which not unfitly is $tiled a $econd Nature) prevailing over the whole World, that touching things men are rather plea$ed with, de- lighted in, and de$irous of tho$e, which, though evil and obnox- ious, are by u$e made familiar to them, than $uch, wherewith, though better, they are not accu$tomed and acquainted. So in like manner, and that chiefly, in <I>Opinions,</I> which when once they are rooted in the Mind, men $tart at, and reject all others what$oever; not only tho$e that are contrary to, but even all that ever $o little di$agree with or vary from theirs, as har$h to the Ear, di$coloured to the Eye, unplea$ant to the Smell, nau$e- ous to the Ta$t, rough to the Touch. And no wonder: For Phy$ical Truths are ordinarily judged and con$idered by men, not according to their E$$ence, but according to the pre$cript of $ome one who$e de$cription or definition of them gaines him Authority among$t the vulgar. Which authority neverthele$s ($ince 'tis no more than humane) ought not to be $o e$teemed, as that that which doth manife$tly appear to the contrary, whether from better Rea$ons lately found out, or from Sen$e it $elf, $hould for its $ake be contemned and $lighted; Nor is Po$terity $o to be confined, but that it may, and dares, not only proceed farther, but al$o bring to light better and truer Experiments than tho$e which have been delivered to us by the Ancients. For the <I>Ge- nius's</I> of the Antients, as in Inventions they did not much $ur- pa$s the Wits of our times; $o for the perfecting of Inventions this Age of ours $eems not only to equal, but far to excell former Ages; Knowledge, whether in the Liberal or Mechanical Arts, daily growing to a greater height. Which A$$ertion might be ea$ily proved, were it not that in $o clear a ca$e, there would be more danger of ob$curing, than hopes of illu$trating it with any farther light.</P> <P>But (that I may not wholly be $ilent in this point) have not the $everal Experiments of Moderns, in many things, $topped the mouth of Venerable Antiquity, and proved many of their great- te$t and weightie$t Opinions, to be vain and fal$e? The Doctrine <foot>of</foot> <p n=>475</p> of the <I>Antipodes</I> by many of the Antients of approved Wi$- dome and Learning was held a Paradox no le$s ab$urd than this Our Opinion of the <I>Earths Motion</I> may $eem to be; as likewi$e that of the <I>Habitablene$$e of the Torrid Zone</I>: Of the$e Opini- ons, the fir$t was accounted unpo$$ible by many, but the latter was ab$olutely denyed by the unanimous con$ent of all: But later Authors (to the great felicity and perpetual Glory of their Age) have, not $o much by Authority, as by accurate diligence and indefatigable $tudy to finde out the truth, pro- ved them both to be undoubtedly true. Thus I affirm that the Antients were deceived, and that in too lightly challenging Credid and Authority for their Inventions, they di$covered too much folly. Here for brevities $ake I pa$s by many Dreams lately detected, both of <I>Ari$totle</I> and other of the antient Philo- $ophers; who in all likelihood if they had dived into the Ob$er- vations of Modern Writers, and under$tood their Rea$ons, would, by changing their Judgements, have given them the precedency, and would have $ub$cribed to their manife$t Truth. Hereby we $ee that we are not to have $o high a re$pect for the Antiens, that whatever they a$$ert $hould be taken upon tru$t, and that Faith $hould be given to their $ayings, as if they were Oracles and Truths $ent down from Heaven. But yet (which indeed is chiefly to be regarded in the$e matters) if any thing be found out that is repugnant to Divine Authority, or to the Sacred Leaves, that were dictated by the Holy Gho$t, and by His In$piration <marg><I>Faith is more certain, than ei- ther Sen$e or Rea- $on.</I></marg> expounded by the Holy Doctors of the Church, in this ca$e not onely Humane Rea$on, but even Sen$e it $elf is to $ubmitt: which, though by all manner of weighty Conditions and circum- $tances it $hould hold forth any thing contrary to Divine Autho- rity, (which indeed is $o plain, that there is no way left to evade the right un er$tanding of it) yet is it to be rejected; and we mu$t conclude our $elves deceived by it, and believe that that is not true which Sen$e and Rea$on repre$ents unto us: For, however we judge of things, we have, both in this and all other ca$es, a more certain knowledge, which proceeds from Divine Faith; as S. <I>Peter</I> hath mo$t excellently expre$t it: Who though with his Sen$es he $aw, and perceived the Glory of our Lord in his Transfiguration, and heard his words manife$ting his great Pow- er, yet neverthele$s all the$e things compared with the Light of Faith, he adds: ^{*}<I>We have al$o a more $ure word of Prophecy,</I> &c. <marg>* 2 Pet. 1. 19.</marg> Wherefore $ince this Opinion of <I>Pythagoras</I> and <I>Copernicus</I> hath entred upon the Stage of the World in $o $trange a Dre$s, and at the fir$t appearance (be$ides the re$t) doth $eem to oppo$e $un- dry Authorities of Sacred Scripture, it hath (this being granted) been ju$tly rejected of all men as a meer ab$urdity.</P> <foot>Ooo 2 But</foot> <p n=>476</p> <P>But yet becau$e the common Sy$teme of the World devi$ed by <I>Ptolomy</I> hath hitherto $atisfied none of the Learned, hereupon a $u$pi<*>ion is ri$en up among$t all, even <I>Ptolemy's</I> followers them- $elves, that there mu$t be $ome other Sy$teme, which is more true than this of <I>Ptolemy</I>; For although the <I>Phœnomena</I> of Cele$tial Bodys may $eem to be generally re$olved by this Hypothe$is, yet they are found to be involved with many difficulties, and refer- red to many devices; as namely of Orbes of $undry Forms and Figures, Epicicles, Equations, Differences, Excentricks, andinnu- merable $uch like fancies and Chymæra's which $avour of the <I>Ens Rationis</I> of Logicians, rather than of any <I>Realem E$$entiam.</I> Of which kinde is that of the <I>Rapid Motion,</I> than which I finde not any thing that can be more weakly grounded, and more ea$i- ly controverted and di$proved: And $uch is that conceit of the ^{*} Heaven void of Stars, moving the inferior Heavens or Orbes: <marg>* Or <I>Primu<*> Mobile.</I></marg> All which are introduced upon occa$ion of the variety of the Motions of Cele$tial Bodyes, which $eemed impo$$ible, by any other way, to be reduced to any certain and determinate Rule. So that the A$$ertors of that common Opinion, freely confe$s, that in de$cribing the Worlds Sy$teme, they cannot as yet di$co- ver, or teach the true Hypothe$is thereof: But that their endea- vours are onely to finde out, among$t many things, what is mo$t agreeable with truth, and may, upon better and more accomo- date Rea$ons, an$wer the Cele$tial <I>Phœnomena.</I></P> <P>Since that, the Tele$cope (an Optick Invention) hath been found out, by help of which, many remarkable things in the Heavens, mo$t worthy to be known, and till then unthought of, were di$- covered by manife$t $en$ation; as for in$tance, That the Moon is Mountainous; <I>Venus</I> and <I>Saturn</I> Tricorporeal; and <I>Jupiter</I> Quadricorporeal: Likewi$e that in the <I>Via Lactea,</I> in the <I>Ple- iades,</I> and in the Stars called <I>Nobulo$œ</I> there are many Stars, and tho$e of the greate$t Magnitude which are by turns adjacent to one another; and in the end it hath di$covered to us, new fixed Stars, new planets, and new Worlds. And by this $ame In$tru- ment it appears very probable, that <I>Venus</I> and <I>Mercury</I> do not move properly about the Earth, but rather about the Sun; and that the Moon alone moveth about the Earth. What therefore can be inferred from hence, but that the Sun doth $tand immo- vable in the Centre, and that the Earth, with the other Cele$tial Orbes, is circumvolved about it? Wherefore by this and many other Rea$ons it appears, That the Opinion of <I>Pythagor as</I> and <I>Copernicus</I> doth not di$agree with A$tronomical and Co$mogra- phical Principles; yea, that it carryeth with it a great likelihood and probability of Truth: Whereas among$t the $o many $eve- ral Opinions, that deviate from the common Sy$teme, and devi$e <foot>others,</foot> <p n=>477</p> others, $uch as were tho$e of <I>Plato, Calippus, Eudoxus</I>; and $ince <marg>* Cardan de re- rum variet. Lib. 1. Cap. 1.</marg> them of <I>Averroe, ^{*} Cardanus, Fraca$torius,</I> and others both Anti- ent and Modern, there is not one found that is more facile, more regularly ahd determinately, accommodated to the <I>Phœnomena</I> and Motions of the Heavens, without <I>Epicycles, Excentrix, Ho- mocentricks</I> Deferents, and the $upputation of the Rapid Motion. And this Hypothe$is hath been a$$erted for true, not onely by <I>Pythagoras,</I> and, after him, by <I>Copernicus,</I> but by many famous men, as namely, <I>Heraclitus,</I> and <I>Ecphantus, Pythagoreans,</I> all the Di$ciples of that Sect, <I>Miceta</I> of <I>Syracu$e, Martianus Capella,</I> and many more. Among$t whom, tho$e (as we have $aid) that have attempted the finding out of New Sy$temes (for they refu- $ed both this of <I>Pythagoras,</I> and that of <I>Ptolemy)</I> are numberle$s: who yet notwith$tanding allowed this Opinion of <I>Pythagoras</I> to carry with it much probability, and indirectly confirmed it; ina$- much as that they rejected the common one as imperfect, defe- <marg>* P. Clavins in ultima $uor. Ope- rum editione.</marg> ctive, and attended with many contradictions and difficulties. Among$t the$e may be numbered Father ^{*} <I>Clavius,</I> a mo$t learn- ed Je$uite; who, although he refutes the Sy$teme of <I>Pythagoras,</I> yet acknowledgeth the Levity of the common Sy$teme, and he ingeniou$ly confe$$eth, that for the removal of difficulties, in which the common Sy$teme will not $erve the turn, A$tronomers are forced to enquire after another Sy$teme, to the di$covery of which, he doth very earne$tly exhort them.</P> <P>Now can there a better or more commodious Hypothe$is be devi$ed, than this of <I>Copernicus,</I>? For <I>t</I>his Cau$e many Mo- dern Authors are induced to approve of, and follow it: but with much hæ$itancy, and fear, in regard that it $eemeth in their Opinion $o to contradict the Holy Scriptures, as that it cannot po$$ibly be reconciled to them. Which is the Rea$on that this Opinion hath been long $uppre$t, and is now entertained by men in a mode$t manner, ad as it were with a veiled Face; according to that advice of the Poet:</P> <P><I>Judicium populi nunquam contemp$eris unus,</I></P> <P><I>Ne nullis place as, dum vis contemnere multos.</I></P> <P>Upon con$ideration of which, (out of my very great love to- wards the Sciences, and my ardent defire to $ee the encrea$e and perfection of them, and the Light of Truth freed from all Er- rours and Ob$curities) I began to argue with my $elf touching this Point after this manner: This Opinion of the <I>Pythagoreans</I> is either true, or fal$e; If fal$e, it ought not to be mentioned, and de$erves not to be divulged: If true, it matters not, though it contradict all, as well Philo$ophers as A$tronomers: And though for its e$tabli$hment and reducement to u$e a new Philo$ophy <foot>and</foot> <p n=>478</p> and A$tronomy, ($ounded upon new Principles and Hypothe$e) $hould be con$tituted: For the Authority of Sacred Scripture will not oppo$e t<*>; neither doth one Truth contradict another. If therefore the Opinion of <I>Pythagoras</I> be true, without doubt God hath di$po$ed and dictated the words of of Holy Writ in $uch a manner, that they may admit an apt $en$e and reconcil<*> tion with that Hypothe$is. Being moved by the$e Rea$ons, and the probability of the $aid Opinion, I thought good to try whe- ther Texts of Sacred Scripture might be expounded according to Theological and Phy$ical Principles, and might be reconciled to it, $o that (in regard that hitherto it hath been held probable) it may in after times, coming without $cruple to be acknowledged for true, advance it $elf, and appear in publick with an uncover- ed Face, without any mans prohibition, and may lawfully and freely hold a Sacred intelligence with Holy Truth, $o earne$tly coveted and commended by good Men. Which de$igne, having hi- <marg><I>The Author fir$t Theologically defende<*>h the Earths Mobili- ty, approved by many of the Mo- derns.</I></marg> therto been undertaken by none that I know, wil, I am per$waded, be very acceptable to the Studious of the$e Learnings, e$pecially to the mo$t Learned <I>Galilœo Galilœi,</I> chief Mathematician to the mo$t Serene Grand Duke of <I>Tu$cany,</I> and <I>John Kepler,</I> chief Mathematician to his Sacred and invincible Maje$ty, the Empe- rour, and to all that Illu$trious, and much to be commended Ac- cademy of the <I>Lynceans</I>; whom, if I mi$take not, are all of this Opinion. Although I doubt not but they, and many other Learned Men might ea$ily have found out the$e or the like Re- conciliations of Scriptural expre$$ions; to whom neverthele$s I have thought fit (in re$pect of that profe$$ion which I have under- taken, upon the faith of my $oul, and the propen$ity that I have towards Truth) to offer that of the Poet,</P> <P><I>Nullius addictus jur are in verba Magi$tri.</I></P> <P>And in te$timony of my e$teem to them and all the Learned, to communicate the$e my thoughts; confidently a$$uring my $elf that they will accept them, with a Candor equal to that where- with I have written them.</P> <P>Therefore to come to the bu$ine$s: All Authorities of Di- vine Writ which $eem to oppo$e this Opinion, are reducible to $ix Cla$$es: The fir$t is of tho$e that affirm the Earth to $tand $till, and not to move: as <I>P$al. 92. He framed the round World $o $ure, that it cannot be moved</I>: Al$o <I>P$al. 104. Who laid the Foundations of the Earth, that it $hould not be removed for ever</I>: And <I>Eccle$ia$tes 1. But the Earth abideth for ever</I>: And others of the like $en$e.</P> <P>The $econd is of tho$e which atte$t the Sun to move, and <foot>Revolve</foot> <p n=>479</p> Revolve about the Earth; as <I>P$al. 19. (b) In them hath be $et a <marg><I>(b) Or</I> In Sole po$uit tabernacu- lum $uum, <I>accor- ding to the Tran- $lation our Au- thor followeth.</I></marg> Tabernacle for the Sun, which cometh forth as a Bridegroom out of his chamber, and rejoyceth as a Gyant to run his Cour$e. It cometh forth from the uttermo$t part of the Heaven, and runneth about unto the end of it again; and there is nothing hid from the heat thereof.</I> And <I>Eccle$ia$t. 1. The Sun ri$eth, and the Sun go- eth down, and ha$teth to the place where be aro$e: it goeth towards the South, and turneth about unto the North.</I> Whereupon the Suns Retrogradation is mentioned as a Miracle, <I>I$aiah 38. The Sun returned ten degrees.</I> And <I>Eccle$ia$ticus 48. In his time the Sun went backward, and lengthened the life of the King.</I> And for this rea$on it is related for a Miracle, in the Book of <I>Jo$buah,</I> that at the Prayers of that great Captain the Sun $tood $till, its motion being forbidden it, by him<I>: Jo$h.</I>10. <I>Sun $tand thou $till upon Gibeon.</I> Now if the Sun $hould $tand $till, and the Earth move about it, its $tation at that time was no Miracle; and if <I>Jo$huah</I> had intended, that the light of the day $hould have been prolonged by the Suns $plendour, he would not have $aid, <I>Sun $tand thou $till,</I> but rather <I>Earth $tand thou $till.</I></P> <P>The third Cla$$is is of tho$e Authorities which $ay, that Hea- ven is <I>above,</I> and the Earth <I>beneath</I>; of which $ort is that place of <I>Joel, chap.</I> 2. cited by S. <I>Peter,</I> in <I>Acts. 2. I will $hew wonders in Heaven above, and $ignes in the Earth beneath,</I> with others of the like purport. Hereupon Chri$t at his Incarnation is $aid to <I>come down from Heaven</I>; and after his Re$urrection to have <I>a$- cended up into heaven.</I> But if the Earth $hould move about the Sun, it would be, as one may $ay, in Heaven, and con$e- quently would rather be <I>above</I> Heaven than <I>beneath</I> it. And this is confirmed; For that the Opinion which placeth the Sun in the Centre, doth likewi$e place <I>Mercury</I> above the Sun, and <I>Venus</I> above <I>Mercury</I>; and the Earth above <I>Venus,</I> together with the Moon, which revolves about the Earth, and therefore the Earth, together with the Moon, is placed in the third Heaven. If therefore in Spherical Bodies, as in the World, <I>beneath</I> $igni- <marg><I>In Spberieall Bodies,</I> Deor$um <I>is the Centre, and</I> Sur$um <I>the Cir- cumference.</I></marg> fies no more than to be neer to the centre, and <I>above,</I> than to approach the Circumference, it mu$t needs follow, that for ma- king good of Theological Po$itions concerning the A$cen$ion and De$cen$ion of Chri$t, the Earth is to be placed in the cen- tre, and the Sun, with the other Heavens in the Circumference; and not according to <I>Copernicus,</I> who$e Hypothe$is inverts this Order: with which one cannot $ee how the true A$cen$ion and De$cen$ion can be con$i$tent.</P> <P>The fourth Cla$$is is of tho$e Authorities which make Hell to be in the Centre of the World, which is the Common Opinion of Divines, and confirmed by this Rea$on, That $ince Hell (ta- <foot>ken</foot> <p n=>480</p> ken in its $trict denomination) ought to be in the lowe$t part of the World, and $ince that in a Sphere there is no part lower then the Centre, Hell $hall be, as it were, in the Centre of the World, which being of a Spherical Figure, it mu$t follow, that <marg><I>Hell is in the centre of the Earth, not of the World.</I></marg> Hell is either in the Sun (fora$much as it is $uppo$ed by this Hy- pothe$is to be in the Centre of the World) or el$e $uppo$ing that Hell is in the Centre of the Earth, if the Earth $hould move about the Sun, it would nece$$arily en$ue, that Hell, together with the Earth, is in Heaven, and with it revolveth about the third Heaven; than which nothing more ab$urd can be $aid or imagi- ned.</P> <P>The fifth Cla$$is, is of tho$e Authorities which alwayes op- <marg><I>Heaven and Earth are always mutually oppo$ed to each other.</I></marg> po$e Heaven to the Earth, and $o again the Earth to Heaven; as if there were the $ame relation betwixt them, with that of the Centre to the Circumference, and of the Circumference to the Centre. But if the Earth were in Heaven, it $hould be on one $ide thereof, and would not $tand in the Middle, and con$equent- ly there would be no $uch relation betwixt them; which never- thele$s do, not only in Sacred Writ, but even in Common Speech, ever and every where an$wer to each other with a mutual Oppo- fition. Whence that of <I>Gene$. 1. In the beginning God created the Heaven and the Earth</I>: and <I>P$al. 115. The Heaven, even the Heavens are the Lords; but the Earth hath he given to the Children of men:</I> and our Saviour in that Prayer which he pre- $cribeth to us, <I>Matth. 6. Thy will be done in Earth, as it is in Heaven:</I> and S. <I>Paul, 1 Corinth. 15. The fir$t man is of the Earth, earthy; the $econd man is of Heaven, heavenly:</I> and <I>Colo$$. 1. By him were all things created that are in Heaven, and that are in Earth</I>: and again, <I>Having made peace through the Blood of his Cro$$e for all things, whether they be things in Earth or things in Heaven:</I> and <I>Chap. 3. Set your affections on things above, not on things on the Earth</I>; with innumerable other $uch like places. Since therefore the$e two Bodies are alwayes mu- tually oppo$ed to each other, and Heaven, without all doubt, referreth to the Circumference, it mu$t of nece$$ity follow, that the Earth is to be adjudged the place of the Centre.</P> <P>The $ixth and la$t Cla$$is is of tho$e Authorities, which (being rather of Fathers and Divines, than of the Sacred Scripture) $ay, That the Sun, after the day of Judgment $hall $tand immoveable <marg><I>After the day of Judgment the Earth $hall $tand immoveable.</I></marg> in the Ea$t, and the Moon in the We$t. Which Station, if the <I>Pythagorick</I> Opinion hold true, ought rather to be a$cribed to the Earth, than to the Sun; for if it be true, that the Earth doth now move about the Sun, it is nece$$ary that after the day of Judgment it $hould $tand immoveable. And truth is, if it mu$t $ub$i$t without motion in one con$tant place, there is no rea$on <foot>why</foot> <p n=>481</p> why it $hould rather $tand in one $ite of that Place than in ano- ther, or why it $hould rather turn one part of it than another to the Sun, if $o be that every of its parts without di$tinction, which is de$titute of the Suns light, cannot choo$e but be di$mal, and much wor$e affected than that part which is illuminated. Hence al$o would ari$e many other ab$urdities be$ides the$e.</P> <P>The$e are the Cla$$es, &c. from which great a$$aults are made again$t the $tructure of the Pythagorick Sy$teme; yet by that time I $hall have fir$t laid down $ix Maximes or Principles, as impregnable Bulwarks erected again$t them, it will be ea$ie to batter them, and to defend the Hypothe$is of <I>Pythagoras</I> from being attaqued by them. Which before I propound, I do pro- fe$s (with that Humility and Mode$ty which becometh a Chri- $tian, and a per$on in Religious Orders) that I do with reverence $ubmit what I am about to $peak to the Judgment of Holy Church. Nor have I undertaken to write the$e things out of any inducements of Temerity, or Ambition, but out of Charity and a De$ire to be auxiliary to my neighbour in his inqui$ition after Truth. And there is nothing in all this Controver$ie maintained by me (that expect to be better in$tructed by tho$e who profe$s the$e Studies) which I $hall not retract, if any per- $ons $hall by $olid Rea$ons & reiterated Experiments, prove $ome other Hypothe$is to be more probable; but yet, until $uch time as they $hall decide the Point, I $hall labour all I can for its $upport.</P> <P>My fir$t and chiefe$t Maxime is this; When any thing is at- tributed in Holy Writ, to God, or to a Creature, thats not be- $eeming to, or incommen$urate with them, it mu$t of nece$$ity be received and expounded one, or more of the four following wayes; Fir$t, it may be $aid to agree with them <I>Metaphorically, and Proportionally, or by Similitude.</I> Secondly, <I>According to our manner of Con$idering, Apprehending, Conceiving, Vnder$tand- ing, Knowing, &c.</I> Thirdly, <I>according to the Opinion of the Vulgar, and the Common way of Speaking:</I> to which Vulgar Speech the Holy Gho$t doth very often with much $tudy acco- modate it $elf. Fourthly, <I>In re$pect of our $elves, and for that he makes him$elf like unto us.</I> Of each of the$e wayes there are the$e examples: God doth not walk, $ince he is Infinite and Im- moveable; He hath no Bodily Members, $ince he is a Pure Act; and con$equently is void of all Pa$$ion of Minde; and yet in Sacred Scripture, <I>Gen. 3. ver$.</I> 8. it is $aid, <I>He walked in the cool of the day</I>: and <I>Job 22. ver$.</I> 14. it is $aid, <I>He walketh in the ^{*} Cir- <marg>* Circa Cardi- nes Cœli.</marg> cuit of Heaven:</I> and in many other places coming, departing, making ha$t is a$cribed to God; and likewi$e Bodily parts, as Eyes, Ears, Lips, Face, Voice, Countenance, Hands, Feet, Bow- els, Garments, Arms; as al$o many Pa$$ions, $uch as Anger, <foot>Ppp Sor-</foot> <p n=>482</p> Sorrow, Repentance, and the like. What $hall we $ay there- fore? Without doubt $uch like Attributes agree with God (to u$e the Schoolmens words <I>Metaphorically, Proportionally, and by Similitude</I>: And touching Pa$$ions, it may be $aid, that God conde$cendeth to repre$ent him$elf after that manner: as for in$tance, <I>The Lord is angry</I>; i.e. <I>He revealeth him$elf as one that is angry: He grieved</I>; i. e. <I>He revealeth him$elf, as one that is $orrowful: It repented him that he had made man</I>; i.e. <I>He $ee- med as one that repented.</I> And indeed all the$e things are <I>Com- parativè ad nos,</I> and in re$pect of us. So God is $aid to be in Heaven, to move in time, to $hew him$elf, to hide him$elf, to ob$erve and mark our $teps; to $eek us, to $tand at the door, to knock at the door; not that he can be contained in a bodily place, nor that he is really moved, nor in time; nor that humane manners or cu$tomes can agree with him, $ave only according to our manner of Apprehen$ion: This Conception of ours orderly di$tingui$heth the$e Attributes in him one from another, when, notwith$tanding, they are one and the $ame with him: This Ap- prehen$ion of ours divideth al$o his actions into $everal times, which, neverthele$$e, for the mo$t part, are produced in one and the $ame in$tant: And this, to conclude, alwayes apprehendeth tho$e things with $ome defect, which, notwith$tanding are in God mo$t perfect. For this rea$on doth the Sacred Scripture expre$s it $elf <I>according to the Vulgar Opinion,</I> whil$t it a$cribes to the Earth Ends and Foundations, which yet it hath not; to the Sea a Depth not to be fathomed; to Death (which is a Pri- vation, and con$equently a Non entity) it appropriates Actions, Motion, Pa$$ions, and other $uch like Accidents, of all which it is deprived, as al$o Epithites and Adjuncts, which really cannot $uit with it: <I>Is not the bitterne$$e of Death pa$t</I>? 1 Sam. 15. 32. <I>Let death come upon them,</I> P$al 6. <I>He hath prepared the In$tru- ments of Death,</I> P$al. 7. 14. <I>Thou rai$e$t me from the gates of Death,</I> P$al. 84. <I>In the mid$t of the $hadow of Death,</I> P$al. 23. <I>Love is $trong as Death,</I> Cant. 8. 9. <I>The Fir$t-Born of Death,</I> Job 18. 13. <I>De$truction and Death $ay, &c.</I> Job 28. 22. And who knows not that the whole Hi$tory of the rich Glutton doth con$i$t of <marg>Luke 16.</marg> the like phra$es of <I>Vulgar Speech</I>? So <I>Eccle$ia$ticus,</I> Chap. 27. ver$. 11. <I>The godly man abideth in wi$dome, as the Sun; but a fool changeth as the Moon</I>; and yet the Moon according to the real truth of the matter no wayes changeth, but abides the $ame for ever, as <I>A$tronomers</I> demon$trate, one half thereof remain- ing alwayes lucid, and the other alwayes opacous. Nor at any time doth this $tate vary in it, unle$$e <I>in re$pect of us,</I> and <I>ac- cording to the opinion of the Vulgar.</I> Hence it is cleer, that the holy Scripture $peaks according to the common form of $peech u- <foot>$ed</foot> <p n=>483</p> $ed among$t the unlearned, and according to the appearance of things, and not according to their true Exi$tence. In like man- ner <I>Gene$.</I> 1. in the de$cription of the Creation of all things, the Light is $aid to be made fir$t of all, and yet it followeth in the Text, <I>And the Evening and the Morning made the fir$t day</I>: and a little after the $everal Acts of the Creation are di$tingui$hed and a$$igned to $everal days, and concerning each of them it is $aid in the Text, <I>And the Evening and the Morning made the $econd day</I>; and then <I>the third day, the fourth day, &c.</I> Hence many doubts ari$e, all which I $hall propound according to the common Sy$teme, that it may appear even from the <I>H</I>ypothe$is of that Sy$teme, that the $acred Scripture $ometimes, for the a- voyding of emergent difficulties, is to be under$tood in a vulgar $en$e and meaning, and in re$pect of us, and not according to the nature of things. Which di$tinction even <I>Ari$totle</I> him$elf <marg>Alia $unt notio- ra nobis, alia, no- tiora natura, vel $ecundum $e, <I>A- r $t. lib. 1. Phy$.</I></marg> $eemeth to have hinted, when he $aith, ^{*} <I>Some things are more intelligible to us; others by nature,</I> or <I>$ecundum $e.</I></P> <P>Fir$t therefore; If the light were made before heaven, then it rolled about without heaven to the making of the di$tinction of Day and Night. Now this is contrary to the very doctrine of the$e men, who affirm that no Cœle$tial Body can be moved unle$$e <I>per accidens,</I> and by the motion of <I>H</I>eaven, <I>and as a knot in a board at the motion of the board.</I> Again, if it be $aid, that the Light was created at the $ame time with <I>H</I>eaven, and began to be moved with <I>H</I>eaven, another doubt ari$eth, that likewi$e oppo$eth the fore$aid common <I>Hypothe$is:</I> For it being $aid, that Day and Night, Morning and Evening were made, that $ame is either in re$pect of the Univer$e, or onely in re$pect of the Earth and us. If $o be that the Sun turning round (according to the <I>Hypothe$is</I> of the Common Sy$teme) doth not cau$e the Night and Day, but only to opacous Bodies which are de$titute of all other light, but that of the Sun, whil$t in their half part (which is their <I>Hemi$phœre)</I> and no more, (for that the Suns light pa$$eth over but one half of an opacous Body, unle$s a ve- ry $mall matter more in tho$e of le$$er bulk) they are illumina- ted by the Suns a$pect, the other half remaining dark and tene- bro$e, by rea$on of a $hadow proceeding from its own Body. Therefore the di$tinction of dayes by the light of heaven, ac- cording to the de$cription of them in the $acred Scriptures, mu$t not be under$tood <I>ab$olutely,</I> and <I>$ecundum $e,</I> and <I>Nature her $elf</I>; but in re$pect of the Earth, and of us its inhabitants, and con$equently <I>$ecundum nos.</I> 'Tis not therefore new, nor unu- $ual in $acred Scripture to $peak of things <I>$ecundum nos,</I> and on- ly <I>in re$pect of us,</I> and <I>$ecundum apparentiam</I>; but not <I>$ecundum $e,</I> and <I>reinaturam,</I> or <I>Ab$olutely</I> and <I>Simply.</I></P> <foot>Ppp 2 And</foot> <p n=>484</p> <P>And if any one would under$tand the$e Days of $acred Scri- pture, not only <I>$ecundum nos,</I> but al$o <I>$ecundum naturam,</I> as circulations of Cœle$tial Light returning to the $elf $ame point from whence it did at fir$t proceed; $o as that there needs no re$pect to be had to Night or to ^{*} Darkne$$e, for which $ole rea- <marg>* Aut ad Umbram</marg> $on we are fain to imbrace the Interpretation of $acred Scripture <I>$ecundum nos</I>; In oppo$ition to this we may thus argue: If the $acred Scripture be under$tood to $peak <I>ab$olutely,</I> of iterated and $ucce$$ive circulations of light, and not <I>re$pectu no$tri,</I> as if the$e words <I>Evening and Morning</I> had never been in$erted, which in their natural acceptation denote the Suns habitude to us and to the Earth: For that the <I>Morning</I> is that time when the Sun be- gins to wax light, and to ri$e above the <I>Horizon</I> in the Ea$t, and become vi$ible in our <I>Hemi$phœre,</I> and <I>Evening</I> is the time in which the Sun declines in the We$t, and approacheth with its light neerer to the other oppo$ite <I>Horizon</I> and <I>Hemi$phœre,</I> which is contiguous to this of ours. But the word <I>Day</I> is a Co- relative to the word <I>Night.</I> From hence therefore it evidently appeareth, that the$e three words <I>Evening, Morning,</I> and <I>Day,</I> cannot be under$tood of a Circulation of Light <I>$ecundum $e,</I> and <I>ab$olutè,</I> but only <I>$ecundum nos,</I> and <I>re$pectu no$tri</I>; and in that $en$e indeed the <I>Morning</I> and <I>Evening</I> do make the <I>Night</I> and <I>Day,</I></P> <P>In like manner, <I>Gen.</I> 1. 16. it is $aid, <I>God made two great Lights; the greater Light to rule the Day, and the le$$er Light to rule the Night, and the Stars.</I> Where both in the Propo$ition and in the $pecification of it, things are $poken which are very di$agreeing with Cœle$tial Bodies. Therefore tho$e words are in that place to be interpreted according to the fore$aid Rules; namely, ac- cording to the third and fourth; $o that they may be $aid to be under$tood <I>according to the $en$e of the vulgar, and the common way of $peaking,</I> which is all one, as if we $hould $ay, <I>$ecundum apparentiam,</I> and <I>$ecundum nos, vel re$pectu no$tri.</I> For fir$t, it is $aid in the Propo$ition, <I>And God made two great Lights</I>; meaning by them the Sun and Moon, whereas according to the truth of the matter the$e are not the Greater Lights; For al- though the Sun may be reckoned among$t the Greater, the Moon may not be $o, unle$s <I>in re$pect of us.</I> Becau$e among$t tho$e that are ab$olutely the Greater, and a little le$$er than the <marg><I>Which are really the great Lights in Heaven.</I></marg> Sun (nay in a manner equal to it) and far bigger than the Moon, we may with great rea$on enumerate <I>Saturn,</I> or $ome of the Fixed Stars of the fir$t Magnitude, $uch as <I>Canopus,</I> (otherwi$e called <I>Arcanar)</I> in the end of a River; or the <I>Little Dog</I> in the mouth of the <I>Great Dog</I>; or the Foot of <I>Orion,</I> called <I>Ri- gel</I>; or his <I>Right $houlder,</I> or any other of that Magnitude. <foot>There-</foot> <p n=>485</p> Therefore the <I>two great Lights</I> are to be under$tood in re$pect of us, and according to vulgar e$timation, and not according to the true and reall exi$tence of $uch Bodies. Secondly, in the $peci- fication of the Propo$ition it is $aid, <I>The greater Light to rule the Day</I>; hereby denoting the Sun; in which the verbal $en$e of Scripture agreeth with the Truth of the Thing; For that the Sun is the Greate$t of all Luminaries, and Globes. But that which followeth immediately after, <I>And the le$$er Light to rule the Night,</I> meaning the Moon, cannot be taken in the true and real $en$e of the words: For the Moon is not the le$$er Light, but <I>Mercury</I>; which is not only much le$$er than the Moon, but al$o than any other Star. And if, again, it be $aid, That the Holy Text doth not $peak of the Stars, but onely of the Luminaries, for that pre$ently after they are mentioned apart, <I>And the Stars</I>; and that what we $ay is true touching the compari$on of the Stars among$t them$elves, but not in re$pect of the Luminaries, name- ly, the Sun and Moon: This reply doth di$cover a man to be utterly ignorant in the$e Studies, and $uch who having not the lea$t $mattering in them, doth conceive an ab$urd and erroneous Opinion of the Cœle$tial Bodies. For the Moon and Sun, con- $idered in them$elves, and as they appear to us, if they $hould be a far greater di$tance from us, than indeed they are, would be no other, nor would appear to us otherwi$e than Stars, as the re$t do in the Firmament. But Great Luminaries they neither <marg><I>The Sun, Moon, and Stars are one & the $ame thing.</I></marg> are, nor $eem to be, $ave only <I>in re$pect of us:</I> And $o, on the other $ide, the Stars, as to them$elves, are no other than $o many Suns and $o many Moons; yet are $o far remote from us, that by rea$on of their di$tance they appear thus $mall, and dim of light, as we behold them. For the greater and le$$er di$tance of heavenly Bodies <I>(cæteris paribus)</I> doth augment and dimini$h their appearance both as to Magnitude and Light. And there- fore the words which follow in that place of <I>Gene$is, And the Stars</I> (as di$tingui$hing the Stars from the Sun and Moon) are to be taken in no other acceptation than that which we have $po- ken of, namely, <I>according to the $en$e of the Vulgar, and the common manner of $peech.</I> For indeed, according to the truth of the matter, all Cœle$tial Bodies, being $hining Globes, are of a va$t bigne$s, to which if we $hould be $o neer as we are to the Moon, they would $eem to us of as great, yea a greater magni- tude than the Moon: As likewi$e on the contrary, if we were as far di$tant from the Sun and Moon, as we are from them, both Moon and Sun would $hew but as $tars to us. And yet the $plendor of the Sun would doubtle$s be greater <I>inten$ivè</I> than that of any other $tar. For, although it $hould be granted that $ome $tars (as tho$e of the Fixed that twinkle) do $hine of them- <foot>$elves</foot> <p n=>486</p> $elves, aud by their own nature, as the Sun, that derives not its light from others (which yet remains undecided and doubtful) and borrow not their light from the Sun; Neverthele$s $ince the brightne$s of none of the $tars may be compared with the Suns $plendour, which was created by God fir$t, and before all other Luminaries, in the highe$t kind of Light, it would therefore notwith$tanding follow, that none of tho$e $tars, although pla- ced in the $ame proximity to us with the Sun, and therefore ap- pearing to us of the $ame Magnitude as the Sun, can be$tow up- on us $o much Light as we receive from the Sun: As on the contrary, the Sun, at the $ame remotene$$e from us as they are, would indeed, as to its Magnitude, appear to us as one of tho$e $tars, but of a $plendour much more <I>inten$e</I> than that of theirs. <marg><I>The Earth is a- nother Moon or Star.</I></marg> So that, now, the Earth is nothing el$e but another Moon or $tar, and $o would it appear to us, if we $hould behold it from a con- venient di$tance <I>on high.</I> And in it might be ob$erved (in that variety of Light and Darkne$s which the Sun produceth in it by making Day and Night) the $ame difference of A$pects that are $een in the Moon, and $uch as are ob$erved in tricorporate <I>Ve- nus</I>; in like manner al$o 'tis very probable that the $ame might be di$cerned in other Planets, which $hine by no light of their own, but by one borrowed from the Sun. What ever there- fore may touching the$e matters be delivered in the $acred Leaves or the common $peech of men, di$$enting from the real truth, it ought (as we have $aid before) ab$olutely to be received and un- der$tood <I>$ecundum vulgi $ententiam, & communem loquendi & concipiendi $tylum.</I></P> <P>And $o, to return to our purpo$e, if, all this con$idered, the <I>Pythagorian</I> opinion be true, it will be ea$ie, according to the $ame Rule, to reconcile the authority of $acred Scriptures with it, however they $eem to oppo$e it, and in particular tho$e of the fir$t and $econd Cla$$is, <I>$cilicet</I> by my fir$t <I>Maxime:</I> For that in tho$e places the holy Records $peak according to our manner of under$tanding, and according to that which appeareth in re$pect of us; <I>For thus it is with tho$e Bodies, in compari$on of us, and</I> <marg><I>Why the Sunne $eemeth to us to move, & not the Earth.</I></marg> <I>as they are de$cribed by the vulgar and commune way of humane Di$cour$e; So that the Earth appears as if it were $tanding $till and immoveable, and the Sun, as if it were circumambient about her.</I> And $o the Holy Scripture is u$ed in the Commune and Vulgar way of $peaking; becau$e in re$pect of our $ight, the Earth $eems rather to $tand fixed in the Centre, and the Sun to circumvolve about it, than otherwi$e: as it happens to tho$e that are putting off from the Banks of a River to whom the $hose $eems to move backwards, and go from them: but they do not perceive (which yet is the truth) that they them$elves go forwards. <foot>Which</foot> <p n=>487</p> Which fallacy of our $ight is noted, and the Rea$on thereof a$- $igned by the Opticks; upon wich, as being $trange to, and be- $ides my purpo$e, I will not $tay) and on this account is <I>Æneas</I> brought in by <I>Virgil,</I> $aying;</P> <marg><I>Æneid.</I> 3.</marg> <head><I>Provehimur portu, terræque urbe$que recedunt.</I></head> <P>But it will not be ami$s to con$ider why the $acred Scripture doth $o $tudiou$ly comply with the opinions of the Vulgar, and why it doth not rather accurately in$truct men in the truth of the matters, and the $ecrets of Nature. The Rea$on is, fir$t, the be- nignity of Divine Wi$dome, whereby it $weetly accomodates it $elf to all things, in proportion to their Capacity and Nature. Whence in Natural Sciences, it u$eth natural and nece$$ary cau- $es, but in Liberal Arts it worketh liberally, upon Generous Per$ons after a $ublime and lofty manner; upon the Common People, familiarly and humbly; upon the Skilful, learnedly; upon the Simple, vulgarly; and $o on every one, according to his condition and quality. Secondly, becau$e it is not its In- tention to fill our mindes in this life with vain and various curi- o$ities, which might occa$ion our doubt and $u$pen$e. For the <marg><I>(a)</I> Eccle$. <I>c. 1. v. ult.</I></marg> truth is, <I>(a) He that increa$eth knowledge, increa$eth $orrow.</I> Moreover it did not only permit, but even decree, thatth e World $hould be very much bu$ied in Controver$ies and Di$pu- tations, and that it $hould be imployed about the uncertainty of <marg><I>(b) Chap. 3. v.</I> 11.</marg> things; according to that $aying of <I>Eccle$ia$tes</I> <I>(b) He hath $et the World in their heart; $o that no man can find out the work that God maketh from the beginning unto the end.</I> And touching tho$e doubts, God will not permit that they $hall be di$covered <marg><I>(c)</I> 1 Cor. <I>c. 4. v.</I> 5</marg> to us before the end of the World: <I>(c) At which time he will bring to light the hidden things of darkne$$e:</I> But Gods onely $cope in the $acred Scripture is to teach men tho$e things which conduce to the attainment of Eternal Life; which having ob- <marg><I>(d)</I> 1 Cor. <I>c. 13. v.</I> 12.</marg> tained, <I>(d) We $hall $ee him face to face: (e) and $hall be</I> <marg><I>(e)</I> 1 John <I>c. 3. v.</I> 2.</marg> <I>like him, for we $hall $ee him as he is.</I> Then $hall he clearly <I>à Priori</I> make known unto us all tho$e Curio$ities, and Dogmati- cal Que$tions, which in this life, <I>(f) in which we $ee through a</I> <marg><I>(f)</I> 1 Cor. <I>c. 13. v.</I> 12.</marg> <I>Gla$$e darkly,</I> could be known by us but imperfectly and <I>à po$te- riori,</I> and that not without much pains and $tudy. For this cau$e the Wi$dome of God, revealed to us in the $acred Leaves, is not $tiled Wi$dome ab$olutely, but <I>(g) Saving Wi$dome</I>; <marg><I>(g)</I> Eccle$ia$t. 15. 3</marg> Its onely end being to lead us to $alvation. And S. <I>Paul</I> preach- ing to the <I>Corinthians,</I> $aith; <I>(h) I determined to know nothing</I> <marg><I>(h)</I> 1 Cor. <I>c. 2. v.</I> 2</marg> <I>among you, $ave Je$us Chri$t, and him crucified:</I> whereas not- with$tanding he was thorowly in$tructed, and profoundly learned <foot>in</foot> <p n=>488</p> in all humane Sciences; but making no account of the$e things he profe$$eth that it was his de$ire to teach them no more but the way to Heaven. Hence is that which God $peaketh to us by <marg><I>(i)</I> I$a. <I>c. 48. v.</I> 17.</marg> <I>I$aiah,</I> <I>(i) Ego Dominus Deus, docens te utilia</I> [<I>I am the Lord thy God which teacheth thee profitable things:</I>] Where the <I>Glo$- $ary</I> addeth, <I>non $ubtilia</I> [not $ubtilties.] For God neither taught us, Whether the <I>Materia Prima</I> of Heaven, and the Elements be the $ame; nor Whether <I>Cominual</I> be compo$ed of Indivi$i- bles, or whether it be divi$ible <I>in infinitum</I>; nor, whether the Elements are formally <I>mixt</I>; nor how many the Cœle$tial Spheres, and their Orbs are; Whether there be Epicycles or Eccentricks; nor the Vertues of Plants and Stones; nor the Na- ture of Animals; nor the Motion and Influence of the Planets; nor the Order of the Univer$e; nor the Wonders of Minerals, and univer$al Nature: but only [<I>utilia:</I>] things profitable, to wit, his Holy Law ordained to the end, that we being put into po$$e$$ion of Ble$$edne$s, might at length be made capable of all perfect knowledge, and the vi$ion of the whole Order and ad- mirable Harmony, as al$o the Sympathy and Antipathy of the Univer$e and its parts, <I>in his Word,</I> wherein all tho$e things $hall mo$t clearly and di$tinctly, then, appear to us, which mean while, in this life, he hath remitted (as far as its ability reacheth) to humane $earch and enquiry: But it was not his purpo$e to determine any thing, directly or indirectly, touching the truth of them. Becau$e as the knowledge thereof would lit- tle or nothing profit Us, but might in $ome ca$es prove prejudi- cial; $o the ignorance thereof can doubtle$s be no detriment, but may in $ome ca$es be very beneficial to us. And therefore by his mo$t admirable Wi$dome it comes to pa$s, that though all things in this World are dubious, uncertain, wavering, and per- plexed; yet his Holy Faith alone is mo$t certain; and although the opinions about Philo$ophical and Doctrinal points be divers, there is in the Church but one Truth of Faith and Salvation. Which Faith, as nece$sary to Salvation, is $o ordered by Divine Providence, that it might not only be indubitable, but al$o un- $haken, $ure, immutable, and manife$t to all men: the infallible Rule of which he hath appointed the Holy Church, that is wa$h- ed with his precious Blood, and governed by his Holy Spirit, to whom belongs our Sanctification, as being his work. This there- <marg>1 The$$. 4.</marg> fore is the Rea$on why God would have Speculative Que$tions, which nothing conduce to our Salvation and Edification, and why the Holy Gho$t hath very often conde$cended to Vulgar Opini- ons and Capacities, and hath di$covered nothing that is $ingular or hidden to us, be$ides tho$e things that pertain to Salvation. So that con$equently it is clear by what hath been $aid, how and <foot>why</foot> <p n=>489</p> why nothing of certainty can be evinced from the fore$aid Au- thorities to the determining of Controver$ies of this Nature; as al$o with what Rea$on from this fir$t <I>Axiome</I> the Objections of the fir$t and $econd Cla$$e are ea$ily an$wered, as al$o any other Authority of $acred Scripture produced again$t the <I>Pythagorian</I> and <I>Copernican</I> Sy$teme $o long as by other proofs it is true.</P> <P>And the Authorities of the $econd Cla$$e in particular by this $ame Maxime, <I>Of the ordinary manner of apprehending things as they appear to us, and after the common way of $peak- ing,</I> may be thus reconciled and expounded; namely, Oftentimes an Agent is commonly, and not improperly $aid to move, (though it have no motion) not becau$e it doth indeed move, but <I>by ex- trin$ick denomination,</I> becau$e receiving its influence and action at the motion of the Subject; the Form and Quality infu$ed to the Subject by the $aid Agent doth likewi$e move. As for ex- ample, a Fire burning in a Chimney is an immoveable Agent, before which a man oppre$t with cold $its to warm him$elf who being warmed on one $ide, turns the other to the Fire, that he may be warmed on that $ide al$o, and $o in like manner he holds every part to the Fire $ucce$$ively, till his whole body be warm- ed. 'Tis clear, that although the Fire do not move, yet at the Motion of the Subject, to wit the Man, who receiveth the heat and action of the Fire, the Form and Quality of its Heat doth move <I>$ingulatim, & per partes,</I> round about the mans body, and alwayes $eeketh out a new place: and $o, though the Fire do not move, yet by rea$on of its effect, it is $aid to go round all the parts of the Mans body, and to warm it, not indeed by a true and real motion of the Fire it $elf, $ince it is $uppo$ed (and that not untruly) not to move, but by the motion to which the Body is excited, out of a de$ire of receiving the heat of the Fire in each of its parts. The $ame may be applied to the Illumina- tion impre$$ed $ucce$$ively on the parts of any Globe, which moves Orbicularly at the a$pect of a $hining immoveable Light. And in the $ame manner may the Sun be $aid to ri$e and $et, and to move above the Earth, although in reality he doth not move, nor $uffer any mutation; that is to $ay, Ina$much as his Light (which effect is the Form and Quality proceeding from him, as the Agent, to the Earth as the Subject) doth $en$ibly glide forwards, by rea$on of the Orbicular motion of the Earth; and doth alwayes be take it $elf to $ome new place of her $urface; upon which ground he is truly $aid <I>($ecundum vnlgarem $ermo- nem)</I> to move above, and revolve about the Earth: Not that the Sun doth move, (for by this Opinion we affirm the Earth to move, that it may receive the Sun one while in one, another while in another part of it) but that at the motion of the Earth <foot>Qqq her</foot> <p n=>490</p> her $elf a contrary way, the Quality diffu$ed into her, and im- pre$$ed upon her by the Sun, namely the Light of the Day is moved, which ri$eth in one part of her, and $ets in another con- trary to that, according to the nature and condition of her motion; And for this rea$on the Sun it $elf by con$equence is $aid to ri$e and $et, (which notwith$tanding <I>ex Hypothe$i</I> $tands immovea- ble) and that no otherwi$e then <I>per donominationem extrin$ecam,</I> as hath been $aid.</P> <P>After this manner the command of <I>Jo$huah, Sun $tand thou</I> <marg>Jo$hua <I>c. 10. ver.</I> 12.</marg> <I>$till,</I> and the Miracle of the Suns ce$$ation of Motion wrought by him, may be $o under$tood, as that not the Solar Body pro- perly, but the Suns $plendour upon the Earth $tood $till; $o that not the Sun it $elf, (being of it $elf before that time immovea- ble) but the Earth that receiveth its $plendour, $tayed her Mo- tion; which, as $he ince$$antly pur$uing her ordinary Motion to- <marg><I>* expected.</I></marg> wards the Ea$t, ^{*} called up the Light of the Sun in the We$t, $o $tanding $till, the Suns light impre$t upon it likewi$e $tood $till. <marg>I$a. <I>c. 38. v.</I> 8.</marg> After the $ame manuer pioportionally is that Text of <I>I$aiah</I> ex- plained, touching the Suns going ten degrees back ward upon the Dial of <I>Ahaz.</I> So (which may $erve for another Example) the Hand being moved about the flame of a burning Candle that $tands $till, the Light moveth on the Hand, that is to $ay, the $aid Hand is illu$trated now in one part, anon in another, when as the Candle it $elf all the while removes not out of its place: whereupon <I>per denominationem extrin$ecam,</I> the $aid Light may be affirmed to ri$e and $et upon the Hand, namely, by the $ole motion of the $aid Hand, the Candle it $elf never moving all the while. And let this $uffice for the explanation of my fir$t Prin- ciple or <I>Maxime,</I> which by rea$on of its difficulty and extraordi- nary weight required $ome prolixity in the handling of it.</P> <P>My $econd Maxime is this, Things both Spiritual and Cor- poreal, Durable and Corruptible, Moveable and Immoveable, have received from God a perpetual, unchangeable, and inviola- ble Law, con$tituting the E$$ence and Nature of every one of them: according to which Law all of them in their own Na- ture per$i$ting in a certain Order and Con$tancy, and ob$erving the $ame perpetual Cour$e, may de$ervedly be $tiled mo$t Stable and Determinate. Thus Fortune (than which there is nothing in the World more incon$tant or fickle) is $aid to be con$tant and unalterable in her continual volubility, vici$$itude, and in- con$tancy, which was the occa$ion of that Ver$e,</P> <head><I>Et $emper con$tans in levitate $ua e$t.</I></head> <P>And thus the motion of Heaven (which by the con$tan Law <foot><I>of</I></foot> <p n=>491</p> of Nature ought to be perpetual) may be $aid to be immutable and immoveable, and the Heavens them$elves to be immovea- bly moved, and Terrene things to be immutably changed, be- cau$e tho$e never cea$e moving, nor the$e changing. By this Prin- ciple or Maxime all difficulties belonging to the fir$t Cla$$is are cleared, by which the Earth is $aid to be $table and immoveable, that is, by under$tanding this one thing, That the Earth, as to its own Nature, though it include in it $elf a local Motion, and that threefold, according to the opinion of <I>Copernicus ($cilicet</I> Diur- <marg><I>Several Motions of the Earth ac- cording to</I> Coper- nicus.</marg> nal, with which it revolveth about its own Centre; Annual, by which it moveth through the twelve Signes of the Zodiack, and the motion of Inclination, by which its Axis is alwayes op- po$ed to the $ame part of the World) as al$o other Species of Mutation, $uch as Generation and Corruption, Accretion and Diminution, and Alteration of divers kinds; yet in all the$e $he is $table & con$tant, never deviating from that Order which God hath appointed her, but moveth continually, con$tantly and im- mutably, according to the $ix before named Species of Motion.</P> <P>My third Maxime $hall be this; When a thing is moved ac- cording to $ome part of it, and not according to its whole, it cannot be $aid to be <I>$imply & ab$olutely</I> moved, but only <I>per acci- dens,</I> for that $tability taken $imply & ab$olutly do rather accord with the $ame. As for example, if a Barrel or other mea$ure of Water be taken out of the Sea, and transferred to another place, the Sea may not therefore <I>ab$olutely & $imply</I> be $aid to be remo- ved from place to place; but only <I>per accidens,</I> and <I>$ecundum quid,</I> that is, according to a part of it, but rather (to $peak $im- ply) we $hould $ay that the Sea cannot be carried or moved out of its proper place,, though as to its parts it be moved, and transfer- red to & again. This Maxime is manife$t of it $elf, and by it may the Authorities be explained which $eem to make for the immo- bility of the Earth in this manner; namely, The Earth <I>per $e & ab$olutè</I> con$idered as to its <I>Whole,</I> is not mutable, $eeing it is neither generated nor corrupted neither increa$ed nor dimini$hed; neither is it altered <I>$ecundum totum,</I> but only <I>$ecundum partes.</I> <marg><I>The Earth Se-</I> cundum Totum <I>is Immutable, though not Immo- vable.</I></marg> Now it plainly appears, that this is the genuine and true Sen$e of what is a$cribed to it out of <I>Eccle$ia$tes, cap. 1. v. 4. One Generation pa$$eth away, and another Generation cometh, but the Earth abideth for ever</I>: as if he $hould $ay; although the Earth, according to its parts, doth generate and corrupt, and is liable to the vici$$itudes of Generation and corruption, yet in reference to its Whole it never generateth nor Corrupteth, but abideth immutable for ever: Like as a Ship, which though it be mended one while in the Sail- yard, another while in the Stern, and afterwards in other parts it yet remains the $ame Ship as it was at fir$t. But tis to be ad- <foot>Qqq 2 verti$ed</foot> <p n=>492</p> vertized, that that Scripture doth not $peak of a Local Motion, but of Mutations of another nature; as in the very $ub$tance, quantity or quality of the Earth it $elf. But if it be $aid, that it is to be under$tood of a Local Motion, then it may be ex- plained by the in$uing Maxime, that is to $ay, a re$pect being had to the natural Place a$$igned it in the Univer$e, as $hall be $hewed by and by.</P> <P>The fourth Axiome is this; That every Corporeal thing, mo- veable or immoveable from its very fir$t Creation, is alotted its proper and natural place; and being drawn or removed from thence, its motion is violent, and it hath a natural tendency to move back thither again: al$o that nothing can be moved from its natural place, <I>$ecundum Totum</I>; For mo$t great and dread$ul mi$chiefs would follow from that perturbation of things in the Univer$e. Therefore neither the whole Earth, nor the whole <marg><I>The Earth can- not</I> Secundum To- tum, <I>remove out of its Natural Place.</I></marg> Water, nor the whole Air can <I>$ecundum totum</I> be driuen or for- ced out of their proper place, $ite, or Sy$teme in the Univer$e, in re$pect of the order and di$po$ition of other mundane Bodies. And thus there is no Star (though Erratick) Orb or Sphere that can de$ert its natural place, although it may otherwi$e have $ome kind of motion. Therefore all things, how moveable $oever, are notwith$tanding $aid to be $table and immoveable in their proper place, according to the fore$aid $en$e, <I>i.e. $ecundum to- tum</I>; For nothing hinders, but that <I>$ecundum partes</I> they may $ome waymove; which motion $hall not be natural, but violent. Therefore the Earth, although it $hould be moveable, yet it might be $aid to be immoveable, according to the precedent Maxime, for that its neither moved in a right Motion nor out of the Cour$e a$$igned it in its Creation for the $tanding Rule of its motion; but keep within its own $ite, being placed in that which is called the Grand Orb, above <I>Venus,</I> and beneath <I>Mars,</I> <marg><I>The Natural Place of the Earth.</I></marg> and being in the middle betwixt the$e (which according to the common opinion is the Suns place) it equally and continually moveth about the Sun, and the two other intermediate Planets, namely <I>Venus</I> and <I>Mercury,</I> and hath the Moon (which is another Earth, but Ætherial, as <I>Macrobius</I> after $ome of the ancient Phi- <marg><I>The Moon is an Ætherial Body.</I></marg> lo$ophers, will have it) about it $elf. From whence, ina$much as $he per$i$teth uniformly in her Cour$e, and never at any time departeth from it, $he may be $aid to be $table and immoveable: and in the $ame $en$e Heaven likewi$e, with all the Elements, may be $aid to be immoveable.</P> <P>The fifth Maxime followeth, being little different from the former. Among$t the things created by God, $ome are of $uch a nature, that their parts may be <I>ab invicem,</I> or by turns, $e- parated from them$elves, and di$-joyned from their Whole; <foot>others</foot> <p n=>493</p> others may not, at lea$t, taken <I>collectively</I>: now tho$e are pe- ri$hable, but the$e perpetual. The Earth therefore $ince it is reckoned among$t tho$e things that are permanent, as hath <marg><I>The Earths Cen- tre keepeth it in its Natural Place.</I></marg> been $aid already, hath its parts, not di$$ipable, nor <I>ab invicem,</I> $eparable from its Centre (whereby its true and proper place is a$$igned it) and from its whole, taken collectively: becau$e ac- cording to its whole it is always pre$erved, compact, united, and cohærent in it $elf, nor can its parts be $eperated from the Cen- tre, or from one another, unle$s it may $o fall out <I>per accidens,</I> and violently in $ome of its parts; which afterwards, the ob$tacle being removed, return to their Natural Station $pontaneou$ly, and without any impul$e. In this Sen$e therefore the Earth is $aid to be Immoveable, and Immutable: yea even the Sea, Aire, Heaven, and any other thing (although otherwi$e moveable) $o long as its parts are not di$$ipable and $eperable, may be $aid to be Immoveable, at lea$t taken <I>collectively.</I> This Principle or Maxim differeth from the precedent only in that this referrs to the parts in order to <I>Place,</I> and this, in order to the Whole.</P> <P>From this Speculation another Secret is di$covered. For hence <marg><I>Gravity and Le- vity of Bodies, what it is.</I></marg> it is manife$t wherein the proper and genuine formality of the Gravity aad Levity of Bodyes con$i$teth; a point which is not $o clearly held forth, nor $o undeniably explained by the Peripate- tick Phylo$ophy. <I>Gravity</I> therefore is nothing el$e according to the Principles of this new Opinion, than a certain power and ap- petite of the Parts to rejoyn with their Whole, and there to re$t as in their proper place. Which Faculty or Di$po$ition is by Divine Providence be$towed not only on the Earth, and Ter- rene Bodies, but, as is believed, on Cœle$tial Bodies al$o, name- <marg><I>All Cœle$tial Bo- dies have Gravity and Levety.</I></marg> ly the Sun, Moon, and Starrs; all who$e parts are by this Impul- $ion connected, and con$erved together, cleaving clo$ely to each other, and on all $ides pre$$ing towards their Centre, until they come to re$t there. From which Concour$e and Compre$$ion a Sphærical and Orbicular Figure of the Cæle$tial Orbes is produ- ced, wherein by this occult Quality naturally incident to each of them they of them$elves $ub$i$t, and are alwayes pre$er- ved. But <I>Levity</I> is the Extru$ion and Exclu$ion of a more te- nuo$e and thin Body from the Commerce of one more Solid and <marg><I>Compre$$ive Ma- tion, proper to Gravity; the Ex- ten$ive, to Levity.</I></marg> den$e, that is Heterogeneal to it, by vertue of Heat. Where- upon, as the Motion of Grave Bodies is <I>Compre$$ive,</I> $o the Mo- tion of Light Bodies is <I>Exten$ive:</I> For its the propperty of Heat to dilate and rarify tho$e things to which it doth apply, conjoine and communicate it $elf. And for this rea$on we find Levity and Gravity not only in re$pect of this our Tere$trial Globe, and the Bodies adjacent to it, but al$o in re$pect of tho$e Bodies which are $aid to be in the Heavens, in which tho$e parts which <foot>by</foot> <p n=>494</p> by rea$on of their proclivity make towards their Centre are Grave, and tho$e that incline to the Circumference Light. And $o in the Sun, Moon, and Starrs, there are parts as well Grave as <marg><I>Heaven is not compo$ed of a fift E$$ence differing from the matter of inferior Bodies.</I></marg> Light. And con$equently Heaven it $elf that $o Noble Body, and of a fifth E$$ence, $hall not be con$tituted of a Matter diffe- rent from that of the Elements, being free from all Mutation in it's Sub$tance, Quantity, and Quality: Nor $o admirable and <marg><I>Nor yet a Solid or den$e Body but Rare.</I></marg> excellent as <I>Ari$totle</I> would make us to believe; nor yet a $olid Body, and impermeable; and much le$$e (as the generality of men verily believe) of an impenetrable and mo$t obdurate Den- $ity: but in it (as this Opinion will have it) Comets may be ge- nerated; and the Sun it $elf, as tis probable, exhaling or attract- ing $undry vapours to the $urface of its Body, may perhaps pro- duce tho$e Spots which were ob$erved to be $o various, and irre- <marg>* Delle Macchie $olarj.</marg> gular in its <I>Di$cus</I>: of which <I>Galilæus</I> in a perticular ^{*} Treati$e hath mo$t excellently and mo$t accurately $poken; in$omuch, that though it were not be$ides my pre$ent purpo$e, yet it is con- venient that I forbear to $peak any thing touching tho$e matters, lea$t I $hould $eem to do that which he hath done before me: But now if there be found in the Sacred Scriptures any Authority contrary to the$e things, it may be $alved by the fore$aid Argu- ments Analogically applyed. And further more it may be $aid, that that Solidity is to be $o under$tood, <I>as that it admits of no vacuum, cleft, or penetration from whence the lea$t vacuity might proceed</I> For the truth is, as that cannot be admitted in bodily Creatures, $o it is likewi$e repugnant to Heaven it $elf, being indeed a Body of its own Nature the mo$t Rare of all o- <marg>* <I>Vnius Corporis fimplicis, unus e$t motus $implex, et huic duæ $pecies, Rectus & Circu- laris: Rectus du- plex à medio, & ad medium; pri- mus levium, ut A- eris & Ignis: $e- cundus gravium, ut Aquæ & Ter- ræ: Circularis, quie$t circa medi- um competit Cœlo, quod neque e$t grave, neque leve.</I> Ari$t. <I>de Cœlo.</I> Lib. 1.</marg> thers, and tenuo$e beyond all Humane Conception, and happly hath the $ame proportion to the Aire, as the Aire to the Water.</P> <P>It is clear al$o from the$e Principles how fal$e the$e words of <I>Ari$totle</I> are, that: <I>Of one $imple Body, there is one $imple Motion</I>; <I>and this is of two kindes, Right and Circular: the Right is two- fold, from the medium, and to the medium; the fir$t of Light Bo- dyes, as the Aire and Fire: the $econd of Grave Bodyes, as the Water and Earth: the Circular, which is about the medium, be- longeth to Heaven, which is neither Grave nor Light</I>: For all this Philo$ophy is now for$aken, and of it $elf grown into di$-e$teem; for though it be received for an unque$tionable truth in this new Opinion, that to a $imple body appertains one only $imple Moti- <marg>* <I>Vide Coperni- cum de Revolutio- nibus Cœle$t.</I></marg> on, yet it granteth no Motion but what is Circular, by which alone a$imple body is con$erved in its naturall Place, and $ub$i$ts in its Unity, and is properly $aid to move <I>in loco</I> [<I>in a place</I>:] whereby <marg><I>Simple Motion peculiar to only Simple Bodies.</I></marg> it comes to pa$s that a Body for this rea$on doth continue to move in it $elf, [<I>or about its own axis</I>;] and although it have a Motion, <foot>yet</foot> <p n=>495</p> yet it abideth $till in the $ame place, as if it were perpetually im- moveable. But right Motion, which is properly <I>ad locum, [to a place]</I> can be a$cribed only to tho$e things which are out of their naturall place, being far from union with one another, and from unity with their whole, yea that are $eperated and divided from it: Which being that it is contrary to the Nature and forme of the Univer$e, it nece$$arily followeth, that right Motion doth in <marg><I>Right Motion belongeth to Im- perfect Bodies, and that are out of their natural Pla- ces.</I></marg> $hort $ute with tho$e things which are de$titute of that perfection, that according to their proper Nature belongeth to them, and which by this $ame right Motion they labour to obtaine, untill they are redintigrated with their Whole, and with one another, and re$tored to their Naturall place; in which at the length, having obtained their perfection, they $ettle and remaine immove- able. Therefore in right Motions there can be no Uniformity, <marg><I>Right Motion cannot be Simple.</I></marg> nor $implicity; for that they vary by rea$on of the uncertaine Levity or Gravity of their re$pective Bodyes: for which cau$e they do not per$evere in the $ame Velocity or Tardity to the end which they had in the beginning. Hence we $ee that tho$e things who$e weight maketh them tend downwards, do de$cend at fir$t with a $low Motion; but afterwards, as they approach neerer and neerer to the Centre, they precipitate more and more $wiftly. And on the other$ide, tho$e things which by rea$on of their light- ne$s are carryed upwards (as this our Terre$triall fire, which is no- thing el$e but a $moak that burneth, and is inkindled into a flame) are no $ooner a$cended on high, but, in almo$t the $elf-$ame mo- ment, they fly and vani$h out of fight; by rea$on of the rare- faction and exten$ion, that they as $oon as they acquire, are freed from tho$e bonds which violently and again$t their own Nature <marg><I>Right Motion is ever mixt with the Circular.</I></marg> kept them under, and deteined them here below. For which rea$on, it is very apparent, that no Right Motion can be called Simple, not only in regard that (as hath been $aid) it is not ^{*} even and uniforme, but al$o becau$e it is mixt with the Circu- <marg>* <I>æquabilis.</I></marg> lar, which lurketh in the Right by an occult con$ent, <I>$cilicet</I> by rea$on of the Natural affection of the Parts to conforme unto their Whole. For when the Whole moveth Circularly, it is re- qui$ite likewi$e that the Parts, to the end that they may be uni- ted to their Whole, (howbeit <I>per accidens</I> they are $ometimes moved with a Right Motion) do move (though not $o appa- rently) with a Circular Motion, as doth their Whole. And thus at length we have evinced that Circular Motion only is Simple, <marg>* <I>Even.</I></marg> Uniform and ^{*} Æquable, and of the $ame tenor [<I>or rate</I>] for that <marg><I>Circular Mo- tion is truly Sim- ple and Perpetual.</I></marg> it is never de$titute of its interne Cau$e: whereas on the contra- ry, Right Motion, (which pertains to things both Heavy and Light) hath a Cau$e that is imperfect and deficient, yea that ari- $eth from Defect it $elf, and that tendeth to, and $eeketh after <foot>no</foot> <p n=>496</p> nothing el$e but the end and termination of it $elf: in regard that Grave and Light Bodies, when once they have attained their proper and Natural Place, do de$i$t from that Motion to which they were incited by Levity and Gravity. Therefore: $ince Cir- <marg><I>Circular Mo- tion belongeth to the Whole Body, and the Right to its parts.</I></marg> cular Motion is proper <I>to the Whole,</I> and Right Motion <I>to the Parts,</I> the$e differences are not rightly referred to Motion, $o as to call one Motion Right, another Circular, as if they were not con$i$tent with one another: For they may be both together, and <marg><I>Circular and Right Motion co- incedent, and may con$i$t together in the $ame Body.</I></marg> that Naturally, in the $ame Body; no le$$e than it is equally Natural for a Man to participate of Sen$e and Rea$on, $eeing that the$e differences are not directly oppo$ite to one another. Hereupon Re$t and Immobility only are oppo$ed to Motion; and not one Species of Motion to another. And for the other differences <I>à medio, ad medium,</I> and <I>circa medium,</I> they are di- $tingui$hed not <I>really,</I> but only <I>formally,</I> as the Point, Line and Superficies, none of which can be without the other two, or without a Body. Hence it appears, that in as much as this Phy- lo$ophy differs from that of <I>Ari$totle,</I> $o in like manner doth this New Co$mographical Sy$tem vary from the Common one, that hath been hitherto received. But this by the way, upon occa$ion of explaining the Fifth Maxim: For as to the truth or fal$hood of the$e foregoing Po$itions (although I conceive them very pro- bable) I am re$olved to determine nothing at pre$ent, neither $hall I make any farther enquiry into them.</P> <P>The Sixth and La$t Maxim is this. Every thing is Simply deno- minated $uch as it is in compari$on of all things, or of many things which make the greater number of that kinde, but not in re$pect of a few which make but the le$$er part of them. As, for in$tance, a Ve$$el $hall not be called ab$olutely Great be- cau$e it is $o whil$t it is compared with two or three others: but it $hall be $aid to be great ab$olutely, and will be $o, if it ex- ceed in magnitude all indivials, or the greater part of them. Nor again $hall a Man be $aid to be ab$olutely Big, becau$e he is big- ger than a Pigmey; nor yet ab$olutely Little, becau$e le$$e than a Gyant: but he $hall be termed ab$olutely Big or Little in com- pari$on of the ordinary Stature of the greater part of Men. Thus the Earth cannot ab$olutely be $aid to be High or Low for that it is found to be $o in re$pect of $ome $mall part of the Univer$e; nor again $hall it be ab$olutely affirmed to be High, being compared to the Centre of the World, or $ome few parts of the Univer$e, more near to the $aid Centre, as is the <I>Sun, Mercury</I> or <I>Venus</I>: <marg><I>The Earth in what $en$e it may <*>b$olutely be $aid <*> be in the lowe$t part of the World.</I></marg> but it $hall receive its ab$olute denomination according as it $hall be found to be in compari$on of the greater number of the Spheres and Bodies of the Univer$e. The Earth therefore, in compari$on of the whole Circuit of the Eighth Sphære which in- <foot>cludeth</foot> <p n=>497</p> cludeth all Corporeal Creatures, and in compari$on of <I>Jupiter, Mars,</I> and <I>Saturn</I> together with the <I>Moon,</I> and much more in compari$on of other Bodies, (if any $uch there be) above the Eighth Sphere and e$pecially the Empyrial Heaven, may be truly $aid to be in the lowe$t place of the World, and almo$t in the Centre of it; nor can it he $aid to be above any of them, except the <I>Sun, Mercury</I> and <I>Venus</I>: So that one may apply unto it the name of an Infime and Low, but not a Supreme or Middle Body. And $o to come down from Heaven, e$pecially the Empyrian, to it (as it is accepted in the De$cent of Chri$t from Heaven to his Holy Incarnation) and from it to go up to Heaven (as in Chri$ts return <marg><I>Chri$t in his Incarnation tru- ly de$cended from Heaven, and in his A$cen$ion tru- ly a$cended into Heaven.</I></marg> to Heaven in his Glorious A$cention) is truly and properly to <I>De$cend</I> from the Circumference to the Centre, and to <I>a$cend</I> from the parts which are neare$t to the Centre of the World to its utmo$t Circumference. This Maxim therefore may ea$ily and according to truth explain Theologicall Propo$itions: and this is $o much the more confirmed, in that (as I have ob$erved) almo$t all Texts of Sacred Scripture which oppo$e the Earth to Heaven, are mo$t conveniently and aptly under$tood of the Em- pyrial Heaven (being the Highe$t of all the Heavens, and Spiritual in re$pect of its end) but not of the inferiour or intermediate Hea- vens, which are a Corporeal, and were framed for the benefit of Corporeal Creatures: and thus when in the Plural Number Heavens are mentioned, then all the Heavens promi$cuou$ly and without di$tinction are to be under$tood, as well the Empyrian it $elf as the Inferiour Heavens. And this Expo$ition indeed any man (that doth but take notice of it) may find to be mo$t true. And $o for this Rea$on the Third Heaveu into which St. <I>Paul</I> <marg>2 Cor. c. 12. v. 3. <I>Whether in the body or out of the body, I cannot tell, The Sunis King, Heart and I amp of the World him- $elf being</I> <G>au arkk<*></G> <I>ab$olutely indepen- dent.</I>)</marg> was wrapt up, by this Maxim may be taken for the Empyrean: if for the the Fir$t Heaven we under$tand that immen$e Space of Erratick and Moveable Bodies illuminated by the Sun, in which are comprehended the Planets, as al$o the Earth moveable, and the Sun immoveable, Who like a King upon his Augu$t Tribu- nal, $its with venerable Maje$ty immoveable and con$tant in Centre of all the Sphæres, and, with his Divine Beames, doth bountifully exhilerate all Cœle$tial Bodies that $tand in need of his vital Light, for which they cravingly wander about him; and doth liberally and on every $ide comfort and illu$trate the Thea- tre of the whole World, and all its parts, even the very lea$t, like an immortal and perpetual Lamp of high and un$peakable va- lue. The Second Heaven $hall be the Starry Heaven, common- ly called the Eighth Sphære, or the Firmament, wherein are all the Fixed Starrs, which according to this Opinion of <I>Pythagoras,</I> is (like as the Sun and Centre) void of all Motion, the Centre and utmo$t Circumference mutually agreeing with each other in <foot>Rrr Immobility</foot> <p n=>498</p> Immobility. And the Third $hall be the Empyrean Heaven, that is the Seat of the Ble$$ed. And in this manner we may come to explain and under$tand that admirable Secret, and profound My- <marg><I>The Ænignsa of Plato.</I></marg> $tery ænigmatically revealed by <I>Plato</I> to <I>Diony$ius</I> of <I>Syracu$e</I>: <marg><I>(a) Circa omni- um Regem $unt omnia. & Secun- da circa Secun- dum, et Tertia circa Tertium: Vide</I> Theodo. de Græc. affect. curat. lib. 2. Steuch. lib. de Parennj. Phi- lo$o.</marg> <I>(a) All things are about the King of all things, Second things about the $econd, and Third things about the Third</I>: For that God being the Centre of Spiritual things, the Sun, of Cor- poreal, Chri$t, of tho$e that are Mixt, or made up of both, things do doubtle$$e depend of that of the$e three Centres that is mo$t corre$pondent and proportionable to them, and the Centre is ever adjudged to be the nobler and worthier place: and therefore in Animals the Heart, in Vegitables the Pith or Kernell wherein the Seed lyeth that con$erveth their perpetuity, and virtually in- cludes the whole Plant, are in the Mid$t, and in the Centre: and thus much $hall $uffice to have hinted at, $ince there may another occa$ion offer it $elf for a larger Explication of the$e things. By this Maxim the Authorities and Arguments of the Third Fourth and Fifth Cla$$es are re$olved.</P> <P>It may be added withall, that even the <I>Sun, Mercury</I> and <I>Ve- nus</I> (that is to $ay in re$pect of the Earth) are to be thought <I>aboue,</I> and not <I>beneath</I> the Earth it $elf, although in re$pect of the Univer$e, yea and al$o ab$olutely, they are <I>below.</I> The rea- $on is, becau$e in re$pect of the Earth they alwayes appear above its Surface: and although they do not environe it, yet by the Motion of the $aid Earth they behold one while one part, another while another part of its Circumference. Since therefore tho$e things which in a Sphærical Body are nearer to the Circumfe- rence and more remote from the Cenrre are $aid to be <I>above,</I> but tho$e that are next adjoyning to the Centre are $aid to be <I>below</I>; it clearly followeth that whil$t the <I>Sun, Mercury</I> and <I>Venus</I> are not only turned towards the Surface and Circumference of the $aid Earth, but are at a very great di$tance without it, $ucce$$ively turned about it, and every way have a view of it, and are very far remote from its Centre, they may, in re$pect of the $aid Earth, be $aid to be <I>above</I> it; as al$o on the other $ide, the Earth in re$pect of them may be $aid to be <I>beneath</I>: howbeit on the con- trary, in re$pect of the Univer$e, the Earth in reality is much higher than they. And thus is $alved the Authority of <I>Eccle$i-</I> <marg>Eccle$. c. 1. 2. 3. <I>and almo$t tho- out.</I></marg> <I>a$tes</I> in many places, expre$$ing tho$e things that are, or are done on the Eeath in the$e words, <I>Which are done, or which are under</I> <marg>* <I>Quod fiunt, vel $unt $ub $ole.</I></marg> <I>the Sun,</I> And in the $ame manner tho$e words are reduced to their true Sen$e wherein it is $aid, That we are <I>under the Sun,</I> and <I>un- der the Moon,</I> whereupon Terrene things are expre$$ed by the name of <I>Sublunary.</I></P> <P>The Sixth Cla$$is threatneth a difficulty which is common as <foot>well</foot> <p n=>499</p> well to this of <I>Copernicus,</I> as to the Vulgar Opinion; $o that they are both alike concerned in the $olution of it: But $o far as it oppo$eth that of <I>Copernicus,</I> its an$wer is ea$y from the Fir$t Maxim.</P> <P>But that which is added in the Fourth Cla$$e, That it follow- eth from this Opinion, that Hell (for that it is included by the Earth, as is commonly held) doth move circularly about the Sun, and in Heaven, and that $o Hell it $elf will be found to be in Heaven; di$covers, in my judgment, nothing but Ignorance and Calumny, that in$inuate the belief of their Arguments ra- ther by a corrupt $en$e of the Words, than by $olid Rea$ons taken from the bo$ome of the Nature of things. For in this place Heaven is no wi$e to be taken for Paradice, nor according to the Sen$e of Common Opinion, but (as hath been $aid above) <marg><I>Heaven accord- ing to Copernicus is the $ame with the mo$t tenuous Æther; but dif- ferent from Para<*> dice, which $ar- pa$$eth all the Heavens.</I></marg> according to the <I>Copernican</I> Hypothe$is, for the $ubtile$t and Pure$t Aire, far more tenuous and rare than this of ours; where- upon the Solid Bodies of the Stars, Moon, and Earth, in their Circular and Ordinary Motions, do pa$$e thorow it, (the Sphære of Fire being by this Opinion taken away.) And as according to the Common Opinion it was no ab$urdity to $ay, That Hell being demerged in the Centre of the Earth and of the World it $elf, hath Heaven and Paradice above and below it, yea and on all $ides of it, and that it is in the middle of all the Cœle$tial Bodies (as if it were po$ited in a more unworthy place) $o, nei- ther in this will it be deemed an Error, if from the other Sy$tem, which differeth not much from the Vulgar one, tho$e or the like things follow as do in that. For both in that of <I>Copernicus,</I> and the Vulgar Hypothe$is, Hell is $uppo$ed to be placed among$t the very dreggs of the Elements, and in the Centre of the Earth it $elf, for the confinement and puni$hment of the damned. There- fore we ought not for want of Rea$ons to trifle away time in vain and impertinent $trife about words, $ince their true Sen$e is clouded then with no ob$curity, and in regard that it is very clear to any man indued with a refined Intellect, and that hath but an indifferent judgment in the Liberal Arts, and e$pecially in the Mathematicks, that the $ame, or not very different Gon- $equences do flow from both the$e Opinions.</P> <P>By the$e Maxims and their Interpretations it appears, that the <I>Pythagorick</I> and <I>Copernican</I> Opinion is $o probable, that its po$$ible it may exceed even the <I>Ptolemaick</I> in probability; and $ince there may be deduced from it a mo$t ordinate Sy$teme, and a mroe admirable and my$terious Hypothe$is of the World than from that of <I>Ptolomy:</I> the Authorities of Sacred Scripture and Theological Tenents in the mean while not oppo$ing it, be- ing opportunely and appo$itely (as I have $hown how they may <foot>Rrr 2 be)</foot> <p n=>500</p> be) reconciled with it: And $ince that by it not only the Phœ- nomena of all the Cœle$tial Bodies are mo$t readily $alved, but al$o many Natural Rea$ons are di$covered, which could not o- therwi$e, (but with extream difficulty) have been found out: And $ince it, la$t of all, doth open a more ea$y way into A$tro- nomy and Phylo$ophy, and rejecteth all tho$e $uperfluous and imaginary inventions produced by A$tronomers to the end only, that they might be able by them to render a rea$on of the $o ma- ny and $o various Motions of the Cœle$tial Orbs.</P> <P>And who knows, but that in that admirable compo$ure of the Candle$tick which was to be placed in the Tabernacle of God, he might out of his extraordinary love to us have been plea$ed to $haddow forth unto us the Sy$teme of the Univer$e, and more <marg>(a) Exod. 25. 31.</marg> e$pecially of the Planets? <I>(a) Thou $halt make a Candle$tick of</I> <marg>(b) <I>My Authour following the vul- gar Tran$lation, which hath an E- ligance in $ome things beyond ours, cites the words thus,</I> Facies Can- delabrum ducti- le de auro mun- di$$imo, Ha$tile ejus, & Calamos, & Sphærulas, ac Lilia, ex ip$o pro- cedentia.</marg> <I>pure Gold,</I> ($aith the Text;) <I>of beaten work $hall it be made: his Shaft, and his Branches, his Bowls, his Knops, and his Flowers (b) $hall be of the $ame.</I> Here are five things de$cribed, the Shaft of the Candle$tick in the midle, the Branches on the $ides, the Bowls, the Knops and the Flowers. And $ince there can be no more Shafts but one, the Branches are immediatly de$cribed in the$e <I>(c)</I> words: <I>Six Branches $hall come out of the $ides of it: three Branches out of the one $ide, and three Branches out of the other $ide:</I> Happly the$e fix Branches may point out to us $ix <I>(d)</I> Heavens, which are moved about the Sun in this order; <I>Saturn,</I> the $lowe$t and mo$t remote of all, fini$heth his cour$e about the <marg>(c) <I>ver$e</I> 12.</marg> Sun thorrow all the twelve Signes of the Zodiack in thirty Years: <marg>(d) <I>or Spheres.</I></marg> <I>Jupiter,</I> being nearer than he, in twelve Years: <I>Mars,</I> being yet <marg>(e) <I>Though our Authour $peaketh here po$itively of nine Months,</I> &c. <I>Fathers are not a- greed about the pe- riod of this planet, nor that of</I> Mercu- ry, <I>as you may $ee at large in</I> Riccio- lus, Almage$t. nov. <I>Tom. 1. part 1. l. 7. $ect. 3. cha. 11. num. 11. page 627. where he maketh</I> Venus <I>to con$um- mate her Revolu- tion in neer 225 dayes, or 7 1/2 Mon. and</I> Mecury <I>in a- bout 88 dayes, or 3 Months: in which he followeth</I> Kepl. <I>in Epitome A$tro- nom. p.</I> 760.</marg> nearer than him, in two Years: The <I>Earth,</I> which is $till nearer than he, doth perform the $ame Revolution, together with the Orbe of the <I>Moon,</I> in the $pace of a Year, that is in Twelve Months: <I>Venus,</I> which is yet nearer than all the$e, in <I>(e)</I> 9 Months: And la$t of all <I>Mercury,</I> who$e vicinity to the Sun is the greate$t of all, accompli$heth its whole conver$ion about the Sun in eighty Dayes. After the de$cription of the $ix Branches, the $acred Text proceeds to the de$cription of the Bowls, the Knops, and the Flowers, $aying, <I>(f) Three Bowls made like unto Almonds, with a Knop and a Flower in one Branch; and three Bowls made like Almonds in the other Branch, with a Knop and a Flower: this $hall be the work of the $ix Branches that come out of the Shaft. And in the Candle$tick $hall be four Bowls made like unto Al- monds, with their Knops and their Flowers: there $hall be a knop under two branches of the $ame, and a Knop under two Branches of the $ame, and a Knop under two Branches of the $ame; which together are $ix Branches, proceeding from one Shaft.</I> The truth <marg>(f) <I>ver$.</I> 33, 34.</marg> is, the $hallowne$$e of my under$tanding cannot fathome the <foot>depth</foot> <p n=>501</p> depth of all the My$teries that are couched in this mo$t wi$e di$po$ure of things: neverthele$$e being amazed, and tran$ported with admiration, I will $ay; Who knows but that tho$e three Bowls like unto Almonds to be repre$ented on each of the Branches of the Candle$tick may $ignifie tho$e Globes which are apter (as is this our Earth) for the receiving than emitting of Influ- ences? Perhaps al$o they denote tho$e Globes of late di$covered by the help of the Optick Tele$cope, which participate with <I>Saturn, Jupiter, Venus,</I> and po$$ibly al$o with the other Planets? Who knows likewi$e, but that there may be $ome occult propor- tion between the$e Globes and tho$e My$terious Knops and Lilies in$inuated unto us in the $acred Scriptures? But this $hall here $uffice to bound humane Pre$umption, and to teach us to ex$pect with an Harpocratick $ilence from Time, the Indice of Truth, a di$covery of the$e My$teries: <I>(g) Solomon</I> made ten <marg>(g) 1 Kings <I>c.</I> 7. <I>v.</I> 49. 2 Chron. <I>c.</I> 4. <I>ver$.</I> 7.</marg> Candle$ticks by the $ame Patern of <I>Mo$es,</I> which he placed, five on one hand and five on another, in the Temple erected by him in honour of the mo$t High God; which very thing doth al$o, without all que$tion, contain mo$t ab$tru$e $igni$ications. More- over, that Apple of the Knowledg of Good and Evil prohibited our fir$t Parents by God is not without a My$tery; which $ome $ay was an Indian Figg. In which the$e things are to be ob$erv- ed: Fir$t, That it is replete with many Kernels, every one of which hath a particular Centre. Secondly, Though of it $elf it be hard and $olid, yet about its Circumference it is of a more rare and tenuou$e $ub$tance; herein re$embling the Earth, which though in its Centre, and tho$e parts which are neare$t to it, it be $tony, Metallick, and compact, yet the nearer one approacheth to the Circumference, its parts are $een to be the more rare and tenuou$e: and withall it hath another body, more rare than its own, namely the Water, above which there is yet another, more $ubtil than all the re$t of inferiour Bodyes, that is to $ay, the Aire,</P> <P>The $ame Repre$entation with that of the Indian Figg is held forth to us by the <I>Malum Punicum,</I> or Pomegranate, with its innumerable poly centrick Stones or Kernels, all which in the parts more remote from their Centre, and nearer approaching towards the Circumference, are of a $ub$tance $o $ubtil and rare, that being but lightly compre$$ed, they in a manner wholly convert into a mo$t tenuo$e Liquor or juice: Of which fruit it plea$ed Divine Wi$dom to make mention, and ordained that its Figure $hould be imbroidered and wrought with a needle in the <I>$acerdotal</I> Garment of <I>Aaron: (h) Beneath</I> ($aith God) <I>upon the hem of it thou</I> <marg>(h) Exod. 28. 33, 34, & 39. v. 24, 25, 26.</marg> <I>$halt make Pomegranates of blew, and of purple, and of $carlet, round about the border thereof; and Bells of gold between them</I> <foot><I>round</I></foot> <p n=>502</p> <I>round about: a golden bell and a pomegranate, a golden bell and a pomegranate, upon the hem of the Robe round about.</I> And that this was a My$tical Repre$entation of the Worlds Effigies, is averred <marg><I>(i)</I> Sap. c. 18. v. 24.</marg> by <I>Solomon,</I> $aying; <I>(i) For in the long (k) Garment that be had on was the (l) whole World; and in the foure rows of the $tones</I> <marg><I>(k)</I> Exod. c. 28. v. 6, 9. 17, 36.</marg> <I>was the Glory of the Fathers graven, and thy Maje$ty in the Di-</I> <marg><I>(l)</I> Or, <I>totus Or- bis Terrarum,</I> as the vulgar Tran$- lation hath it.</marg> <I>adem of his Head.</I></P> <P>The $ame likewi$e is $ignified to us by the Grape, and in like manner by all other Fruits; but e$pecially the Figg, Grape, and Pomegranate: whence the$e three are almo$t alwayes placed to- gether in the Sacred Scriptures. So <I>Numb.</I> 20. the People of I$ra- el complain again$t <I>Mo$es</I> and <I>Aaron: (m) Wherefore have you</I> <marg><I>(m)</I> Numb. c. 20. v. 5.</marg> <I>made us to come up out of Egypt, to bring us into this evil place, where there can grow no Seed, neither is there either Figgs, or Vines, or Pomegranates</I>? Intimating that the$e kinds of Fruits were preferred by them for their excellency before all others. And in <I>Joel (n) The Vine is dryed up, and the Figg-tree langui$h-</I> <marg><I>(n)</I> Joel c. 1. v. 12.</marg> <I>eth, the Pomegranate-trce, the Palm-tree al$o, and the Apple-tree, even all the Trees of the field are withered; becau$e joy is wither- ed away from the Sons of Men.</I> Likewi$e in <I>Haggai: (o) Is the</I> <marg><I>(o)</I> Hagg. c. 2. v. 19.</marg> <I>$eed yet in the Bud? and hath as yet the Vine and the Fig-tree, and the Pomegranate, and the Olive-tree brought forth</I>? In like manner in <I>Deuteronomie</I> the Land of Promi$e is commended to be <I>(p) A Land of Wheat, and Barly, and Vines in which grow,</I> <marg><I>(p)</I> Deut. c. 8. v. 8.</marg> <I>Figg-trees, and Pomegranates, and Olive-trees,</I> &c. And in the Structure of the Temple undertaken by <I>Solomon</I> upon Divine In- <marg><I>(q)</I> 1 Kings c 7. v. 20. & 2 Kings c. 25. v. 17. & 2 Chro. c. 3. v. 15, 16. & c. 4. v. 12. 13. & Jer<*>m. c. 52. v. 21, 22.</marg> $piration the <I>(q)</I> Chapiters of the Pillars were adorned with $eve- ral rowes of Pomegranates: which particular is mentioned, not in one but many places of Holy Writ. Yea and $ometimes acci- dentally and occa$ionally the Holy hath Gho$t ænigmatically re- pre$ented this mo$t admirable and Mo$t Wi$e Sructure of the World, the Order of the Heavens, and the di$po$ure of Crea- tures Spiritual and Corporeal by Emblems, Parables, and Figures, lea$t they $hould be as it were dazled and blinded, by the reful- gent $plendor of $o excellent an Object. Hence we $ee, that in the$e Doctrinal & Dubious Points we may di$cour$e in $uch man- ner by help of the Holy Scripture as is meet for the under$tanding of the Prophets; which $eeing they are very ob$cure, they $hall be fully under$tood, and may be aptly applyed only then when they $hall be fulfilled, and not before: So al$o when once the true Sy$teme of the Univer$e is found out, then, and not till then, the meaning of the$e Figures, and Ænigma's $hall be made known unto us: Thus before the coming of the Son of God had di$co- vered unto us the My$tery of the Holy Trinity, none were able to comprehend or imagine what was concealed under tho$e <foot>words</foot> <p n=>503</p> words; <I>(r) In Principio creavit Elohim Cœlum & Terram:</I> for <marg><I>(r)</I> Gen. c. 1. v. 1</marg> that they did not $ee how the Noun Plural <I>Elohim</I> (which is as much as to $ay <I>Dij,</I> [Gods] $hould be joyned with the Verb Singular, <I>Creavit</I>: But the My$tery of the Unity of E$$ence and Trinity of Per$ons in God being revealed, it was pre$ently known, that the Singular Number, <I>Creavit,</I> had reference to the Unity of E$- $ence, (in regard that the Works of the Trinity <I>ad extra</I> are in- divi$ible) and the Plural, <I>Elohim,</I> to the Per$ons. Who, I pray, in elder times could have found out this My$tery? And thus the Name of God is thrice repeated in <I>P$al. 67. (s) God, even our</I> <marg><I>(s) P</I>$al. 67. v. 6 7.</marg> <I>God $hall ble$$e us, God $hall ble$$e us, &c.</I> Which at fir$t might $eem a Pleona$me, and $uperfluous repetition; but afterwards it was evident that <I>David</I> did there $et out the Benedictions of $e- veral Per$ons implyed, to wit, the Father, Son, and Holy Gho$t. Innumerable Examples of the like kind may be found in the Sa- cred Leaves. Therefore, to conclude, I will $ay with ^{*}<I>David,</I> <marg>* P$al. 92 v. 536.</marg> <I>P$al.</I> 92. <I>Oh Lord how glorious are thy Works! thy thoughts are very deep: an unwi$eman knoweth not, and a fool doth not under$tand the$e things.</I></P> <P>The$e are the particulars that I have thought fit to offer, as a Divine, concerning the not-improbable Opinion of the Mobili- ty of the Earth and Stability of the Sun: which I hope will be acceptable to you, Reverend Sir, out of the love and diligence wherewith you per$ue Virtue and Learning. But (to the end that you may al$o receive an account of my other Studies) I hope very $hortly to publi$h in Print my Second Tome ^{*}<I>Of the In-</I> <marg>* <I>In$titutionu<*> omnium Doctri- narum.</I></marg> <I>$titutions of all Learnings,</I> which $hall containe all the Liberall Arts, as I have already $ignified in that <I>Syntax,</I> and <I>Spicimen</I> by me heretofore put forth, and publi$hed under your Name. The other five following Tomes by me promi$ed (which $hall treat of Phylo$ophy and Theology) are not altogether $o forward, ne- verthele$s they will be $peedily fini$hed. In the mean time there will come forth my Book <I>Concerning ^{*} Oracles,</I> now fini$hed, to- <marg>* <I>De Oraculis.</I></marg> gether with a Treati$e ^{*} <I>Of Artificial Divination.</I> And for a <marg>* <I>De Divinatio- ne artificio$a.</I></marg> pledge thereof, I $end you at this time annexed to this Epi$tle a Tract ^{*} <I>Concerning Natural Co$mological Divination,</I> or of Natu- <marg>* <I>De Divinatio- ne Naturali Co$- mologica.</I></marg> ral Progno$ticks, and Pre$ages of the Changes o$ Weather, and other things which fall within the compa$$e of Natue. God grant you all Happine$$e.</P> <head><I>Mo$t Reverend Sir</I></head> <P><I>NAPLES,</I> from the Covent of the <I>Carmelites,</I> Jan. 6. 1615.</P> <P><I>Your Mo$t Humble Servant</I></P> <P><I>PAOLO ANTONIO FOSCARINI.</I></P> <head>FINIS.</head> <pb> <head><I>Imprimatur,</I> P. ANT. GHIBERT, <I>Vic. Gen.</I> JOANNES LONGUS <I>Can. & Cur. Archiep. Neap.</I> THEOL. <I>Vidit.</I></head> <pb> <head>A TABLE Of the mo$t Ob$ervable PERSONS and MATTERS Mentioned in the FIRST PART Of The Fir$t Tome.</head> <table> <row><col>A</col><col></col></row> <row><col>ABSTACT.</col><col></col></row> <row><col>Things are exactly the $ame in <I>Abstract,</I> as in Concrete.</col><col>185</col></row> <row><col>AIRE.</col><col></col></row> <row><col><*>he part of the <I>Aire</I> inferiour to the Higher Mountains doth follow the Motion of the Earth.</col><col>124</col></row> <row><col><*>he motion of the <I>Aire</I> apt to carry with it light things, but not heavy.</col><col>124</col></row> <row><col><*>he <I>Aire</I> alwayes touching us with the $ame part of it, cannot make us feel it.</col><col>228</col></row> <row><col><*>is more rea$onable that the <I>Aire</I> be commoved by the rugged $urface of the Earth, than by the Cele$tial Motion.</col><col>400</col></row> <row><col><*>is demon$trated, inverting the Argument, that the perpetual Motion of the <I>Aire</I> from Ea$t to We$t, commeth from the Motion of Heaven.</col><col>403</col></row> <row><col>ANIMALS.</col><col></col></row> <row><col><I>Animals, Vide,</I> The Motion of <I>Animals.</I></col><col></col></row> <row><col>The cau$e of the Wearine$$e that attends the Motion of <I>Animals.</I></col><col>244</col></row> <row><col>APOLLONIUS.</col><col></col></row> <row><col><I>Apollonius</I> and Copernicus demon$trate the Re- trogradations of Venus and Mercury.</col><col>311</col></row> <row><col><I>Arguing, Arguments, & Argumentations</I></col><col></col></row> <row><col>Somein <I>Arguing</I> fix in their minds the Conclu- $ion believed by them, and then adapt their Rea$ons to that.</col><col>250</col></row> <row><col>One $ingle Experiment or $ound Demon$trati- on, overthroweth all <I>Arguments</I> meerly pro- bable.</col><col>105</col></row> <row><col>A plea$ant Example $hewing the invalidity of $ome Phi$ical <I>Argumentations.</I></col><col>363</col></row> <row><col>ARISTARCHUS.</col><col></col></row> <row><col>Rea$on and Di$cour$e in <I>Ari$tarchus</I> and Coper- nicus prevailed over manife$t Sen$e.</col><col>301</col></row> <row><col>ARISTOTLE.</col><col></col></row> <row><col><I>Ari$totle</I> maketh the World perfect, becau$eit hath the Threefold Dimen$ion.</col><col>2</col></row> <row><col><I>Ari$t.</I> his Demon$trations to prove the Worlds Dimen$ions to be three, and no more.</col><col>2</col></row> <row><col><I>Ari$totle</I> his Definition of Nature either imper- fect or un$ea$onable.</col><col>7</col></row> <row><col><I>Ari$totle</I> accomodates the Rules of Architecture to the Frame of the World, and not the Frame to the Rules.</col><col>8</col></row> <row><col><I>Ari$totle</I> cannot equivocate, being the Inventer o$ Logick.</col><col>23</col></row> <row><col><I>Ari$totle</I> his Paralogi$me in proving the Earth to be in the centre of the World.</col><col>24</col></row> <row><col><I>Ari$t.</I> Paralogi$me another way di$covered.</col><col>24</col></row> <row><col><I>Ari$totle</I> his Di$cour$e to prove the Incorrupti- bility of Heaven.</col><col>26</col></row> <row><col><I>Ari$totle</I> proveth that Circular Motion hath no Contrary.</col><col>26</col></row> <row><col><I>Ari$totle</I> defective in a$$igning the Cau$es, why the Elements are Generable and Corrup- tible.</col><col>31</col></row> <row><col><I>Ari$iotle</I> would change his opinion, did he $ee the Novelties of our Age.</col><col>37</col></row> <foot>One [<I>Hhh] Ari$t.</I></foot> <pb> <row><col><I>Ari$t,</I> preferres Sen$e before Ratiocination.</col><col>42</col></row> <row><col><I>Ari$totle</I> affirmeth the Heavens alterable, rather then otherwi$e, by his Doctrine.</col><col>42</col></row> <row><col>Requifites to fit a man to Philo$ophate well in the way of <I>Ari$totle.</I></col><col>92</col></row> <row><col>Some of <I>Ari$totles</I> Sectators impaire his Repu- tation, in going about to enhan$e it.</col><col>93</col></row> <row><col>The $ervile Spirit of $ome of <I>Ari$t.</I> followers.</col><col>95</col></row> <row><col>Too clo$e an adherence to <I>Aristotle</I> is blame- able.</col><col>95</col></row> <row><col><I>Ari$totle</I> and <I>Ptolomy</I> argue again$t the Diurnal Motion a$cribed to the Earth.</col><col>97</col></row> <row><col>A Propo$ition that <I>Ari$totle</I> filched from the Ancients, and $omewhat altered.</col><col>99</col></row> <row><col><I>Ari$totle</I> his Arguments for the Earths Quie- $cence and Immobility.</col><col>107</col></row> <row><col><I>Ari$totle</I> were he alive, would either refute his Adver$aries Arguments, or el$e would alter his Opinion.</col><col>113</col></row> <row><col><I>Aristotles</I> fir$t Argument again$t the Earths Mo- tion, is defective in two things.</col><col>121</col></row> <row><col>The Paralogi$me of <I>Aristotle</I> and Ptolomy in $uppo$ing that for known, which is in que- $tion.</col><col>121</col></row> <row><col><I>Ari$totle</I> admitteth that the Fire moveth direct- ly upwards by Nature, and round about, by Participation.</col><col>122</col></row> <row><col><I>Ari$totle</I> and Ptolomy $eem to confute the Earths Mobility again$t tho$e who think that it, ha- ving along time $tood $till, began to move in the time of Pythagoras.</col><col>168</col></row> <row><col><I>Aristotle</I> his errour in affirming falling Grave Bodies to move according to the proportion of their gravities.</col><col>199</col></row> <row><col><I>Ari$totle</I> his Demon$trations to prove the Earth is finite, are all nullified, by denying it to be moveable.</col><col>294</col></row> <row><col><I>Aristotle</I> maketh that Point to be the Centre of the Univer$e, about which all the Cele$tial Spheres do revolve</col><col>294</col></row> <row><col>A que$tion is put, if <I>Ari$t.</I> were forced to receive one of two Propo$itions, that make again$t his Doctrine, which he would admit.</col><col>294</col></row> <row><col><I>Aristotle</I> his Argument again$t the Ancients, who held that the Earth was a Planet.</col><col>344</col></row> <row><col><I>Aristotle</I> taxeth Plato of being over-$tudious of Geometry.</col><col>361</col></row> <row><col><I>Aristotle</I> holdeth tho$e Effects to be miraculous, of which the Cau$es are unknown.</col><col>384</col></row> <row><col>ASTRONOMERS.</col><col></col></row> <row><col><I>A$tronomers</I> confuted by Anti-Tycho.</col><col>38</col></row> <row><col>The principal Scope of <I>A$tronomers</I> is to give a rea$on of Appearances and Phænomena.</col><col>308</col></row> <row><col><I>Actronomers</I> all agree that the greater Magni- tudes of the Orbes is the cau$e of the tardity in their Conver$ions.</col><col>331</col></row> <row><col><I>A$tronomers</I> perhaps have not known what Appearances ought to follow, upon the An- nual Motion of the Earth.</col><col>338</col></row> <row><col><I>Actronomers</I> having omitted to in$tance what al- terations tho$e are, that may be derived from the Annual Motion of the Earth, do thereby te$tifie that they never rightly un- der$tood the $ame.</col><col>343</col></row> <row><col>ASTRONOMICAL.</col><col></col></row> <row><col><I>A$tronomical</I> Ob$ervations wre$ted by Anti-Ty- cho to his own purpo$e.</col><col>39</col></row> <row><col><I>Actronomical</I> In$truments are very $ubject to errour.</col><col>262</col></row> <row><col>ASTRONOMY.</col><col></col></row> <row><col><I>A$tronomy</I> re$tored by Copernicus upon the Suppo$itions of Ptolomy</col><col>308</col></row> <row><col>Many things may remain as yet unob$erved in <I>A$tronomy</I></col><col>415</col></row> <row><col>AUCUPATORIAN.</col><col></col></row> <row><col>An <I>Aucupatorian</I> Problem for $hooting of Birds flying.</col><col>157</col></row> <row><col>AXIOME, or <I>Axiomes.</I></col><col></col></row> <row><col>In the <I>Axiome, Fru$tra fit per plura, &c.</I> the addi- tion of <I>æquœ bene</I> is $uperfluous.</col><col>106</col></row> <row><col>Three <I>Axiomes</I> that are $uppo$ed manife$t.</col><col>230</col></row> <row><col>Certain <I>Axiomes</I> commonly admitted by all Philo$ophers.</col><col>361</col></row> <row><col>B</col><col></col></row> <row><col>BODY and <I>Bodies.</I></col><col></col></row> <row><col>Contraries that corrupt, re$ide not in the $ame <I>Body</I> that corrupteth.</col><col>30</col></row> <row><col>GRAVE BODY; If the Cele$tial Globe were perforated, a <I>Grave Body</I> de$cending by that Bore, would pa$$e and a$cend as far beyond the Centre, as it did de$cend.</col><col>203</col></row> <row><col>The motion of <I>Grave Bodies,</I> Vide <I>Motion.</I></col><col></col></row> <row><col>The Accelleration of <I>Grave Bodies</I> that de$cend naturally, increa$eth from moment to moment.</col><col>205</col></row> <row><col>We know no more who moveth <I>Grave Bodies</I> downwards, than who moveth the Stars round; nor know we any thing of the$e <foot>trudes Cour$es</foot> <pb> Cour$es, more than the Names impo$ed on them by our $elves.</col><col>210</col></row> <row><col>The great Ma$$e of <I>Grave Bodies</I> being tran$- ferred out of their Place, the $eperated parts would follow that Ma$$e.</col><col>221</col></row> <row><col>PENSILE BODY; Every <I>Pen$ile Body</I> carried round in the Circumference of a Circle, ac- quireth of it $elf a Motion in it $elf contrary to the $ame.</col><col>362</col></row> <row><col>CBLESTIAL BODIES neither heavy nor light according to <I>Ari$toile.</I></col><col>23</col></row> <row><col><I>Cele$tial Bodies</I> are Generable and Corruptible becau$e they are Ingenerable aud Incorrup- tible.</col><col>29</col></row> <row><col>Among$t <I>Cele$t. Bodies</I> there is no contrariety.</col><col>29</col></row> <row><col><I>Cele$tial Bodies</I> touch, but are not touched by the Elements.</col><col>30</col></row> <row><col>Rarity and Den$ity in <I>Celectial Bodies,</I> different from Rarity and Den$ity in the Elements.</col><col>30</col></row> <row><col><I>Cele$tial Bodies</I> de$igned to $erve the Earth, need no more but Motion and Light.</col><col>45</col></row> <row><col><I>Cele$tial Bodies</I> wantan interchangeable Opera- tion on each other.</col><col>46</col></row> <row><col><I>Cele$tial Bodies</I> alterable in their externe parts.</col><col>46</col></row> <row><col>Perfect Sphericity why a$cribed to <I>Cele$tial Bo- dies</I> by Peripateticks.</col><col>69</col></row> <row><col>All <I>Celectial Bodies</I> have Gravity and Levity.</col><col>493</col></row> <row><col>ELEMENTARY BODIES; Their propen$i- on to follow the Earth, hath a limited Sphere of Activity.</col><col>213</col></row> <row><col>LIGHT BODIES ea$ier to be moved than heavy, but le$$e apt to con$erve the Motion.</col><col>400</col></row> <row><col>LUMINOUS BODIES; <I>Bodies</I> naturally <I>Lu- minous</I> are different from tho$e that are by na- ture Ob$cure.</col><col>34</col></row> <row><col>The rea$on why <I>Luminous Bodies</I> appear $o much the more enlarged, by how much they are le$$er.</col><col>304</col></row> <row><col>Manife$t Experience $hews that the more <I>Lumi- nous Bodies</I> do much more irradiate than the le$$e Lucid.</col><col>306</col></row> <row><col>SIMPLE BODYES have but one Simple Motion that agreeth with them.</col><col>494</col></row> <row><col>SPHERICAL BODIES; In <I>Spherical Bodies Deor$um</I> is the Centre, and <I>Sur$um</I> the Cir- ference.</col><col>479</col></row> <row><col>BONES.</col><col></col></row> <row><col>The ends of the <I>Bones</I> are rotund, and why.</col><col>232</col></row> <row><col>BUONARRUOTTI.</col><col></col></row> <row><col><I>Buonarruotti</I> a Statuary of admirable ingenuity.</col> <col>86</col></row> <row><col>C</col><col></col></row> <row><col>CANON.</col><col></col></row> <row><col>A $hameful Errour in the Argument taken from the <I>Canon</I>-Bullets falling from the Moons Concave.</col><col>197</col></row> <row><col>An exact Computation of the fall of the <I>Canon</I>- Bullet from the Moons Concave, to the Centre of the Earth.</col><col>198</col></row> <row><col>CELESTIAL</col><col></col></row> <row><col><I>Cele$tial</I> Sub$tances that be Unalterable, and Elementary that be Alterable, nece$$ary in the opinion of <I>Ari$totle.</I></col><col>2</col></row> <row><col>CENTRE.</col><col></col></row> <row><col>The Sun more probably in the <I>Centre</I> of the U- niver$e, than the Earth.</col><col>22</col></row> <row><col>Natural inclination of all the Globes of the World to go to their <I>Centre.</I></col><col>22</col></row> <row><col>Grave Bodies may more rationally be affirmed to tend towards the <I>Centre</I> of the Earth, than of the Univer$e.</col><col>25</col></row> <row><col>CHYMISTS.</col><col></col></row> <row><col><I>Chymi$ts</I> interpret the Fables of Poets to be Se- crets for making of Gold.</col><col>93</col></row> <row><col>CIRCLE, and <I>Circular.</I></col><col></col></row> <row><col>It is not impo$$ible with the Circumference of a $mall <I>Circle</I> few times revolved, to mea$ure and de$cribe a line bigger than any great <I>Cir- cle</I> what$oever.</col><col>222</col></row> <row><col>The <I>Circular Line</I> perfect, according to <I>Ari$totle,</I> and the Right imperfect, and why.</col><col>9</col></row> <row><col>CLARAMONTIUS.</col><col></col></row> <row><col>The Paralogi$me of <I>Claramontius.</I></col><col>241</col></row> <row><col>The Argument of <I>Claramontius</I> recoileth upon him$elf.</col><col>245</col></row> <row><col>The Method ob$erved by <I>Claramontius</I> in confu- ting A$tronomers, and by Salviatus in re- futing him.</col><col>253</col></row> <row><col>CLOUDS.</col><col></col></row> <row><col><I>Clouds</I> no le$$e apt than the Moon to be illumi- nated by the Sun.</col><col>73</col></row> <foot>CA- CON-</foot> <pb> <row><col>CONCLUSION and <I>Conclu$ions.</I></col><col></col></row> <row><col>The certainty of the <I>Conclu$ion</I> helpeth by a re$o- lutive Method to finde the Demon$tration.</col><col>37</col></row> <row><col>The Book of <I>Conclu$io s,</I> frequently mentioned, was writ by Chri$topher Scheiner a Je$uit.</col> <col>195, & 323.</col></row> <row><col>CONTRARIES.</col><col></col></row> <row><col><I>Contraries</I> that corrupt, re$ide not in the $ame Body that corrupteth.</col><col>30</col></row> <row><col>COPERNICAN.</col><col></col></row> <row><col>An$wers to the three fir$t Objections again$t the <I>Copernican Sy$tem.</I></col><col>303</col></row> <row><col>The <I>Copernican Sy$tem</I> difficul to be under$tood, but ea$ie to be effected.</col><col>354</col></row> <row><col>A plain Scheme repre$enting the <I>Copernican Sy- cteme</I> and its con$equences.</col><col>354</col></row> <row><col>The pro$cribing of the <I>Copernican</I> Doctrine, af- ter $o long a Tolleration, and now that it is more than ever followed, $tudied and con- firmed, would be an affront to Truth.</col><col>444</col></row> <row><col>The <I>Copern.</I> Sy$tem admirably agreeth with the Miracle of <I>Jo$huah</I> in the Literal Sen$e.</col><col>456</col></row> <row><col>If Divines would admit of the <I>Copernican</I> Sy- $tem, they might $oon find out Expo$itions for all Scriptures that $eem to make again$t it.</col><col>459</col></row> <row><col>The <I>Copernican</I> Sy$tem rejected by many, out of a devout re$pect to Scripture Authorities.</col><col>461</col></row> <row><col>The <I>Copernican</I> Sy$tem more plainly a$$erted in Scripture than the <I>Ptolomaick.</I></col><col>469</col></row> <row><col>COPERNICANS.</col><col></col></row> <row><col><I>Copernicans</I> are not moved through ignorance of the Arguments on the Adver$e part.</col><col>110</col></row> <row><col><I>Copernicans</I> were all fir$t again$t that Opinion, but the Peripateticks were never on the other $ide.</col><col>110</col></row> <row><col><I>Copernicans</I> too freely admit certain Propo$iti- ons for true, which are doubtful.</col><col>159</col></row> <row><col>He that will be a <I>Copernican</I> mu$t deny his Sen- $es.</col><col>228</col></row> <row><col>A Great Mathematician made a <I>Copernican,</I> by looking into that Doctrine, with a purpo$e to confute it.</col><col>443</col></row> <row><col>COPERNICUS.</col><col></col></row> <row><col><I>Copernicus</I> e$teemeth the Earth a Globe, like to a Planet.</col><col>1</col></row> <row><col>Objections of two Moderne Authours [Schei- ner and Claramontius] again$t <I>Copernicus.</I></col><col>195</col></row> <row><col><I>Copernicus</I> his Opinion overthrows the <I>Criterium</I> of Phylo$ophers.</col><col>223</col></row> <row><col>A gro$le Errour in the Oppo$er of <I>Copernicus,</I> and wherein it appears.</col><col>234, 235, & 236</col></row> <row><col>A $ubtle and withal $imple Argument again$t <I>Copernicus.</I></col><col>234</col></row> <row><col><I>Copernicus</I> his Opponent had but little $tudied him, as appears by another gro$$e Errour.</col><col>235</col></row> <row><col>Its que$tioned whither he under$tood the third Motion a$$igned to the Earth by <I>Copern.</I></col><col>236</col></row> <row><col><I>Copernicus</I> erroneou$ly a$$ignes the $ame Opera- tions to different Natures.</col><col>238</col></row> <row><col>A declaration of the improbability of <I>Copernicus</I> his Opinion.</col><col>301</col></row> <row><col>Rea$on and Di$cour$e in <I>Copernicus</I> and Ari$tar- chus prevailed over Sen$e.</col><col>301</col></row> <row><col><I>Copernicus</I> $peaketh nothing of the $mall Variati- on of Bigne$$e in Venus and Mars.</col><col>302</col></row> <row><col><I>Copernicus</I> per$waded by Rea$ons contrary to Sen$ible Experiments.</col><col>306</col></row> <row><col><I>Copernicus</I> re$tored A$tronomy upon the Suppo- $itions of Ptolomy.</col><col>308</col></row> <row><col>What moved <I>Copernicus</I> to e$tabli$h his Sy- $teme.</col><col>308</col></row> <row><col>Its a great argument in favour of <I>Copernicus,</I> that he obviates the Stations and Retrogradati- ons of the Motions of the Planets.</col><col>309</col></row> <row><col>In$tances Ironically propounded by Scheiner again$t <I>Copernicus.</I></col><col>323</col></row> <row><col><I>Copernicus</I> under$tood not $ome things for want of In$truments.</col><col>338</col></row> <row><col>The grand difficulty in <I>Copernicus</I> his Doctrine, is that which concerns the Phænomena of the Sun and fixed Stars.</col><col>343</col></row> <row><col><I>Copernicus</I> the Re$torer of the Pythagorean Hy- pothe$is, and the Occa$ion of it.</col><col>429</col></row> <row><col><I>Copernicus</I> founded not his Doctrine on Rea$ons depending on Scripture, wherein he might have mi$taken their Sen$e, but upon Natu- ral Conclu$ions and A$tronomical and Ge- ometrical Demon$trations.</col><col>431</col></row> <row><col>CORRUPTIBLE, and <I>Corruptibility.</I></col><col></col></row> <row><col>The perfection of Figure operates in <I>Corruptible Bodies,</I> but not in Eternal.</col><col>69</col></row> <row><col>The Di$paragers of <I>Corruptibility</I> ought to be turned into Statua's.</col><col>37</col></row> <row><col><I>Corruptibility</I> admits of more and le$$e, $o doth not Incorruptibility.</col><col>69</col></row> <row><col>COUNCILS.</col><col></col></row> <row><col>The <I>Councils</I> refu$e to impo$e Natural Conclu- $ions as matters of Faith.</col><col>450</col></row> <foot><I>Coper-</I> DIA-</foot> <pb> <row><col>D</col><col></col></row> <row><col>DIAMONDS.</col><col></col></row> <row><col><I><*>iamonds</I> ground to divers $ides, and why.</col><col>63</col></row> <row><col>DIDACUS.</col><col></col></row> <row><col><I><*>idacus à Stunica</I> reconcileth Texts of Scripture with the Copernican Hypothe$is.</col><col>468</col></row> <row><col>DEFINITIONS.</col><col></col></row> <row><col><I><*>e$initions</I> contain virtually all the Pa$$ions of the things defined.</col><col>87</col></row> <row><col>E</col><col></col></row> <row><col>EARTH.</col><col></col></row> <row><col><*>he <I>Earth</I> Spherical by the Con$piration of its parts to go to its Centre.</col><col>21</col></row> <row><col><*>is ea$ier to prove the <I>Earth</I> to move, than that Corruptibility is made by Contraries.</col><col>27</col></row> <row><col><*>he <I>Earth</I> very Noble, by rea$on of the Mu- tations made therein.</col><col>45</col></row> <row><col><*>he <I>Earth</I> unprofitable and full of Idlene$$e, its Alterations being taken away.</col><col>45</col></row> <row><col><*>he <I>Earth</I> more Noble than Gold and Jewels.</col> <col>45</col></row> <row><col><*>he Cele$tial Bodies de$igned to $erve the <I>Earth,</I> need no more but Motion and Light.</col><col>45</col></row> <row><col><*>he Generations and Mutations that are in the <I>Earth,</I> are all for the Good of Man.</col><col>47</col></row> <row><col><*>rom the <I>Earth</I> we $ee more than half the Lu- nar Globe.</col><col>51</col></row> <row><col><*>even Re$emblances between the <I>Earth</I> and Moon.</col><col>48 to 53</col></row> <row><col><*>he <I>Earth</I> unable to reflect the Suns Rays.</col><col>54</col></row> <row><col><*>he <I>Earth</I> may reciprocally operate on Cele$ti- al Bodies with its Light.</col><col>80</col></row> <row><col><*>ffinity between the <I>Earth</I> and Moon, by rea- $on of their Vicinity.</col><col>81</col></row> <row><col><*>he Motions of the <I>Earth</I> imperceptible to its Inhabitants.</col><col>97</col></row> <row><col><*>he <I>Earth</I> can have no other Motions than tho$e which to us appear commune to all the re$t of the Univer$e, the <I>Earth</I> excepted.</col><col>97</col></row> <row><col><*>he Diurnal Motion $eemeth commune to all the Univer$e, the <I>Earth</I> onely excepted.</col><col>97</col></row> <row><col><*>ri$totle and Ptolomy argue again$t the <I>Earths</I> Diurnal Motion.</col><col>97</col></row> <row><col>The Diurnal Motion of the <I>Earth.</I> Vide <I>Diur- nal Motion.</I></col><col></col></row> <row><col>Seven Arguments to prove the Diurnal Moti- on to belong to the <I>Earth.</I></col><col>99 to 103</col></row> <row><col>The <I>Earth</I> a pendent Body, and equilibrated in a fluid Medium, $eems unable to re$i$t the Rapture of the Diurnal Motion.</col><col>103</col></row> <row><col>Two kinds of Arguments again$t the <I>Earths</I> Motion.</col><col>108</col></row> <row><col>Arguments of Ari$totle, Ptolomy, Tycho, and other per$ons, again$t the <I>Earths</I> Motion.</col> <col>107 & 108</col></row> <row><col>The fir$t Argument again$t the <I>Earths</I> Motion taken from Grave Bodies falling from on high to the Ground.</col><col>108</col></row> <row><col>Which Argument is con$irmed by the Experi- ment of a Body let fall from the Round-top of a Ships Ma$t.</col><col>108</col></row> <row><col>The $econd Argument taken from a Project $hot very high.</col><col>108</col></row> <row><col>The third Argument taken from the Shot of a Canon towards the Ea$t, and towards the We$t.</col><col>108</col></row> <row><col>This Argument is con$irmed by two Shots to- wards the North and South, and two others towards the Ea$t and We$t.</col><col>109</col></row> <row><col>The fourth Argument taken from the Clouds and from Birds.</col><col>113</col></row> <row><col>A fifth Argument taken from the Aire which we feel beat upon us when we run an Hor$e at full $peed.</col><col>114</col></row> <row><col>A $ixth Argument taken from the whirling of Circular Bodies, which hath a faculty to extrude and di$$ipate.</col><col>114</col></row> <row><col>The An$wer to Ari$totles fir$t Argument.</col><col>115</col></row> <row><col>The An$wer to the $econd Argument.</col><col>117</col></row> <row><col>The An$wer to the third Argument.</col><col>120 to 150</col></row> <row><col>An In$tance of the Diurnal Motion of the <I>Earth,</I> taken from the Shot of a Piece of Ordinance perpendicularly, and the An$wers to the $ame, $hewing the Equivoke.</col><col>153, 154</col></row> <row><col>The An$wer to the Argument of the Shots of Canons made towards the North and South.</col><col>158</col></row> <row><col>The An$wer to the Argument taken from the Shots at point blank towards the Ea$t and We$t.</col><col>159</col></row> <row><col>The An$wer to the Argument of the flying of Birds contrary to the Motion of the <I>Earth.</I></col><col>165</col></row> <row><col>An Experiment by which alone is $hewn the Nullity of all the Arguments produced a- gain$t the Motion of the <I>Earth.</I></col><col>165</col></row> <row><col>The Stupidity of $ome that think the <I>Earth</I> be- gan to move, when Pythagoras began to af- firme that it did $o.</col><col>167</col></row> <row><col>A Geometrical Demon$tration to prove the Impo$$ibility of Extru$ion, by means of the <I>Earths</I> Vertigo, in An$wer to the $ixth <foot>on [<I>Iii</I>] At-</foot> <pb> Argument.</col><col>176</col></row> <row><col>Granting the Diurnal Vertigo of the <I>Earth,</I> and that by $ome $udden Stop or Ob$tacle it were Arre$ted, Hou$es, Mountains them$elves, and perhaps the whole Globe, would be $haken in pieces.</col><col>190</col></row> <row><col>Other Arguments of two Modern Authours [Scheiner and. Claramontius] again$t the Copernican Hypothe$is of the <I>Earths</I> Mo- tion.</col><col>195</col></row> <row><col>The fir$t Objection of the Modern Authour [Scheiner] in his Book of Conclu$ions.</col><col>195</col></row> <row><col>The Argument of [Claramontius] again$t the <I>Earths</I> Motion, taken from things falling per- pendicularly, another way an$wered.</col><col>223</col></row> <row><col>The <I>Earths</I> Motion collected from the Stars.</col> <col>229</col></row> <row><col>Argumeuts again$t the <I>Earths</I> Motion, taken <I>ex rerum natura.</I></col><col>230</col></row> <row><col>A Simple Body as the <I>Earth,</I> cannot move with three $everal Motions.</col><col>231</col></row> <row><col>The <I>Earth</I> cannot move with any of the Moti- ons a$$igned it by Copernicus.</col><col>231</col></row> <row><col>An$wers to the Arguments again$t the <I>Earths</I> Motion, token <I>ex rerum natnra.</I></col><col>231</col></row> <row><col>Four Axiomes again$t the Motion of the <I>Earth.</I></col> <col>230 to 232</col></row> <row><col>One onely Principle might cau$e a Plurality of Motions in the <I>Earth.</I></col><col>233</col></row> <row><col>The $ame Argument again$t the Plurality of Motions in the <I>Earth,</I> an$wered by Exam- ples of the like Motions in other Cele$tial Bodies.</col><col>236</col></row> <row><col>A fourth Argument [of Claramontius] again$t the Copernican Hypothe$is of the <I>Earths</I> Mobility.</col><col>239</col></row> <row><col>From the <I>Earths</I> ob$curity, and the $plendor of the fixed Stars, it is argued that it is move- able, and they immoveable.</col><col>239</col></row> <row><col>A fifth Argument [of Claramontius] again$t the Copernican Hypothe$is of the <I>Earths</I> Mobility.</col><col>240</col></row> <row><col>Another difference between the <I>Earth</I> and Ce- le$tial Bodies, taken from Purity and im- purity.</col><col>240</col></row> <row><col>It $eems a Soleci$me, to affirme that the <I>Earth</I> is not in Heaven.</col><col>241</col></row> <row><col>Granting to the <I>Earth</I> the Annual, it mu$t of nece$$ity al$o have the Diurnal Motion a$$i- gned to it.</col><col>300</col></row> <row><col>Di$cour$es more than childi$h, that $erve to keep Fools in the Opinion of the <I>Earths</I> Sta- bility.</col><col>301</col></row> <row><col>The Difficulties removed that ari$e from the <I>Earths</I> moving about the Sun, not $olitari- ly, but in con$ort with the Moon.</col><col>307</col></row> <row><col>The Axis of the <I>Earth</I> continueth alwayes pa- rallel to it $elf, and de$cribeth a Cylindrai- cal Superficies, inclining to the Orb.</col><col>344</col></row> <row><col>The Orb of the <I>Earth</I> never incllneth, but is immutably the $ame.</col><col>345</col></row> <row><col>The <I>Earth</I> approacheth or recedeth from the fixed Stars of the Ecliptick the quantity of the Grand Orb.</col><col>349</col></row> <row><col>If in the fixed Stars one $hould di$cover any Mu- tation, the Motion of the <I>Earth</I> would be undeniable.</col><col>351</col></row> <row><col>Nece$$ary Propo$itions for the better concei- ving of the Con$equences of the <I>Earths</I> Mo- tion.</col><col>354</col></row> <row><col>An admirable Accident depending on the not- inclining of the <I>Earths</I> Axis.</col><col>358</col></row> <row><col>Four $everal Motions a$$igned to the <I>Earth.</I></col><col>362</col></row> <row><col>The third Motion a$cribed to the <I>Earth,</I> is ra- ther a re$ting immoveable.</col><col>363</col></row> <row><col>An admirable interne vertue [or faculty] of the <I>Earths</I> Globe, to behold alwayes the $ame part of Heaven.</col><col>363</col></row> <row><col>Nature, as iu $port, maketh the Ebbing and Flowing of the Sea to prove the <I>Earths</I> Mo- bility.</col><col>379</col></row> <row><col>All Terrene Effects indifferently confirm the Motion or Re$t of the <I>Earth,</I> except the Eb- bing and Flowing of the Sea.</col><col>380</col></row> <row><col>The Cavities of the <I>Earth</I> cannot approach or recede from the Centre of the $ame.</col><col>387</col></row> <row><col>The Hypothe$is of the <I>Earths</I> Mobility taken in favour of the Ebbing and Flowing op- po$ed.</col><col>399</col></row> <row><col>The An$wers to tho$e Objections made again$t the <I>Earths</I> Motion.</col><col>399</col></row> <row><col>The Revolution of the <I>Earth</I> confirmed by a new Argument taken from the Aire.</col><col>400</col></row> <row><col>The vaporous parts of the <I>Earth</I> partake of its Motions.</col><col>400</col></row> <row><col>Another ob$ervation taken from the Ayr, in confirmation of the motion of the <I>Earth.</I></col><col>402</col></row> <row><col>A Rea$on of the continual Motion of the Air and Water may be given by making the <I>Earth</I> moveable, rather then by making it immoveable.</col><col>405</col></row> <row><col>The <I>Earths</I> Mobility held by $undry great Phi- lo$ophers among$t the Antients.</col><col>437 & 468</col></row> <row><col>The Fathers agree not in expounding the Texts of Scripture that are alledged again$t the <I>Earths</I> Mobility.</col><col>450</col></row> <row><col>The <I>Earth</I> Mobility defended by many a- mong$t the Modern Writers.</col><col>478</col></row> <row><col>The <I>Earth</I> $hall $tand $till after the Day of Judgement.</col><col>480</col></row> <row><col>The <I>Earth</I> is another Moon or Star.</col><col>486</col></row> <row><col>The <I>Earths</I> $everal Motions, according to Co- <foot>The pernicus.</foot> <pb> pernicus.</col><col>491</col></row> <row><col>The <I>Earth $ecundum totum</I> is Immutable, though not Immoveable.</col><col>491</col></row> <row><col>The <I>Earths</I> Natural Place.</col><col>492</col></row> <row><col>The <I>Earths</I> Centre keepeth her in her Natural Place.</col><col>493</col></row> <row><col>The <I>Earth,</I> in what Sen$e it may <I>ab$olutely</I> be $aid to be in the lowe$t part of the World.</col><col>496</col></row> <row><col>EBBING and <I>Ebbings.</I></col><col></col></row> <row><col>The fir$t general Conclu$ion of the impo$$ibi- lity of <I>Ebbing</I> and Flowing the Immobility of the Terre$trial Globe being granted.</col><col>380</col></row> <row><col>The Periods of <I>Ebbings</I> and Flowings, Diurnal, Monethly, and Annual.</col><col>381</col></row> <row><col>Varieties that happen in the Diurnal Period of the <I>Ebbings</I> and Flowings.</col><col>382</col></row> <row><col>The Cau$es of <I>Ebbings</I> and Flowings alledged by a Modern Phylo$opher.</col><col>382</col></row> <row><col>The Cau$e of the <I>Ebbing</I> and Flowing a$eribed to the Moon by a certain Prelate.</col><col>383</col></row> <row><col>The Cau$e of the <I>Ebbing, &c.</I> referred by Hye- ronimus Borrius and other Peripateticks, to the temperate heat of the Moon.</col><col>383</col></row> <row><col>An$wersto the Vanities alledged as Cau$es of the <I>Ebbing</I> and Flowing.</col><col>383</col></row> <row><col>Its proved impo$$ible that there $hould natu- rally be any <I>Ebbing</I> and Flowing, the Earth being immoveable.</col><col>386</col></row> <row><col>The mo$t potent and primary Cau$e of the <I>Eb- bing</I> and Flowing.</col><col>390</col></row> <row><col>Sundry accidents that happen in the <I>Ebbings</I> and Flowings.</col><col>391</col></row> <row><col>Rea$ons renewed of the particular Accidents ob$erved in the <I>Ebbings</I> and Flowings.</col><col>393</col></row> <row><col>Second Cau$es why in $everal Seas and Lakes there are no <I>Ebbings</I> and Flowings.</col><col>394</col></row> <row><col>The Rea$on why the <I>Ebbings</I> and Flowings for the mo$t part, are every Six Hours.</col><col>395</col></row> <row><col>The Cau$e why $ome Seas though very long, $uffer no <I>Ebbing</I> and Flowing.</col><col>395</col></row> <row><col><I>Ebbings</I> and Flowings, why greate$t in the Ex- tremities of Gulphs, and lea$t in the middle parts.</col><col>396</col></row> <row><col>A Di$cu$$ion of $ome more Ab$truce Accidents ob$erved in the <I>Ebbing</I> and Flowing.</col><col>396</col></row> <row><col>The <I>Ebbing</I> and Flowing may depend on the Di- urnal Motion of Heaven.</col><col>404</col></row> <row><col>The <I>Ebbing</I> and Flowing cannot depend on the Motion of Heaven.</col><col>405</col></row> <row><col>The Cau$es of the Periods of the <I>Ebbings</I> and Flowings Monethly and Annual, at large a$$igned</col><col>407</col></row> <row><col>The Monethly and Annual alterations of the <I>Ebbings</I> and Flowings, can depend on no- thing, $ave on the alteration of the Additions and Subtractions of the Diurnal Period from the Annual.</col><col>408</col></row> <row><col>Three wayes of altering the proportion of the Additions of the Diurnal Revolutions, to the Annual Motion of the <I>Ebbing</I> and Flow- ing.</col><col>409</col></row> <row><col><I>Ebbings</I> and Flowings are petty things, in compari$on of the va$tne$$e of the Seas, and the Velocity of the Motion of the Terre$trial Globe.</col><col>417</col></row> <row><col>EFFECT and <I>Effects.</I></col><col></col></row> <row><col>Of anew <I>Effect</I> its nece$$ary that the Cau$e be likewi$e new.</col><col>370</col></row> <row><col>The Knowledge of the <I>Effects</I> contribute to the inve$tigation of the Cau$es.</col><col>380</col></row> <row><col>True and Natural <I>Effects</I> follow without diffi- culty.</col><col>387</col></row> <row><col>Alterations in the <I>Effects</I> argue alteration in the Cau$e.</col><col>407</col></row> <row><col>ELEMENTS, <I>and their Motions,</I> Vide MOTION.</col><col></col></row> <row><col>ENCYCLOPEDIA.</col><col></col></row> <row><col>Subtilties fufficiently in$ipid, ironically $poken, and taken from a certain <I>Encyclope<*>ia.</I></col><col>153</col></row> <row><col>EXPERIMENTS.</col><col></col></row> <row><col>Sen$ible <I>Experiments</I> are to be preferred before Humane Argumentations.</col><col>21, 33, 42.</col></row> <row><col>It is good to be very cautious in admitting <I>Ex- periments</I> for true, to tho$e that never tryed them.</col><col>162</col></row> <row><col><I>Experiments</I> and Arguments again$t the Earths Motion, $eem $o far concluding, as they lye under Equivokes</col><col>162</col></row> <row><col>The Authority of Sen$ible <I>Experiments</I> and ne- ce$$ary Demon$trations in deciding of Phy- $ical Controver$ies.</col><col>436</col></row> <row><col>EYE.</col><col></col></row> <row><col>The Circle of the Pupil of the <I>Eye</I> contracteth and enlargeth.</col><col>329</col></row> <row><col>How to finde the di$tance of the Rays Con- cour$e from the Pupil of the <I>Eye.</I></col><col>329</col></row> <row><col>F</col><col></col></row> <row><col>FAITH.</col><col></col></row> <row><col><I>Faith</I> more infallible than either Sen$e of <foot>thing, Rea<*></foot> <pb> Rea$on.</col><col>475</col></row> <row><col>FIRE.</col><col></col></row> <row><col><I>Fire</I> moveth directly upwards by Nature, and round about by Participation, according to Ari$totle.</col><col>122</col></row> <row><col>It is improbable that the Element of <I>Fire</I> $hould be carried round by the Concave of the Moon.</col><col>405</col></row> <row><col>FIGURE and <I>Figures.</I></col><col></col></row> <row><col><I>Figure</I> is not the Cau$e of Incorruptibility, but of Longer Duration.</col><col>66</col></row> <row><col>The perfection of <I>Figure</I> appeareth in Corrup- tible Bodies, but not in the Eternal.</col><col>69</col></row> <row><col>If the Spherical <I>Figure</I> conferred Eternity, all things would be Eternal.</col><col>69</col></row> <row><col>It is more difficult to finde <I>Figures</I> that touch in a part of their Surface, then in one $ole point.</col><col>185</col></row> <row><col>The Circular <I>Figure</I> placed among$t the <I>Postu- lata</I> of Mathematicians.</col><col>186</col></row> <row><col>Irregular <I>Figure</I> and Formes difficult to be in- troduced.</col><col>187</col></row> <row><col>Superficial figures increa$e in proportion dou- ble to their Lines.</col><col>304</col></row> <row><col>FLFXURES.</col><col></col></row> <row><col>The nece$$ity and u$e of <I>Flexures</I> in Animals, for varying of their Motions.</col><col>232</col></row> <row><col>FOSCARINI.</col><col></col></row> <row><col><I>Fo$carini</I> his Reconciling of Scripture Texts with the Copernican <I>Hypothe$is.</I></col><col>473</col></row> <row><col>G</col><col></col></row> <row><col>GENERABILITY.</col><col></col></row> <row><col><I>Generability</I> and Corruptibility are onely a- mong$t Contraries, according to Ari$t.</col><col>26</col></row> <row><col><I>Generability</I> and Alterability are greater perfecti- ons in Mundane Bodies, then the Contrary Qualities.</col><col>44</col></row> <row><col>GEOMETRICAL, and <I>Geometry.</I></col><col></col></row> <row><col><I>Geometrical</I> Demon$trations of the Triple Di- men$ion.</col><col>4</col></row> <row><col><I>Geometrical</I> Exactne$$e needle$$e in Phy$ical Proofs.</col><col>6</col></row> <row><col>Ari$totle taxeth Plato for being too $tudious of <I>Geometry.</I></col><col>334</col></row> <row><col>Peripatetick Phylo$ophers condemne the Stu- dy of <I>Geometry,</I> and why.</col><col>461</col></row> <row><col>GILBERT.</col><col></col></row> <row><col>The Magnetick Phylo$ophy of <I>Will. Gilbert.</I></col><col>364</col></row> <row><col>The Method of <I>Gilbert</I> in his Philo$ophy.</col><col>367</col></row> <row><col>GLOBE.</col><col></col></row> <row><col>Our <I>Globe</I> would have been called Stone, in$tead of Earth, if that name had been given it in the beginning.</col><col>367</col></row> <row><col>GOD.</col><col></col></row> <row><col><I>God</I> and Nature do employ them$elves in caring for Men, as if they minded nothing el$e.</col><col>333</col></row> <row><col>An Example of <I>Gods</I> care of Man-kind, taken from the Sun.</col><col>333</col></row> <row><col><I>God</I> hath given all things an inviolable Law to ob$erve.</col><col>4..</col></row> <row><col>GREAT.</col><col></col></row> <row><col><I>Great</I> and Small, Immen$e, &c. are Relative Terms.</col><col>334</col></row> <row><col>GRAVITY.</col><col></col></row> <row><col><I>Grave</I>; Vide <I>Body.</I></col><col></col></row> <row><col><I>Gravity</I> and Levity, Rarity and Den$ity, are contrary qualities.</col><col>30</col></row> <row><col>Things Grave had being before the Common Centre of <I>Gravity.</I></col><col>221</col></row> <row><col><I>Gravity</I> and Levity of Bodies defined.</col><col>493</col></row> <row><col>GUN and <I>Gunnery.</I></col><col></col></row> <row><col>The Rea$on why a <I>Gun</I> $hould $eem to carry farther towards the We$t than towards the Ea$t.</col><col>148</col></row> <row><col>The Revolution of the Earth $uppo$ed, the Ball in the <I>Gun</I> erected perpendicularly, doth not move by a perpendicular, but an incli- ned Line.</col><col>155</col></row> <row><col>It is ingenuou$ly demon$trated, that, the Earths Motion $uppo$ed, the Shot of Great <I>Guns</I> ought to vary no more than in its Re$t.</col><col>161</col></row> <row><col>The Experiment of a Running Chariot to find out the difference of Ranges in <I>Gunnery.</I></col><col>148</col></row> <row><col>A Computation in <I>Gunnery,</I> how much the Ranges of Great Shot ought to vary from the Mark, the Earths Motion being Granted.</col><col>160</col></row> <foot><I>geome-</I> H</foot> <pb> <row><col>H</col><col></col></row> <row><col>HEAVEN.</col><col></col></row> <row><col><I><*>eaven</I> an Habitation for the Immortal Gods.</col><col>26</col></row> <row><col><I><*>eavens</I> Immutability evident to Sen$e.</col><col>26</col></row> <row><col><I><*>eaven</I> Immutable, becau$e there never was any Mutation $een in it.</col><col>34</col></row> <row><col><*>he cannot ($aith <I>Ari$totle</I>) $peak confident- ly of <I>Heaven,</I> by rea$on of its great di- $tance.</col><col>42</col></row> <row><col><*>he $ub$tance of the <I>Heavens</I> impenetrable, ac- cording to <I>Ari$totle.</I></col><col>54</col></row> <row><col><*>he Sub$tance of <I>Heaven</I> Intangible.</col><col>55</col></row> <row><col><*>$any things may be in <I>Heaven,</I> that are Invi$i- ble to us.</col><col>334</col></row> <row><col><*>here are more Documents in the Open Book of <I>Heaven,</I> than Vulgar Wits are able to Penetrate.</col><col>444</col></row> <row><col><I><*>eaven</I> and Earth ever mutually oppo$ed to each other.</col><col>480</col></row> <row><col><*>hich are really the Greater Lights in <I>Heaven,</I> and which the le$$er.</col><col>484</col></row> <row><col><I><*>eaven</I> is not compo$ed of a fifth E$$ence, differ- ing from the Matter of inferiour Bodies.</col><col>494</col></row> <row><col><I><*>eaven</I> is no Solid or Den$e Body, but Rare.</col><col>494</col></row> <row><col><*>hri$t at his Incarnatiou truly de$cended from <I>Heaven,</I> and at his A$cen$ion truly a$cended into <I>Heaven.</I></col><col>496</col></row> <row><col><*>f the Fir$t, Second and Third <I>Heaven.</I></col><col>497</col></row> <row><col><I><*>eaven</I> in the Sen$e of Copernicus, is the $ame with the mo$t tenuous Æther, but different from Paradice, which excells all the <I>Hea- vens.</I></col><col>499</col></row> <row><col>HELL.</col><col></col></row> <row><col><I>Hell</I> is in the Centre of the Earth, not of the World.</col><col>480</col></row> <row><col>HELIX.</col><col></col></row> <row><col>The <I>Helix</I> about the Cylinder may be $aid to be a Simple Line.</col><col>7</col></row> <row><col>HYPOTHESIS.</col><col></col></row> <row><col>The true <I>Hypothe$is</I> may di$patch its Revoluti- ons in a $horter time in le$$er Circles, than in greater, the which is proved by two Examples.</col><col>410</col></row> <foot>JEST.</foot> <row><col>I</col><col></col></row> <row><col>JEST.</col><col></col></row> <row><col>A <I>Je$t</I> put upon one that offered to $ell a cer- tain Secret of holding Corre$pondence at a Thou$and Miles di$tance.</col><col>79</col></row> <row><col>A <I>Jest</I> of a certain Statuary.</col><col>94</col></row> <row><col>IMPOSSIBILITY and <I>Impo$$ibilities.</I></col><col></col></row> <row><col>Nature attempts not <I>Impo$$ibilities.</I></col><col>10</col></row> <row><col>To $eek what would follow upon an <I>Impo$$ibi- lity</I> is Folly.</col><col>22</col></row> <row><col>INCORRUPTIBILITY.</col><col></col></row> <row><col><I>Incorruptibility</I> e$teemed by the Vulgar, out of their fear of Death.</col><col>45</col></row> <row><col>INFINITY.</col><col></col></row> <row><col>Of <I>Infinity</I> the Parts are not one greater than another, although they are comparatively unequal.</col><col>105</col></row> <row><col>INSTRUMENT and <I>In$truments.</I></col><col></col></row> <row><col><I>In$truments</I> A$tronomical very $ubject to Er- rour.</col><col>262</col></row> <row><col>Copernicus under$tood not $ome things for want of <I>Instruments.</I></col><col>338</col></row> <row><col>A proof of the $mall credit that is to be given to A$tronomical <I>Instruments</I> in Minute Ob- $ervations.</col><col>351</col></row> <row><col>Ptolomy did not confide in an <I>Instruments</I> made by Archimedes.</col><col>352</col></row> <row><col><I>In$truments</I> of Tycho made with great Ex- pence.</col><col>352</col></row> <row><col>What <I>In$truments</I> are mo$t apt for exact Ob$er- vations.</col><col>352</col></row> <row><col>INVENTORS.</col><col></col></row> <row><col>The Fir$t <I>Inventors</I> and Ob$ervers of things ought to be admired.</col><col>370</col></row> <row><col>JOSHUAH.</col><col></col></row> <row><col>The Miracle of <I>Jo$huah</I> in commanding the Sun to $tand $till, contradicts the Ptolomaick Sy$tem.</col><col>456</col></row> <row><col><I>Jo$huahs</I> Miracle admirably agreeth with the Pythagorick Sy$teme.</col><col>457</col></row> <foot>JEST. Vvv IRON.</foot> <pb> <row><col>IRON.</col><col></col></row> <row><col>Its proved that <I>Iron</I> con$i$ts of parts more $ubtil, pure and compact than the Magner.</col><col>370</col></row> <row><col>JUPITER.</col><col></col></row> <row><col><I>Jupiter</I> and Saturn do encompa$$e the Earth, and the Sun.</col><col>258</col></row> <row><col><I>Jupiter</I> an<*>ments le$$e by Irradiation, than the Dog-Star.</col><col>305</col></row> <row><col>K</col><col></col></row> <row><col>KEPLER.</col><col></col></row> <row><col>The Argument of <I>Kepler</I> in favour of Coper- nicus.</col><col>242</col></row> <row><col>An Explanation of the true Sen$e of <I>Kepler,</I> and his Defence.</col><col>243</col></row> <row><col>The feigned An$wer of <I>Kepler</I> couched in an Artificial Irony.</col><col>244</col></row> <row><col><I>Kepler</I> is, with re$pect, blamed.</col><col>422</col></row> <row><col><I>Keplers</I> reconciling of Scripture Texts whith the Copernican Hypothe$is.</col><col>461</col></row> <row><col>KNOW, <I>&c.</I></col><col></col></row> <row><col>The having a perfect <I>Knowledge</I> of nothing, maketh $ome beleeve they under$tand all things.</col><col>84</col></row> <row><col>Gods manner of <I>Knowing</I> different from that of Man.</col><col>87</col></row> <row><col>The great Felicity for which they are to be en- vied, who per$wade them$elves that they <I>Know</I> every thing.</col><col>164</col></row> <row><col>Our <I>Knowledge</I> is a kind of Remini$cence, ac- cording to Plato.</col><col>169</col></row> <row><col>L</col><col></col></row> <row><col>LIGHT.</col><col></col></row> <row><col><I>Light</I> reflected from the Earth into the Moon.</col><col>52</col></row> <row><col>The Reflex <I>Light</I> of uneven Bodies is more uni- ver$al than that of the $mooth, and why.</col><col>62</col></row> <row><col>The more rough Superficies make greater Re- flection of <I>Light</I> than the le$$e rough</col><col>65</col></row> <row><col>Perpendicular Rays of <I>Light</I> illuminate more than the Oblique, and why.</col><col>65</col></row> <row><col>The more Oblique Rays of <I>Light</I> illuminate le$$e, and why,</col><col>65</col></row> <row><col><I>Light</I> or Luminous Bodies appear the brighter in an Ob$cure Ambient.</col><col>74</col></row> <row><col>LINE.</col><col></col></row> <row><col>The <I>Right Line</I> and Circumference of an infi- nite Circle are the $ame thing.</col><col>342</col></row> <row><col>LAWYERS.</col><col></col></row> <row><col>Contentious <I>Lawyers</I> that are retained in an ill Cau$e, keep clo$e to $ome expre$$ion fallen from the adver$e party at unawares.</col><col>324</col></row> <row><col>LOOKING-GLASSES.</col><col></col></row> <row><col>Flat <I>Looking-Gla$$es</I> ca$t forth their Reflection to- wards but one place, but the Spherical eve- ry way.</col><col>39</col></row> <row><col>LYNCEAN.</col><col></col></row> <row><col>The <I>Lyncean</I> Academick the fir$t Di$coverer of the Solar $pots, and all the other Cele$tial Novelties.</col><col>312</col></row> <row><col>The Hi$tory of his proceedings for a long time, about the Ob$ervation of the Solar Spots.</col><col>312</col></row> <row><col>M</col><col></col></row> <row><col>MAGNET.</col><col></col></row> <row><col>Many properties in the <I>Magnet.</I></col><col>367</col></row> <row><col>The <I>Magnet</I> armed takes up more Iron, than when unarmed.</col><col>369</col></row> <row><col>The true cau$e of the Multiplication of Vertue in the <I>Magnet,</I> by means of the Arming.</col><col>370</col></row> <row><col>A $en$ible proof of the Impurity of the <I>Mag- net.</I></col><col>371</col></row> <row><col>The $everal Natural Motions of the <I>Mag- net.</I></col><col>374</col></row> <row><col>Philo$ophers are forced to confe$$e that the <I>Magnet</I> is compounded of Cele$tial Sub$tan- ces, and of Elementary.</col><col>375</col></row> <row><col>The Error of tho$e who call the <I>Magnet</I> a mixt Body, and the Terre$trial Globe, a $imple Body.</col><col>375</col></row> <row><col>An improbable Effect admired by Gilbertus in the <I>Magnet.</I></col><col>376</col></row> <row><col>MAGNETICK <I>Philo$ophy.</I></col><col></col></row> <row><col>The <I>Magnetick Philo$ophy</I> of William Gilbert.</col> <col>364</col></row> <row><col>MAGNITUDE.</col><col></col></row> <row><col>The <I>Magnitude</I> of the Orbs and the Velocity of the Motions of Planets an$wer proporti- <foot>LINE. onably,</foot> <pb> onably, as if de$cended from the $ame place.</col><col>19</col></row> <row><col>Immen$e <I>Magnitudes</I> and Numbers are incom- prehen$ible by our Under$tandings.</col><col>332</col></row> <row><col>MARS.</col><col></col></row> <row><col><I>Mars</I> nece$$arily includeth within its Orb the Earth, and al$o the Sun.</col><col>298</col></row> <row><col><I>Mars</I> at its Oppo$ition to the Sun, $eems $ixty times bigger than towards the Conjuncti- on.</col><col>298</col></row> <row><col><I>Mars</I> makes an hot a$$ault upon the Coperni- can Sy$teme.</col><col>302</col></row> <row><col>MARSILIUS.</col><col></col></row> <row><col><I><*>gnor Cæ$ar Mar$ilius</I> ob$erveth the Meridian to be moveable.</col><col>422</col></row> <row><col>MEDICEAN.</col><col></col></row> <row><col>The time of the <I>Medicean</I> Planets conver$i- ons.</col><col>101</col></row> <row><col>The <I>Medicean</I> Planets are as it were four Moons about <I>Jupiter.</I></col><col>307</col></row> <row><col>MEDITERRAN.</col><col></col></row> <row><col><I>Mediterranean</I> Sea made by the Seperation of Abila and Calpen.</col><col>35</col></row> <row><col><*>he Voyages in the <I>Mediterran</I> from Ea$t to We$t are made in $horter times than from We$t to Ea$t.</col><col>403</col></row> <row><col>MERCURY.</col><col></col></row> <row><col>The Revolution of <I>Mercury</I> concluded to be about the Sun, within the Orb of Venus.</col><col>298</col></row> <row><col><I>Mercury</I> admitteth not of clear Ob$ervati- ons.</col><col>307</col></row> <row><col>MOON.</col><col></col></row> <row><col>The <I>Moon</I> hath no Generation of things, like as we have, nor is it inhabited by Men.</col><col>47</col></row> <row><col><*>n the <I>Moon</I> may be a Generation of things dif- ferent from ours.</col><col>47</col></row> <row><col>There may be Sub$tances in the <I>Moon,</I> very different from ours.</col><col>48</col></row> <row><col>The fir$t re$emblance between the <I>Moon</I> and Earth, which is that of Figure, is proved, by their manner of being illuminated by the Sun.</col><col>48</col></row> <row><col>The $econd re$emblance is the <I>Moons</I> being Opacous, as the Earth.</col><col>48</col></row> <row><col>The third re$emblance is the <I>Moons</I> being Den$e and Mountainous as the Earth.</col><col>49</col></row> <row><col>The fourth re$emblance is the <I>Moons</I> being di- $tingui$hed into two different parts for Cla- rity and Ob$curity, as the Terre$trial Globe into Sea and Land.</col><col>49</col></row> <row><col>The fifth re$emblance is Mutation of Figures in the Earth, like tho$e of the <I>Moon,</I> and made with the $ame Periods.</col><col>49</col></row> <row><col>All the Earth $eeth halfe onely of the <I>Moon,</I> and halfe onely of the <I>Moon</I> $eeth all the Earth</col><col>51</col></row> <row><col>Two Spots in the <I>Moon,</I> by which it is percei- ved that She hath re$pect to the Centre of the Earth in her Motion.</col><col>52</col></row> <row><col>Light reflected from the Earth into the <I>Moon.</I></col><col>52</col></row> <row><col>The $ixth re$emblance is that the Earth and <I>Moon</I> interchangeably illuminate.</col><col>53</col></row> <row><col>The $eventh re$emblance is that the Earth and <I>Moon</I> interchangeably Ecclip$e.</col><col>53</col></row> <row><col>The Secondary Clarity of the <I>Moon</I> e$teemed to be its Native Light.</col><col>54</col></row> <row><col>The Surface of the <I>Moon</I> more $leek then any Looking-Gla$$e.</col><col>55</col></row> <row><col>The eminencies and Cavities in the <I>Moon,</I> are illu- $ions of its Opacous and Per$picuous parts.</col><col>55</col></row> <row><col>The <I>Moons</I> Surface is $harp, as is largely pro- ved.</col><col>57</col></row> <row><col>The <I>Moon,</I> if it it were $leek like a Spherical Looking-Gla$$e, would be invi$ible.</col><col>60 & 62</col></row> <row><col>The apparent Unevenne$$es of the <I>Moons</I> Sur- face aptly repre$ented by Mother of Pearl.</col><col>70</col></row> <row><col>The apparent Unevenne$$es of the <I>Moon</I> cannot be imitated by way of more and le$$e Opa- city, and Per$picuity</col><col>71</col></row> <row><col>The various A$pects of the <I>Moon</I> imitable by any Opacous matter.</col><col>71</col></row> <row><col>Sundry Phænomena from whence the <I>Moons</I> Montuo$ity is argued.</col><col>71</col></row> <row><col>The <I>Moon</I> appears brighter by night, than by day.</col><col>72</col></row> <row><col>The <I>Moon</I> beheld in the day time, is like to a little Cloud.</col><col>72</col></row> <row><col>Clouds are no le$$e apt than the <I>Moon</I> to be il- luminated by the Sun.</col><col>73</col></row> <row><col>A Wall illuminated by the Sun, compared to the <I>Moon,</I> $hines no le$$e than it.</col><col>73</col></row> <row><col>The third reflection of a Wall illuminates more than the fir$t of the <I>Moon.</I></col><col>74</col></row> <row><col>The Light of the <I>Moon</I> weaker than that of the Twy-light.</col><col>74</col></row> <row><col>The $econdary Light of the <I>Moon</I> cau$ed by the Sun, according to $ome.</col><col>76</col></row> <foot>Pa- The</foot> <pb> <row><col>The $econdary Light of the <I>Moon</I> appears in form of a Ring, <I>i. e.</I> bright in the extreme Circumference, and not in the mid$t, and why.</col><col>77</col></row> <row><col>The $econdary Light of the <I>Moon,</I> how it is to be ob$erved.</col><col>78</col></row> <row><col>The <I>Moons</I> Di$cus in a Solar Eclip$e can be $een onely by Privation.</col><col>78</col></row> <row><col>Solidity of the <I>Moons</I> Globe argued from its being Mountainous.</col><col>81</col></row> <row><col>The $econdary Light of the <I>Moon</I> clearer before the Conjunction than after.</col><col>82</col></row> <row><col>The ob$curer parts of the <I>Moon</I> are Plains, and the more bright Mountains.</col><col>83</col></row> <row><col>Long Ledges of Mountains about the Spots of the <I>Moon.</I></col><col>83</col></row> <row><col>There are not generated in the <I>Moon</I> things like to ours, but if there be any Producti- ons, they are very different.</col><col>83</col></row> <row><col>The <I>Moon</I> not compo$ed of Water and Earth.</col><col>83</col></row> <row><col>Tho$e A$pects of the Sun nece$$ary for our Productions, are not $o in the <I>Moon.</I></col><col>83</col></row> <row><col>Natural Dayes in the <I>Moon</I> are of a Moneth long.</col><col>84</col></row> <row><col>To the <I>Moon</I> the Sun declineth with a difference of ten Degrees, and to the Earth of Forty $even Degrees.</col><col>84</col></row> <row><col>There are no Rains in the <I>Moon.</I></col><col>84</col></row> <row><col>The <I>Moon</I> cannot $eperate from the Earth.</col><col>295</col></row> <row><col>The <I>Moons</I> Orbe environeth the Earth, but not the Sun.</col><col>299</col></row> <row><col>The <I>Moon</I> much di$turbeth the Order of the other Planets.</col><col>362</col></row> <row><col>The <I>Moons</I> Motion principally $ought in the Account of Eclip$es.</col><col>416</col></row> <row><col>The <I>Moon</I> is an Æthereal Earth.</col><col>492</col></row> <row><col>MOTION and <I>Motions.</I></col><col></col></row> <row><col><I>Motion</I> of Projects. Vide <I>Projects.</I></col><col></col></row> <row><col>The Conditions and Attributes which differ the Cele$tial and Elementary Bodies depend on the <I>Motions</I> a$$igned them by Ari$totle.</col><col>25</col></row> <row><col>Peripateticks improperly a$$ign tho$e <I>Motions</I> to the Elements for Natural with which they never were moved, and tho$e for Preternatu- ral with which they alwayes move.</col><col>33</col></row> <row><col><I>Motion,</I> as to the things that move thereby, is as if it never were, and $o farre operates, as it relates to things deprived of <I>Motion.</I></col><col>98</col></row> <row><col><I>Motion</I> cannot be made without its moveable Subject.</col><col>104</col></row> <row><col><I>Motion</I> and Re$t principal Accidents in Na- ture.</col><col>112</col></row> <row><col>Two things nece$$ary for the perpetuating of a <I>Motion</I>; an unlimited Space, and an incor- ruptible Moveable.</col><col>117</col></row> <row><col>Di$parity in the <I>Motions</I> of a Stone falling from the Round Top of a Ship, and from the Top of a Tower.</col><col>123</col></row> <row><col>The <I>Motion</I> of grave Pendula might be perpe- tuated, impediments being removed.</col><col>203</col></row> <row><col>Whence the <I>Motion</I> of a Cadent Body is col- lected.</col><col>224</col></row> <row><col>The <I>Motion</I> of the Eye argueth the <I>Motion</I> of the Body looked on.</col><col>224</col></row> <row><col>Different <I>Motions</I> depending on the Fluctuati- on of the Ship.</col><col>226</col></row> <row><col>Our <I>Motion</I> may be either interne, or externe, and yet we never perceive or feelit.</col><col>229</col></row> <row><col>The <I>Motion</I> of a Boat in$en$ible to tho$e that are within it, as to the Sen$e of Feeling.</col><col>229</col></row> <row><col>The <I>Motion</I> of a Boat $en$ible to Sight joyned with Rea$on.</col><col>229</col></row> <row><col>A $imple Body, as the Earth, cannot move with three $everal <I>Motions.</I></col><col>231</col></row> <row><col><I>Motion</I> and Re$t are more different than Right <I>Motion</I> and Circular.</col><col>237</col></row> <row><col>One may more rationally a$cribe to the Earth two intern Principles to the Right and Cir- cular <I>Motion,</I> than two to <I>Motion</I> and Re$t.</col><col>237</col></row> <row><col>The diver$ity of <I>Motions</I> helpeth us to know the Diver$ity of Natures.</col><col>237</col></row> <row><col>Bodies of the $ame kind, have <I>Motions</I> that agree in kinde.</col><col>239</col></row> <row><col>The greatne$$e and $mallne$$e of the Body make a difference in <I>Motion</I> and not in Re$t.</col><col>243</col></row> <row><col>Every pen$ile and librated Body carried round in the Circumference of a Circle acquireth of it $elf a <I>Motion</I> in it $elf equal to the $ame.</col><col>362</col></row> <row><col>Two $orts of <I>Motion</I> in the containing Ve$$el may make the containing Water to ri$e and fall.</col><col>387</col></row> <row><col>An Accident in the Earths <I>Motion</I> impo$$ible to be imitated.</col><col>392</col></row> <row><col>ABSOLUTE MOTION: Things $aid to move according to certain of their parts, and not according to their whole, may not be $aid to move with an Ab$olute <I>Motion,</I> but <I>per accidens.</I></col><col>491</col></row> <row><col>ANIMAL MOTION: The Diver$ity of the <I>Motions</I> of Animals, depend on their Flex- ures.</col><col>232</col></row> <row><col>The Flexures in Animals are not made for vary- ing of their <I>Motions.</I></col><col>232</col></row> <row><col>The <I>Motions</I> of Animals are of one$ort.</col><col>232</col></row> <row><col>The <I>Motions</I> of Animals are all Circular.</col><col>233</col></row> <row><col>Secondary <I>Motion</I> of Animals dependent on the fir$t.</col><col>233</col></row> <foot><I>Mo</I> Ani-</foot> <pb> <row><col>Animals would not grow weary of their <I>Mo- tion,</I> proceeding as that which is a$$igned to the Terre$trial Globe.</col><col>244</col></row> <row><col>The Cau$e of the wearine$$e that attends the <I>Motion</I> of Animals.</col><col>244</col></row> <row><col>The <I>Motion</I> of an Animal is rather to be called Violent than Natural.</col><col>244</col></row> <row><col>ANNUAL MOTION: The Annual <I>Motion</I> of the Earth mu$t cau$e a con$tant and $trong Winde.</col><col>228</col></row> <row><col>The Errour o$ the Antagoni$t of Copernicus is manife$t, in that he declareth that the Annual and Diurnal Motion belonging to the Earth, are both one way, and not contrary.</col><col>235</col></row> <row><col>The Annual <I>Motion</I> of the Earth mixing with the <I>Motions</I> of the other Planets, produce extravagant Appearances.</col><col>296</col></row> <row><col>Re$t, Annual <I>Motion,</I> and the Diurnal, ought to be di$tributed betwixt the Sun, Earth, and Firmament.</col><col>300</col></row> <row><col>Granting to the Earth the Annual, it mu$t of hece$$ity have the Diurnal <I>Motion</I> a$$igned to it.</col><col>300</col></row> <row><col>The $ole Annual <I>Motion</I> of the Earth, cau$eth great inequality in the <I>Motions</I> of the Pla- nets.</col><col>310</col></row> <row><col>A Demon$tration of the inequalities of the three $uperiour Planets dependent on the Annual <I>Motion</I> of the Earth.</col><col>310</col></row> <row><col>The Annual <I>Motion</I> of the Earth mo$t apt to render a rea$on of the Exorbitance of the five Planets.</col><col>312</col></row> <row><col>Argument of Tycho again$t the Annual <I>Moti- on,</I> from the invariable Elevation of the Pole.</col><col>338</col></row> <row><col>Upon the Annual <I>Motion</I> o$ the Earth, alterati- on may en$ue in $ome Fixed Stars, not in the Pole.</col><col>341</col></row> <row><col>The Parallogi$me of tho$e who believe that in the Annual <I>Motion</I> great alterations are to be made about the Elevation of the Fixed Stars, is confuted.</col><col>341</col></row> <row><col>Enquiry is made what mutations, and in what Stars, are to be di$covered by means of the Earths Annual <I>Motion.</I></col><col>342</col></row> <row><col>A$tronomers having omitted to in$tance what alterations tho$e are that may be derived from the Annual <I>Motion</I> of the Earth, do thereby te$tifie that they never rightly un- der$tood the $ame.</col><col>343</col></row> <row><col>The Anuual <I>Motion</I> made by the Centre of the Earth under the Ecliptick, and the Diurnal <I>Motion</I> made by the Earth about its own Centre.</col><col>344</col></row> <row><col>Objections again$t the Earths Annual <I>Motion</I> taken from the Fixed Stars placed in the E- cliptick.</col><col>345</col></row> <row><col>An Indice or Ob$ervation in the Fixed Stars like to that which is $een in the Planets, is an Ar- gument of the Earths Annual <I>Motion.</I></col><col>347</col></row> <row><col>The Suns Annual <I>Motion</I> how it cometh to pa$$e, according to Copernicus.</col><col>355</col></row> <row><col>The Annual and Diurnal <I>Motion</I> are con$i$tent in the Earth.</col><col>362</col></row> <row><col>Three wayes of altering the proportion of the Additions of the Diurnal Revolution to the Annual <I>Motion.</I></col><col>409</col></row> <row><col>The Earths Annual <I>Motion</I> thorow the Ecliptick unequal, by rea$on of the Moons <I>Motion.</I></col><col>413</col></row> <row><col>The Cau$es of the inequality of the Additions and Sub$tractions of the Diurnal Conver$i- on from the Annual <I>Motion.</I></col><col>418</col></row> <row><col>CIRCULAR MOTION: Circular and Right <I>Motion</I> are $imple, as proceeding in $imple Lines.</col><col>6</col></row> <row><col>The Circular <I>Motion</I> is never acquired Natural- ly, unle$$e Right <I>Motion</I> precede it.</col><col>18</col></row> <row><col>Circular <I>Motion</I> perpetually uniforme.</col><col>18</col></row> <row><col>In the Circular <I>Motion</I> every point in the Cir- cumference is the beginning and end.</col><col>20</col></row> <row><col>Circular <I>Motion</I> onely is Uniforme.</col><col>20</col></row> <row><col>Circular <I>Motion</I> may be continued pcrpetu- ally.</col><col>20</col></row> <row><col>Circular <I>Motion</I> onely and Re$t are apt to con- $erve Order.</col><col>20</col></row> <row><col>To the Circular <I>Motion</I> no other <I>Motion</I> is con- trary.</col><col>26</col></row> <row><col>Circular <I>Motions</I> are not contrary, according to Ari$totle.</col><col>100</col></row> <row><col>The <I>Motion</I> of the Parts of the Earth returning to their Whole, may be Circular.</col><col>237</col></row> <row><col>The Velocity in the Circular <I>Motion</I> encrea$eth according to the encrea$e of the Diameter of the Circle.</col><col>242</col></row> <row><col>Circular <I>Motion</I> is truly $imple and perpetu- al.</col><col>495</col></row> <row><col>Circular Motion belongeth to the Whole Bo- dy, and the Right to its Parts.</col><col>496</col></row> <row><col>Circular and Right <I>Motion</I> are coincident, and may con$i$t together in the $ame Body.</col><col>496</col></row> <row><col>COMMON MOTION: A notable In$tance of Sagredus, to $hew the non-operating of Common <I>Motion.</I></col><col>151</col></row> <row><col>An Experiment that $heweth how the Com- mon <I>Motion</I> is imperceptible.</col><col>224</col></row> <row><col>The concurrence of the Elements in a Com- mon <I>Motion</I> imports no more than their con- currence in a Common Re$t.</col><col>239</col></row> <row><col>Common <I>Motion</I> is as if it never were.</col><col>223, 340</col></row> <row><col>COMPRESSIVE MOTION: Compre$$ive <I>Motion</I> is proper to Gravity, Exten$ive to Levity.</col><col>493</col></row> <foot>cliptick Xxx CON-</foot> <pb> <row><col>CONTRARY MOTIONS: An Experi- ment which plainly $hews that two Con- trary <I>Motions</I> may agree in the $ame Move- able.</col><col>363</col></row> <row><col>The parts of a Circle regularly moved about its own Centre, move in diver$e times with Contrary <I>Motions.</I></col><col>389</col></row> <row><col>DESCENDING MOTION: The Inclination of Grave Bodies to the <I>Motion</I> of De$cent, is e- qual to their re$i$tance to the <I>Motion</I> of A$cent.</col><col>191</col></row> <row><col>The Spaces pa$t in the De$cending <I>Motion</I> of the $alling Grave Body, are as the Squares|of their times.</col><col>198</col></row> <row><col>The <I>Motion</I> of De$cent belongs not to the Ter- re$trial Globe, but to its parts.</col><col>362</col></row> <row><col>DIVRNAL MOTION: The Diurnal <I>Motion</I> $eemeth Commune to all the Univer$e, the Earth onely excepted.</col><col>97</col></row> <row><col>Diurnal <I>Motion</I> why it $hould more probably belong to the Earth than to the Re$t of the Univer$e.</col><col>98</col></row> <row><col>The fir$t Di$cour$e to prove that the Diurnal <I>Motion</I> belongs to the Earth.</col><col>99</col></row> <row><col>The Diurnal <I>Motion</I> cau$eth no Mutation among Cele$tial Bodies, but all changes have relati- on to the Earth.</col><col>100</col></row> <row><col>A $econd Confirmation that|the Diurnal <I>Moti- on</I> belongs to the Earth.</col><col>100</col></row> <row><col>A third Confirmation that the Diurnal <I>Motion</I> belongs to the Earth.</col><col>101</col></row> <row><col>A fourth, fi$th, and $ixth Confirmation that the Diurnal <I>Motion</I> belongs to the Eatth.</col><col>102</col></row> <row><col>A$eventh Confirmation that the Diurnal <I>Mo- tion</I> belongs to the Earth.</col><col>103</col></row> <row><col>If the Diurnal <I>Motion</I> $hould alter, the Annual Period would cea$e.</col><col>409</col></row> <row><col>LOCAL MOTION: Local <I>Motion</I> of three kinds, Right, Circular, and Mixt.</col><col>6</col></row> <row><col>An entire and new Science of our Academick [Galileo] concerning Local <I>Motion.</I></col><col>198</col></row> <row><col>MIXT MOTION: Of Mixt <I>Motion</I> we $ee not the part that is Circular, becau$e we pertake thereof.</col><col>218</col></row> <row><col>Ari$totle granteth a Mixt <I>Motion</I> to Mixt Bodies.</col><col>375</col></row> <row><col>The <I>Motion</I> of Mixt Bodies ought to be $uch as may re$ult from the Compo$ition of the <I>Mo- tions</I> of the $imple Bodies compounding.</col><col>375</col></row> <row><col>NATVRAL MOTION: Accelleration of the Natural <I>Motion</I> of Graves is made according to the Odd Numbers beginning at Uni- ty.</col><col>198</col></row> <row><col>Natural <I>Motion</I> changeth into that which is Preter-Natural and Violent.</col><col>212</col></row> <row><col>PROGRESSIVE MOTION: The Progre$$ive <I>Motion</I> may make the Water in a Ve$$el to run to and fro.</col><col>387</col></row> <row><col>RIGHT MOTION: Sometimes Simple, and $ometimes Mixt, according to Ari$totle.</col><col>8</col></row> <row><col>Right <I>Motion</I> impo$$ible in the World exactly Ordinate.</col><col>10</col></row> <row><col>Right <I>Motion</I> Naturally Infinite.</col><col>10</col></row> <row><col>Right <I>Motion</I> Naturally Impo$$ible.</col><col>10</col></row> <row><col>Right <I>Motion</I> might po$$ibly have been in the Fir$t Chaos.</col><col>11</col></row> <row><col>Right <I>Motion</I> is u$eful to reduce into Order things out of Order.</col><col>11</col></row> <row><col>Right <I>Motion</I> cannot naturally be Perpetual.</col><col>20</col></row> <row><col>Right <I>Motion</I> a$$igned to Natural Bodies, to re- duce them to perfect Order, when removed from their Places.</col><col>20</col></row> <row><col>Right <I>Motion</I> of Grave Bodies manife$t to Sen$e.</col><col>22</col></row> <row><col>Right <I>Motion</I> with more rea$on a$cribed to the Parts, than to the whole Elements.</col><col>33</col></row> <row><col>Right <I>Motion</I> cannot be Eternal, and con$e- quently cannot be Natural to the Earth.</col><col>117</col></row> <row><col>Right <I>Motion</I> $eemeth to be wholly excluded in Nature.</col><col>147</col></row> <row><col>With two Right <I>Motions</I> one cannot compo$e Circular <I>Motions.</I></col><col>375</col></row> <row><col>Right <I>Motion</I> belongeth to imperfect Bodies, and that are out of their Natural Places.</col><col>495</col></row> <row><col>Right <I>Motion</I> is not Simple.</col><col>495</col></row> <row><col>Right <I>Motion</I> is ever mixt with the Circular.</col><col>495</col></row> <row><col>SIMPLE MOTION peculiar onely to Simple Bodies.</col><col>494</col></row> <row><col>TERRESTRIAL MOTION collected from the Stars.</col><col>229</col></row> <row><col>The Parts of the Terre$trial Globe accelerate and retard in their <I>Motion.</I></col><col>388</col></row> <row><col>One $ingle Terre$trial <I>Motion</I> $ufficeth not to produce the Ebbing and Flowing.</col><col>421</col></row> <row><col>UNEVEN MOTION may make the Water in a Ve$$el to Run to and fro.</col><col>387</col></row> <row><col>The Mixture of the two <I>Motions</I> Annual and Diurnal, cau$eth the unevenne$$e in the <I>Motion</I> of the parts of the Terre$trial Globe.</col><col>390</col></row> <row><col>MOVE.</col><col></col></row> <row><col>Its que$tionable whether de$cending Bodies <I>Move</I> in a Right Line.</col><col>21</col></row> <row><col>Ari$totles Argument to prove that Grave Bodies <I>Move</I> with an inclination to arrive at the Centre.</col><col>22</col></row> <row><col>Grave Bodies <I>Move</I> towards the Centre of the Centre of the Earth <I>per Accidens.</I></col><col>22</col></row> <row><col>Things for$aking the place which was natural ro them by Creation, are $aid to <I>Move</I> violently, <foot><I>Mo-</I> and</foot> <pb> and naturally tend to return back to the $ame.</col><col>492</col></row> <row><col>MOVEABLE, <I>&c.</I></col><col></col></row> <row><col>A <I>Moveable</I> being in the $tate of Re$t $hall not move unle$$e it have an inclination to $ome particular Place.</col><col>11</col></row> <row><col>The <I>Moveable</I> accellerates its Motion in going towards the Place whither it hath an inclina- tion.</col><col>11</col></row> <row><col>The <I>Moveable</I> departing from Re$t goeth thorow all the Degrees of Tardity.</col><col>11</col></row> <row><col>The <I>Moveable</I> doth not accelerate $ave only as it approacheth near to its terme of Re$t.</col><col>12</col></row> <row><col>To introduce in a <I>Moveable</I> a certain Degree of Velocity, Nature made it to move in a Right Line.</col><col>12</col></row> <row><col>The <I>Moveable</I> departing from Re$t pa$$eth through all the Degrees of Velocity without $taying in any.</col><col>13</col></row> <row><col>The Grave <I>Moveable</I> de$cending, acquireth Impetus $ufficient to re-carry it to the like height.</col><col>13</col></row> <row><col>The Impetus of <I>Moveables</I> equally approaching to the Centre are equal.</col><col>14</col></row> <row><col>Upon an Horizontal Plane the <I>Moveable</I> lyeth $till.</col><col>14</col></row> <row><col>A $ingle <I>Moveable</I> hath but one only Natural Motion, and all the re$t are by participa- tion.</col><col>103</col></row> <row><col>A Line de$cribed by a <I>Moveable</I> in its Natural De$cent, the Motion of the Earth about its own Centre being pre$uppo$ed, would pro- bably be the Circumference of a Circle.</col><col>145</col></row> <row><col>A <I>Moveable</I> falling from the top of a Tower moveth in the Circumference of a Circle.</col><col>146</col></row> <row><col>A <I>Moveable</I> falling from a Tower moveth neither more nor le$$e, then if it had $taid alwayes there.</col><col>146</col></row> <row><col>A <I>Moveable</I> falling from a Tower moveth with an Uniforme not an Accelerate Motion.</col><col>146</col></row> <row><col>The Cadent <I>Moveable,</I> if it fall with a Degree of Velocity acquired in a like time with an Uniform Motion, it $hall pa$$e a $pace double to that pa$$ed with the Accelerate Mo- tion.</col><col>202</col></row> <row><col>Admirable Problems of <I>Moveables</I> de$cending by the Quadrant of a Circle, and tho$e de$cending by all the Chords of the whole Circle.</col><col>412</col></row> <row><col>MUNDANE.</col><col></col></row> <row><col><I>Mundane</I> Bodies were moved in the beginning in a Right Line, and afterwards circularly, according to <I>Plato.</I></col><col>11</col></row> <row><col>N</col><col></col></row> <row><col>NATURAL.</col><col></col></row> <row><col>That which is Violent cannot be Eternall, and that which is Eternal cannot be <I>Natural.</I></col><col>116</col></row> <row><col>NATURE, and <I>Natures.</I></col><col></col></row> <row><col><I>Nature</I> attempts not things impo$$ible to be effected.</col><col>10</col></row> <row><col><I>Nature</I> never doth that by many things which may be done by a few.</col><col>99</col></row> <row><col><I>Nature</I> fir$t made things as $he plea$ed, and afterwards capacitated Mans under$tanding for conceiving of them.</col><col>238</col></row> <row><col>From Common Accidents one cannot know different <I>Natures.</I></col><col>238</col></row> <row><col><I>Natures</I> Order is to make the le$$er Orbes to Cir- culate in $horter times, and the bigger in longer.</col><col>243</col></row> <row><col>That which to us is hard to be under$tood, is with <I>Nature</I> ca$ie to be effected.</col><col>403</col></row> <row><col><I>Nature</I> keeping within the bounds a$$igned her, little careth that her Methods of opperating fall within the reach of Humane Capacity.</col><col>433</col></row> <row><col><I>Natures</I> Actions no le$s admirably di$cover God to us than Scripture Dictions.</col><col>434</col></row> <row><col>NERVES.</col><col></col></row> <row><col>The Original of the <I>Nerves</I> according to Ari$to- tle, and according to Phy$itians.</col><col>91</col></row> <row><col>The ridieulous An$wer of a Phylo$opher deter- mining the Original of the <I>Nerves.</I></col><col>91</col></row> <row><col>O</col><col></col></row> <row><col>OBJECTS.</col><col></col></row> <row><col><I>Objects,</I> the more Vigorous they are in Light, the more they do $eem to encrea$e.</col><col>305</col></row> <row><col>That Remote <I>Objects</I> appear $o $mall is the Defect of the Eye, as is demon$trated.</col><col>337</col></row> <row><col>In <I>Objects</I> far Remote and Luminous, a $mall acce$$ion or rece$$ion is imperceptible.</col><col>350</col></row> <row><col>OPINIONS.</col><col></col></row> <row><col>It's all one, whether <I>Opinions</I> are new to Men, or Men new to <I>Opinions.</I></col><col>77</col></row> <row><col>ORBE, and <I>Orbes.</I></col><col></col></row> <row><col>The greater <I>Orbes</I> make their Conver$ions in <foot>NA- greater</foot> <pb> greater times.</col><col>101 <I>&</I> 331</col></row> <row><col>It's more rational, that the <I>Orbe</I> containing and the Parts contained do move all about one Centre, than about divers.</col><col>295</col></row> <row><col>P</col><col></col></row> <row><col>PASSIONS.</col><col></col></row> <row><col>Infinite <I>Pa$$ions</I> are perhaps but one onely.</col><col>87</col></row> <row><col>PENDULUM, and <I>Pendula.</I></col><col></col></row> <row><col><I>Pendula</I> might have a perpetual Motion, impedi- ments being removed.</col><col>203</col></row> <row><col>The <I>Pendulum</I> hanging at a longer thread maketh its Vibrations more $eldome than the <I>Pendu- lum</I> hanging at a $horter.</col><col>206</col></row> <row><col>The Vibrations of the $ame <I>Pendulum</I> are made with the $ame frequency, whether they be $mall or great.</col><col>206</col></row> <row><col>The cau$e which impedeth the <I>Pendulum,</I> and reduceth it to re$t.</col><col>206</col></row> <row><col>The thread or Chain to which the <I>Pendulum</I> is fa$tened maketh an Arch, and doth not $tretch it $elf $traight out in its Vibrations.</col><col>207</col></row> <row><col>Two particular notable Accidents in the <I>Pendula</I> and their Vibrations.</col><col>411</col></row> <row><col>PERIPATETICK, <I>&c.</I></col><col></col></row> <row><col><I>Peripatetick</I> Phylo$ophy unchangeable.</col><col>42</col></row> <row><col>A brave re$olution of a certain <I>Peripatetick</I> Philo$opher to prove the Right Line to be the $horte$t of all Lines.</col><col>182</col></row> <row><col>The Paralogi$me of the $aid <I>Peripatetick</I> who proveth <I>Ignotum per ignotius.</I></col><col>183</col></row> <row><col>The Di$cour$es of <I>Peripateticks</I> full of Errors and Contradictions.</col><col>376</col></row> <row><col>The <I>Peripateticks</I> per$ecuted Galileo out of envy to his happy Di$coveries in Phylo$ophy.</col><col>427</col></row> <row><col>The <I>Peripateticks</I> in defect of Rea$ons repair to Scripture for Arguments again$t their Adver$aries.</col><col>429</col></row> <row><col>PHYLOSOPHERS.</col><col></col></row> <row><col>It is not ju$t, that tho$e who never. Phylo$ophate, $hould u$urp the title of <I>Phylo$ophers.</I></col><col>96</col></row> <row><col>PHYLOSOPHY.</col><col></col></row> <row><col>The Di$putes and Contradictions of <I>Phylo$ophers</I> may conduce to the benefit of <I>Phylo$ophy.</I></col><col>25</col></row> <row><col>A cunning way to gather <I>Phylo$ophy</I> out of any Book what$oever.</col><col>92</col></row> <row><col>PLANETS.</col><col></col></row> <row><col>The approximation and rece$$ion of the three $uperiour <I>Planets</I> importeth double the Suns di$tance.</col><col>299</col></row> <row><col>The difference of the <I>Tlanets</I> apparent Magni- tude le$$e in Saturn than in Jupiter, and le$$e in Jupiter than in Mars, and why.</col><col>299</col></row> <row><col>The Station, Direction, and Retrogradation of the <I>Planets</I> is known in relation to the fixed Stars.</col><col>347</col></row> <row><col>The particular Structures of the Orbes of the <I>Planets</I> not yet well re$olved.</col><col>416</col></row> <row><col>The <I>Planets</I> places may more certainly be a$$igred by this Doctrine, than by that of Ptolomies great Almage$t.</col><col>469</col></row> <row><col>PLATO.</col><col></col></row> <row><col><I>Plato</I> held, that Humane under$tanding pertook of Divinity, becau$e it under$tood Num- bers.</col><col>3</col></row> <row><col><I>Plato</I> his Ænigma, and the Interpretation of it.</col><col>498</col></row> <row><col>POLE.</col><col></col></row> <row><col>The invariable Elevation of the <I>Pole</I> urged as an Argument again$t the Annual Motion.</col><col>338</col></row> <row><col>An Example to prove that the Altitude of the <I>Pole</I> ought not to vary by means of the Earths Annual Motion.</col><col>340</col></row> <row><col>POWER.</col><col></col></row> <row><col>Of an infinite <I>Power</I> one would think a greater part $hould rather be imployed than a le$$er.</col><col>105</col></row> <row><col>PRINCIPLES.</col><col></col></row> <row><col>By denying <I>Principles</I> in Sciences, any Paradox may be maintained.</col><col>28</col></row> <row><col>Contrary <I>Principles</I> cannot naturally re$ide in the $ame Subject.</col><col>211</col></row> <row><col>PROJECT, <I>&c.</I></col><col></col></row> <row><col>The <I>Project,</I> according to Ari$totle, is not mo- ved by virtue impre$$ed, but by the Me- dium.</col><col>130</col></row> <row><col>Operation of the Medium in continuing the Motion of the <I>Project.</I></col><col>131</col></row> <row><col>Many Experiments and Rea$ons again$t the Motions of <I>Projects</I> a$$igned by Ari$totle.</col><col>132</col></row> <row><col>The Medium doth impede and not conferre the <foot>PLA- Mo-</foot> <pb> Motion of <I>Projests.</I></col><col>134</col></row> <row><col><*>n admirable accident in the Motion of <I>Pro- jects.</I></col><col>135</col></row> <row><col><*>undry curious Problems touching the Motion of <I>Projects.</I></col><col>137</col></row> <row><col><I><*>ojects</I> continue their <I>Motion</I> by a Right Line that follows the direction of the Motion made together with the <I>Projicient,</I> whil$t they were conjoyned therewith.</col><col>154</col></row> <row><col><*>he Motion impre$$ed by the <I>Projicient</I> is onely in a Right Line.</col><col>170</col></row> <row><col><*>he <I>Project</I> moveth by the Tangent of the Cir- cle of the Motion preceeding in the in$tant of Seperation.</col><col>172</col></row> <row><col><*> Grave <I>Project</I> a$$oon as it is $eperated from the <I>Projicient,</I> beginneth to decline.</col><col>173</col></row> <row><col><*>he Cau$e of the <I>Projection</I> encrea$eth not ac- cording to the Proportion of Velocity en- crea$ed by making the Wheel bigger.</col><col>189</col></row> <row><col><*>he Virtue which carrieth Grave <I>Projects</I> up- wards, is no le$$e Natural to them than the Gravity which moveth them down- wards.</col><col>211</col></row> <row><col>PTOLOMY, <I>&c.</I></col><col></col></row> <row><col><*>conveniences that are in the Sy$tem of <I>Pto- lomy.</I></col><col>309</col></row> <row><col><I><*>olomies</I> Sy$tem full of defects.</col><col>476</col></row> <row><col><*>he Learned both of elder and later times di$- $atisfied with the <I>Ptolomaick</I> Sy$tem.</col><col>477</col></row> <row><col>PYTHAGORAS, <I>&c.</I></col><col></col></row> <row><col><I><*>thagorick</I> Mi$tery of Numbers fabulous.</col><col>3</col></row> <row><col><I><*>thagoras</I> offered an Hecatombe for a Geo- metrical Demon$tration which he found.</col><col>38</col></row> <row><col><I><*>ythagoras</I> and many other Ancients enumera- ted, that held the Earths Mobility.</col><col>437 <I>&</I> 468</col></row> <row><col>R</col><col></col></row> <row><col>RAYS.</col><col></col></row> <row><col><*>hining Objects $eem fringed and environed with adventitious <I>Rays.</I></col><col>304</col></row> <row><col>RIST.</col><col></col></row> <row><col><I>Re$t.</I> Vide <I>Motion.</I></col><col></col></row> <row><col><I>Re$t</I> the Infinite degree of Tardity.</col><col>11</col></row> <row><col>RBTROGRADATIONS.</col><col></col></row> <row><col><I>Retrogradations</I> more frequent in Saturn, le$$e fre quent in Jupiter, and yet le$$e in Mars, and why.</col><col>311</col></row> <row><col>The <I>Retrogradations</I> of Venus and Mercury demon$trated by Apollonius and Coper- nicus.</col><col>311</col></row> <row><col>S</col><col></col></row> <row><col>SATURN.</col><col></col></row> <row><col><I>Saturn</I> for its $lowne$$e, and Mercury for its late appearing, were among$t tho$e that were la$t ob$erved.</col><col>416</col></row> <row><col>SCARCITY.</col><col></col></row> <row><col><I>Scarcity</I> and Plenty enhan$e and deba$e the price of all things.</col><col>43</col></row> <row><col>SCHEINER.</col><col></col></row> <row><col>Chri$topher <I>Scheiner</I> the Jefuit his Book of Con- clu$ions confuted.</col><col>78 <I>& 195, & $eq. &</I> 323</col></row> <row><col>A Canon Bullet would $pend more than $ix dayes in falling from the Concave of the Moon to the Center of the Earth, according to <I>Scheiner.</I></col><col>195</col></row> <row><col>Chri$topher <I>Scheiner</I> his Book entituled <I>Apelles po$t Tabulam</I> cen$ured, and di$proved.</col><col>313</col></row> <row><col>The Objections of <I>Scheiner</I> by way of Interro- gation.</col><col>336</col></row> <row><col>An$wers to the Interrogations of <I>Schtiner.</I></col><col>336</col></row> <row><col>Que$tions put to <I>Scheiner,</I> by which the weak- ne$le of his is made appear.</col><col>336</col></row> <row><col>SCIENCES.</col><col></col></row> <row><col>In Natural <I>Sciences</I> the Art of Oratory is of no u$e.</col><col>40</col></row> <row><col>In Natural <I>Sciences</I> it is not nece$$ary to $eek Mathematical evidence.</col><col>206</col></row> <row><col>SCRIPTURE, <I>&c.</I></col><col></col></row> <row><col>The Caution we are to u$e in determining the Sen$e of <I>Scripture</I> in difficult points of Phy- lo$ophy.</col><col>427</col></row> <row><col><I>Scripture</I> $tudiou$ly conde$cendeth to the ap- prehen$ion of the Vulgar.</col><col>432</col></row> <row><col>In dicu$$ing of Natural Que$tions, we ought not to begin at <I>Scripture,</I> but at Sen$ible Experiments and Nece$$ary Demon$tra- tions.</col><col>433</col></row> <row><col>The intent of <I>Scripture</I> is by its Authority to recommend tho$e Truths to our beliefe, which being un-intelligible, could no other wayes be rendered credible.</col><col>434</col></row> <foot>and Yyy <I>Scrip-</I></foot> <pb> <row><col><I>Scripture</I> Authority to be preferred, even in Na- tural Controver$ies to $uch Sciences as are not confined to a Demon$trative Me- thod.</col><col>434</col></row> <row><col>The Pen-men of <I>Scripture,</I> though read in A- $tronomy, intentionally forbear to teach us anything of the Nature of the Stars.</col><col>435</col></row> <row><col>The Spirit had no intent at the Writing of the <I>Scripture,</I> to teach us whether the Earth mo- veth or $tandeth $till, as nothing concerning our Salvation.</col><col>436</col></row> <row><col>Inconveniencies that ari$e from licentious u- $urping of <I>Scripture,</I> to $tuffe out Books that treat of Nat. Arguments.</col><col>438</col></row> <row><col>The Literal Sen$e of <I>Scripture</I> joyned with the univer$al con$ent of the Fathers, is to be re- ceived without farther di$pute</col><col>444</col></row> <row><col>A Text of <I>Scripture</I> ought no le$$e diligently to be reconciled with a Demon$trated Pro- po$ition in Philo$ophy, than with another Text of <I>Scripture</I> $ounding to a contrary Sen$e.</col><col>446</col></row> <row><col>Demon$trated Truth ought to a$$i$t the Com- mentator in finding the true Sen$e of <I>Scrip- ture.</I></col><col>446</col></row> <row><col>It was nece$$ary by way of conde$cen$ion to Vulgar Capacities, that the <I>Scripture</I> $hould $peak of the Re$t and Motion of the Sun and Earth in the $ame manner that it doth.</col><col>447</col></row> <row><col>Not onely the Incapacity of the Vulgar, but the Current Opinion of tho$e times, made the Sacred Writers of the <I>Scripture</I> to ac- commodate them$elves to Popular E$teem more than Truth.</col><col>447</col></row> <row><col>The <I>Scripture</I> had much more rea$on to affirm the Sun Moveable, and the Earth Immove- able, than otherwi$e.</col><col>448</col></row> <row><col>Circum$pection of the Fathers about impo$ing po$itive Sen$es on Doubtful Texts of <I>Scrip- ture.</I></col><col>451</col></row> <row><col>Tis Cowardice makes the Anti-Copernican fly to Scripture Authorities, thinking thereby to affright their Adver$aries.</col><col>455</col></row> <row><col><I>Scripture</I> $peaks in Vulgar and Common Points after the manner of Men.</col><col>462</col></row> <row><col>The intent of <I>Scripture</I> is to be ob$erved in Pla- ces that $eem to affirme the Earths Stabi- lity.</col><col>464</col></row> <row><col><I>Scripture</I> Authorities that $eem to affirm the Mo- tion of the Sun and Stability of the Earth, divided into $ix Cla$$es.</col><col>478</col></row> <row><col>Six Maximes to be ob$erved in Expounding Dark Texts of <I>Scripture.</I></col><col>481</col></row> <row><col><I>Scripture</I> Texts $peaking of things inconveni- ent to be under$tood in their Literal Sen$e, are to be interpreted one of the four wayes named.</col><col>81</col></row> <row><col>Why the Sacred <I>Scripture</I> accommodates it $elf to the Sen$e of the Vulgar.</col><col>487</col></row> <row><col>SEA.</col><col></col></row> <row><col>The <I>Seas</I> Surface would $hew at a di$tance more ob$cure than the Land.</col><col>49</col></row> <row><col>The <I>Seas</I> Reflection of Light much weaker than that of the Earth.</col><col>81</col></row> <row><col>The I$les are tokens of the unevenne$$e of the Bottoms of <I>Seas.</I></col><col>383</col></row> <row><col>SELEUCUS.</col><col></col></row> <row><col>Opinion of <I>Seleucus</I> the Mathematician cen- $ured.</col><col>422</col></row> <row><col>SENSE.</col><col></col></row> <row><col>He who denieth <I>Sen$e,</I> de$erves to be deprived of it.</col><col>21</col></row> <row><col><I>Sen$e</I> $heweth that things Grave move <I>ad Me- dium,</I> and the Light to the Concave.</col><col>21</col></row> <row><col>It is not probable that God who gave us our <I>Sen$es,</I> would have us lay them a$ide, and look for other Proofs for $uch Natural Points as <I>Sen$e</I> $ets before our Eyes.</col><col>434</col></row> <row><col><I>Sen$e</I> and Rea$on le$$e certain than Faith.</col><col>475</col></row> <row><col>SILVER.</col><col></col></row> <row><col><I>Silver</I> burni$hed appears much more ob$cure than the unburni$hed, and why.</col><col>64</col></row> <row><col>SIMPLICIUS.</col><col></col></row> <row><col><I>Simplicius</I> his Declamation.</col><col>43</col></row> <row><col>SOCRATES.</col><col></col></row> <row><col>The An$wer of the Oracle true in judging <I>So- crates</I> the Wi$e$t of his time.</col><col>85</col></row> <row><col>SORITES.</col><col></col></row> <row><col>The Forked Sylogi$me called <G>Sop<*></G></col><col>29</col></row> <row><col>SPEAKING.</col><col></col></row> <row><col>We cannot ab$tract our manner of <I>Speaking</I> from our Sen$e of Seeing.</col><col>461</col></row> <row><col>SPHERE.</col><col></col></row> <row><col>The Motion of 24 hours a$cribed to the Highe$t <foot>named. <I>Sphere</I></foot> <pb> <I>Sphere,</I> di$orders the Period of the Inferi- our.</col><col>102</col></row> <row><col>The <I>Sphere</I> although Material, toucheth the Material Plane but in one point onely.</col><col>182</col></row> <row><col>The Definition of the <I>Sphere.</I></col><col>182</col></row> <row><col>A Demon$tration that the <I>Sphere</I> toucheth the Plane but in one point.</col><col>183</col></row> <row><col>Why the <I>Sphere</I> in ab$tract toucheth the Plane onely in one point, and not the Material in Concrete.</col><col>184</col></row> <row><col>Contact in a Single Point is not peculiar to the perfect <I>Sphere</I> onely, but belongeth to all Curved Figures.</col><col>185</col></row> <row><col>In a Moveable <I>Sphere</I> it $eemeth more rea$ona- ble that its Centre be $table, than any of its parts.</col><col>300</col></row> <row><col>SPHERE of <I>Activity.</I></col><col></col></row> <row><col>The <I>Sphere of Activity</I> greater in Cele$tial Bo- dies than in Elimentary.</col><col>59</col></row> <row><col>STARRY SPHERE.</col><col></col></row> <row><col>Wearine$$e more to be feared in the <I>Starry Sphere</I> than in the Terre$trial Globe.</col><col>245</col></row> <row><col>By the proportion of Jupiter and of Mars, the <I>Starry Sphere</I> is found to be yet more re- mote.</col><col>331</col></row> <row><col>Vanity of tho$e mens di$cour$e, who argue the <I>Starry Sphere</I> to be too va$t in the Coper- nican Hypothe$is.</col><col>335</col></row> <row><col>The whole <I>Starry Sphere</I> beheld from a great di- $tance, might appear as $mall as one $ingle Star.</col><col>335</col></row> <row><col>SPHERICAL.</col><col></col></row> <row><col>The <I>Spherical</I> Figure is ea$ier to be made than any other.</col><col>186</col></row> <row><col><I>Spherical</I> Figures of $undry Magnitudes, may be made with one $ole In$trument.</col><col>187</col></row> <row><col>SPIRIT.</col><col></col></row> <row><col>The <I>Spirit</I> had no intent to teach us whether the Earth moveth or $tandeth $till, as no- thing concerning our Salvation.</col><col>436</col></row> <row><col>SOLAR SPOTS.</col><col></col></row> <row><col><I>Spots</I> generate and di$$olve in the face of the Sun.</col><col>38</col></row> <row><col>Sundry Opinions touching the <I>Solar Spots.</I></col><col>39</col></row> <row><col>An Argument that nece$$arily proveth the <I>So- lar Spots</I> to generate and di$$olve.</col><col>40</col></row> <row><col>A conclu$ive Demon$tration to prove that the <I>Spots</I> are contiguous to the Body of the Sun.</col><col>41</col></row> <row><col>The Motion of the <I>Spots</I> towards the Circum- cumference of the Sun appears $low.</col><col>41</col></row> <row><col>The Figure of the <I>Spots</I> towards the Circumfe- rence of the Suns Di$cus, appear narrow, and why.</col><col>41</col></row> <row><col>The <I>Solar Spots</I> are not Spherical, but flat, like thin plates.</col><col>41</col></row> <row><col>The Hi$tory of the proceedings of the Acade- mian for a long time about the Ob$ervation of the <I>Solas Spots.</I></col><col>312</col></row> <row><col>A conceit that $uddenly came into the mind of our Academian concerning the great con$e- quence that followeth upon the Motion of the <I>Solar Spots.</I></col><col>314</col></row> <row><col>Extravagant Mutations to be ob$erved in the Motions of the <I>Solar Spots</I> foreleen by the Academ<*>ck, in ca$e the Earth had the Annu- al Motion.</col><col>314</col></row> <row><col>The fir$t Accident to be ob$erved in the Moti- on of the <I>Solar Spots,</I> and con$equently all the re$t, explained.</col><col>315</col></row> <row><col>The events being ob$erved were an$werable to the Predictions touching the$e <I>Spots.</I></col><col>318</col></row> <row><col>Though the Annual Motion a$$igned to the Earth, an$wereth to the Phænomena of the <I>Solar Spots,</I> yet doth it not follow by conver- $ion, that from the Phænomena of the <I>Spots</I> one may inferre the Annual Motion to be- long to the Earth.</col><col>319</col></row> <row><col>The Pure Peripatetick Philo$ophers will laugh at the <I>Spots</I> and their Phænomena, as the Illu$ions of the Chri$tals in the Tele- $cope.</col><col>319</col></row> <row><col>The <I>Solar Spots</I> of Galileo.</col><col>494</col></row> <row><col>STAR and <I>Stars.</I></col><col></col></row> <row><col>The <I>Stars</I> infinitely $urpa$$e the re$t of Heaven in Den$ity.</col><col>30</col></row> <row><col>It is no le$$e impo$$ible for a <I>Star</I> to corrupt, than the whole Terre$trial Globe.</col><col>37</col></row> <row><col>New <I>Stars</I> di$covered in Heaven.</col><col>38</col></row> <row><col>The $mall Body of a <I>Star</I> fringed about with Rays, appeareth very much bigger than plain, naked, and in its native Clarity.</col><col>61</col></row> <row><col>An ea$ie Experiment that $heweth the encrea$e in the <I>Stars,</I> by means of the Adventitious Rays.</col><col>305</col></row> <row><col>A <I>Star</I> of the Sixth Magnitude $uppo$ed by Ty- cho and Scheiner an hundred and $ix Millions of times bigger than needs.</col><col>326</col></row> <row><col>A common errour of all A$tronomers touching the Magnitude of the <I>Stars.</I></col><col>326</col></row> <foot>A Venus</foot> <pb> <row><col><*> a fal$e one, none.</col><col>112. 245</col></row> <row><col>TRUTH, and <I>Truths.</I></col><col></col></row> <row><col>Untruths cannot be Demon$trated as <I>Truths</I> are.</col><col>112</col></row> <row><col>The <I>Truth</I> $ometimes gains $trength by Con- tradiction.</col><col>181</col></row> <row><col><I>Truth</I> hath not $o little light as not to be di$co- vered among$t the Umbrages of Fal- $hoods.</col><col>384</col></row> <row><col>TYCHO.</col><col></col></row> <row><col>The Argument of <I>Tycho</I> grounded upon a fal$e Hypothe$is.</col><col>324</col></row> <row><col><I>Tycho</I> and his Followers never attempted to $ee whether there were any Phænomena in the Firmament for or again$t the Annual Mo- tion.</col><col>337</col></row> <row><col><I>Tycho</I> and others argue again$t the Annual Mo- tion, from the invariable Elevation of the Pole.</col><col>338</col></row> <row><col>V</col><col></col></row> <row><col>VELOCITY.</col><col></col></row> <row><col>Vniform <I>Velocity</I> $utable with Circular Mo- tion.</col><col>12</col></row> <row><col>Nature doth not immediately conferre a de- terminate degree of <I>Velocity,</I> although She could.</col><col>12</col></row> <row><col>The <I>Velocity</I> by the inclining plane equal to the <I>Velocity</I> by the Perpendicular, and the Mo- tion by the Perpendicular $wifter than by the inclining plane.</col><col>14</col></row> <row><col><I>Velocities</I> are $aid to be equal, when the Spa- ces pa$$ed are proportionate to their times.</col><col>15</col></row> <row><col>The greater <I>Velocity</I> exactly compen$ates the greater Gravity.</col><col>192</col></row> <row><col>VENUS.</col><col></col></row> <row><col>The Mutation of Figure in <I>Venus</I> argueth its Motion to be about the Sun.</col><col>295</col></row> <row><col><I>Veuus</I> very great towards the Ve$pertine Con- junction, and very $mall towards the Ma- cutine.</col><col>297</col></row> <row><col><I>Venus</I> nece$$arily proved to move about the Sun.</col><col>298</col></row> <row><col>The Phænomena of <I>Venus</I> appear contrary to the Sy$tem of Copernicus.</col><col>302</col></row> <row><col>Another Difficulty rai$ed by <I>Venus</I> again$t Co- pernicus.</col><col>302</col></row> <row><col><I>Venus</I> according to Copernicus either lucid in it $elf, or a tran$parent $ub$tance.</col><col>302</col></row> <row><col>The Rea$on why <I>Venus</I> and Mars do not ap- pear to vary Magnitude $o much as is re- qui$ite.</col><col>303</col></row> <row><col>A $econd Rea$on of the $mall apparent encreale of <I>Venus.</I></col><col>306</col></row> <row><col><I>Venus</I> renders the Errour of A$tronomers in de- termining the Magnitude of Stars inex- cu$eable.</col><col>327</col></row> <row><col>VESSEL.</col><col></col></row> <row><col>Of the Motion of Water in a <I>Ve$$el.</I> Vide <I>Water.</I></col><col></col></row> <row><col>UNDERSTAND, <I>&c.</I></col><col></col></row> <row><col>Man <I>Under$tandeth</I> very much <I>inten$ive,</I> but little <I>exten$ive.</I></col><col>86</col></row> <row><col>Humane <I>Uuder$tanding</I> operates by Ratioci- nation.</col><col>87</col></row> <row><col>UNIVERSE.</col><col></col></row> <row><col>The Con$titution of the <I>Uuiver$e</I> is one of the Noble$t Problems a Man can $tudy.</col><col>187</col></row> <row><col>The Centre of the <I>Univer$e</I> according to Ari- $totle is that Polnt about which the Cele- $tial Spheres do revolve.</col><col>294</col></row> <row><col>Which ought to be accounted the Sphere of the <I>Univer$e.</I></col><col>299</col></row> <row><col>It is a great ra$hne$$e to cen$ure that to be $u- perfluous in the <I>Univer$e</I> which we do not perceive to be made for us.</col><col>334</col></row> <row><col>VURSTITIUS.</col><col></col></row> <row><col>Chri$tianus <I>Vur$titius</I> read certain Lectures touching the Opinion of Copernicus, and what happened thereupon.</col><col>110</col></row> <row><col>W</col><col></col></row> <row><col>WATER.</col><col></col></row> <row><col>He that had not heard of the Element of <I>Water,</I> could never fancie to him$elf Ships and Fi- $hes.</col><col>47</col></row> <row><col>An Experiment to prove the Reflection of <I>Wa- ter</I> lefs bright than that of the Land.</col><col>81</col></row> <row><col>The Motion of the <I>Water</I> in Ebbing and Flow- ing, not interrupted by Re$t.</col><col>251</col></row> <row><col>The vain Argumentation of $ome, to prove the Element of <I>Water</I> to be of a Spherical Superficies.</col><col>377</col></row> <foot><I>Ve-</I> The</foot> <pb> <row><col><*> Progre$$ive and uneven Motion makes the <I><*>ater</I> in a Ve$$el to run to and fro.</col><col>387</col></row> <row><col><*> Several Motions in the conteining Ve$$el, <*>nay make the conteined <I>Water</I> to ri$e and <*>all.</col><col>387</col></row> <row><col><*> <I>Water</I> rai$ed in one end of the Ve$$el re- <*>urneth it $elf to <I>Æquilibrium.</I></col><col>391</col></row> <row><col><*>he $horter Ve$$els the Undulations of <I>Wa- <*>rs</I> are more frequent.</col><col>391</col></row> <row><col><*> greater profundity maketh the Undulati- <*>ns of <I>Water</I> the more frequent.</col><col>391</col></row> <row><col><*>y in narrow places the Cour$e of the <I>Wa- <*>rs</I> is $wifter than in larger.</col><col>396</col></row> <row><col><*> cau$e why in $ome narrow Chanels, we <*>e the Sea-<I>Waters</I> run alwayes one way.</col><col>398</col></row> <row><col><*> <I>Water</I> more apt to con$erve an Impetus <*>onceived than the Air.</col><col>400</col></row> <row><col><*> Motion of the <I>Water</I> dependeth on the <*>otion of Heaven.</col><col>404</col></row> <row><col>WEIGHTS.</col><col></col></row> <row><col><*>que$tionable whether De$cending <I>Weights</I> <*>ove in a Right Line.</col><col>21</col></row> <row><col>WEST.</col><col></col></row> <row><col><*> Cour$e to the <I>West</I> India's ea$ie, the re- <*>rn difficult.</col><col>402</col></row> <row><col>WINDE.</col><col></col></row> <row><col><*>$tant Gales of <I>Winde</I> within the Tropicks <*>low towards the We$t.</col><col>402</col></row> <row><col><I><*>des</I> from the Land, make rough the <*>eas.</col><col>402</col></row> <row><col>WISDOME <I>Divine.</I></col><col></col></row> <row><col><I><*>ine Wi$dome</I> infinitely infinite.</col><col>85</col></row> <row><col><*> Di$cour$es which Humane Rea$on makes in time, the <I>Divine Wi$dom</I> re$olveth in a Moment, that is hath them alwayes pre- $ent.</col><col>87</col></row> <row><col>WIT.</col><col></col></row> <row><col>The <I>Wit</I> of Man admirably acute.</col><col>87</col></row> <row><col>The Pu$ilanimity of Popular <I>Wits.</I></col><col>364</col></row> <row><col>Poctick <I>Wits</I> of two kinds.</col><col>384</col></row> <row><col>WORLD.</col><col></col></row> <row><col><I>World.</I> Vide <I>Univer$e.</I></col><col></col></row> <row><col>The <I>Worlds</I> parts are according to Ari$totle two, Cele$tial and Elementary, contrary to each other.</col><col>6</col></row> <row><col>The <I>World</I> $uppo$ed by the Anthour [Galileo] to be perfectly Ordinate.</col><col>10</col></row> <row><col>The Sen$ible <I>World.</I></col><col>96</col></row> <row><col>It hath not been hitherto proved by any whe- ther the <I>World</I> be finite or infinite.</col><col>293</col></row> <row><col>If the Centre of the <I>World</I> be the $ame with that about which the Planets move, the Sun and not the Earth is placed in it.</col><col>295</col></row> <row><col>WRITING.</col><col></col></row> <row><col>Some <I>Write</I> what they under$tand not, and therefore under$tand not what they <I>Write.</I></col><col>63</col></row> <row><col>The Invention of <I>Writing</I> Stupendious above all others.</col><col>88</col></row> <row><col>Y</col><col></col></row> <row><col>YEAR.</col><col></col></row> <row><col>The <I>Years</I> beginning and ending, which Ptolomy and his Followers could never po$it<*>vely a$- $ign, is exactly determined by the Coper- nican Hypothe$is.</col><col>469</col></row> </table> <head><I>THE END OF THE TABLE.</I></head> <pb> <P>line 31. <I>for</I> where leave, <I>read</I> why omit. p 3, l 32, only for. p 5, l 8, Dimen$ions of a Superfieies, <I>ibid.</I> l 15, <I>for</I> line <*> thread, l <I>u$e.</I> <*> all $ides, <I>r.</I> every way. p 6, l 41, by nece$$ary. p 9, l 15, <I>for</I> Medium, <I>r.</I> Way. l 40. <I>for</I> $ome <I>r.</I> $ave, <I>marg</I> and the. p 10, l 1. farther be l 28. intigrall, l 44, prefixed. p 11, 19, oppo$itely might have. l 10, con$tituted Bodie. l 11, $o are. l 15, would only en$ue. p 12, l 28, <*>p 14, l 25. inclined plane. l 16. p 34, <I>for</I> by <I>r.</I> through. p 17, l 20, beyond T, p 19, l 3, <I>dele</I> of. l 6, <I>for</I> of a$$igning, <I>r.</I> that he $$igned. l 9, <I>for</I> and with the &c. <I>r.</I> and given them the intended inclinations of moving thence towards the Centre. l 15, <I>for</I> Orbes, <*>bes, l 38, <I>for</I> truly <I>r.</I> exactly. p 21, l 36, <I>for</I> another, <I>r.</I> one. p 22, l 19, that <I>contra,</I> l 20. <I>for</I> than contend with you, <I>r.</I> than as being. <*>nced b the $trength of your Rea$ons. p 23, 18. <I>for</I> to di$cover, <I>r.</I> to be $howne, l 17, $uppo$ing. l 22, <I>for</I> follow, <I>r.</I> hit upon. l 52, as <*>ell under$tand. p 24, l 17, <I>for</I> thither <I>r.</I> there. l 33, <I>r.</I> Earth a$cend and de$cend. p 25, l 7, <I>for</I> quality <I>r.</I> power. l <I>ult.</I> part. p 26, <*>ntraries; But of contraries the motions are contrary, l 32, <I>for</I> be$ides <I>r.</I> in ca$e. p 27, l 2, <I>for</I> repeat, <I>r.</I> go over. l 3. <I>for</I> reá$$ume, <I>r.</I> <*> p 29, l <I>ul<*> for</I> $ay <I>r.</I> mean. l 18, re$ide. l 39, <I>for</I> may. <I>r.</I> can. p 32, l 1, <I>for</I> others, <I>r.</I> one. l 15, <I>for</I> the be$t we can whether, <I>r.</I> whats to <*>ne with it in ea$e that. l 37, <I>for</I> they m<I>u</I>$t grant us, <I>r.</I> let it be granted, p 32, l 5, what. l 12, unalterable. p 37, l 28, <I>for</I> confront <I>r.</I> <*>rre. l 37, <I>for</I> bethink, <I>r.</I> think, p 41. l 39, <I>for</I> ob$erve them to be hid, <I>r.</I> ob erve that he hath concealed from you tho$e that are <*>ved. p 42, l 17. which experience and $en$e. l 17, unalterable. p 43, l 15, <I>for</I> more <I>r.</I> le$$e. p 44. l 4, Globes? p 46, l 18, <I>r.</I> liquifie, <I>for</I> them <I>r.</I> it, p 49. l 11, <I>for</I> thing <I>r.</I> thinke. p 50, l 19. <I>and</I> p 51, l 3, Superficies p 53, l 14, Truths. p 54, l 38, be $olid. p 55, l 15, ult. l 36, of the. l 40, <I>omit</I> $o. p 57, l 44, maketh, p 59, l 1, <I>for</I> that <I>r.</I> the. p 61, l 5, tho$e. p 62, l 6, di$per$e. p 64, l 13, evene. l 26, <*>g<*>. l 37, <I>for</I> of, <I>r.</I> in. l 42, acquie$$e. p 65. l 20, happeneth. p 67, l 13, Solutions, l 15, vi$ive. p 69, l 18, wood?. l 31, wayes,. p 70, l 7, <*>u$ion is very good for, l 10, $hould we, l 11, would alter. l 12, <I>for</I> or, <I>r.</I> and, l 22, Propo$itions, l 29, Protubrances, l 41, <I>for</I> their, <I>r.</I> its. <*>l 5, by one <I>dele</I>;. l 12, of the Moon, l 37, <I>for</I> in youes, &c. <I>r.</I> with your Opacity and Per$picuity. p 72, l 20, what time do. <I>for</I> p 74. <I>r.</I> 73. l 26, yea and more. p 76, l 18, <I>Vitellio.</I> p 81, l 6, <I>dele</I> more. p 81, l 25, of a <*>id matter, l 29, <I>for</I> $entence <I>r.</I> Centre. <I>for</I> on <*> p 85, l 39, <I>for</I> make rai$ins, <I>r.</I> make the kernels, p 86, l 29 <I>inten$ivè,</I> p 87, l 17, the which, neither,. p. 88. l. 8. <I>Raffaelio,</I> or <I>a Tiziano</I>?<*> <*>l 20. <I>for</I> rece$$e <I>r.</I> remote. p 91. l 11, curio$ity. p 94. l 41. reputation. p 101, l 18, altercations. p 107, l 29, $tar. p 104, l 32, <*>ally. p 107, l 26, <I>accidens.</I> p 111, l 24, $elfe, l 18, third teime, p. 113, l 21, Guns. p 111, l 3, tran$ported, l. 7, we pa$$e. p 116, l 15, <*>herwi$e <I>r.</I> any wayes. p 118, <I>Marg.</I> rendred, p 128, 35, a$cending?. p 130, l 42, as being the. p 131, l 19, occa$ion, l 30, $eparated. 3, l 11, <I>pendula.</I> p 134, ll 20, arows? p 135, l 12, time to prove it,. p 137, l 2, <I>and</I> 6, <I>for</I> ball, <I>r.</I> bowl. p 142, l 17, <I>dele</I> very. p 143, l 3, l, l 4, liberty. p 144, l 8, find out. p 145, l 21, <I>dele</I>;. p 147, l 9, <I>dele</I> that. p 148, l 7, which $o far exceeds their flight, l 33, is the. l <I>ult.</I> <*>ent of. p 149, l 44, <I>Tycho.</I> p 153, l 21, <I>for</I> that is <I>r.</I> or. p 154. 3. <I>dele</I> of. p 157, l 7, <I>for</I> to the, <I>r.</I> by the. p 161, l 15, fifteen $econds:. <*>g. in its re$t. p 162, l 13, <I>for</I> motion <I>r.</I> Immobility. p. 163, l 30, like. p 164, l 22, $eeing. l 31, <I>for</I> as great as, <I>r</I> no greater than. 166, <*>, poope, l 40, Aufractions. p 175, l 26, $peak. p 177, l 13, contra$t. p 184, l 15, <I>for</I> becau$e it cannot, <I>r.</I> why may it not?. p 185, l 6, 7, is <I>Marg</I> Sphere. p 188. l 19 freind. p 190, l 28. <I>for</I> give leave, <I>r.</I> permit. p 19 i, l 17, <I>for</I> on, <I>r.</I> one. l 31. $ee you. p 193, l 21, <I>Vertigo?</I> <*> <*>7, l 1, $ubject. p 200, l 36, <I>dele</I> that, p 201, l 21, pace, l 35, will profe$$e, with. p 203, l 6, <I>dele</I> $o, l 31, dimini$heth, l 33, degrees. p <*>l 14, <I>dele</I> and. p 206, l 19, and 44. <I>Pendula,</I> p 212, l 14, <I>dele</I> it. p 216, l <I>ult. propieres.</I> p 219, 10, <I>for</I> to the$e that, <I>r.</I> $eeing that to the$e. <*>, l 12, what. p 221, l 2, than an. l 222, 6, us take. p 224, 37, <I>for</I> that to <I>r.</I> that for. 225, 25, invented it, 227, 9, that the 229, l 6, <I>dele</I> <*>. l 18, <I>dele</I> and wee, l 45, <I>for</I> From, <I>r.</I> By. p 230, l 14, in the. p. 232. l 41. augre, p 233, l 22, inarticulate. p 23 4, l 20, <I>for</I> were of, <I>r.</I> <*> with. p 233, l 6, the error, <I>l</I> 14, revolve. p 240, l 40, virtue; among$t. l 41, <I>for Mars, &c. r. Mars.</I> l 44, more $uiting. p 224, l 44, <*>ntrary. p 245, l 10, interfere. p 250, l 44, <I>dele</I> being. p 252, l 26, mainteined. p 254, l 7, it was, l 43, uphold. p 258, l 8, $elfe. p 259, l 9, <*>g p 260, l 35, calculations. p 61, l 16, interfering. l 15. if not. p 264, l 11, $ame that you do, <I>line</I> 18, rather than. p 266, <I>l</I> 8, I make. p 267, <*> <I>Bu$chins.</I> p 266, <I>l 16, r. 40 min. pr.</I> p 272, <I>l 20, r.</I> B G, is 42657. p 272. <I>l</I> 29, of the ^{*} 67^{d} 36. p 274, <I>l</I> 32, every. p 275. <I>l</I> 25, halfe <*> <*>7, <I>l</I> 18, and 37, <I>r.</I> B.D. Chord. p 281, <I>l</I> 30. <I>r.</I> 540. -----540 00. p. 285, <I>l.</I> 8. <I>r.</I> been the, <I>l</I> 38, $ay of: <I>l</I> 39, <I>circà.</I> p 286, <I>l</I> 24, than: BP, PB. <*>g bigger than P D. p 287, <I>l</I> 4, $ee, <I>l</I> 33, 582 ---- 100000. p 288, <I>l</I> 21, <I>r.</I> 276, q. p 289 <I>l</I> 32, $pake. p, 290, <I>for</I> p. 274. <I>r.</I> 290. <I>l</I> 12. they kept. <*>1, <I>l</I> 8, uncertain, <I>l</I> 37, Braces, <I>l</I> 42, breadth. p 292, <I>l</I> 6. <I>dele</I> the other ar. p 244. <I>l</I> 23, Peripateticks -----, p 295. <I>l</I> 1. figure, and morning- <*>7, l 11, oppo$ition, <I>marg r. ve$pertine conjunction.</I> p 298. <I>l</I> 23, argument and. p 301, <I>l</I> 1, your, <I>l</I> 30, are yet p. 304, <I>l</I> 9, and allured, <I>marg- <*>rged $oe.</I> p 305, l 27, we leave. p 306, <I>l</I> 25, it ought. p 307, <I>for</I> 330, <I>r.</I> 307. <I>l</I> 10, digre$$ions, <I>l 16, di$cus. l</I> 32, years, together, with, <I>l</I> 34, <*>ial, <I>l</I> 41, alwayes all <I>lucid.</I> p 308, <I>for</I> 394, <I>r.</I> 30<*>. p 309, <I>for</I> 395, <I>r.</I> 309. <I>l</I> in it. <I>l</I> 34, $ole and $ingle p 310, <I>l</I> 11, CD. DE. EF. p 312. <I>l</I> 19, <*>king off, <I>l</I> 19, matters that. p 314, <I>l 8, &</I> 10, Ecliptick. in 316, <I>l</I> 5, nor, <I>l</I> FG: whereupon p 317, <I>l</I> 25, you next p 319, l 7, circuition, <*>, that he hath. p 220, <I>l</I> 31, extreme Terminator. p 124. <I>l</I> 30, Solar Globe, p 322. <I>l</I> 40. tho$e Phy$ical and. p 324, l 20, $aith that, <I>l</I> 29 <*><I>pernicus</I> $aith. p 329, <I>l</I> 39, this $econd. <I>marg.</I> to be the lame. p. 332, <I>l</I> 24, below it. p 333, <I>l</I> 17, $tar, that. p 335, <I>l</I> 24, $tar now, <I>marg. called <*>ll in.</I> p 338, <I>l</I> 41, Into this p 339, <I>l</I> 17, way?. <I>l</I> 24, now, no not for an. <I>l</I> 28, follow thereupon, <I>l</I> 31, point equidi$tant. p 341, <I>l</I> 10, out <*> <I>l</I> 18, yet the force (which. p 342, <I>l</I> 25, Orbe; $o, p 343, <I>l</I> 21, be $een, <I>l</I> 33, Latitudes and, <I>l</I> 38, ours, <I>l</I> 91, greater varieth. p 344, <I>l</I> 39, and, <*>t. p 345. <I>l</I> 23. <I>Cancer</I> and <I>Capricorn,</I> p 347, <I>l</I> 1, feinedly, <I>l</I> 30, Stars are. p 349, <I>l</I> 35, [in Fig. 9.]. p 350, <I>l</I> 42, knew. p 355. <I>l</I> 12, We$t to $t. p 356, <I>l</I> 15, G N. p 360, <I>l</I> 38, circle l K. p 382, <I>l</I> 30, the propen$ion. <I>marg.</I> librated body. p 363, <I>l</I> 3, Experiment, <I>l</I> 7, ba$on $hall. p 36<*> <*> of <I>uilliam, l</I> 17, tbem as, <I>l</I> 38, that, think. p 366, <I>l</I> 14, <I>and</I> 39, $tived. p 367, <I>l.</I> that this, p 370, <I>l</I> 3, do that for natural, <I>l</I> 12, appli- <*>ion of a per$on to. p 372, <I>l ult.</I> tho$e. p 373, <I>l</I> 17, than if, <I>l</I> 39, Launes, Woods, <I>l</I> 43, whither, p 374, <I>l</I> 16, <I>dele</I> $elfe. p 375, l 28. $treight <*>tion is peculiar, <I>l</I> 39, and an. p 376, <I>l</I> 5, (For, <I>l</I> 6, together, <I>l</I> 12, granted ought, p 380, <I>l</I> 1, <I>dele</I> hath. <I>l</I> 5, a mutual, <I>l</I> 6, <I>Indices, l</I> 45, admit. <*>81, <I>l</I> 36, which with, <I>ibid. dele</I> with. p 382, <I>l</I> 3, place), <I>l</I> 13, extremities. p 384, <I>l</I> 3. write of, <I>l</I> 29, SALU. p 385, <I>l</I> 18, more, in <I>l</I> 13, again$t <*> <I>l</I> 24, $wagg, <I>l</I> 37, reply, p 386, l 16, as if it. p. 387, <I>l</I> 33, Water, conteined. p 389, <I>l</I> 43, at the. p 290 <I>l</I> 22, that the diurnall litle. p 391, <*>, grow even, it, <I>l</I> 13, <I>Æquilibrium,</I> but. <I>p</I> 392, <I>l</I> 35, unitedly, equally. p 394, <I>l</I> 16, velocity, when. p 396, l 11, <I>Sardigna</I> p 397, <I>l</I> 38, returns. <*>99, <I>l</I> 30, is free. p 401, <I>l</I> 10, pound you, <I>l</I> 13, and argument. p 402, <I>l</I> 25, alledged that, <I>l</I> 37, interruptions for. p 405, <I>l</I> 19, contact, <I>l</I> 37, at in a Sea only which, <I>l peault.</I> ordinate. p 205, <I>l</I> 38, concern. p 407, <I>l</I> 3, for $peculation and the, <I>l</I> 8, light, with, <I>l</I> 23, at tho$e, <I>l</I> 28, <*>wings, confifteth, <I>l</I> 42, from the, <I>l penult,</I> $ub$tractions that, <I>l ult.</I> maketh to or from. p 48, <I>l</I> 4, proportion in, <I>l</I> 14, le$$er, $o as that, p <*>9, l 12, $wif, p 411, <I>l</I> 24 circles <I>marg, pendnla,</I> p 412, <I>l</I> 27, $ubtend, p 413, <I>l</I> 14, projected, l 24, con$ume, <I>l</I> 93, is, contracted, <I>l</I> 34, of in the <*>441, <I>l</I> 3, differs, <I>l</I> 5, Moon about, <I>l</I> 21, Orb, by. p 415, <I>l</I> 4, do either with. p 416, <I>l</I> 8, tan, <I>l</I> 11, Excentricks, <I>l</I> 13, apparitions, how, <I>l</I> 33, cliptick divided, <I>l</I> 44, on account. p 417, <I>l</I> 43, on which. p 418, <I>l</I> 4, inequalities. p 419, <I>l</I> 12, <I>dele</I> therefore. p 430, <I>l</I> 33, Anomalies, <*>5, tracts. p 421,, <I>l</I> 4, We$tern. p 423, <I>l</I> 41, <I>dele</I> in. p 425, <I>l</I> 16, GALILEO GALILEI p 428, <I>l</I> 32, the$e. p 430, <I>l</I> 27, from its. p 431, <I>marg. <*>rum, ibid. marg. de iis.</I> p 432, <I>l</I> 39, corporeal. p 433, <I>l</I> 26, <I>dele</I> in, <I>l</I> 37, appearance and. p 435, <I>marg. Cœli e$$e, l</I> 27, Spirit of God who pake by them. p 430, <I>l</I> 34, tatling p 440, <I>l</I> 40, propo$e. <I>p</I> 443, <I>l</I> 2, interfere. <I>p</I> 445, <I>l</I> 34, <I>dele</I> with. <I>p</I> 448, <I>l</I> 14, but. <I>p</I> 449, <I>l</I> 27, make reflection. <*>450, <I>marg. & Sanctœ, l</I> 42, <I>$tood $till,. p</I> 451, <I>l</I> 37, her cur$es. <I>p</I> 453, <I>marg. l</I> 13, <I>evoluerit. p</I> 454, <I>l</I> 29, Lap. Your. <I>marg. l</I> 6, <I>prœ$umptores, <*>atis, l</I> 14, <I>auctoritate non tenentur, ad de$cendendum id, quod levi$$ima temeritate, &. p</I> 45<*>, <I>l</I> 27, or at lea$t the. <I>p</I> 456, l 47, in <I>marg In pist ad Polycarpum. p</I> 463, <I>l</I> 17, Stabil try. <I>p</I> 454, <I>l ult.</I> ri$e, <I>p</I> 468, l 25, motion. <I>p</I> 467, <I>l</I> 12, $eeth the, <I>l</I> 26, Sacred, is the Inqu $ition. <I>p</I> 469, <*>, Alma<*>e$t. <I>p</I> 471, <I>l</I> 28, <I>Si quis. p</I> 475, <I>l</I> 12, <I>Credit. l</I> 19, Antlents. p 476, <I>l</I> 9, Deferents, <I>l</I> 33, and in a word. <I>p</I> 477, <I>l</I> 10, <I>Nicetas. p</I> 478, <I>l</I> 1, <*>ly pothe$es), <I>l</I> 5, <I>dece</I> o$, <I>l</I> 19, <I>Galileo Galilei, l</I> 21, Invin$ible, <I>l</I> 23, who. p 481, <I>l</I> 26, or thats incommen urate, <I>l</I> 33, vulgar mode of, <I>p</I> 48<*> <*>7, grieveth. <I>p</I> 485, <I>l</I> 18, $uch that having, <I>p</I> <*>87, <I>l</I> 3, $tay: and. <I>p</I> 488, <I>l</I> 41, Edification, le$t 3ndeciecd in Holy Scripture. <I>p</I> <*>, <I>l</I> <*>5, <*>lterations. <I>p</I> 492, <I>l</I> 30, keeps, <I>marg Æthereal Earth. p</I> 493, <I>l</I> 17, that that. <I>p</I> 495, <I>l</I> 27, frees them. <I>p</I> 500, <I>marg</I> Autho<*>s are not agreed, <*>582, <I>l</I> 30, Holy Gho$t hath.</P> <pb> <head>MATHEMATICAL COLLECTIONS AND TRANSLATIONS: THE SECOND TOME.</head> <head>THE SECOND PART, Containing,</head> <P>D. BENEDICTUS CASTELLUS, <I>his DISCOURSE of the MENSURATION of RUN- NING WATERS.</I></P> <P><I>His Geometrical DEMONSTRATIONS of the Mea$ure of RUNNING WATERS.</I></P> <P><I>I. His LETTERS and CONSIDERATIONS touching the Draining of FENNS, Diver$ions of RIVERS, &c.</I></P> <P><I>V.</I> D. CORSINUS, <I>His RELATION of the $tate of the Inundations, &c. in the Territories of BOLOGNA, and FERRARA.</I></P> <head>By <I>THOMAS SALUSBURY, E$q.</I></head> <head>LONDON, Printed by WILLIAM LEYBOURNE, MDCLXI.</head> <foot>Aaaa</foot> <pb> <head>OF THE MENSURATION OF RUNNING WATERS.</head> <head>An Excellent Piece <I>Written in ITALIAN</I> BY</head> <head>DON BENEDETTO CASTELLI, Abbot of St. <I>BENEDETTO ALOYSIO,</I> and Profe$$our of the Mathematicks to Pope <I>URBAN VIII.</I> in <I>ROME.</I></head> <head>Engli$hed from the Third and be$t Edition, with the addition of a Second Book not before extant:</head> <head>By <I>THOMAS SALUSBURY.</I></head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, 1661.</head> <foot>Aaaa 2</foot> <pb> <head>THE AUTHOURS EPISTLE TO Pope VRBAN VIII.</head> <P>I lay at the Feet of your Ho- line$$e the$e my Con$ide- rations concerning the MENSURATION OF RUNNING WATERS: Wherein if I $hall have $ucceeded, being a matter $o difficult and unhandled by Wri- ters both Ancient Modern, the di$covery of any thing of truth hath been the Effect of Your Holine$$es Command; and if through inability I have mi$$ed the Mark, the $ame <foot>¶5 Com-</foot> <pb> Command will $erve me for an Excu$e with Men of better Judgment, and more e$peci- ally with Your Holine$$e, to whom I humbly pro$trate my $elf, and ki$$e Your Sacred Feet.</P> <P><I>From ROME.</I></P> <P>Your Holine$$es</P> <P><I>Mo$t humble Servant</I></P> <P><I>BENEDETTO.</I></P> <P>A Monk of <I>Ca$$ino.</I></P> <pb> <head>AN ACCOUNT OF THE Authour and Work.</head> <P>DON BENEDETTO CASTELLI, <I>the famous Authour of the$e en$uing Di$cour$es of the</I> Men$uration of Running Waters, <I>is de$cended from the Wor$hipful FAMILY of the</I> GASTELLII, <I>and took his fir$t breath near to the lake THR A- SIMENVS, (where</I> Hanibal <I>gave a fatal overthrow to the</I> Roman <I>Legions) in that $weet and fertile part of happy</I> ITALY, <I>called the</I> Territory <I>of</I> PERUGIA, <I>a branch of the Dukedome of</I> TUSCANY, <I>which at pre$ent $ubmitteth to the Juri$diction of the Church, as being a part of</I> St. PETER'S Patrimony. <I>His Parents, who were more zealous of the good of his Soul than ob$ervant of the Propen$ion of his Genius, dedicated him (according to the Devotion of that Coun- try) to the Service of the Church; and entered him into the Flou- ri$hing Order of Black-Friers, called from the place Moncks of</I> Monte Ca$ino, <I>and from the Founder</I> Benedictines. <I>Na- ture, that She might con$ummate the Profu$ion of her Fa- vours upon him, $ent him into the World in an Age that was $o ennobled and illuminated with Eminent Scholars in all Kinds of Literature, that hardly any Century $ince the Creation can boa$t the like.</I></P> <foot>§. <I>In</I></foot> <pb> <P>§. <I>In particular, the</I> SCIENCES MATHEMATI- CAL <I>had then got that Fame and E$teem in the Learned World, that all men of Spirit or Quality became either Students in, or Patrons of tho$e Sublime Knowledges. On this occa$ion the Curi- o$ity of our</I> AUTHOUR <I>being awakened, his Active Wit could not endure to be any longer confined to the Slavi$h Tuition of Hermetical Pedagogues; but in concurrence with the Genius of the Age, he al$o betook him$elf to tho$e mo$t Generous and Liberal Studies. His helps in this his de$ign were $o many, and $o extraordinary, that had his Inclination been weaker, or his Apprehen$ion le$$er, he could hardly have failed attaining more than a Common Eminency in the$e Sciences. For be$ides the De- luge of Learned and V$eful Books, which the Pre$$e at that time $ent forth from all parts of</I> EUROPE, <I>he had the good Fortune to fall into the Acquaintance, and under the In$truction of the mo$t Demon$trative and mo$t Familiar Man in the World, the Famous</I> GALILEO<I>: who$e $ucce$$e being no le$$e upon this his</I> Pupil <I>than upon the re$t of tho$e Illu$trious and Ingeni- ous Per$ons that re$orted from all parts to $it under his Admi- rable Lectures, he in a $hort time attained to that Name in the Mathematicks, that he was invited to</I> ROME, <I>Complemen- ted, and Preferred by his then Holine$$e the Eighth</I> URBAN, <I>upon his very fir$t Acce$$ion to the</I> Papacy, <I>which was in the Year</I> 1623.</P> <P>§. <I>This Pope being moved with a Paternal Providence for the Concerns of his Subjects in that part of</I> ITALY <I>about</I> BO- LOGNA, FERRARA, <I>and</I> COMMACHIO, <I>ly- ing between the Rivers of</I> PO <I>and</I> RENO, <I>which is part of</I> Lo Stato della Chie$a, <I>or the Church Patrimony, appoints this our</I> CASTELLI <I>in the Year 1625, to accompany the Right Honourable</I> Mon$ignore GORSINI <I>(a mo$t ob$ervant and intelligent per$on in the$e affaires, and at that time Superinten- dent of the General Draines, and Pre$ident of</I> ROMAGNA) <I>in the Grand Vi$itation which he was then ordered to make con- cerning the di$orders occa$ioned by the Waters of tho$e parts.</I></P> <P>§. CASTELLI, <I>having now an Opportunity to employ, yea more, to improve $uch Notions as he had imbued from the Lectures of his Excellent</I> MASTER, <I>falls to his work with all indu$try: and in the time that his Occa$ions detained him in</I> ROMAGNA <I>he perfected the Fir$t Book of this his Di$- cour$e concerning the</I> Men$uration of Running Waters. <I>He con- fe$$eth that he had $ome years before applyed him$elf to this part of Practical Geometry, and from $everal Ob$ervations collected part of that Doctrine which at this time he put into Method, and which had procured him the Repute of $o much Skill that he began</I> <foot><I>to</I></foot> <pb> <I>to be Courted by $undry Princes, and great Prelates. In particu- lar about the beginning of the Year 1623. and before his Invita- tion to</I> ROME <I>he was employed by Prince</I> Ferdinando I, <I>Grand Duke of</I> TUSCANY, <I>to remedy the Di$orders which at that time happened in the Valley of</I> PISA <I>in the Meadows that lye upon the Banks of</I> Serchio <I>and</I> Fiume Morto: <I>and in the pre- $ence of the Grand Duke, Grand Dutche$$e Mother, the Commi$- $ioners of Sewers, and $undry other Per$ons in a few hours he made $o great a progre$$e in that affair, as gave his Mo$t Serene Highne$$e high $atisfaction, and gained him$elf much Honour.</I></P> <P>§. <I>No $ooner had he in his fore-mentioned Voiage to</I> RO- MAGNA <I>(which was but few Moneths after, in the $ame Year) committed his Conceptions to paper, but he communicated them to certain of his Friends. In which number we finde</I> Signo- re Ciampoli <I>Secretary of the Popes Private Affaires; whom in the beginning of the Fir$t Book he gratefully acknowledgeth to have been contributary, in his Pur$e, towards defraying the charge of Experiments, and in his Per$on, towards the debating and compleating of Arguments upon this Subject. Some few years after the Importunity of Friends, and the Zeal he had for the Publique Good prevailed with him to pre$ent the World with his Fir$t Di$cour$e, accompanied with a Treati$e of the Geometrical Demon$trations of his whole Doctrine. What Reception it found with the Judicious mu$t needs be imagined by any one that hath ob$erved how</I> Novelty <I>and</I> Facility <I>in conjunction with</I> Verity <I>make a Charm of irre$i$table Operation.</I></P> <P>§. New <I>it was, for that no man before him had ever attemp- ted to Demon$trate all the three Dimen$ions, to wit, the Length, Breadth and Profundity, of this Fluid and Current Ele- ment. And he detecteth $uch gro$$e Errours in tho$e few that had untertook to write upon the Subject (of which he in$tan- ceth in</I> Frontinus <I>and</I> Fontana, <I>as tho$e that include the rest) and delivereth $uch $ingular and unheard-of Paradoxes (for $o they $ound in Vulgar Eares) as cannot but procure un$peakable delight to his Reader.</I></P> <P>§. Eafie <I>it is likewi$e and</I> True; <I>and that upon $o Familiar Experiments and Manife$t Demon$trations, that I have oft que- $tioned with my $elf which merited the greater wonder, he, for di$covering, or all men that handled the Argument before him for not di$covering a Doctrine of $uch $trange Facility and Infal- libility. But yet as if our Authour de$igned to oblige the whole World to him by $o excellent a</I> Pre$ent, <I>he $elects a Subject that he knew would be carre$$ed by all per$ons of Nobler Souls, upon the accounts afore-named, and by all Mankind in General, as gratifying them in their much adored Idol</I> Utility. <I>And to ren-</I> <foot><I>der</I></foot> <pb> <I>der his Art the more profitable, he reduceth the lofty, and ea$ie-to- be-mi$taken Speculations of the Theory, into certain and facile Directions for Practice; teaching us how to prevent and repaire the Breaches of Seas, and Inundations of Rivers; to draine and recover Fenns and Marches; to divert, conveigh and di- $tribute Waters for the Flowing and Stercoration of Grounds, $trengthening of Fortifications, $erving of Aquaducts, pre$er- ving of Health (by clean$ing Streets, and $cowring Sewers) and maintaining of Commer$e (by defending Bridges, cleering Ri- vers, and opening Ports and Channels) with innumerable other Benefits of the like nature. And, that I may omit no circum$tance that may recommend my Authour, the Fortune of this his Trea- ti$e hath been $uch, that as if he intended a</I> Plus ultra <I>by it, or as if all men de$paired to out-do it, or la$tly, as if</I> CA- STELLI <I>hath been $o great a</I> Ma$ter <I>that none have pre$u- med to take Pencil in hand for the fini$hing of what he</I> Pour- foild, <I>this $mall Tract like the Arabian Phœnix (of which it is $aid</I> Unica $emper Avis) <I>did for $everal years together continue $ingle in the World, till that to verifie it to be truly</I> Phœnician, <I>it renewed its Age by undergoing a $econd Impre$$ion. And as if this did not make out the Immortal vertue of it, it hath had</I> Anno 1660 <I>a third Circulation, and ri$en in this la$t Edition as it were from the Vrne of its Authour; and that $o improved by the Addition of a $econd part, that it promi$eth to perpetuate his Merits to all Po$terity. To be brief, the meer Fame of this Work re$ounded the Honourable Name of</I> CASTELLI <I>in- to all the Corners of</I> Italy, <I>I may $ay of</I> Europe; <I>in$omuch, that, in hopes to reap great benefit by his Art, the re$pective Grandees of the adjacent Countries courted his Judgment and Advice about their Draining of Fenns, Diver$ion of Rivers, Evacuation of Ports, Preventing of Inundations, &c. So that every Summer he made one or more of the$e Journies or Vi$itati- ons. Particularly, the Senate of</I> Venice <I>con$ulted him about their Lake; to whom he delivered his Opinion in</I> May 1641. <I>and up- on farther thoughts he pre$ented them with another Paper of Con- $iderations the</I> 20 December <I>following. Prince</I> LEOPOLDO <I>of</I> TUSCANY <I>likewi$e reque$ted his Advice in the begin- ning of the en$uing year 1642, which occa$ioned his Letter to Father</I> France$co di San Giu$eppe, <I>bearing date</I> February 1, <I>To which</I> Signore Bartolotti <I>oppo$ing, he writes a $econd Let- ter, directed to one of the Commi$$ioners of Sewers, vindicating his former, and refuting</I> Bartolotti, <I>both which I here give you.</I></P> <P>§. <I>The Preferments which his Merits recommended him unto, were fir$t to be Abbot of</I> Ca$$ino, <I>from which he was removed</I> <foot>Anno</foot> <pb> Anno 1640, <I>or thereabouts, unto the Abbey of</I> Santo Benedet- to Aloy$io; <I>and much about the $ame time preferred to the Dig- nity of Chief Mathematician to his grand Patron Pope</I> URBAN VIII. <I>and Publique Profe$$our of Mathematicks in the Vni- ver$ity of</I> ROME.</P> <P>§. <I>Here a Stop was put to the Carier of his Fortunes, and be- ing fuller of Honour than of Years, was by Death, the Importu- nate Intrerupter of Generous De$igns, prevented in doing that farther Good which the World had good rea$on to promi$e it $elf from $o Profound and Indu$trious a Per$onage, leaving many Friends and Di$ciples of all Degrees and Qualities to lament his lo$$e, and honour his Memory.</I></P> <P>§. <I>His $ingular Virtues and Abilities had gained him the Friend$hip of very many; as to in$tance in $ome, he had con- racted $trict Amity with</I> Mon$ignore Maffei Barberino <I>a Floren- tine, Præfect of the Publique Wayes, and afterwards Pope with the Name of</I> URBAN VIII. <I>as was $aid before; with the above-named</I> Mon$ignore Cor$ini <I>Superintendant of the General Draines: with</I> Mon$ignore Piccolomini <I>Arch-Bi$hop of</I> Siena<I>: with Cardinal</I> Serra: <I>with Cardinal</I> Caponi, <I>who hath $tudied much and writ well upon this Subject; and with Cardinal</I> Gae- tano <I>who frequently con$ulted with him in his de$ign of Drain- ing the Fenns of</I> ROMAGNA. <I>Moreover Prince</I> LEO- POLDO, <I>and his Brother the Grand Duke had very great kindne$$e for him; which $peaks no $mall attractions in him, con$idering him as a favourite of the Family of</I> Barberini, <I>be- tween whom and the Hou$e of</I> Medeci <I>there is an inveterate Fewd. Among$t per$ons of a lower Quality he acknowledgeth</I> Signore Ciampoli <I>the Popes Secretary,</I> Sig. Ferrante Ce$arini, Sig. Giovanni Ba$adonna <I>Senator of</I> Venice; <I>and I find menti- oned</I> Sig. Lana, Sig. Albano, Padre Serafino, Pad. France$co de San. Giu$eppe, <I>and many others.</I></P> <P>§. <I>The Works in which he will $urvive to all $ucceeding Ages are fir$t His $olid and $ober Confutation of the Arguments of</I> Signore Lodovico dell Columbo, <I>and</I> Signore Vincentio di Gratia <I>again$t the Tract of</I> Galileo Delle co$e che $tanno $opra Aqua, <I>wherein he vindicates bis $aid</I> Ma$ter <I>with a Gratitude that Tutors very rarely reap from the pains they take in Culti- vating their Pupils. This Apology was fir$t Printed</I> Anno 1615. <I>and was a $econd time publi$hed, as al$o tho$e of his Antago- ni$ts, among$t the Works of</I> GALILEO, <I>$et forth by the Learned</I> Viviani 1656. <I>He hath likewi$e writ $everal other curious Pieces, as I am informed by the mo$t Courteous</I> Carolo Manole$$i <I>of</I> Bologna; <I>among$t others an excellent Treati$e concerning</I> Colours, <I>which he putteth me in hopes to $ee printed</I> <foot><I>very</I></foot> <pb> <I>very $peedily. And la$t of all the$e Di$cour$es and Reflections upon the</I> Men$uration of Running Waters, <I>with the addition of a Second Book, three Epi$tles, and four Con$iderations upon the $ame Argument, which conduce much to Illu$trate his Do- ctrine and Facilitate the Practice of it; and which with a Rela- tion of</I> Mon$ignore Cor$ini, <I>make the $econd part of my Fir$t Tome.</I></P> <P>§. <I>I might here $ally forth into the Citation of $undry Au- thours of Good Account, that have tran$mitted his Character to Po$terity, but $hall confine my $elf to onely two; the one is of his</I> Ma$ter, <I>the other of his</I> Scholar; <I>than whom there can- not be two more competent Judges of his Accompli$hments. To begin with his</I> Ma$ter, <I>the Quick-$ighted, and truly Lyncean</I> GALILEO, <I>who $peaking of his Abilities in A$tronomy $aith</I> <marg><I>(a)</I>Nella continu- atione dell Nun- tio $iderio.</marg> <I>(a)</I> Che la felicità del $uo ingegno non la fà bi$ogno$a dell' o- pera $uo. <I>And again, $ubmitting a certain Demon$tration, which he intended to divulge, to the Judgment of this our Abbot, he</I> <marg><I>(b)</I> Lettera al P. Abbate D. B. Ca- $telli D'Arcetro; li. 3. Decemb. 1639.</marg> <I>writes to him in this manner: (b)</I> Que$to lo comunico a V. S. per lettera prima che ad alcun altro, con attenderne principal- mente il parer $uo, e doppo quello de' no$tri Amici di$co$ti, conpen$iero d' inviarne poi altre Copie ad altri Amici d' Italia, e di Francia, quando io ne venga da lei con$igliato: e qui pre- gandola a farci parte d' alcuna delle $ue peregrine $peculationi; con $inceri$$imo affetto, &c. <I>And the mo$t acute Mathematician</I> Signore Evangeli$ta Terricelli, <I>late Profe$$our to the Grand Duke in immediate Succe$$ion after</I> GALILEO, <I>maketh this</I> <marg><I>(c)</I> De Motu A- quarum. Lib. 2. Prop. 37. p. 191.</marg> <I>Honourable and Grateful Mention of him, and his Book: (c)</I> O- mitto magnum illum nutantis Maris motum; Prætereo etiam omnem Fluminum, Aquarumque Currentium tum men$urum, tum u$um, quarum omnis doctrina reperta primum fuit ab Abbate BENEDICTO CASTELLIO Preceptore meo. Scrip$it ille Scientiam $uam, & illam non $olum demon$tra- tione, verum etiam opere confirmavit, maxima cum Princi- pum & populorum utilitatate, majore cum admiratione Phylo- $ophorum. Extat illius Liber, vere aureus.</P> <P>§. <I>I have onely two particulars more to offer the Engli$h Rea- der: The one concerns the Book, and it is this, That after the general Aprobation it hath had in</I> Italy, <I>I cannot but think it de$erveth the $ame Civil Entertainment with us, in regard that it cometh with no le$$e</I> Novelty, Facility, Verity, and Utility <I>to us than to tho$e whom the Authour favoured with the Original. Our Rivers and Sewers through Publique Di$tractions and Pri- vate Incroachments are in great di$order, as tho$e Channels for iu$tance which formerly were Navigable unto the very Walls of</I> <foot>York</foot> <pb> York <I>and</I> Salisbury, <I>&c: Our Ports are choaked and ob$tructed by Shelfes and Setlements: Our Fenns do in a great part lie wa$te and unimproved: Now all the$e may be (and, as I find by the Confe$$ion of $ome who$e Practi$es upon the Copy of the Fir$t Book onely of our Authour hath got them both Money and Repu- tation, in part have been) remedied by the Ways and Means he here $ets down. The truth is the Argument hath been pa$t over with an Vniver$al Silence; $o that to this day I have not $een any thing that hath been written Demon$tratively and with Ma- thematical Certainty concerning the $ame, $ave onely what this Learned Prelate hath delivered of his Own Invention in the$e Treati$es: who yet hath $o fully and plainly handled the Whole Doctrine, that I may affirm his Work to be every way ab$olute. It mu$t be confe$t the Demon$tration of the Second Propo$ition of the Second Book did not well plea$e the Authour, and had he lived he would have $upplyed that defect, but being prevented by Death, the Reader mu$t content him$elf with the Mechanical Proof that he giveth you of the truth of $o Excellent a Con- clu$ion.</I></P> <P>§. <I>The other particular that I am to offer is, that out of my de- $ire to contribute what lyeth in me to the compleating of this Piece for Engli$h Practice, I have exeeded my promi$e not onely in gi- ving you the Second and following Books which were not extant at the time of tendring my Overtures, but al$o in that I have added a Map or Plat of all the Rivers, Lakes, Fenns, &c. mentioned thorow out the Work. And if I have not kept touch in point of Time, let it be con$idered that I am the Tran$lator and not the Printer. To conclude, according to your acceptance of the$e my endeavours, you may expect $ome other Tracts of no le$$e Profit and Delight.</I> Farewell.</P> <head><I>T. S.</I></head> <pb> <head>ERRATA of the <I>$econd</I> PART of the <I>fir$t</I> TOME.</head> <P>In PREFACE, I cad <I>Ferdinando II.</I> ibid. <I>l' Aqua.</I></P> <P>PAGE 2. LINE 26, <I>for</I> mu$t <I>read</I> much. P. 3. l. 22, <I>r.</I> and let. l. 25. <I>r.</I> water, from l. 41. <I>r.</I> Tappe, (<I>as every where el$e).</I> Page 4. l. 18. <I>r</I> cords. Page 6. l. 9. <I>r.</I> acquire, or. Page 9. l. 1. <I>r.</I> irreperable. P. 10. l. 13. <I>r.</I> di$$imboguement. <I>For</I> Page 17. <I>r.</I> P. 15. P. 15. l. 27, <I>r.</I> in. l. 36, <I>r.</I> is as. l. 38, <I>r. Panaro.</I> P. 17. l. 12, <I>Giulio.</I> l. 17. <I>r.</I> Mea$urers. l. 25, <I>r.</I> mea$ured it,. <I>r.</I> nece$$arily. P. 23. l. 19. <I>r.</I> for help. <I>for</I> Page 31. <I>r.</I> P. 32. P. 24. l. 14, <I>r.</I> to. l. 17, <I>r.</I> namly, of the. l. 23, <I>r.</I> ea$ie. P. 25. l. 38. <I>r.</I> Cock. p. 29. l. 7. <I>r.</I> la$ted,. p. 31. l. 32. <I>r.</I> Soe. p. 41. l. 20. <I>r.</I> to the line. p. 48. l. 19. <I>r.</I> us the ^{*}. id. <I>Figure fal$e</I> p. 52. l. 30, and 31. <I>for</I> Theorem <I>r.</I> Propo$ition. p. 53. l. 29. <I>r.</I> again. p. 57. l. 19, <I>r.</I> $ame if. l. 44. <I>r.</I> bodily. p. 58. l. 9, <I>r.</I> gathering. l. 40. <I>omit</I>;. p. 60. l. 2. <I>omit,</I> if. p. 65. l. 1. <I>r.</I> tide <I>dele</I>;. p. 66. l. 35. <I>r.</I> Stoppage of. p. 68. l. 12, <I>for</I> Lords the <I>r.</I> Lords. l. <I>ult. for</I> they <I>r.</I> it. p. 69. l. 14. <I>r.</I> to one. <I>id.</I> carried <I>dele</I> to. p. 71. l. 20, <I>r.</I> and that. l. 25, <I>r.</I> Braces; it. l. 29. <I>r.</I> Braces. l. 44, <I>r.</I> the <I>Brent.</I> p. 72. l. 23. <I>r. Serene Highne$$e.</I> p. 73. l. 24, <I>r.</I> deliberation:. l. 26, <I>for</I> $umme <I>r.</I> Moddel. l. 40. <I>r.</I> Months. p. 79. l. 18. <I>r.</I> that into. p. 82. l. 22. <I>dele</I>;. p. 85. l. 9, 10. <I>dele</I> a free drame. p. 88. l. 5. <I>r.</I> Palmes. p. 89. l. 8. <I>r.</I> Princes. p. 92. l. 3. <I>r.</I> Di$- cour$es. p. 93. l. 31. <I>r.</I> Tautologie. p. 94. l. 9. <I>r.</I> miracle;. p. 97. l. 13. <I>r,</I> weighty. p. 101. l. 21. <I>r.</I> Marrara. p. 107. l. 28, <I>r.</I> Patrimony. l. 40, <I>r.</I> above. p. 111. l. 16. <I>r.</I> $aid. <I>For</I> p. 432. <I>r.</I> p. 114. p. 114. l. 35. <I>r.</I> of 200, l. 41. <I>r.</I> clo$ed. p. 115. l. 29. <I>r.</I> con$tant;.</P> <p n=>1</p> <head>OF THE MENSURATION OF Running Waters.</head> <head><I>LIB.</I> I.</head> <P>What, and of how great moment the confi- deration of <I>Motion</I> is in natural things, is $o manife$t, that the Prince of <I>Peri- pateticks</I> pronounced that in his Schools now much u$ed Sentence: <I>Ignorato mo- tu, ignoratur natura.</I> Thence it is that true Philo$ophers have $o travailed in the contemplation of the Cele$tial motions, and in the $peculation of the motions of Animals, that they have arrived to a wonderful height and $ub- limity of under$tanding. Under the $ame Science of <I>Motion</I> is comprehended all that which is written by Mechanitians con- cerning Engines moving of them$elves, <I>Machins</I> moving by the force of Air, and tho$e which $erve to move weights and im- men$e magnitudes with $mall force. There appertaineth to the Science of <I>Motion</I> all that which hath been written of the alteration not onely of Bodies, but of our Minds al$o; and in $um, this ample matter of <I>Motion</I> is $o extended and di- lated, that there are few things which fall under mans no- tice, which are not conjoyned with <I>Motion,</I> or at lea$t de- pending thereupon, or to the knowledge thereof directed; and of almo$t every of them, there hath been written and compo$ed by $ublime wits, learned Treati$es and In$tructions. <foot>Bbbb And</foot> <p n=>2</p> And becau$e that in the years pa$t I had occa$ion by Order of our Lord Pope <I>Vrban</I> 8. to apply my thoughts to the motion of the Waters of Rivers, (a matter difficult, mo$t important, and little handled by others) having concerning the $ame ob$erved $ome particulars not well ob$erved, or con$idered till now, but of great moment both in publick and private affairs; I have thought good to publi$h them, to the end that ingenious $pirits might have occa$ion to di$cu$$e more exactly then hitherto hath been done, $o nece$$ary and profitable a matter, and to $upply al$o my defects in this $hort and difficult Tractate. Difficult I $ay, for the truth is, the$e knowledges, though of things next our $en$es, are $ometimes more ab$truce and hidden, then the knowledge of things more remote; and much better, and with greater exqui$it- ne$s are known the motions of the Planets, and Periods of the Stars, than tho$e of Rivers and Seas: As that $ingular light of Philo$ophie of our times, and my Ma$ter <I>Signore Galileo Galilei</I> wi$ely ob$erveth in his Book concerning the Solar $pots. And to proceed with a due order in Sciences, I will take $ome $uppo- $itions and cognitions $ufficiently clear; from which I will after- wards proceed to the deducing of the principal conclu$ions. But to the end that what I have written at the end of this di$cour$e in a demon$trative and Geometrical method, may al$o be under- $tood of tho$e which never have applyed their thoughts to the $tudy of Geometry; I have endeavoured to explain my conceit by an example, and with the con$ideration of the natural things them$elves, mu$t after the $ame order in which I began to doubt in this matter; and have placed this particular Treati$e here in the beginning, adverting neverthele$s, that he who de$ires more full and ab$olute $olidity of Rea$ons, may overpa$s this prefatory di$cour$e, and onely con$ider what is treated of in the demon$tra- tions placed towards the end, and return afterwards to the con$i- deration of the things collected in the <I>Corollaries</I> and Appendices; which demon$trations notwith$tanding, may be pretermitted by him that hath not $een at lea$t the fir$t $ix Books of the Elements of Euclid; $o that he diligently ob$erveth that which fol- loweth.</P> <P>I $ay therefore, that having in times pa$t, on divers occa$i- ons heard $peak of the mea$ures of the waters of Rivers, and Fountains, $aying, $uch a River is two or three thou$and feet of water; $uch a $pring-water is twenty, thirty, or forty inches, <I>&c.</I> Although in $uch manner I have found all to treat thereof in word and writing, without variety, and as we are wont to $ay, <I>con$tanti $ermone,</I> yea even Arti$ts and Ingeneers, as if it were a thing that admitted not of any doubt, yet how$oever I re- mained $till infolded in $uch an ob$curity, that I well knew I un- <foot>der$tood</foot> <p n=>3</p> der$tood nothing at all, of that which others pretended full and clearly to under$tand. And my doubt aro$e from my frequent ob$ervation of many Trenches and Channels, which carry water to turn Mills, in which Trenches, and Channels, the water being mea$ured, was found pretty deep; but if afterwards the $ame water was mea$ured in the fall it made to turn the Wheel of the Mill, it was much le$$e, not amounting often to the tenth part, nor $ometimes to the twentieth, in$omuch, that the $ame running water came to be one while more, another while le$s in mea$ure, in divers parts of its Channel; and for that rea$on this vulgar manner of mea$uring running Waters, as indeterminate and uncertain, was by me ju$tly $u$pected, the mea$ure being to be de- terminate, and the $ame. And here I freely confe$$e that I had fin- gular help to re$olve this difficulty from the excellent & accurate way of di$cour$ing, as in allother matters, $o al$o in this, of the Right Honourable and Truly Noble Signior <I>Ciampoli,</I> Secretary of the Popes $ecret affairs. Who moreover, not $paring $or the co$ts of the $ame, generou$ly gave me occa$ion a few years pa$t to try by exact experiments that which pa$t concerning this particular. And to explain all more clearly with an example; we $uppo$e a Ve$$el filled with Water, as for in$tance a Butt, which is kept full, though $till water runneth out, and the Water run out by two Taps equal of bigne$$e, one put in the bottom of the Ve$$el, and the other in the upper part; it is manife$t that in the time wherein from the upper part $hall i$$ue a determinate mea$ure of water $rom the inferiour part there $hall i$$ue four, five, and many more of the $ame mea$ures, according to the difference of the height of the Taps, and the di$tance of the upper Tap from the Superfici- es and level of the water of the Ve$$el: and all this will alwayes follow, though, as hath been $aid, the Taps be equal, and the water in di$charging keep the $aid Taps alwayes full. Where fir$t we note, that, although the mea$ure of the Taps be equal, never- thele$$e there i$$ueth from them in equal times unequal quantities of water, And if we $hould more attentively con$ider this bu$i- ne$$e, we $hould find, that the water by the lower Tap, run- neth and pa$$eth with much greater velocity, then it doth by the upper, whatever is the rea$on. If therefore we would have $uch a quantity of Water di$charge from the upper tap, as would di$charge from the neather in the $ame time, it is plain, that either the upper Taps mu$t be multiplyed in $uch $ort, that $o many more Taps in number be placed above than below, as the neather tap $hall be more $wift than the upper, or the upper Tap made $o much bigger than the nether, by how much that be- neath $hall be more $wift than that above; and $o then in equal times, the $ame quantity of Water $hall di$charge from the upper, as doth from the neather part.</P> <foot>Bbbb 2 I</foot> <p n=>4</p> <P>I will declare my $elf by another example. If we $hould ima- gine, that two cords or lines of equal thickne$s, be drawn through two holes of equal bore; but $o that the fir$t pa$s with quadruple velocity to the $econd: It is manife$t, that if in a determinate time, we $hall by the fir$t bore have drawn four Ells of the line, in the $ame time, by the $econd hole we $hall have drawn but one Ell of cord onely; and if by the fir$t there pa$$e twelve Ells, then through the $econd there $hall pa$$e onely three Ells; and in $hort the quantity of cord $hall have the $ame proportion to the cord, that the volocity hath to the velocity. And therefore we de$iring to compen$ate the tardity of the $econd cord, and main- taining the $ame tardity to draw through the $econd hole as much cord as through the fir$t, it will be nece$$ary to draw through the $econd bore four ends of cord; $o that the thickne$s of all the cords by the $econd hole, have the $ame proportion to the thick- ne$s of the cord which pa$$eth onely by the fir$t, as the velocity of the cord by the fir$t hole hath reciprocally to the velocity of the codrs by the $econd hole. And thus its clear, that when there is drawn through two holes equal quantity of cords in equal time, but with unequal velocity, it will be nece$$ary, that the thickne$s of all the four cords $hall have the $ame reciprocal proportion to the thickne$s of the $wifter cord, that the velo- city of the $wifter cord hath to the velocity of the $lower. The which is verified likewi$e in the fluid Element of Water.</P> <P>And to the end that this principal fundamental be well under- $tood, I will al$o note a certain ob$ervation made my me in the Art of Wyer-drawing, or $pinning Gold, Silver, Bra$s, and Iron, and it is this; That $uch Artificers de$iring more and more to di$gro$$e and $ubtillize the $aid Metals, having would about a <I>R</I>ocket or Barrel, the thread of the Metal, they place the Roc- ket in a frame upon a $tedfa$t Axis, in $uch $ort that the Rocket may turn about in it $elf; then making one end of the thread to pa$$e by force through a Plate of Steel pierced with divers holes, greater and le$$er, as need requireth, fa$tning the $ame end of the thread to another Rocket, they wind up the thread, which pa$- $ing through a bore le$s than the thickne$$e of the thread, is of force con$trained to di$gro$$e and $ubtillize. Now that which is inten$ly to be ob$erved in this bu$ine$s, is this, That the parts of the thread before the hole, are of $uch a thickne$$e, but the parts of the $ame thread after it is pa$$ed the hole, are of a le$$er thick- ne$$e: and yet neverthele$$e the ma$$e and weight of the thread which is drawn forth, is ever equal to the ma$$e and weight of the thread which is winded up. But if we $hould well con$ider the mat- ter, we $hould finde, that the thicker the thread before the hole is, than the thread pa$$ed the hole, the greater reciprocally is the <foot>velocity</foot> <p n=>5</p> velocity of the parts of the thread pa$$ed the hole, than the volo- city of the parts before the hole: In$omuch that if <I>verbi gratia</I> the thickne$$e of the thread before the hole, were double to the thickne$$e after the hole, in $uch ca$e the velocity of the parts of the thread pa$$ed the hole, $hould be double to the velocity of the parts of the thread before the hole; and thus the thickne$$e compen$ates the velocity, and the velocity compen$ates the thick- ne$$e. So that the $ame occurreth in the $olid Metals of Gold, Silver, Bra$s, Iron, &c. that eveneth al$o in the fluid Element of Water, and other liquids, namely, That the velocity beareth the $ame proportion to the velocity, that the thickne$$e of the Me- tal, or Water, hath to the thickne$$e.</P> <P>And therefore granting this di$cour$e, we may $ay, that as of- ten as two Taps with different velocity di$charge equal quanti- ties of Water in equal times, it will be nece$$ary that the Tap le$$e $wift be $o much greater, and larger, than the Tap more $wift, by how much the $wifter $uperates in velocity the $lower; and to pronounce the Propo$ition in more proper terms, we $ay; That if two Taps of unequal velocity, di$charge in equal times equal quantities of Water, the greatne$$e of the fir$t $hall be to the greatne$$e of the $econd, in reciprocal proportion, as the ve- locity of the $econd to the velocity of the fir$t. As for example, if the fir$t Tap $hall be ten times $wifter than the $econd Tap, it will be nece$$ary, that the $econd be ten times bigger and larger than the fir$t; and in $uch ca$e the Taps $hall di$charge equall quantities of water in equal times; and this is the principal and mo$t important point, which ought to be kept alwayes in minde, for that on it well under$tood depend many things profitable, and worthy of our knowledge.</P> <P>Now applying all that hath been $aid neerer to our purpo$e, I con$ider, that it being mo$t true, that in divers parts of the $ame River or Current of running water, there doth always pa$$e equal quantity of Water in equal time (which thing is al$o demon- $trated in out fir$t Propo$ition) and it being al$o true, that in di- vers parts the $ame River may have various and different veloci- ty; it follows of nece$$ary con$equence, that where the River hath le$$e velocity, it $hall be of greater mea$ure, and in tho$e parts, in which it hath greater velocity, it $hall be of le$$e mea- $ure; and in $um, the velocity of $everal parts of the $aid River, $hall have eternally reciprocall and like proportion with their mea$ures. This principle and fundamental well e$tabli$h- ed, that the $ame Current of Water changeth mea$ure, accor- ding to its varying of velocity; that is, le$$ening the mea$ure, when the velocity encrea$eth, and encrea$ing the mea$ure, when the velocity decrea$eth; I pa$$e to the con$ideration of many <foot>par-</foot> <p n=>6</p> particular accidents in this admirable matter, and all depending on this $ole Propo$ition, the $en$e of which I have oft repeated, that it might be well under$tood.</P> <head><I>COROLLARIE</I> I.</head> <P>And fir$t, we hence conclude, that the $ame Streams of a Torrent, namely, tho$e $treams which carry equal quantity of Water in equal times, make not the $ame depths or mea$ures in the River, in which they enter, unle$$e when in the entrance in- to the River they acquire; or to $ay better, keep the $ame velo- city; becau$e if the velocicities acquired in the River $hall be different, al$o the mea$ures $hall be diver$e; and con$equently the depths, as is demon$trated.</P> <head><I>COROLLARIE</I> II.</head> <P>And becau$e $ucce$$ively, as the River is more and more full, it is con$tituted ordinarily in greater & greater velocity: hence it is that the $ame $treams of the Torrent, that enter into the Ri- ver, make le$$e and le$$e depths, as the River grows more and more full; $ince that al$o the Waters of the Torrent being en- tered into the River, go acquiring greater and greater velocities, and therefore dimini$h in mea$ure and height.</P> <head><I>COROLLARIE</I> III.</head> <P>We ob$erve al$o, that while the main River is $hallow, if there fall but a gentle rain, it $uddenly much increa$eth and ri$eth; but when the River is already $welled, though there fall again a- nother new violent $hower, yet it increa$eth not at the $ame rate as before, proportionably to the rain which fell: which thing we may affirm particularly to depend on this, that in the fir$t ca$e, while the River is low, it is found al$o very $low, and there- fore the little water which entereth into it, pa$$eth and runs with little velocity, and con$equently occupieth a great mea$ure: But when the River is once augmented, by new water being al$o made more $wift, it cau$eth the great Flood of water which fal- leth, to bear a le$$e mea$ure, and not to make $uch a depth.</P> <head><I>COROLLARIE</I> IV.</head> <P>From the things demon$trated is manife$t al$o, that whil$t a Torrent entereth into a River, at the time of Ebbe, then the Torrent moveth with $uch a certain velocity, what ever it be, <foot>pa$$ing</foot> <p n=>7</p> pa$$ing by its extreame$t parts, wherewith it communicateth with the River; in which parts, the Torrent being mea$ured, $hall have $uch a certain mea$ure: but the River $welling and ri$ing, al$o tho$e parts of the Torrent augment in greatne$$e and mea- $ure, though the Torrent, in that in$tant, di$-imbogue no more water than it did before: $o that the River being $welled, we are to con$ider two mouths of the $ame Torrent, one le$$e be- fore the ri$ing, the other greater after the ri$ing, which mouths di$charge equal quantities of water in equal times; therefore the velocity by the le$$er mouth $hall be greater than the velocity by the greater mouth; and thus the Torrent $hall be retarded from its ordinary cour$e.</P> <head><I>COROLLARIE</I> V.</head> <P>From which operation of Nature proceedeth another effect worthy of con$ideration; and it is, that the cour$e of the water retarding, as hath been $aid in tho$e ultimate parts of the Tor- rent, if it $hall happen that the Torrent grow torbid and mud- dy, and its $treame be retarded in $uch a degree, that it is not able to carry away tho$e minute grains of Earth, which com- po$e the muddine$$e; in this ca$e the Torrent $hall clear away the mud, and carry away the Sand at the bottome of its own Chanel, in the extream parts of its mouth, which rai$ed and voided Sand, $hall again afterwards be carried away, when the River abating, the Torrent $hall return to move with its primitive velocity.</P> <head><I>COROLLARIE</I> VI.</head> <P>Whil$t it is demon$trated, that the $ame water hath different mea$ures in its Chanel or cour$e, according as it varieth in velocity; $o that the mea$ure of the water is alwayes greater, where the velocity is le$$er; and on the contrary, the mea$ure le$$er, where the velocity is greater: from hence we may mo$t ele- gantly render the rea$on of the u$ual Proverb, <I>Take heed of the $till waters:</I> For that if we con$ider the $elf $ame water of a River in tho$e parts, wherein it is le$s $wift, and thence called <I>$till</I> or <I>$mooth</I> water, it $hall be, of nece$$ity, of greater mea$ure than in tho$e parts, in which it is more $wift, and therefore ordi- narily $hall be al$o more deep and dangerous for pa$$engers; whence it is well $aid, <I>Take heed of the $till Waters</I>; and this $aying hath been $ince applied to things moral.</P> <foot><I>CO.</I></foot> <p n=>8</p> <head><I>COROLLARIE</I> VII.</head> <P>Likewi$e, from the things demon$trated may be concluded, that the windes, which $top a <I>R</I>iver, and blowing again$t the Current, retard its cour$e and ordinary velocity $hall nece$$arily amplifie the mea$ure of the $ame River, and con$equently $hall be, in great part, cau$es; or we may $ay, potent con-cau$es of making the extraordinary inundations which Rivers u$e to make. And its mo$t certain, that as often as a $trong and continual wind $hall blow again$t the Current of a River, and $hall reduce the water of the River to $uch tardity of motion, that in the time wherein before it run five miles, it now moveth but one, $uch a River will increa$e to five times the mea$ure, though there $hould not be added any other quantity of water; which thing indeed hath in it $omething of $trange, but it is mo$t certain, for that look what proportion the waters velocity before the winde, hath to the velocity after the winde, and $uch reciprocally is the mea- $ure of the $ame water after the winde, to the mea$ure before the winde; and becau$e it hath been $uppo$ed in our ca$e that the velocity is dimini$hed to a fifth part, therefore the mea$ure $hall be increa$ed five times more than that, which it was before.</P> <head><I>COROLLARIE</I> VIII.</head> <P>We have al$o probable the cau$e of the inundations of <I>Tyber,</I> which befel at <I>Rome,</I> in the time of <I>Alexander</I> the Sixth, & of <I>Clement</I> the Seventh; which innundations came in a $erene time, and without great thaws of the Snows; which therefore much puzzled the wits of tho$e times. But we may with much pro- bability affirm, That the River ro$e to $uch a height and excre$- cence, by the retardation of the Waters dependant on the boi$trous and con$tant Winds, that blew in tho$e times, as is no- red in the memorials.</P> <head><I>COROLLARIE.</I> IX.</head> <P>It being mo$t manife$t, that by the great abundance of Water the Torrents may increa$e, and of them$elves alone exorbitantly $well the River; and having demon$trated that al$o without new Water, but onely by the notable retardment the River ri$eth and increa$eth in mea$ure, in proportion as the velocity decrea$eth: hence it is apparent, that each of the$e cau$es being able of it $elf, and $eparate from the other to $well the River; when it $hall happen that both the$e two cau$es con$pire the augmentation of <foot>the</foot> <p n=>9</p> the River, in $uch a ca$e there mu$t follow very great and irre- pable innundations.</P> <head><I>COROLLARIE</I> X.</head> <P>From what hath been demon$trated, we may with facility re- $olve the doubt which hath troubled, and $till po$eth the mo$t diligent, but incautelous ob$ervers of Rivers, who mea$uring the Streams and Torrents which fall into another River; as tho$e for in$tance, which enter into the <I>Po,</I> or tho$e which fall into <I>Ti- ber</I>; and having $ummed the total of the$e mea$ures, and con- ferring the mea$ures of the Rivers and Brooks, which fall into <I>Tiber,</I> with the mea$ure of <I>Tiber,</I> and the mea$ures of tho$e which di$imbogue into <I>Po,</I> with the mea$ure of <I>Po,</I> they find them not equal, as, it $eems to them, they ought to be, and this is becau$e they have not well noted the mo$t important point of the varia- tion of velocity, and how that it is the mo$t potent cau$e of won- derfully altering the mea$ures of running Waters; but we mo$t facilly re$olving the doubt, may $ay that the$e Waters dimini$h the mea$ure, being once entered the principal Channel, becau$e they increa$e in velocity.</P> <head><I>COROLLARIE</I> XI.</head> <P>Through the ignorance of the force of the velocity of the Wa- ter, in altering its mea$ure, & augmenting it when the velocity dimini$heth; and dimini$hing it when the velocity augmenteth: The Architect <I>Giovanni Fontana,</I> endeavoured to mea$ure, and and to cau$e to be mea$ured by his Nephew, all the Brooks and Rivers which di$charged their Waters into <I>Tiber,</I> at the time of the Innundation; which happened at <I>Rome</I> in the year 1598, and publi$hed a $mall Treati$e thereof, wherein he $ummeth up the mea$ures of the extraordinary Water which fell into <I>Tiber,</I> and made account that it was about five hundred Ells more than ordinary; and in the end of that Treati$e concludeth, that to re- move the Innundation wholly from <I>Rome,</I> it would be nece$$ary to make two other Channels, equal to that at pre$ent, and that le$$e would not $uffice; and finding afterwards that the whole Stream pa$$ed under the Bridge <I>Quattro-Capi,</I> (the Arch where- of is of a far le$s mea$ure then five hundred Ells) concludeth, that under the $aid Bridge pa$t a hundred fifty one Ells of Water compre$$ed, (I have $et down the preci$e term of compre$t Wa- ter, written by <I>Fontana</I>) wherein I finde many errors.</P> <P>The fir$t of which is to think that the mea$ures of the$e Wa- ters compre$$ed in the Channels of tho$e Brooks and Rivers, <foot>Ccco $hould</foot> <p n=>10</p> $hould maintain them$elves the $ame in <I>Tiber,</I> which by his leave, is mo$t fal$e, when ever tho$e waters reduced into <I>Tiber,</I> retain not the $ame velocity which they had in the place in which <I>Fon- tana</I> and his Nephew mea$ured them: And all this is manife$t from the things which we have above explained; for, if the Wa- ters reduced into <I>Tiber</I> increa$e in velocity, they decrea$e in mea- $ure; and if they decrea$e in velocity, they increa$e in mea- $ure.</P> <P>Secondly, I con$ider that the mea$ures of tho$e Brooks and Rivers, which enter into <I>Tiber</I> at the time of Innundation, are not between them$elves really the $ame, when their velocities are not equal, though they have the $ame names of Ells and Feet; for that its po$$ible that a di$inboguement of ten Ells requadrated (to $peak in the phra$e of <I>Fontana</I>) of one of tho$e Brooks, might di$charge into <I>Tiber</I> at the time of Innundation, four, ten, and twenty times le$s Water, than another mouth equal to the fir$t in greatne$s, as would occur when the fir$t mouth were four, ten, or twenty times le$s $wift than the $econd. Whereupon, whil$t <I>Fontana</I> $ummes up the Ells and Feet of the mea$ures of tho$e Brooks and Rivers into a total aggregate, he commits the $ame error with him, which would add into one $umme diver$e moneys of diver$e values, and diver$e places, but that had the $ame name; as if one $hould $ay ten Crowns of <I>Roman</I> money, four Crowns of Gold, thirteen Crowns of <I>Florence,</I> five Growns of <I>Venice,</I> and eight Crowns of <I>Mantua,</I> $hould make the $ame $umme with forty Crowns of Gold, or forty Crowns of <I>Mantua.</I></P> <P>Thirdly, It might happen that $ome River or Current in the parts nearer <I>Rome,</I> in the time of its flowing, did not $end forth more Water than ordinary; and however, its a thing very clear, that whil$t the $tream came from the $uperior parts, that $ame Brook or River would be augmented in mea$ure, as hath been noted in the fourth <I>Corollary</I>; in $uch $ort, that <I>Fontana</I> might have inculcated, and noted that $ame River or Current as con- curring to the Innundation, although it were therein altogether unconcerned.</P> <P>Moreover, in the fourth place we mu$t note, That it might $o fall out, that $uch a River not onely was unintere$$ed in the Innundation, though augmented in mea$ure, but it might I $ay happen, that it was in$trumental to the a$$waging the Innunda- tion, by augmenting in the mea$ure of its own Channel; which matter is $ufficiently evident; for if it be $uppo$ed that the Ri- ver in the time of flood, had not had of it $elf, and from its pro- per $prings more Water than ordinary, its a thing certain, that the Water of <I>Tiber</I> ri$ing and increa$ing; al$o that River, to le- vel it $elf with the Water of <I>Tiber,</I> would have retained $ome of <foot>its</foot> <p n=>11</p> its Waters in its own Chanel, without di$charging them into <I>Ty- ber,</I> or el$e would have ingorged and $wallowed (if I may $o $ay) $ome of the water of <I>Tyber</I>; and in this ca$e, at the time of In- undation, le$$e abundance of water would have come to <I>Rome,</I> and yet neverthele$$e the mea$ure of that River would have been increa$ed.</P> <P>Fifthly, <I>Fontana</I> deceiveth him$elf, when he concludeth, that to remove the Inundation from <I>Rome,</I> it would be nece$$ary to make two other Chanels of Rivers, that were as large as that, which is the pre$ent one, and that le$s would not $uffice, which, I $ay, is a fallacy: and to convince him ea$ily of his errour, it $ufficeth to $ay, that all the Streams being pa$$ed under the Bridge <I>Quattro-Capi,</I> as he him$elf atte$ts, a Channel would $uffice on- ly of the capacity of the $aid Bridge, provided that the water there might run with the $ame velocity, as it did under the Bridge at the time of Inundation; and on the contrary, twenty Cur- rents of capacity equal to the pre$ent one, would not $uffice, if the water $hould run with twenty times le$s velocity, than it made at the time of the Inundation.</P> <P>Sixthly, to me it $eemeth a great weakne$$e to $ay, that there $hould pa$$e under the Bridge <I>Quattro-Capi,</I> an hundred fifty one ells of water compre$$ed; for that I do not under$tand that wa- ter is like Cotton or Wool, which matters may be pre$t and trod, as it happeneth al$o to the air, which receiveth compre$$ion in $uch $ort, that after that in $ome certain place a quantity of air $hall be reduced to its natural con$titution; and having taken up all the $aid place, yet neverthele$$e compre$$ing the fir$t Air with force and violence, it is reduced into far le$s room, and will admit four or $ix times as much air, as before, as is experimen- tally ^{*} $een in the Wind-Gun, invented in our dayes by <I>M. Vin,</I> <marg>* And as is at large demon$trated by that mo$t excel- lent and lonour- able per$onage Mr. <I>Botle</I> in the indu- $trious experiment of his Pneumatical Engine.</marg> <I>cenzo Vincenti</I> of <I>Vrbin,</I> which property of the Air of admit- ting conden$ation, is al$o $een in the portable Fountains of the $ame <I>M. Vincenzo:</I> which Fountains $pirt the Water on high, by force of the Air compre$$ed, which whil$t it $eeks to reduce its $elf to its natural con$titution, in the dilation cau$eth that vi- olence. But the water can never, for any thing I know, crowd, or pre$s $o, as that if before the compre$$ion it held or po$$e$t a place, being in its natural con$titution, I believe not, I $ay, that it is po$$ible, by pre$$ing and crowding to make it po$$e$s le$s room, for if it were po$$ible to compre$s the Water, and make it to oc- cupy a le$s place, it would thence follow, that two Ve$$els of e- qual mea$ure, but of unequal height, $hould be of unequal capa- city, and that $hould hold more water which was higher; al$o a Cylinder, or other Ve$$el more high than broad, would containe more water erected, than being laid along; for that being erect- <foot>Cccc 2 ed,</foot> <p n=>12</p> ed, the water put therein would be more pre$$ed and crowded.</P> <P>And therefore, in our ca$e, according to our principles we will $ay, that the water of that Stream pa$seth all under the $aid Bridge <I>Quattro-Capi,</I> for that being there mo$t $wift, it ought of con$equence to be le$s in mea$ure.</P> <P>And here one may $ee, into how many errours a man may run through ignorance of a true and real Principle, which once known and well under$tood, takes away all mi$ts of doubting, and ea- $ily re$olveth all difficulties.</P> <head><I>COROLLARIE.</I> XII.</head> <P>Through the $ame inadvertency of not regarding the variation of velocity in the $ame Current, therea re committed by Ingi- neers and Learned men, errours of very great moment (and I could thereof produce examples, but for good rea$ons I pa$s them over in $ilence) when they think, and propo$e, by deriving new Channels from great Rivers, to dimini$h the mea$ure of the water in the River, and to dimini$h it proportionally, according to the mea$ure of the Water which they make to pa$s through the Channel, as making <I>v.g</I> a Channel fifty foot broad, in which the derived water is to run wa$te, ten foot deep, they think they have dimini$hed the mea$ure of the Water in the River five hun- dred feet, which thing doth not indeed $o fall out; and the rea- $on is plain; for that the Chanel being derived, the re$t of the main River, dimini$heth in velocity, and therefore retains a grea- ter mea$ure than it had at fir$t before the derivation of the Cha- nel; and moreover, if the Chanel being derived, it $hall not con$erve the $ame velocity which it had at fir$t in the main Ri- ver, but $hall dimini$h it, it will be nece$$ary, that it hath a grea- ter mea$ure than it had before in the River; and therefore to accompt aright, there $hall not be $o much water derived into the Channel, as $hall dimini$h the River, according to the quanti- ty of the water in the Channel, as is pretended.</P> <head><I>COROLLARIE</I> XIII.</head> <P>This $ame con$ideration giveth me occa$ion to di$cover a mo$t ordinary errour, ob$erved by me in the bu$ine$$e of the wa- ter of <I>Ferara,</I> when I was in tho$e parts, in $ervice of the mo$t Reverend and Illu$trious Monfignor <I>Cor$ini</I>; the $ublime wit of whom hath been a very great help to me in the$e contemplations; its very true, I have been much perplexed, whether I $hould commit this particular to paper, or pa$$e it over in $ilence, for that I have ever doubted, that the opinion $o common and <foot>more-</foot> <p n=>13</p> moreover confirmed with a mo$t manife$t experiment, may not onely make this my conjecture to be e$teemed far from true, but al$o to di$credit with the World the re$t of this my Treati$e: Neverthele$$e I have at la$t re$olved not to be wanting to my $elf, and to truth in a matter of it $elf, and for other con$e- quences mo$t important; nor doth it $eem to me requi$ite in difficult matters, $uch as the$e we have in hand, to refigne our $elves to the common opinion, $ince it would be very $trange if the multitude in $uch matters $hould hit on the truth, nor ought that to be held difficult, in which even the vulgar do know the truth and right; be$ides that I hope morever to prove all in $uch $ort, that per$ons of $olid judgment, $hall re$t fully per$waded, $o that they but keep in mind the principal ground and foundation of all this Treati$e; and though that which I will propo$e, be a par- ticular, as I have $aid, pertaining onely to the intere$ts of <I>Ferara</I>; yet neverthele$$e from this particular Doctrine well under$tood, good judgement may be made of other the like ca$es in general.</P> <P>I $ay then, for greater per$pecuity, and better under$tanding of the whole, That about thirteen miles above <I>Ferara,</I> near to <I>Stellata,</I> the main of P<I>o,</I> branching it $elf into two parts, with one of its Arms it cometh clo$e to <I>Ferara,</I> retaining the name of the P<I>o</I> of <I>Ferara</I>; and here again it divideth it $elf into two other branches, and that which continueth on the right hand, is called the P<I>o</I> of <I>Argenta,</I> and of <I>Primaro</I>; and that on the left the P<I>o</I> of <I>Volana.</I> But for that the bed of the P<I>o</I> of <I>Ferara</I> being here- tofore augmented and rai$ed, it followeth that it re$teth wholly deprived of the Water of the great P<I>o,</I> except in the time of its greater $welling; for in that ca$e, this P<I>o</I> of <I>Ferara</I> being re- $trained with a Bank near to <I>Bondeno,</I> would come al$o in the overflowings of the main P<I>o,</I> to be free from its Waters: But the Lords of <I>Ferara</I> are wont at $uch time as the P<I>o</I> threateneth to break out, to cut the bank; by which cutting, there di$- gorgeth $uch a Torrent of Water, that it is ob$erved, that the main P<I>o</I> in the $pace of $ome few hours abateth near a foot, and all per$ons that I have $poken with hitherto, moved by this ex- periment, think that it is of great profit and benefit to keep ready this Vent, and to make u$e of it in the time of its fullne$$e. And indeed, the thing con$idered $imply, and at the fir$t appearance, it $eemeth that none can think otherwi$e; the rather, for that many examining the matter narrowly, mea$ure that body of Water which runneth by the Channel, or Bed of the P<I>o</I> of <I>Fera- ra,</I> and make account, that the body of the Water of the great P<I>o,</I> is dimini$hed the quantity of the body of the Water which runneth by the P<I>o</I> of <I>Ferara.</I> But if we well remember what hath been $aid in the beginning of the Treati$e, and how much <foot>the</foot> <p n=>14</p> the variety of the velocities of the $aid Water importeth, and the knowledge of them is nece$$ary to conclude the true quantity of the running Water, we $hall finde it manife$t, that the benefit of this Vent is far le$$e than it is generally thought: And mereover, we $hall finde, if I deceive not my $elf, that there follow from thence $o many mi$chiefs, that I could greatly incline to believe, that it were more to the purpo$e wholly to $top it up, than to maintain it open: yet I am not $o wedded to my opinion, but that I am ready to change my judgement upon $trength of better rea$ons; e$pecially of one that $hall have fir$t well under$tood the beginning of this my di$cour$e, which I frequently inculcate, becau$e its ab$olutely impo$$ible without this adverti$ement to treat of the$e matters, and not commit very great errours.</P> <P>I propo$e therefore to con$ideration, that although it be true, that whil$t the water of the main P<I>o</I> is at its greate$t height, the Bank and Dam then cut of the P<I>o</I> of <I>Ferara,</I> and the $uperior waters having a very great fall into the Channel of <I>Ferara,</I> they precipitate into the $ame with great violence and velocity, and with the $ame in the beginning, or little le$$e, they run towards the P<I>o</I> of <I>Volana,</I> and of <I>Argenta</I> on the $ea coa$ts; yet after the $pace of $ome few hours, the P<I>o</I> of <I>Ferara</I> being full, and the $u- perior Waters not finding $o great a diclivity there, as they had at the beginning of the cutting, they fall not into the $ame with the former velocity, but with far le$$e, and thereby a great deal le$$e Water begins to i$$ue from the great P<I>o</I>; and if we dili- gently compare the velocity at the fir$t cutting, with the velocity of the Water after the cutting made, and when the P<I>o</I> of <I>Ferara</I> $hall be full of Water, we $hall finde perhaps that to be fifteen or twenty times greater than this, and con$equently the Water which i$$ues from the great P<I>o,</I> that fir$t impetuo$ity being pa$t, $hall be onely the fifteenth or twentieth part of that which i$$ued at the beginning; and therefore the Waters of the main P<I>o</I> will return in a $mall time almo$t to the fir$t height. And here I will pray tho$e who re$t not wholly $atisfied with what hath been $aid, that for the love of truth, and the common good, they would plea$e to make diligent ob$ervation whether in the time of great Floods, the $aid Bank or Dam at <I>Bondeno</I> is cut, and that in few hours the main P<I>o</I> dimini$heth, as hath been $aid about a foot in its heigh<*>; that they would ob$erve I $ay, whether, a day or two being pa$t, the Waters of the main P<I>o</I> return almo$t to their fir$t height; for if this $hould follow, it would be very clear, that the benefit which re$ulteth from this diver$ion or Vent, is not $o great as is univer$ally pre$umed; I $ay, it is not $o great as is pre$umed; becau$e, though it be granted for true, that the Waters of the main P<I>o,</I> abate at the beginning of <foot>the</foot> <p n=>17</p> the Vent, yet this benefit happens to be but temporary and for a few hours: If the ri$ing of P<I>o,</I> and the dangers of breaking forth were of $hort duration, as it ordinarily befalleth in the overflow- ings of Torrents, in $uch a ca$e the profit of the Vent would be of $ome e$teem: But becau$e the $wellings of P<I>o</I> continue for thirty, or $ometimes for forty dayes, therefore the gain which re$ults from the Vent proveth to be incon$iderable. It remain- eth now to con$ider the notable harms which follow the $aid Sluice or Vent, that $o reflection being made, and the profit and the detriment compared, one may rightly judge, and choo$e that which $hall be mo$t convenient. The fir$t prejudice therefore which ari$eth from this Vent or Sluice, is; That the Channels of <I>Ferara, Primaro,</I> and <I>Volana</I> filling with Water, all tho$e parts from <I>Bondeno</I> to the Sea $ide are allarmed and endangered thereby. Secondly, The Waters of the P<I>o</I> of <I>Primaro</I> having free ingre$$e into the upper Valleys, they fill them to the great damage of the Fields adjacent, and ob$truct the cour$e of the ordinary Trenches in the $ame Valleys; in$omuch that all the care, co$t, and labour about the draining, and freeing the upper Valleys from Water, would al$o become vain and ineffectual. Thirdly, I con$ider that the$e Waters of the P<I>o</I> of <I>Ferara</I> being pa$$ed downwards towards the Sea, at the time that the main P<I>o</I> was in its greater excre$cences and heights, it is manife$t by expe- rience, that when the great P<I>o</I> dimini$heth, then the$e Waters pa$$ed by the P<I>o</I> of <I>Ferara</I> begin to retard in their cour$e, and finally come to turn the current upwards towards <I>Stellata,</I> re$ting fir$t iu the intermediate time, almo$t fixed and $tanding, and therefore depo$ing the muddine$$e, they fill up the Channel of the River or Current of <I>Ferara.</I> Fourthly and la$tly, There followeth from this $ame diver$ion another notable damage, and it is like to that which followeth the breaches made by Rivers; near to which breaches in the lower parts, namely below the breach, there is begot in the Channel of the River, a certain ridge or $helf, that is, the bottom of the River is rai$ed, as if $ufficiently manife$t by experience; and thus ju$t in the $ame manner cutting the Bank at <I>Bondeno,</I> there is at it were a breach made, from which followeth the ri$ing in the lower parts of the main P<I>o,</I> being pa$t the mouth of <I>Pamaro</I>; which thing, how pernitious it is, let any one judge that under$tandeth the$e matters. And therefore both for the $mall benefit, and $o many harms that en$ue from maintain- ing this diver$ion, I $hould think it were more $ound advice to keep that Bank alwaies whole at <I>Bondeno,</I> or in any other conve- nient place, and not to permit that the Water of the Grand P<I>o</I> $hould ever come near to <I>Ferara.</I></P> <foot><I>CO-</I></foot> <p n=>16</p> <head><I>COROLLARIE</I> XIV.</head> <marg>* <I>Arte$ia.</I></marg> <P>In the Grand Rivers, which fall into the Sea, as here in <I>Italy Po, Adige,</I>^{*} and <I>Arno,</I> which are armed with Banks again$t their excre$cencies, its ob$erved that far from the Sea, they need Banks of a notable height; which height goeth afterwards by degrees dimini$hing, the more it approacheth to the Sea-coa$ts: in $uch $ort, that the P<I>o,</I> di$tant from the Sea about fifty or $ixty miles at <I>Ferara,</I> $hall have Banks that be above twenty feet higher than the ordinary Water marks; but ten or twelve miles from the Sea, the Banks are not twelve feet higher than the $aid ordinary Water-marks, though the breadth of the River be the $ame, $o that the excre$cence of the $ame Innundation happens to be far greater in mea$ure remote from the Sea, then near; and yet it $hould $eem, that the $ame quantity of Water pa$$ing by every piace, the River $hould need to have the $ame altitude of Banks in all places: But we by our Principles and fundamentals may be able to render the rea$on of that effect, and $ay; That that exce$$e of quantity of Water, above the ordinary Water, goeth alwaies acquiring greater velocity; the nearer it approach- eth the Sea, and therefore decrea$eth in mea$ure, and con$equenly in height. And this perhaps might have been the cau$e in great part, why the <I>Tyber</I> in the Innundation <I>Anno</I> 1578. i$$ued not forth of its Channel below <I>Rome</I> towards the Sea.</P> <head><I>COROLLARIE</I> XV.</head> <P>From the $ame Doctrine may be rendred a mo$t manife$t rea- $on why the falling Waters go le$$ening in their de$cent, $o that the $ame falling Water, mea$ured at the beginning of its fall, is greater, and bigger, and afterwards by degrees le$$eneth in mea$ure the more it is remote from the beginning of the fall. Which dependeth on no other, than on the acqui$ition, which it $ucce$$ively makes of greater velocity; it being a mo$t fami- liar conclu$ion among Philo$ophers, that grave bodies falling, the more they remove from the beginning of their motion, the more they acquire of $wiftne$$e; and therefore the Water, as a grave body, falling, gradually velocitates, and therefore de- crea$eth in mea$ure, and le$$eneth.</P> <head><I>COROLLARIE</I> XVI.</head> <P>And on the contrary, the $pirtings of a Fountain of Water, which $pring on high, work a contrary effect; namely <foot>in</foot> <p n=>17</p> in the beginning they are $mall, and afterwards become greater and bigge; and the rea$on is mo$t manife$t, becau$e in the be- ginning they are very $wift, and afterwards gradually relent their impetuo$ity, and motion, $o that in the beginning of the excur$ion that they make, they ought to be $mall, and after- wards to grow bigger, as in the effect is $een.</P> <head>APPENDIX. I.</head> <P>Into the errour of not con$idering how much the different velocities of the $ame running water in $everal places of its current, are able to change the mea$ure of the $ame water, and to make it greater, or le$$e, I think, if I be not deceived, that <I>Ginlio Frontino</I> a noble antient Writer, may have faln in the Second Book which he writ, of the Aqueducts of the City of <I>Rome</I>: Whil$t finding the mea$ure of the Water ^{*}<I>Commentaries</I> le$$e than it was <I>in erogatione 1263. Quinaries,</I> he <marg>+ <I>Commentarius</I> beareth many $en- $es, but in this place $ignifieth a certain Regi$ter of the quantities of the Waters in the $everal publique A- qu ducts of <I>Rome</I>; which word I find frequently u$ed in the Law-books of antient Civilians: Andby errogation we are to under- $tand the di$tribu- tion or delivering out of tho$e $tores of Water.</marg> thought that $o much difference might proceed from the negligence of the Mea$ures; and when afterwards with his own indu$try he mea$ured the $ame water at the beginnings of the Aqueducts, finding it neer 10000. <I>Quinaries</I> bigger than it was <I>in Commenta- riis</I> he judged, that the overplus was imbeziled by Mini$ters and Partakers; which in part might be $o, for it is but too true, that the publique is almo$t alwayes defrauded; yet neverthele$$e, I verily believe withal, that be$ides the frauds of the$e Officers, the velocities of the water in the place wherein <I>Frontino</I> mea$u- red, it might be different from tho$e velocities, which are found in other places before mea$ured by others; and there- fore the mea$ures of the waters might, yea ought nec$$arily to be diffcrent, it having been by us demon$trated, that the mea- $ures of the $ame running water have reciprocal proportion to their velocities. Which <I>Frontino</I> not well con$idering, and find- ing the water <I>in Commentariis 12755. Quinaries in erogati- one</I> 14018, and in his own mea$ure <I>ad capita ductuum,</I> at the head of the fountain 22755. <I>Quinaries,</I> or thereabouts, he thought, that in all the$e places there pa$t different quantities of water; namely, greater at the fountain head then that which was <I>in Erogatione,</I> and this he judged greater than that which was <I>in Commentariis.</I></P> <head>APPENDIX II.</head> <P>Alike mi$take chanced lately in the Aqueduct of <I>Acqua- Paola,</I> which Water $hould be 2000 Inches, and $o many effectively ought to be allowed; and it hath been given in <foot>Dddd $o</foot> <p n=>18</p> $o to be by the Signors of <I>Bracciano</I> to the <I>Apo$tolick-Chamber</I>; and there was a mea$ure thereof made at the beginning of the Aqueduct; which mea$ure proved afterwards much le$$e and $hort, con$idered and taken in <I>Rome,</I> and thence followed di$- contents and great di$orders, and all becau$e this property of Running-Waters, of increa$ing in mea$ure, where the velocity decrea$ed; and of dimini$hing in mea$ure, where the velocity augmented, was not lookt into.</P> <head>APPENDIX III.</head> <P>Alike errour, in my judgement, hath beeen committed by all tho$e learned men, which to prevent the diver$ion of the <I>Reno</I> of <I>Bologna</I> into P<I>o</I> by the Channels, through which it at pre$ent runneth, judged, that the <I>Reno</I> being in its greater excre$cence about 2000 feet, and the P<I>o</I> being near 1000 feet broad, they judged, I $ay, that letting the <I>Reno</I> into P<I>o,</I> it would have rai$ed the Water of P<I>o</I> two feet; from which ri$e, they concluded afterwards mo$t exorbitant di$orders, either of extraordinary Inundations, or el$e of immen$e and intolera- ble expences to the people in rai$ing the Banks of P<I>o</I> and <I>Reno,</I> and with $uch like weakne$$es, often vainly di$turbed the minds of the per$ons concerned: But now from the things demon$tra- ted, it is manife$t, That the mea$ure of the <I>Reno</I> in <I>Reno,</I> would be different from the mea$ure of <I>Reno</I> in P<I>o</I>; in ca$e that the velocity of the <I>Reno</I> in P<I>o,</I> $hould differ from the velocity of <I>Reno</I> in <I>Reno,</I> as is more exactly determined in the fourth Pro- po$ition.</P> <head>APPENDIX IV.</head> <P>No le$s likewi$e are tho$e Ingeneers and Arti$ts deceived, that have affirmed, That letting the <I>Reno</I> into P<I>o,</I> there would be no ri$e at all in the Water of P<I>o</I>: For the truth is, That letting <I>Reno</I> into P<I>o,</I> there would alwaies be a ri$ing; but $ometimes greater, $ometimes le$$e, as the P<I>o</I> $hall have a $wifter or $lower Current; $o that if the P<I>o</I> $hall be con$tituted in a great velocity, the ri$e will be very $mall; and if the $aid P<I>o</I> $hall be $low in its cour$e, then the ri$e will be notable.</P> <head>APPENDIX V.</head> <P>And here it will not be be$ides the purpo$e to adverti$e, That the mea$ures, partments, and di$tributions of the Waters of Fountains, cannot be made exactly, unle$s there be con- <foot>$idered</foot> <p n=>19</p> fidered, be$ides the mea$ure, the velocity al$o of the Water; which particular not being thorowly ob$erved, is the cau$e of continual mi$cariages in $uch like affairs.</P> <head>APPENDIX VI.</head> <P>Like con$ideration ought to be had with the greater diligence, for that an errour therein is more prejudicial; I $ay, ought to be had by tho$e which part and divide Waters; for the watering of fields, as is done in the Territories of <I>Bre$cia, Ber- gama, Crema, Pavia, Lodigiano, Cremona,</I> and other places: For if they have not regard to the mo$t important point of the variation of the velocity of the Water, but onely to the bare Vulgar mea$ure, there will alwaies very great di$orders and pre- judices en$ue to the per$ons concerned.</P> <head>APPENDIX VII.</head> <P>It $eemeth that one may ob$erve, that whil$t the Water run- neth along a Channel, Current, or Conduit, its velocity is retarded, withheld, and impeded by its touching the Bank or $ide of the $aid Channel or Current; which, as immoveable, not following the motion of the Water, interrupteth its velocity: From which particular, being true, as I believe it to be mo$t true, and from our con$iderations, we have an occa$ion of di$- covering a very nice mi$take, into which tho$e commonly fall who divide the Waters of Fountains. Which divi$ion is wont to be, by what I have $een here in <I>Rome,</I> performed two wayes; The fir$t of which is with the mea$ures of like figures, as Cir- cles, or Squares, having cut through a Plate of metal $everal Circles or Squares, one of half an inch, another of one inch, another of two, of three, of four, <I>&c.</I> with which they after- wards adju$t the Cocks to di$pence the Waters. The other manner of dividing the Waters of Fountains, is with rectangle paralellograms, of the $ame height, but of different Ba$es, in $uch $ort likewi$e, that one paralellogram be of half an inch, another of one, two, three, <I>&c.</I> In which manner of mea$uring and dividing the Water, it $hould $eem that the Cocks being placed in one and the $ame plain, equidi$tant from the level, or $uperior $uperficies of the water of the Well; and the $aid mea$ures be- ing mo$t exactly made, the Water ought con$equently al$o to be equally divided, and parted according to the proportion of the mea$ures. But if we well con$ider every particular, we $hall finde, that the Cocks, as they $ucce$$ively are greater, di$charge alwaies more Water than the ju$t quantity, in compari$on of <foot>Dddd 2 the</foot> <p n=>20</p> the le$$er; that is, to $peak more properly, The Water which pa$$eth through the greater Cock, hath alwaies a greater pro- portion to that which pa$$eth through the le$$er, than the greater Cock hath to the le$$er. All which I will declare by an exam- ple.</P> <P>Let there be $uppo$ed for more plainne$s two Squares; (the $ame may be under$tood of Circles, and other like Figures) The fir$t Square is, as we will $uppo$e, quadruple to the other, and the$e Squares are the mouths of two Cocks.; one of four inches, the other of one: Now its manife$t by what hath been $aid, that the Water which pa$$eth by the le$s Cock, findeth its velocity impeded in the circumference of the Cock; which impediment <fig> is mea$ured by the $aid circumfe- rence. Now it is to be con$ider- ed, that if we would have the Wa- ter which pa$$eth through the greater Cock, to be onely qua- druple to that which pa$$eth through the le$$e, in equal $paces of time, it would be nece$$ary, that not onely the capacity and the mea$ure of the greater Cock be quadruple to the le$$er Cock, but that al$o the impediment be quadrupled. Now in our ca$e it is true, That the belly and mouth of the Cock is quadrupled, and yet the impediment is not quadrupled, but is onely doubled; $eeing that the circumference of the greater Square, is onely double to the circumference of the le$ier Square; for the greater circumference containeth eight of tho$e parts, of which the le$$er containeth but four, as is ma- nife$t by the de$cribed Figure; and for that cau$e there $hall pa$s by the greater Cock, above four times as much Water, as $hall pa$s by the le$$er Cock.</P> <P>The like errour occurreth al$o in the other manner of mea$u- ring the Water of a Fountain, as may ea$ily be collected from what hath been $aid and ob$erved above.</P> <head>APPENDIX VIII.</head> <P>The $ame contemplation di$covereth the errour of tho$e Architects, who being to erect a Bridge of $undry Arches over a River, con$ider the ordinary breadth of the River; which being <I>v. g.</I> fourty fathom, and the Bridge being to con$i$t of four Arches, it $ufficeth them, that the breadth of all the four Arches taken together, be fourty fathom; not con$idering that in the ordinary Channel of the River, the Water hath onely two impediments which retard its velocity; namely, the touching and gliding along the two $ides or $hores of the River: but <foot>the</foot> <p n=>21</p> the $ame water in pa$$ing under the Bridge, in our ca$e meeteth with eight of the $ame impediments, bearing, and thru$ting upon two $ides of each Arch (to omit the impediment of the bottom, for that it is the $ame in the River, and under the Bridge) from which inadvertency $ometimes follow very great di$orders, as quotidian practice $hews us.</P> <head>APPENDIX IX.</head> <P>It is al$o worthy to con$ider the great and admirable benefit that tho$e fields receive, which are wont to drink up the Rain- water with difficulty, through the height of the water in the principal Ditches; in which ca$e the careful Husbandman cutteth away the reeds and ru$hes in the Ditches, through which the waters pa$s; whereupon may be pre$ently $een, $o $oon as the reeds and ru$hes are cut, a notable Ebb in the level of the water in the Ditches; in$omuch that $ometimes it is ob$erved, that the water is abated after the $aid cutting a third and more, of what it was before the cutting. The which effect $eemingly might de- pend on this, That, before tho$e weeds took up room in the Ditch, and for that cau$e the water kept a higher level, and the $aid Plants being afterwards cut and removed, the water came to abate, po$$e$$ing the place that before was occupied by the weeds: Which opinion, though probable, and at fir$t $ight $a- tisfactory, is neverthele$s in$ufficient to give the total rea$on of that notable abatement which hath been $poken of: But it is ne- ce$$ary to have recour$e to our confideration of the velocity in the cour$e of the water, the chiefe$t and true cau$e of the vari- ation of the mea$ure of the $ame Running-Water; for, that multitudes of reeds, weeds, and plants di$per$ed through the cur- rent of the Ditch, do chance notably to retard the cour$e of the water, and therefore the mea$ure of the water increa$eth; and tho$e impediments removed, the $ame water gaineth velocity, and therefore decrea$eth in mea$ure, and con$equently in height.</P> <P>And perhaps this point well under$tood, may be of great profit to the fields adjacent to the <I>Pontine</I> Fens, and I doubt not but if the River <I>Ninfa,</I> and the other principal Brooks of tho$e Territories were kept well clean$ed from weeds, their waters would be at a lower level, and con$equently the drains of the fields would run into them more readily; it being alwayes to be held for undoubted, that the mea$ure of the water before the clean$ing, hath the $ame proportion to the mea$ure after clean- $ing, that the velocity after the clean$ing hath to the velocity before the clean$ing: An dbecau$e tho$e weeds being clean$ed <foot>away,</foot> <p n=>22</p> away, the cour$e ef the water notably increa$eth, it is therefore nece$$ary that the $aid water abate in mea$ure, and become lower.</P> <head>APPENDIX. X.</head> <P>We having above ob$erved $ome errors that are commit- ted in di$tributing the waters of Fountains, and tho$e that $erve to water fields; it $eemeth now fit, by way of a clo$e to this di$cour$e, to adverti$e by what means the$e divi- $ions may be made ju$tly and without error. I therefore think that one might two $everal wayes exqui$itly divide the water of Fountains; The fir$t would be by diligently examining, Fir$t, how much water the whole Fountain di$chargeth in a determi- nate time, as for in$tance: How many Barrels, or Tuns it carri- eth in a $et time; and in ca$e you are afterwards to di$tribute the water, di$tribute it at the rate of $omany Barrels or Tuns, in that $ame time; and in this ca$e the participants would have their punctual $hares: Nor could it ever happen to $end out more water, than is reckoned to be in the principal Fountain; as befel <I>Giulio Frontino,</I> and as al$o it frequently happeneth in the Mo- dern Aqueducts, to the publick and private detriment.</P> <P>The other way of dividing the $ame waters of a Fountain, is al$o $ufficiently exact and ea$ie, and may be, by having one one- ly $ize for the Cock or Pipe, as $uppo$e of an inch, or of half an inch; and when the ca$e requireth to di$pence two, three, and more inches, take $o many Cocks of the $aid mea$ure as do eva- cuate the water, which is to be emitted; and if we are to make u$e onely of one greater Cock, we being to place one to di$- charge for example four inches; and having the former $ole mea- $ure of an inch, we mu$t make a Cock that is bigger, its true, than the Cock of one inch; but not $imply in a quadruple propor- tion, for that it would di$charge more than ju$t $o much water, as hath been $aid above; but we ought to examine diligently how much water the little Cock emitteth in an hour; and then enlarge, and contract the greater Cock, $o, that it may di$- charge four times as much water as the le$$er in the $ame time; and by this means we $hall avoid the di$order hinted in the $eventh Appendix. It would be nece$$ary neverthele$s, to ac- commodate the Cocks of the Ci$tern $o, that the level of the water in the Ci$tern may alwayes re$t at one determinate mark above the Cock, otherwi$e the Cocks will emit $ometimes greater, and $ometimes le$$e abundance of water: And becau$e it may be that the $ame water of the Fountain may be $ometimes more abundant, $ometimes le$s; in $uch ca$e it will be nece$$ary <foot>to</foot> <p n=>23</p> to adju$t the Ci$tern $o, that the exce$s above the ordinary wa- ter, di$charge into the publick Fountains, that $o the particular participants may have alwayes the $ame abundance of water.</P> <head>APPENDIX XI.</head> <P>Much more difficult is the divi$ion of the waters which $erve to water the fields, it not being po$$ible to ob$erve $o commodiou$ly, what quantity of water the whole Ditch $ends forth in one determinate time, as may be done in Fountains: Yet neverthele$s, if the $econd propo$ition by us a little below demon$trated, be well under$tood, there may be thence taken a very $afe and ju$t way to di$tribute $uch waters. The Propo$ition therefore by us demon$trated is this: If there be two Sections, (namely two mouths of Rivers) the quantity of the water which pa$$eth by the fir$t, hath a proportion to that which pa$$eth by the $econd, compounded of the proportions of the fir$t Section to the $econd, and of the velocity through the fir$t, to the velocity through the $econd: As I will declare for example by help of practice, that I may be under$tood by all, in a matter $o important. Let the two mouths of the Rivers be A, and B, and let <fig> the mouth A be in mea$ure and content thirty two feet, and the mouth B, eight feet. Here you mu$t take notice, that it is not alwayes true, that the Water which pa$$eth by A, hath the $ame proportion to that which pa$$eth by B, that the mouth A hath to the mouth B; but onely when the velocityes by each of tho$e pa$$ages are equal: But if the velocityes $hall be unequal, it may be that the $aid mouths may emit equal quantity of Water in equal times, though their mea$ure be un- equal; and it may be al$o, that the bigger doth di$charge a great- er quantity of Water: And la$tly, it may be, that the le$s mouth di$chargeth more Water than the greater; and all this is mani- fe$t by the things noted in the beginning of this di$cour$e, and by the $aid $econd Propo$ition. Now to examine the propor- tion of the Water that pa$$eth by one Ditch, to that which pa$- $eth by another, that this being known, the $ame Waters and mouths of Ditches may be then adju$ted; we are to keep ac- count not onely of the greatne$s of the mouths or pa$$ages of the Water, but of the velocity al$o; which we will do, by fir$t find- ing two numbers that have the $ame proportion between them- <foot>$elves,</foot> <p n=>24</p> $elves, as have the mouths, which are the numbers 32 and 8 in our example: Then this <fig> being done, let the velocity of the Water by the pa$$a- ges A and B, be examined (which may be done keeping account what $pace a piece of Wood, or other body that $wimmeth, is carried by the $tream in one determinate time; as for in$tance in 50 pul$es) and then work by the golden Rule, as the velocity by A, is to the velocity by B, $o is the number 8, to another number, which is 4. It is clear by what is demon$tra- ted in the $aid $econd Propo$ition, that the quantity of water, which pa$$eth by the mouth A, $hall have the $ame proportion of that which pa$$eth by the mouth B, that 8 hath to 1. Such pro- portion being compo$ed of the proportions of 32 to 8, and of 8 to 4; namely, tothe greatne$s of the mouth A, to the greatne$s of the mouth B, and of the velocity in A, to the velocity in B. This being done, we mu$t then contract the mouth which di$chargeth more then its ju$t quantity of water, or enlarge the other which di$char- geth le$s, as $hal be mo$t commodious in practice, which to him that hath under$tood this little that hath been delivered, will be very afie.</P> <head>APPENDIX XII.</head> <P>The$e opperations about Water, as I have hitherto on $un- dry occa$ions ob$erved, are involved in $o many difficul- ties, and $uch a multiplicity of mo$t extravagant accidents, that it is no marvel if continually many, and very important er- rours be therein committed by many, and even by Ingeneers them$elves, and Learned-men; and becau$e many times they concern not onely the publique, but private intere$ts: Hence it is, that it not onely belongeth to Arti$ts to treat thereof, but very oft even the vulgar them$elves pretend to give their judgement therein: And I have been troubled many times with a nece$$ity of treating, not onely with tho$e, which either by practice, or particular $tudy, under$tood $omewhat in the$e matters; but al$o with people wholly void of tho$e notions, which are nece$$ary for one that would on good grounds di$cour$e about this particular; and thus many times have met with more difficulty in the thick skulls of men, than in precipitous Torrents, and va$t Fennes. And in particular, I had occafion $ome years pa$t to go $ee the Gave or Emi$$ary of the Lake of <I>Perugia,</I> made many years agon by <I>Braccio Fortobraccio,</I> but for that it was with great ruines by Time decayed, and rendred unu$eful, it was repaired with in- <foot>du$try</foot> <p n=>25</p> du$try truly heroicall and admirable, by Mon$ignor <I>Maffei Bar- herino,</I> then Prefect for the Wayes, and now Pope. And being nece$$itated, that I might be able to walk in the Cave, and for other cau$es, I let down the Sluices of the $aid Cave, at the mouth of the Lake: No $ooner were they $topt, but a great many of the people of the Towns and Villages coa$ting upon the <I>L</I>ake flocking thither, began to make grievous complaints, that if tho$e Sluices were kept $hut, not onely the Lake would want its due Vent, but al$o the parts adjacent to the Lake would be over flown to their very great detriment. And becau$e at fir$t appea- rance their motion $eemed very rea$onable, I found my $elf hard put to it, $eeing no way to per$wade $uch a multitude, that the prejudice which they pretended I $hould do them by keeping the Sluices $hut for two dayes, was ab$olutely in$en$ible; and that by keeping them open, the Lake did not ebb in the $ame time $o much as the thickne$s of a $heet of Paper: And therefore I was nece$$itated to make u$e of the authority I had, and $o followed my bu$ine$s as cau$e required, without any regard to that Rab- ble tumultuou$ly a$$embled. Now when I am not working with Mattock or Spade, but with the Pen and Di$cour$e, I intend to demon$trate clearly to tho$e that are capable of rea$on, and that have well under$tood the ground of this my Treati$e, that the fear was altogether vain which tho$e people conceited. And therefore I $ay, that the Emi$$ary or Sluice of the Lake of <I>Peru- gia,</I> $tanding in the $ame mannner as at pre$ent, and the water pa$$ing thorow it with the $ame velocity as now; to examine how much the Lake may abate in two days $pace, we ought to con$ider, what proportion the $uperficies of the whole Lake hath to the mea$ure of the Section of the Emi$$ary, and afterwards to infer, that the velocity of the water by the Emi$$ary or Sluice, $hall have the $ame proportion to the abatement of the Lake, and to prove thorowly and clearly this di$cour$e, I intend to demon$trate the following Propo$ition.</P> <P>Suppo$e a Ve$$el of any bigne$$e, and that it hath an Emi$$ary or Cock, by which it di$chargeth its water. And look what pro- portion the $uper$icies of the ve$$el hath to the mea$ure of <fig> the $ection of the cock, $uch pro- portion $hall the velocity of the Water in the Cock have to the abatement of the Lake Let the Ve$$el be A B C D, H I L B, through which the Water runneth, the $uperficies of the Water in the Ve$$el A D, and the $ection of the Cock H L: and let the Water in the Ve$$el be $uppo$ed to have falne in one determinate time from A to F. <foot>Eeee I</foot> <p n=>26</p> I $ay that the proportion of the $uperficies of the Ve$$el A D is in proportion to the mea$ure of the $ection of the Emi$$ary H L, as the velocity of the Emi$$ary or Cock to the line A F; which is manife$t, for that the Water in the Ve$sel moving by the line A F; as far as F, and the whole ma$s of Water A G di$charging it $elf, and in the $ame time the $ame quantity of Water being di$charged by the $ection of the Emi$$ary H L; it is nece$$ary by what I have demon$trated in the third Propo$ition, and al$o explained in the beginning of this Treati$e, that the ve- locity by the Emi$$ary or Cock be in proportion to the velocity of the abatement, as the $uperficies of the Ve$$el to the mea- $ure of the $ection of the Emi$$ary, which was to be demon- $trated.</P> <P>That which hath been demon$trated in the Ve$$el, falls out ex- actly al$o in our Lake of <I>Perugia,</I> and its Emi$sary; and becau$e the immen$ity of the $uperficies of the Lake is in proportion to the $uperficies of the Emi$sary or Sluice, as many millions to one, as may be ea$ily calculated; it is manife$t, that $uch abate- ment $hall be imperceptible, and almo$t nothing, in two dayes $pace, nay in four or $ix: and all this will be true, when we $uppo$e that for that time there entreth no other Water into the Lake from Ditches or Rivolets, which falling into the Lake would render $uch abatement yet le$s.</P> <P>Now we $ee, that it's nece$sary to examine $uch abatements and ri$ings, with excellent rea$ons, or at lea$t, with accurate ex- periments, before we re$olve and conclude any thing; and how farre the vulgar are di$tant from a right judgment in $uch matters.</P> <head>APPENDIX XIII.</head> <P>For greater confirmation of all this which I have $aid, I will in$tance in another like ca$e, which al$o I met with here- tofore, wherein, for that the bu$ine$s was not rightly un- der$tood, many di$orders, va$t expences, and con$iderable mi$- chiefs have followed. There was heretofore an Emi$sary or Sluice made to drain the Waters, which from Rains, Springs, and Rivolets fall into a Lake; to the end, the $hores adjoyning on the Lake, $hould be free from the overflowing of the Waters; but becau$e perhaps the enterprize was not well managed and carried on, it fell out, that the Fields adjacent to the $aid Chanel could not drain, but continued under water; to which di$orders a pre$ent remedy hath been u$ed, namely, in a time convenient to $top up the Sluice, by meanes of certain Floodgates kept on purpo$e for that end; and thus abating the Level of the Water <foot>in</foot> <p n=>27</p> in the Emi$$ary, in the $pace of three or four dayes, the Fields have been haply drained. But on the other part, the proprietors bordering on the Lake oppo$ed this, grievou$ly complaining, that whil$t the Floodgates are $hut, and the cour$e of the Water of the Sluice hindered, the Lake overflowes the Lands adjacent, by meanes of the Rivers that fell into it, to their very great damage; and $o continuing their $uits, they got more of vexation than $a- tisfaction. Now, being asked my opinion herein, I judged it requi$ite ($ince the point in controver$ie was about the ri$ing and falling of the Lake) that the $aid abatement, when the Floodgates are open, and increa$e when they are $hut $hould be exactly mea$ured, and told them, that it might be ea$ily done at a time when no extraordinary Waters fell into the Lake, neither of Rain, or otherwi$e; and the Lake was undi$turbed by winds that might drive the Water to any $ide, by planting neer to an I$let, which is about the middle of the Lake, a thick po$t, on which $hould be made the marks of the Lakes ri$ing and falling for two or three dayes. I would not, at that time, pawn, or re- $olutely declare, my judgment, in regard I might be, by divers accidents mi$led. But this I told them, that (by what I have demon$trated, and particularly that which I have $aid above touching the Lake of <I>Perugia</I>) I inclined greatly to think, that the$e ri$ings and fallings would prove imperceptible, and incon$iderable; and therefore, that in ca$e experience $hould make good my rea$on, it would be to no purpo$e for them to continue di$puting and wrangling, which cau$eth, (according to the Proverb) <I>A great deal of cry,</I> but produceth not much <I>Wool.</I></P> <P>La$tly, it importing very much to know what a Rain conti- nued for many dayes can do in rai$ing the$e Lakes, I will here in- $ert the Copy of a Letter, which I writ formerly to <I>Signior Ga- lilæo Galilæi,</I> chief Philo$opher to the Grand Duke of <I>Tu$cany,</I> wherein I have delivered one of my conceits in this bu$ine$$e, and it may be, by this Letter, I may, more $trongly, confirm what I have $aid above.</P> <foot>Eeee 2 <I>The</I></foot> <p n=>28</p> <P><I>The Copy of a Letter to</I> Signore GALILÆO GALILÆI, <I>Chief Philo$opher to the mo$t Serene Great Duke of TVSCANY.</I></P> <P><I>Worthy and mo$t Excellent</I> SIR,</P> <P>In $atisfaction of my promi$e, in my former Letters of repre$enting unto you $ome of my Con$iderations made upon the Lake <I>Thra$imeno,</I> I $ay, That in times pa$t, being in <I>Perugia,</I> where we held our General Convention, having under$tood that the Lake <I>Thra$imeno,</I> by the great drought of many Moneths was much abated, It came into my head, to go privately and $ee this novelty, both for my particular $atisfaction, as al$o that might I be able to relate the whole to my Patrons, upon the certitude of my own $ight of the place. And $o being come to the Emi$$ary of the Lake, I found that the Level of the Lakes $urface was ebbed about five Ro- man Palmes of its wonted watermark, in$omuch that it was lower than the tran$ome of the mouth of the Emi$$ary, by the length of ---------------------------- this de$cribed line, and there- fore no Water i$$ued out of the Lake, to the great prejudice of all the places and villages circumjacent, in regard that the Wa- ter which u$ed to run from the $aid Lake turned 22 Mills, which not going, nece$$itated the inhabitants of tho$e parts to go a dayes journey and more, to grinde upon the <I>Tiber.</I> Being retur- ned to <I>Perugia,</I> there followed a Rain, not very great, but con- $tant, and even, which la$ted for the $pace of eight hours, or thereabouts; and it came into my thoughts to examine, being in <I>Perugia,</I> how much the Lake was increa$ed and railed by this Rain, $uppo$ing (as it was probable enough) that the Rain had been univer$al over all the Lake; and like to that which fell in <I>Perugia,</I> and to this purpo$e I took a Gla$$e formed like a Cy- linder, about a palme high, and half a palme broad; and having put in water $nfficient to cover the bottome of the Gla$$e, I no- ted diligently the mark of the height of the Water in the Gla$$e, and afterwards expo$ed it to the open weather, to receive the Raine-water, which fell into it; and I let it $tand for the $pace of an hour; and having ob$erved that in that time the Wa- ter was ri$en in the Ve$$el the height of the following line---, I con$idered that if I had expo$ed to the $ame rain $uch other ve$- $els equal to that, the Water would have ri$en in them all accor- ding to that mea$ure: And thereupon concluded, that al$o in all <foot>the</foot> <p n=>29</p> the whole extent of the Lake, it was nece$$ary the Water $hould be rai$ed in the $pace of an hour the $ame mea$ure. Yet here I con$idered two difficulties that might di$tutb and altar $uch an effect, or at lea$t render it inob$erveable, which afterwards well weighed, and re$olved, left me (as I will tell you anon) in the conclu$ion the more confirmed; that the Lake ought to be in- crea$ed in the $pace of eight hours, that the rain la$ted eight times that mea$ure. And whil$t I again expo$ed the Gla$s to re- peat the experiment, there came unto me an Ingeneer to talk with me touching certain affairs of our Mona$tary of <I>Perugia,</I> and di$cour$ing with him, I $hewed him the Gla$s out a<*> my Cham- ber-window, expo$ed in a Court-yard; and communicated to him my fancy, relacing unto him all that I had done. But I $oon perceived that this brave fellow conceited me to be but of a dull brain, for he $milling $aid unto me; Sir, you deceive your $elf: I am of opinion that the Lake will not be increa$- ed by this rain, $o much as the thickne$$e of a ^{*} <I>Julio.</I> <marg>* A Coyn of Pope <I>Julius</I> worth $ix pence.</marg> Hearing him pronounce this his opinion with freene$s and confidence, I urged him to give me $ome rea$on for what he $aid, a$$uring him, that I would change my judgement, when I $aw the $trength of his Arguments: To which he an$wered, that he had been very conver$ant about the Lake, and was every day upon it, and was well a$$ured that it was not at all increa$ed. And importuning him further, that he would give me $ome rea$on for his $o thinking, he propo$ed to my con$ideration the great drought pa$$ed, and that that $ame rain was nothing for the great parching: To which I an$wered, I believe Sir that the $ur- face of the Lake, on which the rain had fallen was moi$tned; and therefore $aw not how its drought, which was nothing at all, could have drunk up any part of the rain. For all this he per- $i$ting in his conceit, without yielding in the lea$t to my allega- tion; he granted in the end (I believe in civility to me) that my rea$on was plau$ible and good, but that in practi$e it could not hold. At la$t to clear up all, I made one be called, and $ent him to the mouth of the Emi$$ary of the Lake, with order to bring me an exact account, how he found the water of the Lake, in re$pect of the Tran$ome of the Sluice. Now here, Signore <I>Galilo,</I> I would not have you think that I had brought the matter in hand to concern me in my honour; but believe me (and there are witne$$es of the $ame $till living) that my me$$en- ger returning in the evening to <I>Perugia,</I> he brought me word, that the water of the Lake began to run through the Cave; and that it was ri$en almo$t a fingers breadth above the Tran$ome: In$omuch, that adding this mea$ure, to that of the lowne$s of the $urface of the Lake, beneath the Tran$ome before the rain, <foot>it</foot> <p n=>30</p> it was manife$t that the ri$ing of the Lake cau$ed by the rain, was to a hair tho$e four fingers breadth that I had judged it to be. Two dayes after I had another bout with the Ingeneer, and re- lated to him the whole bu$ine$s, to which he knew not what to an$wer.</P> <P>Now the two difficulties which I thought of, able to impede my conclu$ion, were the$e following: Fir$t, I con$idered that it might be, that the Wind blowing from the $ide where the Sluice $tood, to the Lake-ward; the mole and ma$s of the Wa- ter of the Lake might be driven to the contrary $hore; on which the Water ri$ing, it might be fallen at the mouth of the Emi$$a- ry, and $o the ob$ervation might be much ob$cured. But this difficulty wholly vani$hed by rea$on of the Aires great tranqui- lity; which it kept at that time, for no Wind was $tirring on any $ide, neither whil$t it rained, nor afterwards.</P> <P>The $econd difficulty which put the ri$ing in doubt, was, That having ob$erved in <I>Florence,</I> and el$ewhere, tho$e Ponds into which the rain-water, falling from the hou$e, is conveyed through the Common-$hores: And that they are not thereby ever filled, but that they $wallow all that abundance of water, that runs into them by tho$e conveyances which $erve them with water; in$omuch that tho$e conveyances which in time of drought maintain the Pond, when there comes new abundance of water into the Pond, they drink it up, and $wallow it: A like effect might al$o fall out in the Lake, in which there being many veins (as it is very likely) that maintain and feed the Lake; the$e veins might imbibe the new addition of the Rain-water, and $o by that means annuall the ri$ing; or el$e dimini$h it in $uch $ort, as to render it inob$ervable. But this difficulty was ea$ily re$olved by con$idering my Treati$e of the mea$ure of Running-Waters; fora$much as having demon$trated, that the abatement of a Lake beareth the reciprocal proportion to the velocity of the Emi$$a- ry, which the mea$ure of the Section of the Emi$$ary of the Lake, hath to the mea$ure of the $urface of the Lake: making the calculation and account, though in gro$s; by $uppo$ing that its veins were $ufficiently large, and that the velocity in them were notable in drinking up the water of the Lake; yet I found never- thele$s, that many weeks and moneths would be $pent in drink- ing up the new-come abundance of water by the rain, $o that I re$ted $ure, that the ri$ing would en$ue, as in effect it did.</P> <P>And becau$e many of accurate judgement, have again cau$ed me to que$tion this ri$ing, $etting before me, that the Earth be- ing parched by the great drought, that had $o long continued, it might be, that that Bank of Earth which environed the brink of the Lake, being dry, and imbibing great abundance of Water <foot>from</foot> <p n=>31</p> from the increa$ing Lake, would not $uffer it to increa$e in height: I $ay therefore, that if we would rightly con$ider this doubt here propo$ed, we $hould, in the very con$ideration of it, $ee it re$olved; for, it being $uppo$ed that that li$t or border of Banks which was to be occupied by the increa$e of the Lake, be a Brace in breadth quite round the Lake, and that by rea$on of its dryne$s it $ucks in water, and that by that means this propor- tion of water co-operates not in rai$ing of the Lake: It is ab$o- lutely nece$$ary on the other hand, that we con$ider, That the Circuit of the water of the Lake being thirty miles, as its com- monly held, that is to $ay, Ninety thou$and Braces of <I>Florence</I> in compa$s; and therefore admitting for true, that each Brace of this Bank drink two quarts of water, and that for the $pieading it require three quarts more, we $hall finde, that the whole agre- gate of this portion of water, which is not imployed in the rai$ing of the Lake, will be four hundred and fifty thou$and Quarts of water; and $uppo$ing that the Lake be $ixty $quare miles, three thou$and Braces long, we $hall finde, that to di$pence the water po$$e$t by the Bank about the Lake, above the total $urface of the Lake, it ought to be $pread $o thin, that one $ole quart of water may over-$pread ten thou$and $quare Braces of $urface: $uch a thinne$s, as mu$t much exceed that of a leaf of beaten Gold, and al$o le$s than that skin of water which covers the Bub- bles of it: and $uch would that be, which tho$e men would have $ub$tracted from the ri$ing of the Lake: But again, in the $pace of a quarter of an hour at the beginning of the rain, all that Bank is $oaked by the $aid rain, $o that we need not for the moi$tning of it, imploy a drop of that water which falleth into the Lake. Be$ides we have not brought to account that abun- dance of water which runs in time of rain into the Lake, from the $teepne$s of the adjacent Hills and Mountains; which would be enough to $upply all our occa$ions: So that, neither ought we for this rea$on to que$tion our pretended ri$ing. And this is what hath fallen in my way touching the con$ideration of the <I>Thra$imenian</I> Lake.</P> <P>After which, perhaps $omewhat ra$hly, wandring beyond my bounds, I proceeded to another contemplation, which I will re- late to you, hoping that you will receive it, as collected with the$e cautions requi$ite in $uch like affairs; wherein we ought not too po$itively to affirm any thing of our own heads for cer- tain, but ought to $ubmit all to the $ound and $ecure delibera- tion of the Holy Mother-Church, as I do this of mine, and all others; mo$t ready to change my judgement, and conform my $elf alwaies to the deliberations of my Superiors. Continu- <foot>ing</foot> <p n=>32</p> ing therefore my above-$aid conceit about the ri$ing of the wa- ter in the gla$s tried before, it came into my minde, that the forementioned rain having been very gentle, it might well be, that if there $hould have faln a Rain fifty, an hundred, or a thou- $and times greater than this, and much more inten$e (which would in$ue as oft as tho$e falling drops were four, $ive or ten times bigger than tho$e of the above-mentioned rain, keeping the $ame number) in $uch a ca$e its manife$t, that in the $pace of an hour the Water would ri$e in our Gla$s, two, three, and perhaps more Yards or Braces; and con$equently, if $uch a Raine $hould fall upon a Lake, that the $aid Lake would ri$e, according to the $ame rate: And likewi$e, if $uch a Rain were univer$all, over the whole Terre$triall Globe, it would nece$$arily, in the $pace of an hour, make a ri- $ing of two, or three braces round about the $aid Globe, And becau$e we have from Sacred Records, that in the time of the Deluge, it rained fourty dayes and fourty nights; namely, for the $pace of 960 houres; its clear, that if the $aid Rain had been ten times bigger than ours at <I>Perugia,</I> the ri$ing of the Waters above the Terre$trial Globe would reach and pa$s a mile higher than the tops of the Hills and Mountains that are upon the $uperficies of the Earth; and they al$o would concur to increa$e the ri$e. And therefore I conclude, that the ri$e of the Waters of the Deluge have a rational congruity with natural Di$cour$es, of which I know very well that the eternal truths of the Divine leaves have no need; but however I think $o clear an agreement is worthy of our con$ideration, which gives us occa- $ion to adore and admire the greatne$$e of God in his mighty Works, in that we are $ometimes able, in $ome $ort, to mea$ure them by the $hort Standard of our Rea$on.</P> <P>Many Le$$ons al$o may be deduced from the $ame Doctrine, which I pa$$e by, for that every man of him$elf may ea$ily know them, having once $tabli$hed this Maxime; That it is not po$$i- ble to pronounce any thing, of a certainty, touching the quantity of Running Waters, by con$idering only the $ingle vulgar mea- $ure of the Water wichout the velocity; and $o on the contrary, he that computes only the velocity, without the mea$ure, $hall commit very great errours; for treating of the mea$ure of Run- ning Waters, it is nece$$ary, the water being a body, in handling its quantity, to con$ider in it all the three dimen$ions of breadth, depth, and length: the two fir$t dimen$ions are ob$erved by all in the common manner, and ordinary way of mea$uring Running Waters; but the third dimen$ion of length is omitted; and hap- ly $uch an over$ight is committed, by rea$on the length of Run- <foot>ning</foot> <p n=>33</p> ning Water is reputed in $ome $en$e infinite, in that it never cea- $eth to move away, and as infinite is judged incomprehen$ible; and $uch as that there is no exact knowledge to be had thereof; & $o there comes to be no account made thereof; but if we $hould make $trict reflection upon our con$ideration of the velocity of Water, we $hould find, that keeping account of the $ame, there is a reckoning al$o made of the length; fora$much as whil$t we $ay, the Water of $uch a Spring runs with the velocity of pa$$ing a thou$and or two thou$and paces an hour: this in $ub$tance is no other than if we had $aid, $uch a Fountain di$chargeth in an hour a Water of a thou$and or two thou$and paces long. So that, albeit the total length of Running water be incomprehen- $ible, as being infinite, yet neverthele$$e its rendered intelligible by parts in its velocity. And $o much $ufficeth to have hinted about this matter, hoping to impart on $ome other occa$ion other more accurate Ob$ervations in this affair.</P> <head><I>LAVS DEO.</I></head> <fig> <foot>Ffff</foot> <pb> <head>GEOMETRICAL DEMONSTRATIONS OF THE MEASURE OF Running Waters.</head> <head>BY D. BENEDETTO CASTELLI, Abbot of CASSINA, and Mathematician to P. <I>VRBAN. VIII.</I></head> <head>DEDICATED <I>To the mo$t Illu$trious, and mo$t Excellent Prince</I></head> <head>DON THADDEO BARBERINI, PRINCE OF PALESTRINA, AND GENERAL of the HOLY CHURCH.</head> <head><I>LONDON,</I> Printed <I>Anno Domini,</I> MDCLXI.</head> <p n=>37</p> <head>OF THE MENSURATION OF Running Waters.</head> <head>SUPPOSITION I.</head> <P>Let it be $uppo$ed, that the banks of the Rivers of which we $peak be erected perpendicular to the plane of the up- per $uperficies of the River.</P> <head>SUPPOSITION II.</head> <P>We $uppo$e that the plane of the bottome of the River, of which we $peak is at right angles with the banks.</P> <head>SUPPOSITION III.</head> <P>It is to be $uppo$ed, that we $peak of Rivers, when they are at ebbe, in that $tate of $hallowne$$e, or at flowing in that $tate of deepne$$e, and not in their tran$ition from the ebbe to the flowing, or fr m the flowing to the ebbe.</P> <head><I>Declaration of Termes.</I></head> <head>FIRST.</head> <P>If a River $hall be cut by a Plane at right angles to the $urface of the water of the River, and to the banks of the River, that $ame dividing Plane we call the Section of the River; and this Section, by the Suppo$itions above, $hall be a right angled Parallelogram.</P> <head>SECOND.</head> <P>We call tho$e Sections equally Swift, by which the water runs with equal velocity; and more $wift and le$s $wift that Section of another, by which the water runs with greater or le$$e velocity.</P> <foot>AX</foot> <p n=>38</p> <head>AXIOME I.</head> <P>Sections equal, and equally $wift, di$charge equal quantities of Water in equal times.</P> <head>AXIOME II.</head> <P>Sections equally $wift, and that di$charge equal quantity of Water, in equal time, $hall be equal.</P> <head>AXIOME III.</head> <P>Sections equal, and that di$charge equal quantities of Water in equal times, $hall be equally $wift.</P> <head>AXIOME IV.</head> <P>When Sections are unequal, but equally $wift, the quanti- ty of the Water that pa$$eth through the fir$t Section, $hall have the $ame proportion to the quantity that pa$- $eth through the Second, that the fir$t Section hath to the $econd Section. Which is manife$t, becau$e the velocity being the $ame, the difference of the Water that pa$$eth $hall be according to the difference of the Sections.</P> <head>AXIOME V.</head> <P>If the Sections $hall be equal, and of unequal velocity, the quantity of the Water that pa$$eth through the fir$t, $hall have the $ame proportion to that which pa$$eth through the $econd, that the velocity of the fir$t Section, $hall have to the velocity of the $econd Section. Which al$o is manife$t, becau$e the Sections being equal, the difference of the Water which pa$$eth, dependeth on the velocity.</P> <head><I>PETITION.</I></head> <P>A Section of a River being given, we may $uppo$e another equal to the given, of different breadth, heigth, and ve- locity.</P> <foot>PRO.</foot> <p n=>37</p> <head>PROPOSITION I.</head> <P><I>The Sections of the $ame River di$charge equal quan- tities of Water in equal times, although the Secti- ons them$elves he unequal.</I></P> <P>Let the two Sections be A and B, in the River C, running from A, towards B; I $ay, that they di$charge equal quan- tity of Water in equal times; for if greater quantity of Wa- ter $hould pa$s through A, than pa$$eth through B, it would <fig> follow that the Water in the intermediate $pace of the River C, would increa$e continually, which is manife$tly fal$e, but if more Water $hould i$$ue through the Section B, than entreth at the Section A, the Water in the intermediate $pace C, would grow continually le$s, and alwaies ebb, which is likewi$e fal$e; therefore the quantity of Water that pa$$eth through the Secti- on B, is equal to the quantity of Water which pa$$eth through the Section A, and therefore the Sections of the $ame River di$- charge, <I>&c.</I> Which w s to be demon$trated.</P> <head>PROPOSITION II.</head> <P><I>In two Sections of Rivers, the quantity of the Water which pa$$eth by one Section, is to that which pa$- $eth by the $econd, in a Proportion compounded of the proportions of the fir$t Section to the $econd, and of the velocitie through the first, to the velocitie of the $econd.</I></P> <P>I Et A, and B be two Sections of a River; I $ay, that the quantity of Water which pa$$eth through A, is to that which pa$$eth through B, in a proportion compounded of the pro- portions of the fir$t Section A, to the Section B; and of the velo- city through A, to the velocity through B: Let a Section be <p n=>40</p> $uppo$ed equal to the Section A, in magnitude; but of velocity equal to the Section B, and let it be G, and as the Section A is <fig> to the Section B, $o let the line F be to the line D; and as the velocity A, is to the velocity by B, $o let the line D be to the line R: Therefore the Water which pa$$eth thorow A, $hall be to that which pa$$eth through G (in regard the Sections A and G are of equal bigne$s, but of unequal velocity) as the velocity through A, to the velocity through G; But as the velocity through A, is to the velocity through G, $o is the velocity through A, to the velocity through B; namely, as the line D, to the line R: therefore the quantity of the Water which pa$$e the through A, $hall be to the quantity which pa$$eth through G, as the line D is to the line R; but the quantity which pa$$eth through G, is to that which pa$$eth through B, (in regard the Sections G, and B, are equally $wift) as the Section G to the Se- ction B; that is, as the Section A, to the Section B; that is, as the line F, to the line D: Therefore by the equal and perturbed proportionality, the quantity of the Water which pa$$eth through A, hath the $ame proportion to that which pa$$eth through B, that the line F hath to the line R; but F to R, hath a proportion compounded of the proportions of F to D, and of D to R; that is, of the Section A to the Section B; and of the velocity through A, to the velocity through B. Therefore al$o the quantity of Water which pa$$eth through the Section A, $hall have a propor- tion to that which pa$$eth through the Section B, compounded of the proportions of the Section A, to the Section B; and of the velocity through A, to the velocity through B: And therefore in two Sections of Rivers, the quantity of Water which pa$$eth by the fir$t, <I>&c.</I> which was to be demon$trated.</P> <head><I>COROLLARIE.</I></head> <P>The $ame followeth, though the quantity of the Water which pa$$eth through the Section A, be equal to the quantity of Water which pa$$eth through the Section B, as is manife$t by the $ame demon$tration.</P> <foot>PROPO-</foot> <p n=>41</p> <head>PROPOSITION III.</head> <P><I>In two Sections unequal, through which pa$s equal quantities of Water in equal times, the Sections have to one another, reciprocal proportion to their velocitie.</I></P> <P>Let the two unequal Sections, by which pa$s equal quantities of Water in equal times be A, the greater; and B, the le$$er: I $ay, that the Section A, $hall have the $ame Proportion to the Section B, that reciprocally the velocity through B, hath to the velocity through A; for $uppo<*>ing that as the Water that pa$$eth through A, is to that which pa$$eth through B, $o is the <fig> line E to the line F: therefore the quantity of water which pa$- $eth through A, being equal to that which pa$$eth through B, the line E $hall al$o be equal to the line F: Suppo$ing moreover, That as the Section A, is to the Section B, $o is the line F, to the line G; and becau$e the quantity of water which pa$$eth through the Section A, is to that which pa$$eth through the Section B, in a proportion compo$ed of the proportions of the Section A, to the Section B, and of the velocity through A, to the velocity through B; therefore the line E, $hall be the line to F, in a proportion compounded of the $ame proportions; namely, of the proportion of the Section A, to the Section B, and of the ve- locity through A, to the velocity through B; but the line E, hath to the line G, the proportion of the Section A, to the Section B, therefore the proportion remaining of the line G, to the line F, $hall be the proportion of the velocity through A, to the velocity through B; therefore al$o the line G, $hall be to the line E, as the velocity by A, to the velocity by B: And conver$ly, the ve- locity through B, $hall be to the velocity through A, as the line E, to the line G; that is to $ay, as the Section A, to the Section B, and therefore in two Sections, &c. which was to be demon$trated.</P> <foot>Gggg <I>COROL-</I></foot> <p n=>42</p> <head><I>COROLLARIE.</I></head> <P>Hence it is manife$t, that Sections of the $ame River (which are no other than the vulgar mea$ures of the River) have betwixt them$elves reciprocal proportions to their veloci- ties; for in the fir$t Propo$ition we have demon$trated that the Sections of the $ame River, di$charge equal quantities of Water in equal times; therefore, by what hath now been demon$trated the Sections of the $ame River $hall have reciprocal proportion to their velocities; And therefore the $ame running water chan- geth mea$ure, when it changeth velocity; namely, increa$eth the mea$ure, when it decrea$eth the velocity, and decrea$eth the mea$ure, when it increa$eth the velocity.</P> <P>On which principally depends all that which hath been $aid above in the <I>Di$cour$e,</I> and ob$erved in the <I>Corollaries</I> and <I>Ap- pendixes</I>; and therefore is worthy to be well under$tood and heeded.</P> <head>PROPOSITION IV.</head> <P><I>If a River fall into another River, the height of the fir$t in its own Chanel $hall be to the height that it $hall make in the $econd Chanel, in a proportion compounded of the proportions of the breadth of the Chanel of the $econd, to the breadth of the Chanel of the fir$t, and of the velocitie acquired in the Chanel of the $econd, to that which it had in its proper and first Chanel.</I></P> <P>Let the River A B, who$e height is A C, and breadth C B, that is, who$e Section is A C B; let it enter, I $ay, into a- nother River as broad as the line E F, and let it therein make the ri$e or height D E, that is to $ay, let it have its Section in the River whereinto it falls D E F; I $ay, that the height A C hath to the height D E the proportion compounded of the pro- portions of the breadth E F, to the breadth C B, and of the ve- locity through D F, to the velocity through A B. Let us $up- po$e the Section G, equal in velocity to the Section A B, and in breadth equal to E F, which carrieth a quantity of Water e- qual to that which the Section A B carrieth, in equal times, and con$equently, equal to that which D F carrieth. Moreover, as the breadth E F is to the breadth C B, $o let the line H be to <foot>the</foot> <p n=>43</p> the line I; and as the velocity of D F is to the velocity of A B, $o let the line I be to the line L; becau$e therefore the two Sections A B and G are equally $wift, and di$charge equal quan- tity of Water in equal times, they $hall be equal Sections; and <fig> therefore the height of A <I>B</I> to the height of G, $hall be as the breadth of G, to the breadth of A <I>B,</I> that is, as E F to C <I>B,</I> that is, as the line H to the line I: but becau$e the Water which pa$$eth through G, is equal to that which pa$$eth through D E F, therefore the Section G, to the Section D E F, $hall have the re- ciprocal proportion of the velocity through D E F, to the velo- city through G; but al$o the height of G, is to the height D E, as the Section G, to the Section D E F: Therefore the height of G, is to the height D E, as the velocity through D E F, is to the velocity through G; that is, as the velocity through D E F, is to the velocity through A <I>B</I>; That is, finally, as the line I, to the line L; Therefore, by equal proportion, the height of <I>A B,</I> that is, A C, $hall be to the height D E; as H to L, that is, com- pounded of the proportions of the breadth E F, to the breadth C <I>B,</I> and of the velocity through D F, to the velocity through A <I>B</I>: So that if a River fall into another River, &c. which was to be demon$trated.</P> <foot>Gggg 2 PROPO-</foot> <p n=>44</p> <head>PROPOSITION V.</head> <P><I>If a River di$charge a certain quantitie of Water in a certain time; and after that there come into it a Flood, the quantity of Water which is di$char- ged in as much time at the Flood, is to that which was di$charged before, whil$t the River was low, in a proportion compounded of the proportions of the velocity of the Flood, to the velocity of the first Water, and of the height of the Flood, to the height of the first Water.</I></P> <P>Suppo$e a River, which whil$t it is low, runs by the Section AF; and after a Flood cometh into the $ame, and runneth through the Section D F, I $ay, that the quantity of the Wa- ter which is di$charged through D F, is to that which is di$charged <fig> through A F, in a proportion compounded of the proportions of the velocity through D F, to the velocity through A F, and of the height D <I>B,</I> to the height A <I>B</I>; As the velocity through DF is to the velocity through A F, $o let the line R, to the line S; and as the height D <I>B</I> is to the height A <I>B,</I> $o let the line S, to the line T; and let us $uppo$e a Section L M N, equal to D F in height and breadth; that is L M equal to D <I>B,</I> and M N equal to <I>B F</I>; but let it be in velocity equal to the Section A F, there- fore the quantity of Water which runneth through D F, $hall be to that which runneth through LN, as the velocity through DF, is to the velocity through L N, that is, to the velocity through <I>A F</I>; and the line R being to the line S, as the velocity through D <I>F,</I> to the velocity through <I>A F</I>; therefore the quantity which runneth through D <I>F,</I> to that which runneth through L N, $hall have the proportion of R to S; but the quantity which runneth through L N, to that which runneth through <I>A F,</I> (the Sections <foot>being</foot> <p n=>45</p> being equally $wift) $hall be in proportion as the Section <I>L</I> N, to the Section A F; that is, as D B, to A B; that is as the line S, to the line T: Therefore by equal proportion, the quantity of the water which runneth through D F, $hall be in proportion to that which runneth through A F, as R is to T; that is, compounded of the proportions of the height D B, to the height A B, and of the velocity through <I>D F,</I> to the velocity through <I>A F</I>; and therefore if a River di$charge a certain quantity, <I>&c.</I> which was to be de- mon$trated.</P> <head>ANNOTATION.</head> <P>The $ame might have been demon$trated by the $econd Propo$ition above demon$trated, as is manife$t.</P> <head>PROPOSITION VI.</head> <P><I>If two equal $treams of the $ame Torrent, fall into a River at divers times, the heights made in the Ri- ver by the Torrent, $hall have between them- $elves the reciprocal proportion of the velocities acquired in the River.</I></P> <P>Let A and B, be two equal $treams of the $ame Torrent, which falling into a River at divers times, make the heights C D, and F G; that is the $tream A, maketh the height C D, and the $tream B, maketh the height F G; that is, Let their Sections in the River, into which they are fallen, be C E, and FH; I $ay, that the height C D, $hall be to the height F G, in reciprocal proportion, as the velocity through F H, to the ve- locity through C E; for the quantity of water which pa$$eth through A, being equal to the quantity which pa$$eth through B, in equal times; al$o the quantity which pa$$eth through C E, $hall <fig> be equal to that which pa$$eth through F H: And therefore the proportion that the Section C E, hath to the Section F H; $hall be the $ame that the velocity through F H, hath to the velocity through C E; But the Section C E, is to the Section F H, as C D, to F G, by rea$on they are of the $ame breadth: Therefore C D, $hall be to F G, in reciprocal proportion, as the velocity through F H, is to the velocity through C E, and therefore if two equal $treams of the $ame Torrent, <I>&c.</I> which was to be de- mon$trated.</P> <p n=>47</p> <head>OF THE MENSURATION OF Running Waters.</head> <head><I>Lib.</I> II.</head> <P>Having, in the clo$e of my Treati$e of the Men$uration of Running Waters promi$ed to declare upon another occa$ion other par- ticulars more ob$cure, and of very great concern upon the $ame argumement: I now do perform my promi$e on the occa$ion that I had the pa$t year 1641. to propound my thoughts touching the $tate of the Lake of <I>Venice,</I> a bu$ine$s certainly mo$t important, as being the concernment of that mo$t noble and mo$t admirable City; and indeed of all <I>Italy,</I> yea of all <I>Europe, A$ia, & Africa</I>; & one may truly $ay of all the whole World. And being to proceed according to the method nece$$ary in Sciences, I wil propo$e, in the fir$t place certain Definitions of tho$e Terms whereof we are to make u$e in our Di$cour$e: and then, laying down certain Principles we will demon$trate $ome Problemes and Theoremes nece$$ary for the under$tanding of tho$e things which we are to deliver; and moreover, recounting $undry ca$es that have happened, we will prove by practice, of what utility this contemplation of the Mea$ure of Running Waters is in the more important affairs both Publique and Private.</P> <head>DEFINITION I.</head> <P>Two Rivers are $aid to move with equal velocity, when in e- qual times they pa$$e $paces of equal length.</P> <head>DEFINITION II.</head> <P>Rivers are $aid to move with like velocity, when their propor- tional parts do move alike, that is, the upper parts alike to the upper, and the lower to the lower; $o that if the upper part of one River $hall be more $wift than the upper part of ano- ther; then al$o the lower part of the former $hall be more $wift than the part corre$pondent to it in the $econd, proportionally.</P> <foot><I>DEFI-</I></foot> <p n=>48</p> <head>DEFINITON III.</head> <P>To mea$ure a River, or running Water, is in our $en$e to finde out how many determinate mea$ures, or weights of Water in a given time pa$$eth through the River, or Channel of the Water that is to be mea$ured.</P> <head>DEFINITION IV.</head> <P>If a Machine be made either of Brick, or of Stone, or of Wood, $o compo$ed that two $ides of the $aid Machine be placed at right angles upon the ends of a third $ide, that is $uppo$ed to be placed in the bottom of a River, parallel to the Horizon, in $uch a manner, that all the water which runneth through the $aid River, pa$$eth thorow the $aid Machine: And if all the water coming to be diverted <fig> that runneth through the $aid River, the upper $uperficies of that third $ide placed in the bottom do remain uncovered and dry, and that the dead water be not above it; This $ame Machine $hall be <marg>* Or Sluice.</marg> called by us ^{*} REGULATOR: And that third $ide of the Machine which $tandeth Horizontally is called the bottom of the Regulator; and the other two $ides, are called the banks of the Regulator; as is $een in this fir$t Figure: A B C D, $hall be the Regulator; B C the bottom; and the other two $ides A B, and C D are its banks.</P> <head>DEFINITION V.</head> <P>By the quick height, we mean the Perpendicular from the upper $uperficies of the River, unto the upper $uperficies of the bot- tom of the Regulator; as in the foregoing Figure the line. G H.</P> <head>DEFINITION VI.</head> <P>If the water of a <I>R</I>iver be $uppo$ed to be marked by three $ides of a Regulator, that Rightangled Parallelogram compre- hended between the banks of the Regulator, and the bottom, and the $uperficies of the Water is called a Section of the River.</P> <foot>ANNOTA-</foot> <p n=>49</p> <head>ANNOTATION.</head> <P>Here it is to be noted, that the River it $elf may have $undry and divers heights, in $everal parts of its Chanel, by rea$on of the various velocities of the water, and its mea$ures; as hath been demon$trated in the fir$t book.</P> <head>SUPPOSITION I.</head> <P>It is $uppo$ed, that the Rivers equal in breadth, and quick height, that have the $ame inclination of bed or bottom, ought al$o to have equal velocities, the accidental impediments being removed that are di$per$ed throughout the cour$e of the water, and ab$tracting al$o from the external windes, which may velo- citate, and retard the cour$e of the water of the River.</P> <head>SUPPOSITION II.</head> <P>Let us $uppo$e al$o, that if there be two Rivers that are in their beds of equal length, and of the $ame inclination, but of quick heights unequal, they ought to move with like velocity, according to the $en$e explained in the $econd definition.</P> <head>SUPPOSITION III.</head> <P>Becau$e it will often be requi$ite to mea$ure the time exactly in the following Problems, we take that to be an excellent way to mea$ure the time, which was $hewed me many years $ince by <I>Signore Galilæo Galilæi,</I> which is as followeth.</P> <P>A $tring is to be taken three Roman feet long, to the end of which a Bullet of Lead is to be hanged, of about two or three ounces; and holding it by the other end, the Plummet is to be removed from its perpendicularity a Palm, more or le$s, and then let go, which will make many $wings to and again, pa$$ing and repa$$ing the Perpendicular, before that it $tay in the $ame: Now it being required to mea$ure the time that is $pent in any what- $oever operation, tho$e vibrations are to be numbred, that are made whil$t the work la$teth; and they $hall be $o many $econd minutes of an hour, if $o be, that the $tring be three Roman feet long, but in $horter $trings, the vibrations are more frequent, and in longer, le$s frequent; and all this $till followeth, whether the Plummet be little or much removed from its Perpendicularity, or whether the weight of the Lead be greater or le$$er.</P> <P>The$e things being pre-$uppo$ed, we will lay down $ome fa- <foot>Hhhh miliar</foot> <p n=>50</p> miliar Problems, from which we $hall pa$s to the Notions and que$tions more $ubtil and curious; which will al$o prove profi- table, and not to be $leighted in this bu$ine$s of Waters.</P> <head>PROPOSITION I. PROBLEME I.</head> <P><I>Achanel of Running-Water being given, the breadth of which pa$sing through a Regulator, is three Palms; and the height one Palm, little more or le$s, to mea$ure what water pa$$eth through the Regulator in a time given.</I></P> <P>Fir$t, we are to dam up the Chanel; $o that there pa$s not any water below the Dam; then we mu$t place in the $ide of the Chanel, in the parts above the Regulator three, or four, or five Bent-pipes, or Syphons, according to the quantity of the water that runneth along the Chanel; in $uch $ort, as that they may drink up, or draw out of the Chanel all the water that the Cha- nel beareth (and then $hall we know that the Syphons drink up all the water, when we $ee that the water at the Dam doth nei- ther ri$e higher, nor abate, but alwaies keepeth in the $ame Le- vel.) The$e things being prepared, taking the In$trument to mea$ure the time, we will examine the quantity of the water that i$$ueth by one of tho$e Syphons in the $pace of twenty vibrations, and the like will we do one by one with the other Syphons; and then collecting the whole $umme, we will $ay, that $o much is the water that pa$$eth and runneth thorow the Regulator or Chanel (the Dam being taken away) in the $pace of twenty $e- cond minutes of an hour; and calculating, we may ea$ily reduce it to hours, dayes, months, and years: And it hath fallen to my turn to mea$ure this way the waters of Mills and Fountains, and I have been well a$$ured of its exactne$s, by often repeating the $ame work.</P> <head>CONSIDERATION.</head> <P>And this method mu$t be made u$e of in mea$uring the waters, that we are to bring into Conducts, and carry into Cities and Ca$tles, for Fountains; and that we may be able afterwards to divide and $hare them to particular per$ons ju$tly; which will prevent infinite $uits and controver$ies that every day happen in the$e matters..</P> <foot>PROPO-</foot> <p n=>51</p> <head>PROPOSITION II. THEOREM I.</head> <P><I>If a River moving with $uch a certain velocitie through its Regulator, $hall have a given quick height, and afterwards by new water $hall increa$e to be double, it $hall al$o increa$e double in ve- locitie.</I></P> <P>Let the quick height of a River in the Regulator A B C D, be the perpendicular F B, and afterwards, by new water that is added to the River, let the water be $uppo$ed to be rai$- ed to G, $o that G B may be double to E B. I $ay, that all the water G C $hall be double in velocity to <fig> that of E C: For the water G F, having for its bed the bottom E F, equally in- clined as the bed B C, and its quick height G E being equal to the quick height E C, and having the $ame breadth B C, it $hall have of it $elf a velocity e- qual to the velocity of the fir$t water F C: but becau$e, be$ides its own moti- on, which is imparted to it by the motion of the water E C, it hath al$o over and above its own motion, the motion of E C. And becau$e the two waters G C, and E C, are alike in velocity, by the third Suppo$ition; therefore the whole water G C $hall be double in velocity to the water E C; which was that which we were to demon$trate.</P> <P><I>This demon$tration is not here in$erted, as perfect, the Authour ha- ving by $everal letters to his friends confe$$ed him$elf un$atisfi- ed therewith; and that he intended not to publi$h the</I> Theorem <I>without a more $olid demon$tration, which he was in hope to light upon. But being overtaken by Death, he could not give the fini$hing touch either to this, or to the rest of the $econd Book. In con$ideration of which, it $eemed good to the Publi$her of the $ame, rather to omit it, than to do any thing contrary to the mind of the Authour. And this he hints, by way of adverti$ement, to tho$e that have Manu$cript Copies of this Book, with the $aid de- mon$tration. For this time let the Reader content him$elf with the knowledge of $o ingenious and profitable a Conclu$ion; of the truth of which he may, with $mall expence and much plea$ure, be a$$ured by means of the experiment to be made in the $ame man- ner, with that which is laid down in the $econd Corollary of</I> <foot>Hhhh 2 <I>the</I></foot> <p n=>52</p> <I>the fourth</I> Theorem <I>of this, with its Table, and the u$e there- of annexed.</I></P> <head><I>COROLLARIE</I></head> <P>Hence it followeth, that when a River increa$eth in quick height by the addition of new water, it al$o increa$eth in ve- locity; $o that the velocity hath the $ame proportion to the velo- city that the quick height hath to the quick height; as may be demon$trated in the $ame manner.</P> <head>PROPOS. III. PROBLEME II.</head> <P><I>Achanel of Water being given who$e breadth exceeds not twenty Palms, or thereabouts, and who$e quick beight is le$s than five Palms, to mea$ure the quantity of the Water that runneth thorow the Chanel in a time given.</I></P> <P>Place in the Chanel a Regulator, and ob$erve the quick height in the $aid Regulator; then let the water be turned away from the Chanel by a Chanellet of three or four Palms in breadth, or thereabouts: And that being done, mea$ure the quantity of the water which pa$$eth thorow the $aid Chanellet, as hath been taught in the $econd Propo$ition; and at the $ame time ob$erve exactly how much the quick height $hall be abated in the greater Chanel, by means of the diver$ion of the Chancl- let; and all the$e particulars being performed, multiply the quick height of the greater Chanel into it $elf, and likewi$e multiply into it $elf the le$$er height of the $aid bigger Chanel, and the le$$er $quare being taken, from the greater, the remainder $hall have the $ame proportion to the whole greater $quare, as the wa- ter of the Chanellet diverted, hath to the water of the bigger Chanel: And becau$e the water of the Chanellet is known by the Method laid down in the fir$t Theorem, and the terms of the Theorem being al$o known, the quantity of the water which run- neth thorow the bigger Chanel, $hall be al$o known by the Gol- den <I>R</I>ule, which was that that was de$ired to be known. We will explain the whole bu$ine$s by an example.</P> <P>Let a Chanel be, for example, 15 Palms broad, its quick height before its diver$ion by the Chanellet $hall be $uppo$ed to be 24 inches; but after the diver$ion, let the quick height of the Chanel be onely 22 inches. Therefore the greater height to the le$$er, is as the number 11. to 12. But the $quare of 11. is 121, and the $quare of 12. is 144, the difference between the $aid le$$er <foot>$quare</foot> <p n=>53</p> $quare and the greater is 23. Therefore the diverted water, is to the whole water, as 23. to 144: which is well near as 1 to 6 6/23: and that is the proportion that the quantity of the water which runneth through the Chanellet $hall have, to all the water that runneth thorow the great Chanel. Now if we $hould finde by the Rule mentioned above in the fir$t Propo$ition, that the quantity of the water that runneth through the Chanellet, is <I>v. g.</I> an hundred Barrels, in the $pace of 15 $econd minutes of an hour, it is manife$t, that the water which runneth through the great Chanel in the $aid time of 35 min. $ec. $hall be about 600 Barrels.</P> <head><I>The $ame operation performed another way.</I></head> <P>And becau$e very often in applying the Theory to Practice it happeneth, that all the nece$$ary particulars in the The- ory cannot $o ea$ily be put in execution; therefore we will here add another way of performing the $ame Problem, if it $hould chance to happen that the Chanellet could not commodiou$ly be diverted from the great Chanel, but that it were ea$ier for the water of another $maller Chanel to be brought into the greater Chanel; which water of the $maller Chanel might be ea$ily mea- $ured, as hath been $hewen in the fir$t Probleme; or in ca$e that there did fall into a greater Chanel, a le$$er Chanel that might be diverted and mea$ured. Therefore I $ay in the fir$t ca$e, If we would mea$ure the quantity of the water that runneth in a certain time thorow the greater Chanel, into which another le$$er Chanel that is mea$urable may be brought, we mu$t fir$t exactly mea$ure the Chanellet, and then ob$erve the quick height of the greater Chanel, before the introduction of the le$$er; and having brought in the $aid Chanellet, we mu$t agnin find the propor- tion that the water of the Chanellet hath to all the water of the great Ghanel; for the$e terms of the proportion being known, as al$o the quantity of the water of the Chanellet, we $hall al$o come to know the quantity of the water that runneth thorow the great Chanel. It is likewi$e manife$t, that we $hall obtain our intent, if the ca$e were that there entered into the great Chanel, another le$$er Chanel that was mea$urable, and that might be diverted.</P> <head>CONSIDERATION.</head> <P>It would be nece$$ary to make u$e of this Doctrine in the di- $tribution of the waters that are imploy'd to overflow the fields, as is u$ed in the <I>Bre$ciau, Cremone$e, Bergama$e, Lodigian, Mila-</I> <foot><I>ne$e,</I></foot> <p n=>54</p> <I>ne$e</I> territories, and many other places, where very great $uits and differences ari$e, which not being to be determined with in- telligible rea$ons, come oftentimes to be decided, by force of armes; and in$tead of flowing their Grounds with Waters, they cruelly flow them with the $hedding of humane blood, impiou$ly inverting the cour$e of Peace and Ju$tice, $owing $uch di$orders and feuds, as that they are $ometimes accompanied with the ru- ine of whole Cities, or el$e unprofitably charge them with vain, and $ometimes prejudicial expences.</P> <head>PROPOS. IV. THEOR. II.</head> <P><I>If a River increa$e in quick height, the quantitie of Water which the River di$chargeth after the in- crea$e, hath the Proportion compounded of the Proportions of the Quick height to the Quick height, and of the velocity to the velocity.</I></P> <P>Let there be a River, which whil$t it is low, runneth thorow the Regulator D F, with the Quick height A B, and after- wards let a Flood come; and then let it run with the height D B, I $ay, that the quantity of the Water that is di$charged through D F, to that which di$chargeth through A F, hath the proportion compounded of the proportions of the velocity through D F to the velocity through A F, and of the height D B to the height A B. As the velocity through D F is to the velocity through A F, $o let the line R be to the line S; and as the height D B is to the height A B; $o let the line S be to the <fig> line T. And let a Section be $uppo$ed L M N equal to the Section D F in height and length, but let it be in velocity equal to the Section AF. Therefore the quantity of the Water that run- neth through D F to that which runneth through L N, $hall be <foot>as</foot> <p n=>55</p> as the velocity through D F, to the velocity of L N, that is, to the velocity through L N, that is, to the velocity through <I>A F.</I> therefore the quantity of Water which runneth through D <I>F,</I> to that which pa$$eth through L N, $hall have the proportion that R hath to S; but the quantity of the Water that runneth through L N, to that which runneth through <I>A F</I>; (the Sections being equally $wift) $hall have the proportion that the Section L N hath to the Section A F, that is, that the height <I>B</I> D hath to the height <I>B</I> A, that is, that S hath to T. Therefore, by equal proportion, the quantity of the Water which runneth by D F, to that which runneth by A F, $hall have the proportion of R to T, that is, $hall be compounded of the proportions of the height D <I>B,</I> to the height A <I>B</I>; and of the velocity through D F, to the velocity through A F. And therefore if a River increa$e in quick height, the quantity of the Water that runneth after the increa$e, to that which runneth before the increa$e, hath the proportion compounded, &c. Which was to be demon$trated.</P> <head><I>COROLLARIE I.</I></head> <P>Hence it followeth, that we having $hewn, that the quantity of the Water which runneth, whil$t the River is high, to that which ran, whil$t it was low, hath the proportion compounded of the velocity to the velocity, and of the height to the height. And it having been demon$trated, that the velocity to the velo- city is as the height to the height; it followeth, I $ay, that the quantity of the Water that runneth, whil$t the River is high, to that which runneth, whil$t it is low, hath duplicate proportion of the height to the height, that is, the proportion that the $quares of the heights have.</P> <head><I>COROLLARIE II.</I></head> <P>Vpon which things dependeth the rea$on of that which I have $aid, in my $econd Con$ideration, that if by the diver$ion of 5/9 of the Water that entereth by the Rivers into the Moor or Fen, the Water be abated $uch a mea$ure, that $ame $hall be only one third of its whole height; but moreover diverting the 4/9, it $hall abate two other thirds, a mo$t principal point; and $uch, that its not having been well under$tood, hath cau$ed very great di$orders, and there would now, more than ever, follow extream dammage, if one $hould put in execution the diver$ion of the <I>Sile</I> and other Rivers; and it is manife$t, that in the $ame manner, wherewith it hath been demon$trated, that the quantity of the Water increa$ing quadruple, the height would increa$e onely <foot>double</foot> <p n=>56</p> double, and the quantity increa$ing nonuple, the height increa- $eth triple; $o that, by adding to units all the odde numbers, ac- cording to their Series, the heights increa$e according to the na- tural progre$$ion of all the numbers, from units. As for exam- ple, there pa$$ing thorow a Regulator $uch a certain quantity of Water in one time; adding three of tho$e mea$ures, the quick height is two of tho$e parts, which at fir$t was one; and con- tinuing to adde five of tho$e $aid mea$ures, the height is three of tho$e parts which at fir$t were one; and thus adding $even, and then nine, and then 11. and then 13, &c. the heights $hall be 4. then 5, then 6. then 7, &c. And for the greater facility of the Work, we have de$cribed the following Table, of which we will declare the u$e: The Table is divided into three Series or Pro- gre$$ions of Numbers: the fir$t Series containeth all the Num- bers in the Natural Progre$$ion, beginning at a Unit, and is called the Series of the Heights; the $econd containeth all the odde numbers, beginning at an unit, and is called the Series of the Additions: the third containeth all the $quare numbers, begin- ning at an unit, and is called the Series of Quantity.</P> <table> <row><col>Heights.</col><col>1</col><col>2</col><col>3</col><col>4</col><col>5</col><col>6</col><col>7</col><col>8</col><col>9</col><col>10</col><col>11</col></row> <row><col>Additions.</col><col>1</col><col>3</col><col>5</col><col>7</col><col>9</col><col>11</col><col>13</col><col>15</col><col>17</col><col>19</col><col>21</col></row> <row><col>Quantities.</col><col>1</col><col>4</col><col>9</col><col>16</col><col>25</col><col>36</col><col>49</col><col>64</col><col>81</col><col>100</col><col>121</col></row> </table> <head><I>The u$e of the afore-mentioned Table.</I></head> <P>Fir$t, if we $uppo$e the whole quick height of a River of Run- ning Water to be divided into any number of equal parts, at plea$ure, and would abate the $ame one fift, by means of a divi- $ron; let there be found in the Table in the Series of heights the number 5. the denominator of the part which the River is to a- bate, and take the number that is immediately under it in the row of Additions, which is 9. which let be $ub$tracted from the number 25. placed underneath the $ame in the row of Quanti- ties, the remainder 16. $ignifieth that of the 25. parts of Water that ran in the River, whil$t it was 5 mea$ures high, there do onely run 16. parts; $o that to make it abate 1/5 it is nece$$ary to take 9/25 from the Water that the whole River did carry; $o that with $ub$tracting $omewhat more than one third of the Water of the River, it is abated but only one fift.</P> <P>2. And thus, in the $econd place, if on the contrary, one would know how much water is to be added to the $aid River to make it increa$e one fift more in height, $o as that it may run in the <foot>Regulator</foot> <p n=>57</p> Regulator 6. of tho$e parts high; of which it ran before but 5. let 6 be found in the row of heights, and let the number 11. $tand- ing under the $ame be taken and added to the number 25. that is placed under the number 9. in the Additions, and 5. in the heights, and you $hall have 36; which is the quantity of the water that runneth with the height of the River, when it is high 6 of tho$e parts, whereof it was before but 5.</P> <P>3. But if it $hould be de$ired, to know how much water it is requi$ite to add to make the River ri$e $o, as that it may run in height 8. of tho$e parts of which before it ran but 5; one ought to take the $um of the number of the Series of Additions $tanding under 8. 7. and 6, which are 15. 13. and 11. that is, 39. and this $hall be the $umme that mu$t be added to 25: So that to make the River to run 8. of tho$e parts in height, of which it before did run 5, it will be nece$$ary to add 39. of tho$e parts, of which the River before was 25.</P> <P>4. Likewi$e the $ame Table giveth the quantity of water that runneth from time to time through a River, that increa$eth by the addition of new water to the $ame in one of its heights, the quantity of its water be known. As for example: If we knew that the River in one minute of an hour di$chargeth 2500. of tho$e mea- $ures of water, and runneth in height 5. parts in the Regulator, and afterwards $hould $ee that it runneth 8 Palms high, finding in the row of quantity the number placed under 8. which is 64. we would $ay that the River heightned, carrieth of water 64. of tho$e parts whereof it carried before but 25; and becau$e before it carried 2500. mea$ures, by the Golden Rule we will $ay, that the River carrieth 6400. of tho$e mea$ures, of which before it carried 2500.</P> <P>In this progre$s of Nature, is one thing really curious, and that at fir$t $ight $eemeth to be $omewhat Paradoxal, that we pro- ceeding ordinately in the diver$ions and additions, with additi- ons and diver$ions $o unequal, the abatings do notwith$tanding alwaies prove equal, and $o do the ri$ings: And who would ever think that a River in height, <I>v. g.</I> 10. Palms, running and carry- ing an hundred mea$ures in a minute of an hour, is to abate but one Palm, onely by the diver$ion of 19. of tho$e mea$ures; and then again, that the bui$ine$s cometh to that pa$s, that it abateth likewi$e a Palm by the diver$ion of three onely of tho$e mea$ures, nay, by the diver$ion of but one mea$ure? and yet it is mo$t certain: And this truth meets with $o manife$t proofs in experi- ence, that it is very admirable! And for the full $atisfaction of tho$e, who not being able to comprehend $ubtil demon$trati- ons, desire to be clearly inform'd by the matters of fact, and to $ee with their bobily eyes, and touch with their hands, what their under$tanding and rea$on cannot reach unto: I will hear add another very ea$ie way to reduce all to an experiment, the <foot>Iiii which</foot> <p n=>58</p> which may be made in little, in great, or in very great; of which I make u$e frequently, to the admiration of $uch as $ee it.</P> <P>I prepared an hundred Siphons, or, if you will, bowed Pipes, all equal; and placed them at the brim of a Ve$$el, wherein the water is kept at one and the $ame level (whether all the Syphons work, or but a certain number of them) the mouths by which the water i$$ueth being all placed in the $ame level, parallel to the Horizon; but lower in level than the water in the Ve$$el; and gathered all the water falling from the Syphons into another Ve$$el $tanding lower than the former, I made it to run away thorow a Chanel, in $uch manner inclined, that wanting water from the Syphons, the $aid Chanel remained quite dry.</P> <P>And this done, I mea$ured the quick height of the Chanel with care, and afterwards divided it exactly into 10 equal parts, and cau$ing 19. of tho$e Syphons to be taken away, $o that the Chanel did not run water, $ave onely with 81 of tho$e Syphons, I again ob$erved the quick height of the water in the $ame $ite ob$erved before, and found that its height was dimini$hed pre- ci$ely the tenth part of all its fir$t height; and thus continuing to take away 17. other Syphons, the height was likewi$e dimini$h- ed 1/1. of all its fir$t quick height; and trying to take away 15. Syphons, then 13, then 11, then 9, then 7, then 5, and then 3. alwaies in the$e diver$ions, made in order as hath been $aid, there en$ued $till an abatement of 1/1. of the whole height.</P> <P>And here was one thing worthy of ob$ervation, that the water encrea$ing in [<I>or through</I>] the Chanel, its quick height was diffe- rent in different $ites of the Chanel, that is $till le$$er, the more one approached to the Out-let; notwith$tanding which the abate- ment followed in all places proportionably, that is in all its $ites the fir$t part of the height of that $ite dimini$hed: And more- over the water i$$ued from the Chanel, and dilated into a broader cour$e, from which likewi$e having divers Out-lets and Mouths; yet neverthele$s in that breadth al$o the quick heights $ucce$$ive- ly varied and altered in the $ame proportions. Nor did I here de$i$t my ob$ervation, but the water being dimini$hed, that i$$u- ed from the Syphons, and there being but one of them left that di$charged water; I ob$erved the quick height that it made in the above-$aid $ites, (the which was likewi$e 1/1. of all the fir$t height) there being added to the water of that Syphon, the water of three other Syphons; $o that all the water was of 4 Syphons, and con$equently quadruple to the fir$t Syphon; but the quick height was onely double, and adding five Siphons, the quick height became triple, and with adding $even Syphons, the height increa$ed quadruple; and $o by adding of 9. it increa$ed quin- tuple, and by adding of 11. it increa$ed $extuple, and by ad- <foot>ding</foot> <p n=>59</p> ding of 13. it increa$ed $eptuple, and by adding of 15. octuple, and by adding of 17. nonuple, and la$tly by adding 19. Syphons; $o that all the water was centuple to the water of one Syphon, yet neverthele$s the quick height of all this water was onely de- cuple to the fir$t height conjoyned by the water that i$$ued from one onely Syphon.</P> <P>For the more clear under$tanding of all which, I have made the following Figure; in which we have the mouth A, that maintaineth the water of the Ve$$el B C in the $ame level; though it continually run; to the brim of the Ve$$el are put 25. Sy- phons (and there may be many more) divided into 5 Cla$$es, D E F G H, and the fir$t D, are of one onely Syphon; the $econd E, of three Syphons; the third F, of five; the fourth G, of 7; the fifth H, of 9; and one may $uppo$e the $ixth of 11, the $eventh of 13 Syphons, and $o of the other Cla$$es, all containing in con- $equent odd numbers $ucce$$ively (we are content to repre$ent in the Figure no more but the five forenamed Cla$$es to avoid con- fu$ion) the gathered water D E F G H, which runneth thorow the Chanel I K L, and falleth into the out-let M N O P; and $o much $ufficeth for the explanation of this experiment.</P> <fig> <foot>Iiii 2 PROP.</foot> <p n=>60</p> <head>PROPOS. V. PROB. III.</head> <P><I>Any River of any bigne$s, if being given to examine the quantity of the Water that runneth thorow the River in a time a$$igned.</I></P> <P>By what we have $aid already in the two preceding Pro- blems, we may al$o re$olve this that we have now before us; and it is done, by diverting in the fir$t place from the great River a good big mea$urable Chanel, as is taught in the $econd Probleme, and ob$erving the abatement of the River, cau$ed by the diver$ion of the Chanel; and finding the proporti- on that the Water of the Chanel hath to that of the River, then let the Water of the Chanel be mea$ured by the $econd Pro- bleme, and work as above, and you $hall have your de$ire.</P> <head>CONSIDERATION. I.</head> <P>And although it $eemeth as if it might prove difficult, and almo$t impo$$ible to make u$e of the Regulator number, if one be about to mea$ure the water of $ome great River, and con$equently would be impo$$ible, or at lea$t very difficult to reduce the Theory of the fir$t Probleme into practice: Yet ne- verthele$s, I could $ay that $uch great conceits of mea$uring the water of a great River, are not to come into the minds of any but great Per$onages, and potent Princes; of whom it is expected for their extraordinary concerns, that they will make the$e kinde of enquiries; as if here in <I>Italy</I> it $hould be of the Rivers <I>Tyber, Velino, Chiana, Arno, Serchio, Adice,</I> in which it $eemeth real- ly difficult to apply the <I>R</I>egulator, to finde exactly the quick height of the <I>R</I>iver: But becau$e in $uch like ca$es $ometimes it would turn to account to be at $ome charge, to come to the exact and true knowledge of the quantity of water which that <I>R</I>iver carrieth, by knowledge whereof, other greater di$- bur$ments might afterwards be avoided, that would oft times be made in vain; and prevent the di$gu$ts, which $ometimes happen among$t Princes: Upon this ground I think it will be well to $hew al$o the way how to make u$e of the <I>R</I>egulator in the$e great <I>R</I>ivers; in which if we will but open our eyes, we $hall meet with good ones, and tho$e made without great co$t or labour, which will $erve our turn.</P> <P>For upon $uch like <I>R</I>ivers there are Wears, or Lockes made, <foot>to</foot> <p n=>61</p> to cau$e the Waters to ri$e, and to turn them for the $ervice of Mills, or the like. Now in the$e Ca$es it is $ufficient, that one erect upon the two extreames of the Weare two Pila$ters either of Wood or Brick, which with the bottome of the Weare do compo$e our Regulator, wherewith we may make our de$ired operation, yea the Chanel it $elf diverted $hall $erve, without making any other diver$ion or union. And in brief, if the bu- fine$$es be but managed by a judicious per$on, there may wayes and helps be made u$e of, according to occa$ion, of which it would be too tedious to $peak, and therefore this little that hath been hinted $hall $u$$ice.</P> <head>CONSIDERATION II.</head> <P>From what hath been declared, if it $hall be well under- $tood, may be deduced many benefits and conveniences, not onely in dividing of Running Waters for infinite u$es that they are put to in turning of Corne-Mills, Paper-Mills, Gins, Powder-Mills, Rice-Mills, Iron Mills, Oil-Mills, Saw- ing-Mills, Mirtle-Mills, Felling-Mills, Fulling-Mills, Silk-Mills, and $uch other Machines; but al$o in ordering Navigable Cha- nels, diverting Rivers and Chanels of Waters, or terminating and limiting the $izes of Pipes for Fountains: In all which af- fairs there are great errours committed, to the lo$$e of much expence, the Chanels and Pipes that are made, $ometimes not being $ufficient to carry the de$igned Waters, and $ometimes they are made bigger than is nece$$ary; which di$orders $hall be avoided, if the Engineer be advi$ed of the things above$aid: and in ca$e that to the$e Notions there be added the knowledge of Philo$ophy and Mathematicks, agreeable to the $ublime Di$co- veries of <I>Signore Galilæo,</I> and the further improvement thereof by <I>Signore Evangeli$ta Torricelli,</I> Mathematician to the Grand <I>Duke of Tu$cany,</I> who hath $ubtilly and admirably handled this whole bu$ine$$e of Motion, one $hall then come to the know- ledge of particular notions of great curio$ity in the Theoricks, and of extraordinary benefit in the Practicks that daily occur in the$e bu$ine$$es.</P> <P>And to $hew, in effect, of what utility the$e Notions are, I have thought fit to in$ert, in this place, the Con$iderations by me made upon the Lake of <I>Venice,</I> and to repre$ent, at large, by the experience of the la$t year 1641. the mo$t Se- rene <I>Erizzo,</I> then Duke of the $aid Republique. Being therefore at <I>Venice,</I> in the year afore$aid, I was reque$ted by the mo$t Illu$trious and mo$t Excellent <I>Signore Giovanni Ba$a-</I> <foot><I>donna</I></foot> <p n=>62</p> <I>donna,</I> a Senatour of great worth and merit, that I would inge- nuou$ly deliver my opinion touching the $tate of the Lake of <I>Venice</I>; and after I had di$cour$ed with his Honour $eve- ral times, in the end I had order to $et down the whole bu$ine$$e in writing, who having afterwards read it privately, the $aid <I>Signore</I> imparted the $ame, with like privacy, to the mo$t Serene PRINCE, and I received order to repre$ent the $ame to the full <I>Colledge,</I> as accordingly I did in the Moneth of <I>May,</I> the $ame year, and it was as followeth.</P> <fig> <p n=>63</p> <head>CONSIDER ATIONS Concerning the LAKE OF VENICE. BY</head> <head>D. BENEDETTO CASTELLI, Abbot of S. <I>Benedetto Aloy$io,</I> Mathematician to Pope <I>VR BAN VIII.</I> and Profe$$or in ROME.</head> <head><I>CONSIDERATION I.</I></head> <P>Though the principal cau$e be but one onely, that in my judgment threatneth irreparable ruine to the Lake of <I>Venice,</I> in the pre$ent $tate in which it now $tands; Yet neverthele$$e, I think that two Heads may be con$idered. And this Con$ideration may peradven- ture $erve us for to facilitate and explain the opportune remedies, though not to render the $tate of things ab$olutely unchangeable and eternal: an enterprize impo$$ible, and e$pecially in that which having had $ome beginning, ought likewi$e nece$$arily to have its end; or at lea$t to prevent the danger for many hundreds of years; and po$$ibly it may, in the mean time, by the mutation it $elf be brought into a better condition.</P> <P>I $ay therefore, that the pre$ent di$order may be con$idered under two Heads; One is the very notable di$covery of Land that is ob$erved at the time of low Water, the which, be$ides the ob$tructing of Navigation in the Lake and al$o in the Chanels, doth likewi$e threaten another mi$chief and di$order <foot>worthy</foot> <p n=>64</p> worthy of very particular con$ideration, which is, That the Sun drying up that mudde, e$pecially in the times of hot Summers, doth rai$e thence the putrified and pernicious vapours, fogs, and exhalations that infect the Air, and may render the City unha- bitable.</P> <P>The $econd Head is the great Stoppage that daily is grow- ing in the Ports, e$pecially of <I>Venice,</I> at <I>Malamoco</I>; concerning which matters I will hint certain general points, and then will proceed to the more particular and important affairs.</P> <P>And fir$t, I $ay, that I hold it altogether impo$$ible to effect any thing, though never $o profitable, which doth not bring with it $ome mi$chief; and therefore the good and the hurt ought to be very well weighed, and then the le$$e harmful part to be im- braced.</P> <P>Secondly, I propo$e to con$ideration, that the $o notable di$- covery of Earth & Mud, hath not been long ob$erved, as I under- $tand, from old per$ons that can remember pa$$ages for fifty years pa$t; which thing being true, as to me it $eemeth mo$t true, it $hould appear that it could not but be good to reduce matters to that pa$$e that they were at formerly, (laying a$ide all affection or pa$$ion that $elf-flattering minds have entertained for their own conceits) or at lea$t it $hall be nece$$ary $peedily to con$ult the whole.</P> <P>Thirdly, I hold that it is nece$$ary to weigh, whether from the fore$aid di$covery of Land, it followeth, that onely the Earth ri- $eth, as it is commonly thought by all, without di$pute; or whe- ther the Waters are abated and faln away; or el$e whether it proceedeth from both the one and other cau$e. And here it would be $ea$onable to enquire, what $hare the $aid cau$es may have, each con$idered apart in the fore$aid effect. For, in the fir$t ca$e, if the Earth have been rai$ed, it would be nece$$ary to con$ider of taking it down, and removing it: But if the Wa- ters have failed or abated, I believe that it would be extreamly ne- ce$$ary to re$tore and rai$e them: And if both the$e rea$ons have con$pired in this effect, it will be nece$$ary to remedy them each apart. And I do, for my part, think, that the $o notable appea- rance of Shelves at the time of low Water, proceeds principally from the decrea$e and abatement of the Waters, which may confidently be affirmed to need no other proof, in regard that the <I>Brent</I> hath been actually diverted which did formerly di$charge its Water into the Lake.</P> <P>As to the other point of the great Stoppage of Ports, I hold, that all proceedeth from the violence of the Sea, which being $ometimes di$turbed by windes, e$pecially at the time of the wa- ters flowing, doth continually rai$e from its bottome immen$e <foot>heaps</foot> <p n=>65</p> heaps of $and, carrying them by the tide; and force of the waves into the Lake; it not having on its part any $ttength of current that may rai$e and carry them away, they $ink to the bottom, and $o they choke up the Ports. And that this effect happeneth in this manner, we have mo$t frequent experiences thereof along the Sea-coa$ts: And I have ob$erved in <I>Tu$cany</I> on the <I>Roman- $hores,</I> and in the Kingdom of of <I>Naples,</I> that when a river fal- leth into the Sea, there is alwaies $een in the Sea it $elf, at the place of the rivets out-let, the re$emblance, as it were, of an half-Moon, or a great $helf of $ettled $and under water, much higher then the re$t of the $hore, and it is called in <I>Tu$cany, il Cavallo</I>; and here in <I>Venice, lo Scanto</I>: the which cometh to be cut by the current of the river, one while on the right $ide, another while on the left, and $ometimes in the mid$t, according as the Wind fits. And a like effect I have ob$erved in certain little Rillets of water, along the Lake of <I>Bol$ena</I>; with no other difference, $ave that of $mall and great.</P> <P>Now who$o well con$idereth this effect, plainly $eeth that it proceeds from no other, than from the contrariety of the $tream of the River, to the <I>impetus</I> of the Sea waves; $eeing that great abundance of $and which the Sea continually throws upon the $hore, cometh to be driven into the Sea by the $tream of the river; and in that place where tho$e two impediments meet with equal force, the $and $etleth under water, and thereupon is made that $ame Shelf or <I>Cavallo</I>; the which if the river carry water, and that any con$iderable $tore, it $hall be thereby cut and broken; one while in one place, and another while in ano- ther; as hath been $aid, according as the Wind blows: And through that Chanel it is that Ve$$els fall down into the Sea, and again make to the river, as into a Port. But if the Water of the river $hall not be continual or $hall be weak, in that ca$e the force of the Sea-Wind $hall drive $uch a quantity of $and into the mouth of the Port, and of the river, as $hall wholly choak it up. And hereupon there are $een along the Sea-$ide, very many Lakes and Meers, which at certain times of the year abound with waters, and the Lakes bear down that enclo$ure, and run into the Sea.</P> <P>Now it is nece$$ary to make the like reflections on our Ports of <I>Venice, Malamocco, Bondolo,</I> and <I>Chiozza</I>; which in a certain $en$e are no other than Creeks, mouths, and openings of the $hore that parts the Lake from the main Sea; and therefore I hold that if the Waters in the Lake were plentiful, they would have $trength to $cowr the mouths of the Ports thorowly, & with great force; but the Water in the Lake failing, the Sea will with- out any oppo$al, bring $uch a drift of $and into the Ports; that if <foot>Kkkk it</foot> <p n=>66</p> it doth not wholly choke them up, it $hall render them at lea$t unprofitable, and impo$$ible for Barks and great Ve$$els.</P> <P>Many other con$iderations might be propounded concerning the$e two heads of the $toppage of the Ports, and of the appea- rance of the Ouze and Mud in the Lakes, but $o much $hall $uf- fice us to have hinted, to make way for di$cour$ing of the opera- tions about the oportune remedies.</P> <P>Yet before that I propound my opinion, I $ay, That I know very well that my propo$al, at fir$t $ight, will $eem ab$urd and in- convenient; and therefore, as $uch, will perhaps be rejected by the mo$t: and $o much the rather, for that it will prove directly contrary to what hath hitherto been, and as I hear, is intended to be done. And I am not $o wedded to my opinions, but that I do con$ider what others may judge thereof: But be it as it will, I am obliged to $peak my thoughts freely, and that being done, I will leawe it to wi$er men than my $elf; when they $hall have well con$idered my rea$ons, to judge and deliberate of the <I>quid agendum:</I> And if the $entence $hall go again$t me, I appeal to the mo$t equitable and inexorable Tribunal of Nature, who not caring in the lea$t to plea$e either one party or another, will be alwaies a punctual and inviolable executrix of her eternal De- crees, again$t which neither humane deliberations, nor our vain de$ires; $hall ever have power to rebell. I added by word of mouth that which followeth.</P> <P>Though your Highne$s intere$t your $elf in this Noble Col- ledge, and cau$e it to be confirmed in the ^{*} Senate by univer$al <marg>* In <I>Pregadi,</I> a particular Coun- cil, the Senators of which have great Authority.</marg> Vote, that the Winds do not blow, that the Sea doth not fluctuate, that the Rivers do not run; yet $hall the Winds be alwaies deaf, the Sea $hall be con$tant in its incon$tancy, and the Rivers mo$t ob$tinate: And the$e $hall be my Judges, and to their determi- nation I refer my $elf.</P> <P>By what hath been $aid, in my opinion, that is made very clear and manife$t, which in the beginning of this di$cour$e I glanced at; namely, That the whole di$order, although it be divided into two heads, into the di$covery of the Mud, and of the $toppage Ports, yet neverthele$s, by the application of one onely remedy, and that in my e$teem very ea$ie, the whole $hall be removed: And this it is; That there be re$tored into the Lake as much Water as can be po$$ible, and in particular from the upper parts of <I>Venice,</I> taking care that the Water be as free from Mud as is po$$ible. And that this is the true and real remedy of the prece- dent di$orders, is manife$t: For in the pa$$age that this Water $hall make thorow the Lakes, it $hall of it $elf by degrees clear the Chanels in $undry parts of them, according to the currents that it $hall $ucce$$ively acquire, and in this manner being di$- <foot>per$ed</foot> <p n=>67</p> per$ed thorow the Lake, it $hall maintain the waters in the $ame, and in the Chanels much higher, as I $hall prove hereafter; a thing that will make Navigation commodious; and that, which moreover is of great moment in our bu$ine$$e; tho$e Shelves of Mud which now di$cover them$elves at the time of Low- Waters $hall be alwayes covered, $o that the putrefaction of the Air $hall al$o be remedied.</P> <P>And la$tly, this abundance of Water being alwayes to di$- charge it $elf into the Sea by the Ports, I do not doubt, but that their bottomes will be $coured. And that the$e effects mu$t fol- low, Nature her $elf $eemeth to per$wade, there remaining onely one great doubt, whether that abundance of Water that $hall be brought into the Lake may be really $ufficient to make the Wa- ters ri$e $o much as to keep the Shelves covered, and to facilitate Navigation, which ought to be at lea$t half a ^{*} Brace, or there- <marg>* A <I>Venice</I> Brace is 11/16 of our yard.</marg> abouts. And indeed it $eemeth at fir$t $ight to be impo$$ible, that the $ole Water of the ^{*} <I>Brent</I> let into the Lake, and di$- <marg>* A River of that name.</marg> per$ed over the $ame, can occa$ion $o notable an height of water; and the more to confirm the difficulties, one might $ay, reducing the rea$on to calculation, that in ca$e the <I>Brent</I> were 40. Bra- ces broad, and two and an half high, and the breadth of the Lake were 20000. Braces, it would $eem nece$$ary that the height of the water of the <I>Brent</I> dilated and di$tended thorow the Lake would be but onely 1/200 of a Brace in height, which is imperceptible, and would be of no avail to our purpo$e; nay more, it being very certain that the <I>Brent</I> runneth very muddy and foul, this would occa$ion very great mi$chief, filling and contracting the Lake, and for that rea$on this remedy ought, as pernicious, to be totally excluded and condemned.</P> <P>I here confe$$e that I am $urprized at the forme of the Argu- ment, as if I were in a certain manner convinced, that I dare not adventure to $ay more, or open my mouth in this matter; but the $trength it $elf of the Argument, as being founded upon the means of Geometrical and Arithmetical Calculation, hath opened me the way to di$cover a very crafty fraud that is couch- ed in the $ame Argument, which fraud I will make out to any one that hath but any in$ight in <I>Geometry</I> and <I>Arithmetick.</I> And as it is impo$$ible, that $uch an argument $hould be produced by any but $uch as have ta$ted of the$e, in $uch affairs, mo$t pro- fitable, and mo$t nece$$ary Sciences; $o do not I pretend to make my $elf under$tood, $ave onely by $uch, to whom I will evince $o clearly, as that more it cannot be de$ired, the errour and fraud wherein tho$e Ancients and Moderns have been, and alwayes are intangled, that have in any way yet handled this matter of con$idering the Mea$ure and Quantity of the Waters that move. <foot>Kkkk 2 And</foot> <p n=>68</p> And $o great is the e$teem that I have for that which I am now about to $ay touching this particular, that I am content that all the re$t of my Di$cour$e be rejected; provided, that that be per- fectly under$tood, which I am hereafter to propo$e, I holding and knowing it to be a main Principle, upon which all that is founded that can be $aid either well or hand$omely on this parti- cular. The other Di$cour$es may have an appearance of being probable, but this hits the mark as full as can be de$ired, arriving at the highe$t degree of certainty.</P> <P>I have, $eventeen years $ince, as I repre$ented to the mo$t Se- rene Prince, and to the Right Honourable the Pre$ident of the Lords the Commi$$ioners of the ^{*}Sewers, written a Treati$e of the <marg>* <I>I. Savii dell' Acque,</I> a particu- lar Council that take care of the Lakes and other Aquatick affairs.</marg> Mea$ure of the waters that move, in which I Geometrically de- mon$trate and declare this bu$ine$$e, and they who $hall have well under$tood the ground of my Di$cour$e, will re$t fully $a- tisfied with that which I am now about to propo$e: But that all may become rhe more ea$ie, I will more briefly explicate and declare $o much thereof as I have demon$trated in the Di$cour$e, which will $uffice for our purpo$e: And if that $hould not be enough, we have alwayes the experiment of a very ea$ie and cheap way to clear up the whole bu$ine$$e. And moreover I will take the boldne$$e to affirm, that in ca$e there $hould not for the pre$ent any deliberation be made concerning this affair, ac- cording to my opinion; yet neverthele$$e it will be, at $ome time or other; or if it be not, things will grow wor$e and wor$e.</P> <P>For more clear under$tanding, therefore, it ought to be known, that it being required, as it is generally u$ed, to mea$ure the wa- ters of a River, its breadth and its depth is taken, and the$e two dimen$ions being multiplied together, the product is affirmed to be the quantity of that River: As for example, if a River $hall be 100. feet broad, and 20. feet high, it will be $aid, that that River is 2000 feet of Water, and $o if a Ditch $hall be 15. feet broad, and 5. feet high, this $ame Ditch will be affirmed to be 75. feet of Water: And this manner of mea$uring Running Water hath been u$ed by the Ancients, and by Moderns, with no other difference, $ave onely that $ome have made u$e of the Foot, others of the Palme, others of the Brace, and others of other mea$ures.</P> <P>Now becau$e that in ob$erving the$e Waters that move, I fre- quently found, that the $ame Water of the $ame River was in $ome $ites of its Chanel pretty big, and in others much le$$e, not arriving in $ome places to the twentieth, nor to the hundreth part of that which it is $een to be in other places; therefore this vulgar way of mea$uring the Waters that move, for that they did <foot>not</foot> <p n=>69</p> not give me a certain and $table mea$ure and quantity of Water, began de$ervedly to be $u$pected by me, as difficult and defective, being alwayes various, and the mea$ure, on the contrary, being to be alwayes determinate, and the $ame; it is therefore written, that <I>Pondus & Pondus, Men$ura & Men$ura, utrumque abomi- nabile e$t apud Deum,</I> Exod. I con$idered that in the Terri- tory of <I>Bre$cia,</I> my native Countrey, and in other places, where Waters are divided to overflow the Grounds, by the like way of mea$uring them, there were committed grievous and mo$t impor- tant errours, to the great prejudice of the Publique and of Pri- vate per$ons, neither they that $ell, nor they that buy under- $tanding the true quantity of that which is $old and bought: In regard that the $ame $quare mea$ure, as is accu$tomed in tho$e parts, a$$igned one particular per$on, carried to $ometimes above twice or thrice as much water, as did the $ame $quare mea$ure a$- $igned to another. Which thing proveth to be the $ame incon- venience, as if the mea$ure wherewith Wine and Oil is bought and $old, $hould hold twice or thrice as much Wine or Oil at one time as at another. Now this Con$ideration invited my minde and curio$ity to the finding out of the true mea$ure of Running Waters. And in the end, by occa$ion of a mo$t important bu- $ine$$e that I was imployed in $ome years $ince, with great in- ten$ene$$e of minde, and with the $ure direction of <I>Geometry,</I> I have di$covered the mi$take, which was, that we being upon the bu$ine$$e of taking the mea$ure of the Waters that move, do make u$e of two dimen$ions onely, namely, breadth and depth, keep- ing no account of the length. And yet the Water being, though running, a Body, it is nece$$ary in forming a conceit of its quan- tity, in relation to another, to keep account of all the three Di- men$ions, that is of length, breadth, and depth.</P> <P>Here an objection hath been put to me, in behalf of the ordi- nary way of mea$uring Running Waters, in oppo$ition to what I have above con$idered and propo$ed: and I was told, Its true, that in mea$uring a Body that $tands $till, one ought to take all the three Dimen$ions; but in mea$uring a Body that continually moveth, as the Water, the ca$e is not the $ame: For the length is not to be had, the length of the water that moveth being infi- nite, as never fini$hing its running; and con$equently is incom- prehen$ible by humane under$tanding, and therefore with rea$on, nay upon nece$$ity it cometh to be omitted.</P> <P>In an$wer to this, I $ay, that in the above$aid Di$cour$e, two things are to be con$idered di$tinctly; Fir$t, whether it be po$$ible to frame any conceit of the quantity of the Body of the Water with two Dimen$ions onely. And $econdly, whether this length be to be found. As to the fir$t, I am very certain that no man, let <foot>him</foot> <p n=>70</p> him be never $o great a Wit, can never promi$e to frame a con- ceit of the quantity of the Body of Water, without the third Dimen$ion of length: and hereupon I return to affirm, that the vulgar Rule of mea$uring Running water is vain and erroneous. This point being agreed on, I come to the $econd, which is, Whe- ther the third Dimen$ion of length may be mea$ured. And I $ay, that if one would know the whole length of the water of a Fountain or River, thereby to come to know the quantity of all the Water, it would prove an impo$$ible enterprize, nay the knowing of it would not be u$eful. But if one would know how much water a Fountain, or a River carrieth in a determinate time of an hour, of a day, or of a moneth, &c. I $ay, that it is a very po$$ible and profitable enquiry, by rea$on of the innumerable benefits that may be derived thence, it much importing to know how much Water a Chanel carrieth in a time given; and I have demon$trated the $ame above in the beginning of this Book; and of this we $tand in need in the bu$ine$$e of the Lake, that $o we may be able to determine how much $hall be the height of the <I>Brent,</I> when it is $pread all over the Lake: For the three dimen- $ions of a Body being given, the Body is known; and the quan- tity of a Body being given, if you have but two dimen$ions, the third $hall be known. And thus diving farther and farther into this Con$ideration, I found that the Velocity of the cour$e of the water may be an hundred times greater or le$$er in one part of its Chanel than in another. And therefore although there $hould be two mouths of Waters equal in bigne$$e; yet neverthele$s it might come to pa$$e, that one might di$charge an hundred or a thou$and times more water than another: and this would be, if the water in one of the mouths $hould run with an hundred or a thou$and times greater velocity, than the other; for that it would be the $ame as to $ay, that the $wifter was an hundred or a thou$and times longer, than the $lower: and in this manner I di$covered that to keep account of the velocity, was the keeping account of the Length.</P> <P>And therefore it is manife$t, that when two Mouths di$charge the $ame quantity of Wa r in an equal velocity, it is nece$$ary that the le$s $wift Mouth be $o much bigger than the more $wift; as the more $wift exceedeth in velocity the le$s $wift; as for example.</P> <P>In ca$e two Rivers $hould carry equal quantity of water in equal times, but that one of them $hould be four times more $wift than the other, the more $low $hould of nece$$ity be four times more large. And becau$e the $ame River in any part thereof alwaies di$chargeth the $ame quantity of Water in equal times (as is demon$trated in the fir$t Propo$ition of the fir$t <foot>Book</foot> <p n=>71</p> <marg>* He here intends the Demon$trati- ons following, at the end of the fir$t Book</marg> Book^{*} of the mea$ure of Running Watets;) but yet doth not run thorowout with the $ame velocity: Hence it is, that the vul- gar mea$ures of the $aid River, in divers parts of its Chanel, are alwaies divers; in$omuch, that if a River pa$$ing through its cha- nel had $uch velocity, that it ran 100 Braces in the 1/60 of an hour- and afterwards the $aid River $hould be reduced to $o much tardi, ty of motion, as that in the $ame time it $hould not run more than one Brace, it would be nece$$ary that that $ame River $hould be- come 100. times bigger in that place where it was retarded; I mean, 100. times bigger than it was in the place where it was $wifter. And let it be kept well in mind, that this point rightly under$tood, will clear the under$tanding to di$cover very many accidents worthy to be known. But for this time let it $uffice, that we have onely declared that which makes for our purpo$e, referring apprehen$ive and $tudious Wits to the peru$al of my aforenamed Treati$e; for therein he $hall finde profit and delight both together.</P> <P>Now applying all to our principal intent, I $ay, That by what hath been declared it is manife$t, that if the <I>Brent</I> were 40. Bra- ces broad, and 2 1/2 high, in $ome one part of its Chanel, that after- wards the $ame Water of the <I>Brent</I> falling into the Lake, andpa$- $ing thorow the $ame to the Sea, it $hould lo$e $o much of its ve- locity, that it $hould run but one Brace, in the time wherein whil$t it was in its Chanel at the place afore$aid, it ran 100. Bra- ces. It would be ab$olutely nece$$ary, that increa$ing in mea- $ure, it $hould become an hundred times ^{*} thicker; and therefore <marg>* Deeper.</marg> if we $hould $uppo$e that the Lake were 20000. Braces, the <I>Brent</I> that already hath been $uppo$ed in its Chanel 100. Braces, being brought into the Lake, $hould be 100. times 100. Brates; that is, $hall be 10000. Braces in thickne$s, and con$equently $hall be in height half a Brace; that is, 100/200 of a Brace, and not 1/2. of a Brace, as was concluded in the Argument.</P> <P>Now one may $ee into what a gro$s errour of 99. in 100. one may fall through the not well under$tanding the true quantity of Running Water, which being well under$tood, doth open a direct way to our judging aright in this mo$t con$iderable affair.</P> <P>And therefore admitting that wich hath been demon$trated, I fay, that I would (if it did concern me) greatly encline to con- $ult upon the returning of the <I>Brent</I> again into the Lake: For it being mo$t evident, that the <I>Brent</I> in the Chanel of its mouth, is much $wifter than the <I>Brent</I> being brought into the Lake, it will certainly follow thereupon, that the thickne$s of the Water of <I>Brent</I> in the Lake, $hall be $o much greater than that of <I>Brent</I> in <I>Brent,</I> by how much the <I>Bront</I> in <I>Brent</I> is $wifter than thh <I>Brent</I> in the Lake.</P> <foot>1. From</foot> <p n=>72</p> <P>1. From which operation doth follow in the fir$t place, that the Lake being filled and increa$ed by tbe$e Waters, $hall be more Navigable, and pa$$ible, than at pre$ent we $ee it to be.</P> <P>2. By the current of the$e Waters, the Chanels will be $cour- ed, and will be kept clean from time to time.</P> <P>3. There will not appear at the times of low-waters $o many Shelves, and $uch heaps of Mud, as do now appear.</P> <P>4. The Ayr will become more whole$om, for that it $hall not be $o infected by putrid vapours exhaled by the Sun, $o long as the Miery Ouze $hall be covered by the Waters.</P> <P>5. La$tly, in the current of the$e advantagious Waters,, which mu$t i$$ue out of the Lake into the Sea, be$ides tho$e of the Tyde, the Ports will be kept $coured, and clear: And this is as much as I $hall offer for the pre$ent, touching this weighty bui$ine$s; al- waies $ubmitting my $elf to $ounder judgements.</P> <P>Of the above-$aid Writing I pre$ented a Copy at <I>Venice,</I> at a full Colledge, in which I read it all, and it was hearkned to with very great attention; and at la$t I pre$ented it to the Duke, and left $ome Copies thereof with $undry Senators, and went my way, promi$ing with all inten$ene$s to apply my pains with reiterated $tudies in the publick $ervice; and if any other things $hould come into my minde, I promi$ed to declare them $incerely, and $o took leave of <I>His $erenity,</I> and that Noble Council. When I was returned to <I>Rome,</I> this bu$ine$s night and day continually run- ning in my mind, I hapned to think of another admirable and mo$t important conceit, which with effectual rea$ons, confirmed by exact operations, I with the Divine a$$i$tance, made clear and manife$t; and though the thing at fir$t $ight $eemed to me a mo$t extravagant Paradox, yet notwith$tanding, having $atisfied my $elf of the whole bu$ine$s, I $ent it in writing to the mo$t Illu$tri- ous and mo$t Noble <I>Signore Gio. Ba$adonna</I>; who after he had well con$idered my Paper, carried it to the Council; and after that tho$e Lords had for many months maturely con$idered thereon, they in the end re$olved to $u$pend the execution of the diver$ion which they had before con$ulted to make of the River <I>Sile,</I> and of four other Rivers, which al$o fall into the Lake; a thing by me blamed in this $econd Paper, as mo$t prejudicial, and harmful. The writing $pake as followeth.</P> <foot>A</foot> <p n=>73</p> <head>CONSIDERATIONS Concerning the LAKE OF VENICE.</head> <head><I>CONSIDERATION II.</I></head> <P>If the di$cour$ing well about the truth of things, Mo$t Serene Prince, were as the carrying of Burdens, in which we $ee that an hundred Hor$es carry a greater weight than one Hor$e onely; it would $eem that one might make more account of the opinion of many men, than of one alone; But becau$e that di$cour$ing more re$embleth running, than carrying Burdens, in which we $ee that one Barb alone runneth fa$ter than an hundred heavy-heel'd Jades; therefore I have ever more e$teemed one Conclu$ion well managed, and well con$idered by one under$tanding man, although alone, than the common and Vulgar opinions; e$pecially, when they concern ab$truce and arduous points: Nay in $uch ca$es the opinions moulded and framed by the mo$t ignorant and $tupid Vulgar, have been ever $u$pected by me as fal$e, for that it would be a great wonder if in difficult matters a common capacity $hould hit upon that which is hand$om, good, and true. Hence I have, and do hold in very great veneration the $umme of the Government of the mo$t Serene, and eternal Republick of <I>Venice</I>; which although, as being in nature a Common-wealth, it ought to be governed by the greater part; yet neverthele$s, in arduous affairs, it is alwaies directed by the Grave Judgement of few, and not judged blindly <foot>Llll by</foot> <p n=>74</p> by the <I>Plebeian</I> Rout. Tis true, that he that propoundeth Pro- po$itions far above the reach of common capacity, runneth a great hazard of being very often condemned without further Pro- ce$s, or knowledge of the Cau$e; but yet for all that, the truth is not to be de$erted in mo$t weighty affairs, but ought rather to be explained in due place and time with all po$$ible per$picuity; that $o being well under$tood, and con$idered, it may come after- wards for the Common good to be embraced.</P> <P>This which I $peak in general, hath often been my fortune in very many particulars, not onely when I have kept within the bounds of meer $peculation, but al$o when I have chanced to de- $cend to Practice, and to Operations: and your Highne$s know- eth very well what befel me the la$t Summer 1641. when in obe- dience to your Soveraign Command, I did in full Colledge repre- $ent my thoughts touching the $tate of the Lake of <I>Venice</I>; for there not being $uch wanting, who without $o much as vouch- $afing to under$tand me, but having onely had an inkling, and bad apprehen$ion of my opinion, fell furiou$ly upon me, and by violent means both with the Pen and Pre$s, full of Gall, did abu$e me in reward of the readine$s that I had expre$t to obey and $erve them: But I was above mea$ure encouraged and plea$ed, to $ee that tho$e few who vouch$afed to hear me, were all either thorowly per$waded that my opinion was well grounded, or at lea$t $u$pended their prudent verdict to more mature deliberati- on. And though at the fir$t bout I chanced to propo$e a thing that was totally contrary to the mo$t received and antiquated opinion, and to the re$olutions and con$ultations taken above an hundred years ago: Moved by the$e things, and to $atisfie al$o to the promi$e that I had made of tendering unto them what $hould farther offer it $elf unto me touching the $ame bu$ine$s; I have re$olved to pre$ent to the Throne of your Highne$s, another Con$ideration of no le$s importance, which perhaps at fir$t $ight will appear a $tranger Paradox; but yet brought to the Te$t and Touch-$tone of experience, it $hall prove mo$t clear and evident. If it $hall be accounted of, $o that it $ucceedeth to the benefit of your Highne$s, I $hall have obtained my defire and intent: And if not, I $hall have $atisfied my $elf, and $hall not have been wanting to the Obligation of your mo$t faithful Servant, and na- tive $ubject.</P> <P>That which I propounded in the Mouths pa$s, touching the mo$t important bu$ine$s of the Lake, though it did onely expre$- ly concern the point of the diver$ion of the Mouth of the Lake, already made and put in execution; yet it may be under$tood and applyed al$o to the diver$ion under debate, to be made of the other five Rivers, and of the <I>Sile</I> in particular.</P> <foot>Now</foot> <p n=>75</p> <P>Now touching this, I had the fortune to offer an admirable accident that we meet with when we come to the effect, which I verily believe will be an utter ruine to the Lake of <I>Ve- nice.</I></P> <P>I $ay therefore, that by diverting the$e five Rivers that re- main, although their water that they di$charge for the pre$ent in- to the Lake is not all taken together 4/5 parts of what the <I>Brent</I> alone did carry, yet neverthele$$e the abatement of the water of the Lake which $hall en$ue upon this la$t diver$ion of four parts, which was the whole water, $hall prove double to that which hath happened by the diver$ion of <I>Brent</I> onely, although that the <I>Brent</I> alone carried five parts of that water, of which the Rivers that are to be diverted carry four: A wonder really great, and altogether unlikely; for the reducing all this Propo$ition to be under$tood, is as if we $hould $ay, that there being given us three Rivers, of which the fir$t di$chargeth five parts, the $econd three, and the third one, and that from the diver$ion of the fir$t, there did follow $uch a certain abatement or fall; from the taking away of the $econd there ought to follow al$o $o much more abatement; And la$tly, from the withdrawing of the third the water ought to fall $o much more, which is wholly impo$$ible: And yet it is mo$t certain, and be$ides the demon- $tration that per$wades me to it, which I $hall explain in due time, I can $et before your eyes $uch an experiment as is not to be denied by any one, although ob$tinate: and I will make it plainly $een and felt, that by taking away only four parts of the five, which $hall have been taken away, the abatement proveth double to the abatement en$uing upon the diverting fir$t of the five onely; which thing being true, as mo$t certainly it is, it will give us to under$tand how pernicious this diver$ion of five Rivers is like to prove, if it $hall be put in execution.</P> <P>By this little that I have hinted, and the much that I could $ay, let your Highne$$e gather with what circum$pection this bu- $ine$$e ought to be managed, and with how great skill he ought to be furni$hed who would behave him$elf well in the$e difficult affairs.</P> <P>I have not at this time explained the demon$tration, nor have I $o much as propounded the way to make the Experiment, that I am able to make in confirmation of what I have $aid, that $o by $ome one or others mi$-apprehending the Demon$tration, and maiming the Experiment, the truth may not happen to $hine with le$$e clarity than it doth, when all mi$ts of difficulty are re- moved: and if $o be, no account $hould be made of the Rea$ons by me alledged, and that men $hould $hut their eyes again$t the Experiments that without co$t or charge may be made, I do de- <foot>Llll 2 clare</foot> <p n=>76</p> clare and prote$t that there $hall follow very great dammages to the Fields of the main Land, and extraordinary $ummes $hall be expended to no purpo$e. The Lake undoubtedly will become almo$t dry, and will prove impa$$ible for Navigation, with a manife$t danger of corrupting the Air: And in the la$t place there will unavoidably en$ue the choaking and $toppage of the Ports of <I>Venice.</I></P> <P>Upon the 20th. of <I>December,</I> 1641. I imparted this my $econd Con$ideration to the mo$t Excellent <I>Signore Ba$adonna,</I> pre$en- ting him with a Copy thereof among$t other Writings, which I have thought good to in$ert, although they $eem not to belong directly to our bu$ine$$e of the Lake.</P> <head>The way to examine the MUD and SAND that entereth and remaineth in the LAKE of <I>VENICE.</I></head> <head><I>To the mo$t Excellent</I></head> <head>SIGNORE GIO. BASADONNA.</head> <P>Two very con$iderable Objections have been made a- gain$t my opinion concerning the Lake of <I>Venice:</I> One was that, of which I have $poken at large in my fir$t Con$ideration, namely, that the <I>Brents</I> having been taken out of the Lake, cannot have been the occa$ion of the notable fall of the Waters in the Lake, as I pretend, and con$equently, that the turning <I>Brent</I> into the Lake would be no con$iderable reme- dy, in regard that the water of <I>Brent,</I> and the great expan$ion of the Lake over which the water of <I>Brent</I> is to diffu$e and $pread being con$idered, it is found that the ri$e proveth in- $en$ible.</P> <P>The $econd Objection was, that the <I>Brent</I> is very muddy, and therefore if it $hould fall muddy into the Lake, the Sand would $ink and fill up the $ame.</P> <P>Touching the fir$t Query, enough hath been $aid in my fir$t Con$ideration, where I have plainly di$covered the deceipt of the Argument, and $hewn its fallacy; It remaineth now to examine <foot>the</foot> <p n=>77</p> the $econd: to which in the fir$t place I $ay, that one of the fir$t things that I propo$ed in this affair was, that I held it impo$$ible to do any act, though never $o beneficial, that was not al$o ac- companied by $ome inconvenience and mi$chief; and therefore we are to con$ider well the profit, and the lo$$e and prejudice; and they both being weighed, we $hall be able to choo$e the le$- $er evil: Secondly, I admit it to be mo$t true, that <I>Brent</I> is at $ome times muddy, but it is al$o true, that for the greater part of the year it is not muddy. Thirdly, I do not $ee nor under$tand what $trength this objection hath, being taken $o at large, and in general; and methinks that it is not enough to $ay, that the <I>Brent</I> runneth muddy, and to a$$ert that it depo$eth its Muddi- ne$$e in the Lake, but we ought moreover to proceed to particu- lars, and $hew how much this Mud is, and in what time this choaking up of the Ports may be effected. For the Rea$ons are but too apparent and particular, that conclude the ruine of the Lake, and that in a very $hort time, (for mention is made of dayes) the Waters diver$ion being made, and moreover we have the circum$tance of an Experiment, the $tate of things be- ing ob$erved to have grown wor$e $ince the $aid diver$ion. And I have demon$trated, that in ca$e the Diver$ion of the <I>Sile</I> and the other Rivers $hould be put in execution, the Lake would in a few dayes become almo$t dry; and the Ports would be lo$t, with other mi$chievous con$equences. But on the other $ide, al- though that we did grant the choaking of them, we may very probably $ay, that it will not happen, $ave onely in the $ucce$$ion of many and many Centuries of years. Nor can I think it pru- dent coun$el to take a re$olution and imbrace a De$igne now, to obtain a benefit very uncertain, and more than that, which only $hall concern tho$e who are to come very many Ages after us, and thereby bring a certain inconvenience upon our $elves, and upon our children that are now alive and pre$ent.</P> <P>Let it be alledged therefore, (although I hold it fal$e) that by the diver$ions of the Rivers the Lake may be kept in good con- dition for $everal years to come.</P> <P>But I $ay confidently, and hope to demon$trate it; That the Diver$ions will bring the Lake, even in our dayes, to be almo$t dry, and at lea$t will leave $o little water in it, that it $hall cea$e to be Navigable, and the Ports $hall mo$t infallibly be choaked up. I will therefore $ay upon experience, in an$wer to this Ob- jection, that it is very nece$$ary fir$t well to di$cour$e, and ratio- nally to particularize and a$certain the be$t that may be this point of the quantity of this $inking Mud or Sand.</P> <P>Now I fear I $hall make my $elf ridiculous to tho$e, who mea- $uring the things of Nature with the $hallowne$$e of their brains <foot>do</foot> <p n=>78</p> do think that it is ab$olutely impo$$ible to make this enquiry, and will $ay unto me, <I>Quis men$us e$t pugillo aquas, & terram palmo ponderavit</I>? Yet neverthele$s I will propound a way whereby, at lea$t in gro$s, one may find out the $ame.</P> <P>Take a Ve$$el of Cylindrical Figure, holding two barrels of water, or thereabouts; and then fill it with the water of <I>Brent,</I> at its Mouth or Fall into the Lake; but in the Lake at the time that the <I>Brent</I> runneth muddy, and after it hath begun to run muddy for eight or ten hours, to give the mud time to go as far as S. <I>Nicolo,</I> to i$$ue into the Sea; and at the $ame time take another Ve$$el, like, and equal to the fir$t, and fill it with the wa- ter of the Lake towards S. <I>Nicolo,</I> (but take notice that this ope- ration ought to be made at the time when the waters go out, and when the Sea is calm) and then, when the waters $hall have $etled in the afore$aid Ve$$els, take out the clear water, and con- $ider the quantity of Sand that remains behind, and let it be $et down, or kept in mind: And I am ea$ily induced to think, that that $hall be a greater quantity of Sand which $hall be left in the fir$t Ve$$el, than that left in the $econd Ve$$el. Afterwards when the <I>Brent</I> $hall come to be clear, let both the operations be repeated, and ob$erve the quantity of Sand in the afore$aid Ve$- $els; for if the Sand in the fir$t Ve$$el $hould be mo$t, it would be a $ign, that in the revolution of a year the <I>Brent</I> would depo$e Sand in the Lake: And in this manner one may calculate to a $mall matter what proportion the Sand that entreth into the Lake, hath to that which remains: And by that proportion one may judge how expedient it $hall be for publick benefit. And if at $everal times of the year you carefully repeat the $ame operati- ons, or rather ob$ervations, you would come to a more exact knowledge in this bu$ine$s: And it would be good to make the $aid operations at tho$e times, when the Lake is di$turbed by $trong high Winds, and made muddy by its own Mud, rai$ed by the commotion of the Waters.</P> <P>This notion would give us great light, if the $ame ob$ervations $hould be made towards the Mouth of <I>Lio,</I> at $uch time as the waters flow and ebb, in calm $ea$ons; for $o one $hould come to know whether the waters of the Lake are more thick at the going out, than at the entrance. I have propounded the foregoing way of mea$uring Sands and Mud, to $hew that we are not $o generally, and incon$iderately to pronounce any $entence, but proceed to $tricter inquiries, and then deliberate what $hall be mo$t expedient to be done. Others may propo$e more exqui- $ite examinations, but this $hall $erve me for the pre$ent.</P> <P>I will add onely, that if any one had greater curio$ity (it would be profitable to have it) in inve$tigating more exactly the quan- <foot>tity</foot> <p n=>79</p> tity of the Water that entereth into the Lake, by the means that I have $hewen in the beginning of this Book: When he $hall have found the proportion of the quantity of water to the quan- tity of Sand or Mud, he $hall come to know how much Sand the <I>Brent</I> $hall leave in the Lake in the $pace of a year. But to perform the$e things, there are required per$ons of di$cretion, and fidelity, and that are imployed by publick Order; for there would thence re$ult eminent benefit and profit.</P> <P><I>Here are wanting</I> LETTERS <I>from $everal per$ons.</I></P> <head>To the Reverend Father, <I>France$co di</I></head> <head>S. GIUSEPPE.</head> <P>In execution of the command that you laid upon me in your former Letters, by order from the mo$t Serene, my Lord, <I>Prince Leopold</I>; that I $hould $peak my judgment concern- ing the di$imboguement of the River called <I>Fiume morto,</I> whe- ther it ought to be let into the Sea, or into <I>Serchio</I>; I $ay, that I chanced 18. years $ince to be pre$ent, when the $aid Mouth was opened into the Sea, and that of <I>Serchio</I> $topt; which work was done to remedy the great Innundation that was made in all that Country, and Plain of <I>Pi$a,</I> that lyeth between the River <I>Arno,</I> and the Mountains of <I>S. Giuliano,</I> and the River <I>Serchio</I>; which Plain continued long under water, in$omuch that not onely in the Winter, but al$o for a great part of the Summer, tho$e fields were overflowed; and when that the Mouth of <I>Fiume morto</I> was effectually opened into the Sea, the place was pre$ently freed from the waters. and drained, to the great $atisfaction of the Owners of tho$e Grounds. And here I judge it worth your notice, that for the generality of tho$e that po$$e$s e$tates in tho$e parts, they de$ired that the Mouth of <I>Fiume morto</I> might $tand open to the Sea, and tho$e who would have it open into <I>Serchio,</I> are per$ons that have no other concernment there, $ave the hopes of gaining by having the di$po$e of Commi$$ions, and the like, &c,</P> <P>But for the more plain under$tanding of that which is to be $aid, it mu$t be known, That the re$olution of opening the $aid Mouth into <I>Serchio,</I> was taken in the time of the Great Duke <I>Ferdinando</I> the fir$t, upon the $ame motives that are at this time again propo$ed, as your Letters tell me, Since that, it manife$t- ly appearing, that <I>Fiume morto</I> had, and hath its Mouth open to the Sea, the Plain hathbeen kept dry; and it being al$o true, that <foot>the</foot> <p n=>80</p> the fury of the South, and South-We$t-Winds carryed $uch abundance of $and into the Mouth, or Out-let of <I>Fiume morto,</I> that it wholly $topt it up: e$pecially when the waters on <I>Pi$a</I> $ide were low and $hallow, And they think, that turning the Lake of <I>Fiume morto</I> into <I>Serchio,</I> and the <I>Serchio</I> maintaining continually its own Mouth with the force of its waters open to the Sea, and con$equently al$o <I>Fiume morto,</I> they would have had the Out-let clear and open; and in this manner they think, that the Plain of <I>Pi$a</I> would have been freed from the waters. The bu- $ine$s pa$$eth for current, at fir$t $ight; but experience proveth the contrary, and Rea$on confirmeth the $ame: For the height of the water of tho$e Plains, was regulated by the height of the waters in the Mouth of <I>Fiume morto</I>; that is, The waters at the Mouth being high, the waters al$o do ri$e in the fields; and when the waters at the Mouth are low, the waters of the fields do like- wi$e abate: Nor is it enough to $ay, That the Out-let or Vent of <I>Fiume morto</I> is continual, but it mu$t be very low: Now if <I>Fiume morto</I> did determine in <I>Serchio,</I> it is manife$t that it would determine high; for <I>Serchio</I> terminating in the Sea, when ever it more and more aboundeth with water, and ri$eth, it is ne- ce$lary that al$o <I>Fiume morto</I> hath its level higher, and con$e- quently $hall keep the waters in the Plains higher. Nay, it hath happened $ometimes (and I $peak it upon my own $ight) that <I>Fiume morto</I> hath rever$ed its cour$e upwards towards <I>Pi$a</I>; which ca$e will ever happen, when$oever the <I>Pi$an</I> waters chance to be lower than the level of tho$e of <I>Serchio</I>; for in that ca$e the waters of <I>Serchio</I> return back upon the Plains thorow <I>Fiume morto</I> in $uch $ort, that the Muddine$$es, and the <I>Serchio</I> have been ob$erved to be carried by this return as farr as the Walls of <I>Pi$a</I>; and then before $uch time as $o great waters can be a$- $waged, which come in with great fury, and go out by little and little, there do pa$s very many days, and moneths, nay $ome- times one being never able to find the waters of <I>Serchio,</I> when at the $hallowe$t, $o low as the Sea in level; (which is the lowe$t place of the waters) it thence doth follow, that the wa- ters of <I>Fiume morto</I> $hould never at any time of the year, $o long as they determine in <I>Serchio,</I> be $o low, as they come to be when the $ame <I>Fiume morto</I> determineth in the Sea. Tis true indeed, that the Mouth of <I>Fiume morto,</I> opened into the Sea, is $ubject to the inconvenience of being $topt up by the force of Winds: But in this ca$e, it is nece$$ary to take $ome pains in opening it; which may ea$ily be done, by cutting that Sand a little which $tayeth in the Mouth, after that the Wind is laid; and it is enough if you make a Trench little more than two Palms in breadth; for the water once beginning to run into it, it will in a few hours carry <foot>that</foot> <p n=>81</p> that Sand away with it, and there will en$ue a deep and broad Trench that will drain away all the water of the Plains in very lit- tle time. And I have found by practice, that there having been a great quantity of Sand driven back, by the fury of the South- We$t-Wind, into the Mouth of <I>Fiume morto,</I> I having cau$ed the little gutter to be made in the Morning, $omewhat before Noon, a Mouth hath been opened of 40. Braces wide, and notably deep, in$omuch that the water, which before had incommoded all the Champian ran away in le$s than three dayes, and left the Coun- try free and dry, to the admiration of all men. There was pre- $ent upon the place, at this bu$ine$s, on the $ame day that I opened the Mouth, the mo$t Serene great Duke, the mo$t Serene Arch-Dutche$s Mother, all the Commi$$ioners of Sewers, with many other Per$ons and Pea$ants of tho$e parts; and they all $aw very well, that it was never po$$ible that a little Bark of eight Oars, which was come from <I>Legorn</I> to wait upon the great Duke, $hould ever be able to ma$ter the Current, and to make up into <I>Fiume morto</I>; and his Highne$s, who came with an intent to cau$e the $aid Mouth towards the Sea to be $topt; and that into <I>Serchio</I> to be opened, changed his judgement, giving order that it $hould be left open towards the Sea, as it was done. And if at this day it $hall return into <I>Serchio,</I> I am very certain that it will be nece$$ary to open it again into the Sea. And there was al$o charge and order given to a per$on appointed for the pur- po$e, that he $hould take care to open the $aid Mouth, as hath been $aid upon occa$ion. And thus things have $ucceeded very well unto this very time. But from the middle of <I>October,</I> until this fir$t of <I>February,</I> there having continued high South, and South-We$t-Winds, with frequent and abundant Rains; it is no wonder that $ome innundation hath happened; but yet I will affirm, that greater mi$chiefs would have followed, if the Mouth had been opened into <I>Serchio.</I> This which I have hitherto $aid, is very clear and intelligible to all $uch as have but competent in- $ight, and indifferent skill in the$e affairs. But that which I am now about to propo$e farther, will, I am very certain, be under- $tood by your $elf, but it will $eem $trange and unlikely to many others. The point is, that I $ay, That by rai$ing the level of <I>Fiume morto,</I> one half Brace, onely at its Mouth, (it will peni- penitrate into <I>Serchio</I> farther than it would into the Sea) it $hall cau$e the waters to ri$e three, or perhaps more Braces upon the fields towards <I>Pi$a,</I> and $till more by degrees as they $hall recede farther from the Sea-$ide; and thus there will follow very great Innundations, and con$iderable mi$chiefs. And to know that this is true, you are to take notice of an accident, which I give warning of in my di$cour$e of the Mea$ure of Running Waters: <foot>Mmmm where</foot> <p n=>82</p> where al$o I give the rea$on thereof, ^{*} <I>Coroll.</I> 14. The ac- cident is this, That there coming a Land-Flood, for example, into <I>Arno,</I> which maketh it to ri$e above its ordinary Mouth wthin <I>Pi$a,</I> or a little above or below the City $ix or $even Bra- ces; this $ame height becometh alwaies le$$er and le$$er, the more we approach towards the Sea-$ide; in$omuch, that near to the Sea the $aid River $hall be rai$ed hardly half a Brace: Whence it followeth of nece$$ary con$equence, that $hould I again be at the Sea-$ide, and knowing nothing of what hapneth, $hould $ee the River <I>Arno</I> rai$ed by the acce$$ion of a Land-flood, one third of a Brace; I could certainly infer, that the $ame River was rai$ed in <I>Pi$a</I> tho$e $ame $ix or $even Braces. And that which I $ay of <I>Arno,</I> is true of all Rivers that fall into the Sea. Which thing being true, it is nece$$ary to make great account of every $mall ri$ing, that <I>Fiume morto</I> maketh towards the Sea-$ide by fal- ling into <I>Serchio.</I> For although the ri$ing of <I>Fiume morto,</I> by being to di$gorge its Waters into <I>Serchio,</I> towards the Sea, were onely a quarter of a Brace; we might very well be $ure, that fart from the Sea, about <I>Pi$a,</I> and upon tho$e fields the ri$e $hall be much greater, and $hall become two or three Braces: And be- cau$e the Countrey lyeth low, that $ame <*>i$e will cau$e a conti- nual Innundation of the Plains, like as it did before; I cau$ed the Mouth to be opened into the Sea. And therefore I conclude that the Mouth of <I>Fiume morto,</I> ought by no means to be opened into <I>Serchio</I>; but ought to be continued into the Sea, u$ing all diligence to keep it open after the manner afore$aid, $o $oon as ever the Wind $hall be laid. And if they $hall do otherwi$e, I confidently affirm, that there will daily follow greater damages; not onely in the Plains, but al$o in the whole$omne$s of the Air; as hath been $een in times pa$t. And again, It ought with all care to be procured, that no waters do by any means run or fall from the Trench of <I>Libra,</I> into the Plain of <I>Pi$a,</I> for the$e Waters being to di$charge into <I>Fiume morto,</I> they maintain it much higher than is imagined, according to that which I have de- mon$trated in my con$ideration upon the $tate of the Lake of <I>Venice.</I> I have $aid but little, but I $peak to you, who under- $tandeth much, and I $ubmit all to the mo$t refined judgment of our mo$t Serene Prince <I>Leopold,</I> who$e hands I be$eech you in all humility to ki$s in my name, and implore the continuance of his Princely favour to me; and $o de$iring your prayers to God for me, I take my leave.</P> <P><I>Rome</I> 1. Feb. 1642.</P> <P><I>Your mo$t affectionate Servant,</I></P> <P>D. BENEDETTO CASTELLI.</P> <foot>The</foot> <p n=>83</p> <head>The an$wer to a Letter written by BAR- TOLOTTI, touching the difficultyes ob$erved.</head> <head><I>The former part of the Letter is omitted, and the di$cour$e beginneth at the fir$t Head.</I></head> <P>And fir$t I $ay, Whereas I $uppo$e that the level of the <I>Ser- chio</I> is higher than that of <I>Fiume morto</I>; this is mo$t true, at $uch time as the waters of <I>Fiume morto</I> are di$charged in- to the Sea; but I did never $ay that things could never be brought to that pa$s, as that the level of <I>Fiume morto</I> $hould be higher than <I>Serchio</I>: and $o I grant that it will follow, that the waters of <I>Fiume morto</I> $hall go into <I>Serchio,</I> and its very po$$ible, that the Drain of <I>Fiume morto</I> into <I>Serchio</I> may be continuate; and I far- ther grant, that its po$$ible, that the <I>Serchio</I> doth never di$gorge thorow <I>Fiume morto</I> towards <I>Pi$a</I>; Nay, I will yet farther grant that it might have happened, that <I>Fiume morto</I> might have had $uch a fall into <I>Serchio,</I> as would have $ufficed to have turned Mills: But then I add withall, that the Plains of <I>Pi$a,</I> and the City it $elf mu$t be a meer Lake.</P> <P>2. <I>Signore Bartololti</I> $aith confidently, that when the Sea $wel- leth by the South-We$t, or other Winds, the level of <I>Serchio</I> in the place marked A in the Platt, di$tant about 200. Braces, ri$eth very little: But that <I>Fiume morto</I> in D, and in E, many miles more up into Land ri$eth very much, and that certain Fi$hermen confirm this, and $hew him the $ignes of the ri$ing of the Water. I grant it to be very true, and I have $een it with my own eyes: But this cometh to pa$s, when the Mouth of <I>Fiume morto</I> is $topt up by the Sea; as I $hall $hew by and by. And this ri$ing near the Sea-$ide, is of no con$iderable prejudice to the fields. And this is as much as I find to be true in the a$$ertion of <I>Signore Bar- tolotti,</I> (without his confirming it by any other proof; as indeed it needs none) That the level of <I>Fiume morto</I> ri$eth in E, and ma- ny miles farther upwards it ri$eth much; nor did I ever affirm the contrary.</P> <P>3. Concerning the difficulty of opening the Mouth of <I>Fiume morto</I> into the Sea, that which <I>Il Ca$tellano</I> $aith is mo$t certain; namely, That at the entrance upon the opening of the Mouth, it is nece$$ary to make a deep Trench: But I $ay, that at that time it is difficult to open it, unle$s upon great occa$ions; for that the <foot>Mmmm 2 difficulty</foot> <p n=>84</p> difficulty proceedeth from the waters of <I>Fiume morto</I> being low, and the fields drained.</P> <P>4. As to the particular of the Cau$es that you tell me men pre$s $o much unto the mo$t <I>Serene Grand Duke,</I> and to the Prince, I have not much to $ay, becau$e it is not my profe$$ion; nor have I con$idered of the $ame: Yet I believe, that when the Prince and his Highne$$e $ee the benefit of his People and Sub- jects in one $cale of the Ballance, and the accomodation of Hunt$men in the other, his Highne$$e will incline to the profit of his $ubjects; $uch have I alwayes found his Clemency and Noblene$$e of minde. But if I were to put in my vote upon this bu$ine$$e, I would $ay, that the points of Spears, and the mouths of Guns, the yelping of Dogs, the wilyne$$e of Hunt$- men, who run thorow and narrowly $earch all tho$e Woods, Thickets and Heathes, are the true de$troyers of Bucks and Boares, and not a little Salt-water, which $etleth at la$t in $ome low places, and $preadeth not very far. Yet neverthele$$e, I will not enter upon any $uch point, but confine my $elf $olely to the bu$ine$$e before me.</P> <P>5. That Experiment of joyning together the water of <I>Fiume morto,</I> and that of <I>Serchio</I> by a little trench to $ee what advan- tage the Level E hath upon the Level I, doth not give me full $atisfaction, taken $o particularly, for it may come to pa$$e, that $ometimes E may be higher, and $ometimes A lower, and I do not que$tion but that when <I>Serchio</I> is low, and <I>Fiume morto</I> full of Water, the level of <I>Fiume morto</I> will be higher than that of <I>Serchio.</I> But <I>Serchio</I> being full, and <I>Fiume morto</I> $cant of Wa- ter, the contrary will follow, if the Mouth $hall be opened to the Sea. And here it $hould $eem to me, that it ought to be con$idered, that there is as much advantage from E to the Sea through the little Trench opened anew into <I>Serchio,</I> as from E to the Sea by the Mouth of <I>Fiume morto.</I> But the difficulty (which is that we are to regard in our ca$e) is, that the cour$e of the Waters thorow the Trench is three times longer than the cour$e of the Mouth of <I>Fiums morto,</I> as appeareth by the Draught or Plat which you $ent me, which I know to be very exactly drawn, for that the $ituation of tho$e places are fre$h in my memory. Here I mu$t give notice, that the waters of <I>Fiume morto</I> determi- ning thorow the Trench in <I>Serchio</I> (the waters of which <I>Fiume morto</I> are, for certain, never $o low as the Sea) their pendency or declivity $hall, for two cau$es, be le$$e than the pendency of tho$e waters through the Mouth towards the Sea, that is, becau$e of the length of the line through the Trench, and becau$e of the height of their entrance into <I>Serchio,</I> a thing which is of very great import in di$charging the waters which come $uddenly, as <foot>he</foot> <p n=>85</p> he $hall plainly $ee, who $hall have under$tood my Book of the Mea$ure of Running Waters And this was the Rea$on why all the Countrey did grow dry upon the opening of the Mouth into the Sea. And here I propo$e to con$ideration that which the Pea- $ants about <I>Pi$a</I> relate, namely, That the Water in the Fields doth no con$iderable harm by continuing there five or $ix, yea, or eight dayes. And therefore the work of the Countrey is to o- pen the Mouth of <I>Fiume morto,</I> in $uch manner, that the Water being come, they may have the Trench free and ready, when that the Water cometh it may have a free drain, and may not $tay there above eight or nine dayes, for then the overflowings be- come hurtful. It is to be de$ired al$o, that if any Propo$ition is produced touching the$e affairs, it might be propounded the mo$t di$tinctly that may be po$$ible, and not con$i$t in generals, e$pe- cially when the Di$pute is of the ri$ings, of velocity, of tardity, of much and little water; things that are all to be $pecified by mea$ures.</P> <P>6. Your Letter $aith, in the next place, that <I>Signore Barto- lotti</I> confe$$eth, that if the Mouth of the <I>Fiume morto</I> might al- wayes be kept open, it would be better to let it continue as it is: the which, that I may not yield to him in courte$ie, I confe$$e, for the keeping it $topt on all $ides would be a thing mo$t per- nicious. But admitting of his confe$$ion I again reply, that <I>Fi- ume morto</I> ought not to be let into <I>Serchio,</I> but immediately in- to the Sea; becau$e although $ometimes the Mouth to Sea- wards be $topt up, yet for all that, the rai$ing of the Bank above the Plains (which is all the bu$ine$$e of importance) $hall be ever le$$er, if we make u$e of the Mouth leading to the Sea, than u- $ing that of <I>Serchio.</I></P> <P>7. I will not omit to mention a kinde of $cruple that I have concerning the po$ition of <I>Sign. Bartolotti,</I> that is, where he $aith that the two Mouths A and D are equal to the like Mouths into the Sea; Now it $eems to me, that the Mouth A of <I>Fiume morto</I> into <I>Serchio</I> is ab$olutely within <I>Serchio,</I> nor can it be made low- er, and is regulated by the height of <I>Serchio</I>: But the Mouth of <I>Fiume morto</I> terminates, and ought to be under$tood to ter- minate in the Sea it $elf, the lowe$t place. And this I believe was very well peroeived by <I>Sig. Bartolotti,</I> but I cannot tell why he pa$t it over without declaring it: and we $ee not that the Mouth D falleth far from the Sea, which Mouth ought to be let into the Sea it $elf, and $o the advantage of the <I>M</I>outh into the Sea more clearly appeareth.</P> <P>8. That which <I>Sig. Bartolotti</I> addeth, that when it is high Waters, at $uch time as the Waters are out, and when Winds choak up <I>Fiume morto,</I> they not only retard it, but return the <foot>cour$e</foot> <p n=>86</p> cour$e of the Waters upwards very lea$urely, per$wadeth me more readily to believe that <I>Sig. Bartolotti</I> knoweth very well, that the Mouth of <I>Fiume morto</I> let into <I>Serchio</I> is hurtful: for by this he acknowledgeth that the Mouth towards the Sea doth in $uch $ort drain the Countrey of the Waters, as that they be- come very low; and therefore upon every little <I>impetus</I> the wa- ters turn their cour$e: And from the motions, being exceeding $low, is inferred, that the abundance of Sea-water that com- eth into <I>Fiume morto,</I> is $o much as is believed, and as <I>Sig. Bat- tolotti</I> affirmeth.</P> <P>9. After that <I>Sig. Bartolotti</I> hath $aid what he promi$eth a- bove, namely, that when the Windes blowing $trongly do $top up <I>Fiume morto,</I> and not onely retard but turn the cour$e up- wards, the time being Rainy, and the Mouth of <I>Fiume morto</I> $hut up, the Waves of the Sea pa$$e over the Bank of <I>Fiume morto</I>; at that time, $aith <I>Signore Bartolotti,</I> the Champain $hall know the benefit of <I>Fiume morto</I> di$charged into <I>Serchio,</I> and the mouth A $hall $tand alwayes open; and <I>Fiume morto</I> may alwayes con- $tantly run out, as al$o the Rains and Rain-waters, although the hurtful Tempe$t $hould la$t many dayes, &c. And I reply, that all the Art con$i$ts in this; for the benefit of tho$e Fields doth not depend on, or con$i$t in $aying, that <I>Fiume morto</I> is alwayes open, and <I>Fiume morto</I> draineth continually; But all the bu$i- ne$$e of profit lyeth and con$i$teth in maintaining the Waters low in tho$e Plaines, and tho$e Ditches, which $hall never be ef- fected whil$t the World $tands, if you let <I>Fiume morto</I> into <I>Ser- chio</I>; but yet it may, by opening the mouth into the Sea: and $o much rea$on and nature proveth, and (which importeth) Ex- perience confirmeth.</P> <P>10. In the tenth place I come to con$ider the an$wer that was made to another Propo$ition in the Letter which I writ to Father <I>France$co,</I> which prudently of it $elf alone might $erve to clear this whole bu$ine$$e. I $aid in my Letter, That great account is to be made of every $mall ri$ing and ebbing of the Waters neer to the Sea in <I>Fiume morto,</I> for that the$e ri$ings and fallings, although that they be $mall neer to the Sea-$ide, yet ne- verthele$$e, they operate and are accompanied by notable ri$ings and fallings within Land, and far from the Sea-$ide, and I have declared by an example of <I>Arno,</I> in which a Land-flood falling, that made it increa$e above its ordinary height within <I>Pi$a</I> $ix or $even Braces, that this height of the $ame Flood becometh $till le$$er, the neerer we approach to the Sea-coa$ts. Nor $hall the $aid River be rai$ed hardly half a Brace; whereupon it nece$$- rily followeth, that if I $hould return to the Sea-$ide, and not knowing any think of that which happeneth at <I>Pi$a,</I> and $eeing <foot>the</foot> <p n=>87</p> the River <I>Arno</I> rai$ed by a Land-flood half a Brace, I might con- fidently affirm the $aid River to be rai$ed in <I>Pi$a</I> tho$e $ix or $e- ven Braces, &c. From $uch like accidents I conclude in the $ame Letter, that it is nece$$ary to make great account of every little ri$e that <I>Fiume morto</I> $hall make towards the Sea. Now cometh <I>Bartolotti</I> (and perhaps becau$e I knew not how to expre$s my $elf better, under$tandeth not my Propo$ition) and $peaketh that which indeed is true, but yet be$ides our ca$e: Nor have I ever $aid the contrary; and withall doth not apply it to his purpo$e. Nay I $ay, that if he had well applyed it, this alone had been a- ble to have made him change his opinion. And becau$e he $aith, that I $aid, that it is true, when the abatement proceedeth from $ome cau$e above, as namely by Rain, or opening of Lakes; But when the cau$e is from below, that is, by $ome $top, as for in$tance $ome Fi$hers Wears or Locks, or $ome impediment re- mote from the Sea, although at the Level it $hall ri$e $ome Braces where the impediment is, yet that ri$ing $hall go upwards; and here he fini$heth his Di$cour$e, and concludeth not any thing more. To which I $ay fir$t, that I have al$o $aid the $ame in the Propo$ition, namely, that a Flood coming (which maketh <I>Arno</I> to ri$e in <I>Pi$a</I> $ix or $even Braces (which I take to be a $uperiour cau$e whether it be Rain or the opening of Lakes, as be$t plea- $eth <I>Bartolotti</I>) in $uch a ca$e I $ay, and in no other (for towards the Sea-coa$ts it $hall not cau$e a ri$ing of full half a Brace; and therefore $eeing <I>Arno</I> at the Sea-$ide to be rai$ed by a Flood, whe- ther of Rain, or of opening of Lakes half a Brace) it may be inferred, that at <I>Pi$a</I> it $hall be rai$ed tho$e $ix or $even Braces; which variety, well con$idered, explaineth all this affair in favour of my opinion: For the ri$ing that is made by the impediment placed below, of Fi$hing Weares and Locks, operateth at the be- ginning, rai$ing the Waters that are neer to the impediment; and afterwards le$s and le$s, as we retire upwards from the im- pediment: provided yet that we $peak not of a Flood that com- meth by acce$$ion, but onely of the ordinary Water impeded. But there being a new acce$$ion, as in our ca$e, then the Water of the Flood, I $ay, $hall make a greater ri$ing in the parts $uperi- our, far from the impediment; and the$e impediments $hall come to be tho$e that $hall overflow the Plains, as happened eighteen or nineteen years ago, before the opening of <I>Fiume morto</I> into the Sea, The $ame will certainly follow, if <I>Fiume morto</I> be let into <I>Serchio.</I> Here I could alledge a very pretty ca$e that befell me in <I>la ^{*} Campagna di Roma,</I> neer to the Sea- <marg>* The Countrey or Province lying round the City, heretofore called <I>Lati u<*>s.</I></marg> $ide. where I drained a Bog or Fen, of the nature of the Wa- ters of <I>Pi$a,</I> and I $ucceeded in the enterprize, the Waters in their $ite towards the Sea abating only three Palmes, and yet in the <foot>Fen</foot> <p n=>88</p> Fen they fell more than fifteen Palmes. But the bu$ine$$e would be long, and not $o ea$ily to be declared, and I am cer- tain that <I>Sig. Bartolotti</I> having con$idered this, would alter his judgment, and withall would know that remitting that impedi- ment anew, which I had left for le$$e than three Palmes towards the Sea, the Waters in the Fen would return with the fir$t Floods and Raines to the $ame height as before, as likewi$e <I>Fiume morto</I> will do if it $hall be let again into <I>Serchio.</I></P> <P>Here I intreat your Honour to do me the favour to importune <I>P. France$co</I> in my behalf, that he would be plea$ed to deelare my meaning in the afore$aid Letter to <I>Sig. Bartolotti,</I> for I hope that if he will under$tand this point, he will be no longer $o te- nacious in his opinion.</P> <P>Next that the$e Lords in the Commi$$ion of Sewers, with the Right Honourable the Marque$$e of S. <I>Angelo,</I> and your Honour do approve of my judgment, doth very much rejoyce me; but becau$e that I know that they do it not in de$ign to complement me, but onely to $erve his Highne$s our Grand Duke, I freely profe$s that I will pretend no farther obligations from them there- in, than I account my $elf to owe to tho$e who$e opinions are contrary to mine, for that I know that they have the $ame end. The definitive $entence of this whole bu$ine$s is, that they give the$e Plains, the$e Draines, and the$e Waters farre fetcht ap- pellations.</P> <P>11. As to the quantity of the Water that <I>Fiume morto</I> di$- chargeth into the Sea, there are very great di$putes about it, and I have been pre$ent at $ome of them. But let your Honour be- lieve me, that as this is not continual, but only during a few dayes, $o it will never be of any great prejudice to the$e Fields; and if your Lord$hip would be a$certained thereof, you may plea$e to go to <I>Fiume morto</I> at about a mile's di$tance from the Sea, in the time of the$e $trong Windes, and ob$erve the cur- rent from thence upwards, for you $hall finde it extream $low, and con$equently will know that the quantity of the Water that is repuls'd is very $mall. And this $eems to be contradicted by the rule of Ri$ings proceeding from cau$es below, which occa$ion no con$iderable alteration far from the Sea.</P> <P>I am nece$$itated to go to morrow out of <I>Rome</I> with his Emi- nence Cardinal <I>Gaetano</I> about certain affairs touching Waters, therefore I $hall not farther inlarge, but for a clo$e to this tedious Di$cour$e, I conclude in few words, that <I>Fiume morto</I> is by no means to be let into <I>Serchio,</I> nor are there any means intermedi- ate cour$es to be taken, for they will alwayes be prejudicial; but <I>Fiume morto</I> is to be di$charged immediately into the Sea. When it is $topt up by the fury of the Sea waves, I affirm that it is a <foot>$ign</foot> <p n=>89</p> $ign that there is no need of opening it; and if there be any oc- ca$ion to open it, it is ea$ily done. As for the re$t your Lord$hip may plea$e to keep account of all the particulars that occur, for the memory of things pa$t is our Tutre$$e in tho$e that are to come. If occa$ion $hall offer, I intreat you to bow humbly in my name to His Highne$s the Grand Duke, and the mo$t Serene Prince <I>Leopold</I>; and to attend the $ervice of Their Highne$$es, for you $erve I rinces of extraordinary merit; And to whom I my $elf am al$o exceedingly obliged. In the controver$ies that ari$e re$pect the pious end of $peaking the Truth, for then every thing will $ucceed happily. I ki$s the hands of <I>Padre France$co,</I> of <I>Sig. Bartolotti,</I> and of your Lord$hip.</P> <P><I>Rome, 14. March</I> 1642.</P> <P><I>Your Honours</I></P> <P><I>most Obliged Servant</I></P> <P>D. <I>BENEDETTO CASTELLI.</I></P> <P>Vpon this occa$ion I will here in$ert a Di$cour$e that I made upon the Draining and improvement of the <I>Pontine Fens,</I> for that I think that what$oever may be done well and to pur- po$e in this matter hath ab$olute dependance on the perfect know- ledge of that $o important Propo$ition, by me demon$trated and explained in my Treati$e of the <I>Men$uration</I> of <I>Running Wa- ters,</I> namely, That the $ame water of a River doth continually change Mea$ures, according as it altereth and changeth the ve- locity of its cour$e; $o that the mea$ure of the thickne$$e of a River in one Site, to the mea$ure of the $ame River in another Site, hath the $ame proportion reciprocally that the velocity in this $ite hath to the velocity in the fir$t $ite. And this is a Truth $o con$tant and unchangeable, that it altereth not in the lea$t point on any occurrences of the Waters that change: and being well under$tood, it openeth the way to the knowledge of $undry adverti$ements in the$e matters, which are all re$olved by this $ole Principle; and from it are derived very con$iderable be- nefits; and without the$e it is impo$$ible to do any thing with ab$olute perfection</P> <foot>Nnnn A</foot> <p n=>91</p> <head>A CONSIDERATION Upon the DRAINING OF THE Pontine Fenns. BY</head> <head>D. BENEDETTO CASTELLI, Abbot of S. BENEDETTO ALOISIO, and Profe$$or of the <I>Mathematicks</I> to P. <I>Urban</I> VIII. in the Univer$ity of <I>ROME.</I></head> <head><I>CONSIDERATION</I> III.</head> <P>Among$t the enterprizes by me e$teemed, if not ab- $olutely impo$$ible, , at lea$t exceeding difficult, one was that famous one of Draining the <I>Pontine Fenns</I>; and therefore I was thorowly re$olved never to apply my minde thereunto, although by my Patrons I $hould be commanded to the $ame: accounting that it was an occa$ion rather of lo$ing repu- tation by the mi$carriage of the attempt, than of gaining fame by reducing things to a better pa$s then they now are at. Yet never- thele$s, having of late years ob$erved the place, and $ailed through tho$e Chanels, and tho$e Waters; after I had made $ome reflection thereupon, I thought that the enterprize was not $o difficult as I had at fir$t conceited it to be; and I am the more confirmed in this opinion, upon the ind<*>ement of that which I have written <foot>Nnnn 2 Geo-</foot> <p n=>92</p> Geometrically in my Treati$e of the Men$uration of Running Waters; $o that talking with $everal per$ons, I adventured to affirm, in di$coures, that this improvement might po$$ibly be brought into a good e$tate.</P> <P>Now I have re$olved to $et down my thoughts in writing, and to honour this my Paper with the Noble Name of your Lord$hip, to render it the more credible and con$picuous at the fir$t view, if it $hould chance that the Subject I treat of, were not of $uch moment, as that it did de$erve to be valued for any other rea$on. Pardon me, Sir, if I have been too bold, and continue me in the number of your Servants.</P> <P>The enterprize of Draining a great part of the Territories of the <I>Pontine Fenns,</I> hath been undertaken both in the time of the antient <I>Romans,</I> and la$t of all, in our days; yea in the late times by <I>Sixtus</I> V. I do not doubt in the lea$t, but that it will be po$$ible yet to reduce things to a very good pa$s; and if I be not mi$taken, with a very $mall charge in compari$on of the profit that would be received from tho$e rich Grounds. This improvement was of great expence in the time of <I>Sixtus Quintus,</I> but by rea- $on the thing was not rightly under$tood, there were made many Drains; a great part of which were unprofitable and vain: and among$t $o many operations, there hapned $ome to be made that $ucceeded, as was de$ired; but not being under$tood, they were held in no account; and thus the bu$ine$s being neglected, the waters are returned into the $ame $tate as they were at fir$t, be- fore the improvement. Here I have by familiar di$cour$es with my friends, explained this enterprize undertaken by <I>Six- tus</I> V. and haply al$o by $ome more antient, with the example of the Fable of <I>Orilo,</I> in <I>Ario$to.</I> This Mon$ter was made up with $uch enchantment, that men fought with him alwayes in vain; for though in the Combate he were cut in pieces, tho$e divided Members pre$ently re-united, and returned to the fight more fierce then ever. But the <I>Paladine A$tolfo</I> coming to undertake him, after a long di$pute, at the end he cut his head $heer off from the $houlders at one blow; and nimbly alighting from his Hor$e, took the Mon$trous head, and mounting again, as he rid away he fell to $have the Pole of that Mon$ter, and $o he lo$t the Lock of Hair, in which alone the enchantment lay; and then the horrible Head in an in$tant manife$ted $igns of death, and the trunk which ran, $eeking to reunite to it anew, gave the la$t ga$p, and in this manner the enchantment ended. The Book of Fate $erved admirably to the <I>Paladine,</I> whereby he came to under- $tand that Charm; for by $having his whole head, the enchanted hairs came to be cut off among$t the re$t: In the $ame manner, I $ay, that it hath $ometimes happened in Draining tho$e Fields; <foot>for</foot> <p n=>93</p> for that among$t $o many tryals as have been made, that al$o was light upon, on which the improvement and remedy to the di$order did depend. And to us my fore-named Treati$e $hall $erve for a Rule, which being well under$tood, $hall make us to know wherein con$i$teth, and whereon dependeth this mi$carri- age, and con$equently it will be ea$ie to apply thereunto a $ea$o- nable remedy.</P> <P>And fir$t I $ay, That there is no doubt but that the waters continue $o high on tho$e Plains becau$e they are $o high in the principal River, which ought to receive them, and carry them into the Sea. Now the Cau$es of the height of the River, may in my judgement be reduced to one alone; which is that by me $o often mentioned for the mo$t Potent one, and declared in my afore-named Tractate; to wit, The tardity of the motion of the waters, which doth alwayes infallibly, and preci$ely cau$e the $elf $ame Running Water to change the mea$ure of its thickne$s at $uch a rate, that the more it encrea$eth in velocity, the more it decrea$eth in mea$ure; and the more it decrea$eth in velocity, the more it encrea$eth in mea$ure: As for example; If a River run in $uch a place with the velocity of moving a mile in the $pace of an hour, and afterwards the $ame River in another place doth encrea$e in velocity, $o as to make three miles an hour; that $ame River $hall dimini$h in thickne$s two thirds: And on the contrary, If it $hall dimini$h in velocity $o, as that it runneth but half a mile in the $ame time, it $hall encrea$e the double in thickne$s and mea$ure. And in a word, look what proportion the velocity in the fir$t place, hath to the velocity in the $econd, and $uch hath reciprocally the mea$ure of the thickne$s in the $econd place, to the mea$ure in the fir$t; as I have clearly demon- $trated in my Treati$e: Which I repeat $o frequently, that I fear the Profe$$ors of Polite Learning will charge me with Tua- tologie, and vain Repetition. But I am $o de$irous in this mo$t important point to be well under$tood, becau$e it will then be ea$ie to comprehend all the re$t; and without this it is impo$$ible (I will not $ay difficult, but ab$olutely impo$$ible) to under$tand, or ever to effect any thing to purpo$e. And the better to ex- plain the example, let it be $uppo$ed, <fig> That the water of a River A D, runneth high at the level of A F, with $uch a certain velocity; and let it, by the $ame water, be velocitated three times more; I $ay, that it will abate 1/3, and $hall $tand at the level in B E; and if it $hall more veloci- tate, it will abate the more at the Sea; But if it $hould retard <foot>more</foot> <p n=>94</p> more than it did at the level AF, it would ri$e yet more above the $aid level A F; although that the $elf $ame quantity of water runneth all the while. By the above-named $olid Principle I re$olve extravagant Problems in my Treati$e, and a$$ign the Rea- $ons of admirable effects of Running Waters: But as for what concerneth our purpo$e of the <I>Pontine Fenns,</I> we have the Cau- $es very plain and clear; for which, by the trampling of Cattle which pa$s thorow the <I>Draining River,</I> the waters abate $o nota- bly, that it is as it were a miracle for tho$e Reeds, Flags, and Weeds that $pring up, encrea$e, and $pread all over the River, $top and impede that velocity of the waters which they would have by means of their declivity. But that pa$$age of tho$e Bea$ts, treading down tho$e Weeds unto the bottom of the River, in $uch $ort, as that they no longer hinder the Current of the Water; and the $ame Waters increa$ing in their cour$e, they do dimi- ni$h in mea$ure and height; and by this meanes the Ditches of the Plains empty into the $ame $ucce$sfully, and leave them free from Waters, and Drained. But the$e Weeds in a $hort time $prouting up anew, and rai$ing their $talkes thorow the body of the Waters, they reduce things to the $ame evil $tate, as before, retarding the velocity of the Water, ma- king it to increa$e in height, and perhaps do occa$ion grea- ter mi$chiefs; $eeing that tho$e many knots which each plant $hoots forth, begets a greater multitude of Stalks, which much more incumbering the Water of the River, are a greater impe- diment unto its velocity, and con$equently make the height of the waters to encrea$e $o much the more, and do more mi$chief than before.</P> <P>Another head to which the$e harms may be reduced, but pro- ceeding from the $ame Root, which hath a great part in this di$order, is the impediment of tho$e Wears in the River which are made by heightning the bed of the $ame, for placing of fi$h- ing-nets; of which <I>Pi$caries</I> I reckoned above ten, when I made a voyage thorow tho$e waters to <I>Sandolo.</I> And the$e Fi$hing- Wears are $uch impediments, that $ome one of them makes the water of the River in the upper part to ri$e half a Palm, and $ometimes a whole Palm, and more; $o that when they are all gathered together, the$e impediments amount to more than $even, or po$$ibly than eight Palms.</P> <P>There concurreth for a third mo$t Potent Cau$e of the waters continuing high in the evacuating, or Draining Chanel, and con- $equently on the Plains; The great abundance of water that i$$u- eth from <I>Fiume Si$to,</I> the waters of which do not keep within its Banks when they are abundant; but encrea$ing above its Chanel, they unite with tho$e of the Evacuator, and di$per$ing thorow <foot>the</foot> <p n=>95</p> the Fens are rai$ed with great prejudice, and much grea- ter than is conceived, according to what hath been demon- $trated in the Second Con$ideration upon the <I>Lake of Venice.</I> Nor is it to any purpo$e to $ay, that if we $hould mea$ure all the Waters that disimbogue from <I>Fiume Si$to,</I> and gather them into one $umme, we $hould not finde them to be $uch, as that they $hall be able to make the Waters of the Fens to increa$e, by rea$on of the great expan$ion of them, over which that body of water is to di$tend: for to this in$tance we an$wer wich that which we have given notice of in the Fir$t Con- $ideration touching the <I>Lake of Venice,</I> treating of the abate- ment that is cau$ed by the <I>Brent</I> let into the Lake. And more- over, if I $hall adde thereto that which I write in the Second Con$ideration, it will be very apparent how greatly harmfull and prejudicial the$e excurfions of Waters from <I>Fiume Si$to</I> may be, which are not kept under, and confined within the River: Therefore, proceeding to the provi$ions, and ope- rations that are to be accounted Principall, I reduce them to three Heads.</P> <P>In the fir$t place it is nece$$ary to throw down tho$e Weares, and to take the Pi$ciaries quite away, ob$erving a Maxime, in my judgment, infallible, that Fi$hing and Sowing are two things that can never con$i$t together; Fi$hing being on the Water, and Sowing on land.</P> <P>Secondly, it will be nece$$ary to cut under Water in the bot- tome of the River tho$e Weeds and Plants that grow and in- crea$e in the River, and leave them to be carried into the Sea by the Stream; for by this means the$e Reeds $hall not $pring up and di$tend along the bottome of the River, by means of the Bea$ts treading upon them; And the $ame ought to be done often, and with care, and mu$t not be delaied till the mi$- chief increa$e, and the Champain Grounds be drowned, but one ought to order matters $o, as that they may not drown. And I will affirm, that otherwi$e this principal point would be- come a mo$t con$iderable inconvenience.</P> <P>Thirdly, it is nece$$ary to make good the Banks of <I>Fiume Si$to</I> on the left hand, and to procure that tho$e Waters may run in the Chanel, and not break forth. And it is to be noted, that it is not enough to do one or two of tho$e things, but we are to put them all in execution; for omitting any thing, the whole machine will be out of tune, and $poiled. But proceeding with due care, you $hall not only Drain the <I>Pontine Fens,</I> but by means of this la$t particular the Current of <I>Fiums Sisto</I> $hall $cowr its own Chanel of its $elf, even to the carrying part of it away: and haply with this abundance of water that it $hall <foot>bear</foot> <p n=>96</p> bear, the Mouth <I>della Torre</I> may be opened, and kept open into the Sea. And it would, la$t of all, be of admirable bene- fit to clean$e <I>Fiume Sisto</I> from many Trees and Bu$hes where- with it is overgrown.</P> <P>And with this I conclude, that the Improvement or Drain po$$ible to be made con$i$teth in the$e three particulars. Fir$t, in taking away the Fi$hing Weares, leaving the Cour$e of the Waters free. Secondly, in keeping the Principal Rivers clear from Weeds and Plants. Thirdly, in keeping the water of <I>Fiume Sisto</I> in its own Chanel. All which are things that may be done with very little charge, and to the manife$t benefit of the whole Country, and to the rendering the Air whol$omer in all tho$e Places adjoyning to the <I>Pon- tine Fens.</I></P> <fig> <p n=>97</p> <head>A CONSIDERATION Upon the DRAINING Of the Territories of Bologna, Ferrara, AND Romagna.</head> <head>BY D. BENEDETTO CASTELLI, Abbot of S. BENEDETTO ALOISIO, <I>Mathematician</I> to P. <I>Vrban</I> VIII. and Profe$$or in the Univer$ity of <I>ROME.</I></head> <P>The weghty bu$ine$$e of the Draining of the Territories of <I>Bologna, Ferrara,</I> and <I>Romagna</I> having been punctually handled and declared in writing from the excellent memory of the Right Ho- nourable and Noble <I>Mon$ignore Cor$ini,</I> who was heretofore Deputed Commi$- $ary General, and Vi$itor of tho$e Wa- ters; I am not able to make $uch ano- ther Di$cour$e upon the $ame Subject, but will only $ay $ome- what for farther confirmation of that which I have $aid in this Book upon the <I>Lake of Venice,</I> upon the <I>Pontine Fens,</I> and up- on the Draining of tho$e Plains of <I>Pi$a,</I> lying between the Ri- vers <I>Arno</I> and <I>Serchio</I>; whereby it is manife$t, that in all the <foot>Oooo afore-</foot> <p n=>98</p> aforementioned Ca$es, and in the pre$ent one that we are in hand with, there have, in times pa$t, very gro$$e Errours been com- mitted, through the not having ever well under$tood the true mea$ure of Running waters; and here it is to be noted, that the bu$ine$$e is, that in <I>Venice,</I> the diver$ion of the waters of the Lake, by diverting the <I>Brent</I> was debated, and in part executed, without con$ideration had how great abatement of water might follow in the Lake, if the <I>Brent</I> were diverted, as I have $hewn in the fir$t Con$ideration upon this particular, from which act there hath in$ued very bad con$equences, not only the difficulty of Navigation, but it hath infected the whol$omne$$e of the Air, and cau$ed the $toppage of the Ports of <I>Venice.</I> And on the contrary, the $ame inadvertency of not con$idering what ri$ing of the Water the <I>Reno,</I> and other Rivers being opened into the Val- leys of <I>Bologna</I> and <I>Ferrara,</I> might cau$e in the $aid Valleys, is the certain cau$e that $o many rich and fertile Fields are drown- ed under water, converting the happy habitations and dwellings of men into mi$erable receptacles for Fi$hes: Things which doubtle$$e would never have happened, if tho$e Rivers had been kept at their height, and <I>Reno</I> had been turn'd into <I>Main-Po,</I> and the other Rivers into that of <I>Argenta,</I> and of <I>Volano.</I> Now there having $ufficient been $poken by the above-named <I>Mon$ig. Cor$ini</I> in his Relation, I will only adde one conceit of my own, which after the Rivers $hould be regulated, as hath been $aid, I verily believe would be of extraordinary profit, I much doubt in- deed that I $hall finde it a hard matter to per$wade men to be of my mind, but yet neverthele$s I will not que$tion, but that tho$e, at lea$t, who $hall have under$tood what I have $aid and demon- $trated concerning the manners and proportions, according to which the abatements and ri$ings of Running waters proceed, that are made by the Diver$ions and Introductions of Waters, will apprehend that my conjecture is grounded upon Rea$on. And although I de$cend not to the exactne$$e of particulars, I will open the way to others, who having ob$erved the requi$ite Rules of con$idering the quantity of the waters that are intro- duced, or that happen to be diverted, $hall be able with punctu- ality to examine the whole bu$ine$$e, and then re$olve on that which $hall be expedient to be done.</P> <P>Reflecting therefore upon the fir$t Propo$ition, that the Ri$ings of a Running Water made by the acce$$ion of new water into the River, are to one another, as the Square-Roots of the quantity of the water that runneth; and con$equently, that the $ame cometh to pa$s in the Diver$ions: In$omuch, that a River running in height one $uch a certain mea$ure, to make it encrea$e double in height, the water is to be encrea$ed to three times as <foot>much</foot> <p n=>99</p> much as it ran before; $o that when the water $hall be quadru- ple, the height $hall be double; and if the water were centuple, the height would be decuple onely, and $o from one quantity to another: And on the contrary, in the Diver$ions; If of the 100. parts of water that run thorow a River, there $hall be di- verted 19/160, the height of the River dimini$heth onely 1/10, and con- tinuing to divert 17/100, the height of the River abateth likewi$e 1/10, and $o proceeding to divert 15/100 and then 13/100, and then 11/100, and then 9/100, and then 7/100, and then 5/100, and then 3/106, alwaies by each of the$e diver$ions, the height of the Running Water di- mini$heth the tenth part: although that the diver$ions be $o une. qual. Reflecting I $ay upon this infallible Truth, I have had a conceit, that though the <I>Reno</I> and other Rivers were diverted from the Valleyes, and there was onely left the <I>Chanel of Navi- gation,</I> which was onely the 1/20 part of the whole water that fal- leth into the Valleys; yet neverthele$s, the water in tho$e $ame Valleyes would retain a tenth part of that height that became conjoyned by the concour$e of all the Rivers: And therefore I $hould think that it were the be$t re$olution to maintain the <I>Gha- nel of Navigation</I> (if it were po$$ible) continuate unto the P<I>o</I> of <I>Ferrara,</I> and from thence to carry it into the P<I>o</I> of <I>Volano</I>; for be$ides that it would be of very great ea$e in the Navigation of <I>Bologna,</I> and <I>Ferrara,</I> the $aid water would render the P<I>o</I> o$ <I>Volano</I> navigable as far as to the very Walls of <I>Ferrara,</I> and con- $equently the Navigation would be continuate from <I>Bologna</I> to the Sea-$ide.</P> <P>But to manage this enterprize well, it is nece$$ary to mea$ure the quantity of the Water that the Rivers di$charge into the Val- leys, and that which the <I>Chanel of Navigation</I> carryeth, in man- ner as I have demon$trated at the beginning of this Book; for this once known, we $hall al$o come to know, how profitable this di- ver$ion of the <I>Chanel of Navigation</I> from the Valleys is like to prove; which yet would $till be unprofitable, if $o be that all the Rivers that di$charge their waters into the Valleys, $hould not $ir$t be Drained, according to what hath been above ad- verti$ed.</P> <P>Abbot CASTELLI, <I>in the pre$ent con$ideration referring himfelf to the Relation of</I> Mon$ig. Cor$ini, <I>grounded upon the Ob- $ervations and Precepts of the $aid Abbot; as is $een in the pre- $ent Di$cour$e. I thought it convenient for the compleating of the Work of our Aulhour, upon the$e $ubjects, to in$ert it in this place.</I></P> <foot>Oooo2 A</foot> <p n=>100</p> <head>A Relation of the Waters in the Territories of <I>Bologna</I> and <I>Ferrara.</I> BY</head> <head>The Right Honourable and Illu$trious, <I>Mon$ig- nore</I> CORSINI, a Native of <I>Ju$cany,</I> Su- perintendent of the general DRAINS, and Pre$ident of <I>Romagna-</I></head> <P>The <I>Rheno,</I> and other Brooks of <I>Romagna,</I> were by the advice of <I>P. Ago$tino Spernazzati</I> the Je$uite, towards the latter end of the time of <I>Pope Clement</I> VIII. notwith- $tanding the oppo$ition of the <I>Bologne$i,</I> and others concerned therein, diverted from their Chanels, for the more commodious clean$ing of the P<I>o</I> of <I>Ferrara,</I> and of its two Branches of <I>Prima- ro,</I> and <I>Volano</I>; in order to the introducing the water of the <I>Main-Po</I> into them, to the end that their wonted Torrents being re$tored, they might carry the Muddy-water thence into the Sea, and re$tore to the City the Navigation which was la$t, as is ma- nife$t by the Brief of the $aid <I>Pope Clement,</I> directed to the <I>Car- dinal San Clemence,</I> bearing date the 22. of <I>Augu$t,</I> 1604.</P> <P>The work of the $aid clean$ing, and introducing of the $aid P<I>o,</I> either as being $uch in it $elf, or by the contention of the <I>Cardinal Legates</I> then in the$e parts; and the jarrings that hap- ned betwixt them, proved $o difficult, that after the expence of va$t $umms in the $pace of 21. years, there hath been nothing done, $ave the rendring of it the more difficult to be effected.</P> <P>Interim, the Torrents with their waters, both muddy and clear, have damaged the Grounds lying on the right hand of the P<I>o</I> of <I>Argenta,</I> and the <I>Rheno</I> tho$e on its Banks; of which I will $peak in the fir$t place, as of that which is of greater impor- tance, and from which the principal cau$e of the mi$chiefs that re$ult from the re$t doth proceed.</P> <marg>* Or Lord$hip.</marg> <P>This <I>Rbeno</I> having overflowed the ^{*} Tennency of <I>Sanmartina,</I> in circumference about fourteen miles given it before, and part of that of <I>Cominale</I> given it afterwards, as it were, for a recepta- cle; from whence, having depo$ed the matter of its muddine$s, it i$$ued clear by the Mouths of <I>Ma$i,</I> and of <I>Lievaloro,</I> into the P<I>o</I> of <I>Primaro,</I> and of <I>Volano</I>; did break down the encom- <foot>pa$$ing</foot> <p n=>101</p> pa$$ing Bank or Dam towards S. <I>Martino,</I> and that of its new Chanel on the right hand neer to <I>Torre del Fondo.</I></P> <P>By the breaches on this $ide it $treamed out in great abun- dance from the upper part of <I>Cominale,</I> and in the parts about <I>Raveda, Pioggio, Caprara, Chiare di Reno, Sant' Ago$tino, San Pro$pero, San Vincenzo,</I> and others, and made them to become incultivable: it made al$o tho$e places above but little fruitful, by rea$on of the impediments that their Draines received, finding the Conveyances called <I>Riolo</I> and <I>Scor$uro,</I> not only filled by <I>la Motta</I> and <I>la Belletta,</I> but that they turned backwards of them- $elves.</P> <P>But by the Mouths in the inclo$ing Bank or Dam at <I>Borgo di</I> S. <I>Martino</I> i$$uing with violence, it fir$t gave ob$truction to the ancient Navigation of <I>la Torre del la Fo$$a,</I> and afterwards to the moderne of the mouth of <I>Ma$i,</I> $o that at pre$ent the Com- merce between <I>Bologna</I> and <I>Ferrara</I> is lo$t, nor can it ever be in any durable way renewed, whil$t that this exceeds its due bounds, and what ever moneys $hall be imployed about the $ame $hall be without any equivalent benefit, and to the manife$t <marg>* The Popes Exchequer.</marg> and notable prejudice of the ^{*} Apo$tolick Chamber.</P> <P>Thence pa$$ing into the Valley of <I>Marzara,</I> it $welleth high- er, not only by the ri$ing of the water, but by the rai$ing of the bottome, by rea$on of the matter $unk thither after Land- floods, and dilateth $o, that it covereth all the Meadows there- abouts, nor doth it receive with the wonted facility the Drains of the upper Grounds, of which the next unto it lying under the wa- ters that return upwards by the Conveyances, and the more re- mote, not finding a pa$$age for Rain-waters that $ettle, become either altogether unpro$itable or little better.</P> <P>From this Valley, by the Trench or Ditch of <I>Marzara,</I> or of <I>la Duca</I> by <I>la Buova,</I> or mouth of <I>Ca$taldo de Ro$$i,</I> and by the new pa$$age it falleth into the P<I>o</I> of <I>Argenta,</I> which being to re- ceive it clear, that $o it may $ink farther therein, and receiving it muddy, becau$e it hath acquired a quicker cour$e, there will ari$e a very contrary effect.</P> <P>Here therefore the $uperficies of the water keeping high, until it come to the Sea, hindereth the Valleys of <I>Ravenna,</I> where the River <I>Senio,</I> tho$e of <I>San Bernardino</I> where <I>Santerno</I> was turned, tho$e of <I>Buon' acqui$to,</I> and tho$e of <I>Marmorto,</I> where the <I>Idice, Quaderna, Sellero</I> $all in, from $wallowing and taking in their Waters by their u$ual In-lets, yet many times, as I my $elf have $een in the <I>Vi$itation,</I> they drink them up plentifully, whereupon, being conjoyned with the muddine$$e of tho$e Ri- vers that fall into the $ame, they $well, and dilate, and overflow $ome grounds, and deprive others of their Drains in like manner <foot>as</foot> <p n=>102</p> as hath been $aid of that of <I>Marrara,</I> in$omuch that from the Point of S. <I>Giorgio,</I> as far as S. <I>Alberto</I> all tho$e that are between the Valleys and P<I>o</I> are $poiled, of tho$e that are between Valley and Valley many are in a very bad condition, and tho$e that are $ome con$iderable $pace above not a little damnified.</P> <P>In fine, by rai$ing the bottom or $and of the Valleys, and the bed of <I>Reno,</I> and the too great repletion of the P<I>o</I> of <I>Primaro</I> with waters, the Valleys of <I>Comacchio</I> (on which $ide the Banks are very bad) and ^{*} <I>Pole$ine di</I> S. <I>Giorgio</I> are threatned with a <marg>+ <I>Pole$ine</I> is a plat of Ground al- mo$t $urrounded with Bogs or wa- ters, like an I$land</marg> danger, that may in time, if it be not remedied, become irrepa- rable, and at pre$ent feeleth the incommodity of the Waters, which penetrating thorow the pores of the Earth do $pring up in the $ame, which they call <I>Purlings,</I> which is all likely to redound to the prejudice of <I>Ferrara,</I> $o noble a City of <I>Italy,</I> and $o im- portant to the <I>Eccle$ta$tick State.</I></P> <P>Which particulars all appear to be atte$ted under the hand of a Notary in the <I>Vi$itation</I> which I made upon the command of His Holine$$e, and are withall known to be true by the ^{*}<I>Ferrare$t</I> <marg>* People of <I>Fer- rara.</I></marg> them$elves, of whom (be$ides the reque$t of the <I>Bologne$i</I>) the greater part beg compa$$ion with $undry <I>Memorials,</I> and reme- dies, a$well for the mi$chiefs pa$t, as al$o for tho$e in time to come, from which I hold it a duty of Con$cience, and of Cha- rity to deliver them.</P> <P>Pope <I>Clement</I> judged, that the $ufficient means to effect this was the $aid Introduction of the <I>Main Po</I> into the Chancl of <I>Ferrara</I>; a re$olution truly Heroical, and of no le$$e beauty than benefit to that City, of which I $peak not at pre$ent, be- cau$e I think that there is need of a readier and more acco- modate remedy.</P> <P>So that I $ee not how any other thing can be $o much con$ide- rable as the removal of <I>Reno,</I> omitting for this time to $peak of <marg>* In Chanels made by hand.</marg> ^{*} inclo$ing it from Valley to Valley untill it come to the Sea, as the Dukes of <I>Ferrara</I> did de$ign, fora$much as all tho$e <I>Ferra- re$i</I> that have intere$t in the <I>Pole$ine di</I> S. <I>Giorgio,</I> and on the right hand of the P<I>o</I> of <I>Argenta</I> do not de$ire it, and do, but too openly, prote$t again$t it; and becau$e that before the Chanel were made as far as the Sea, many hundreds of years would be $pent, and yet would not remedy the dammages of tho$e who now are agrieved, but would much increa$e them, in regard the Valleys would continue $ubmerged, the Drains $topped, and the other Brooks ob$tructed, which would of nece$$ity drown not a few Lands that lie between Valley and Valley; and in fine, in regard it hath not from <I>San Martina</I> to the Sea for a $pace of $if- ty miles a greater fall then 19, 8, 6, feet, it would want that force which they them$elves who propound this project do require it to <foot>have</foot> <p n=>103</p> have, that $o it may not depo$e the matter of the muddine$s when it is intended to be let into <I>Volana.</I></P> <P>So that making the Line of the bottome neer to <I>Vigarano,</I> it would ri$e to tho$e prodigious termes that they do make bigger, and they may thence expect tho$e mi$chiefs, for which they will not admit of introducing it into the $aid P<I>o</I> of <I>Volana.</I></P> <P>Among$t the wayes therefore that I have thought of for effect- ing that $ame remotion, and which I have cau$ed to be viewed by skilful men that have taken a level thereof, (with the a$$i$tance of the venerable Father, <I>D. Benedetto Ca$telli</I> of <I>Ca$ina,</I> a man of much fidelity and hone$ty, and no le$s expert in $uch like affairs touching waters, than perfect in the <I>Mathematick</I> Di$ciplines) two onely, the re$t being either too tedious, or too dangerous to the City, have $eemed to me worthy, and one of them al$o more than the other, to offer to your Lord$hip.</P> <P>The one is to remit it into the Chanel of <I>Volana,</I> thorow which it goeth of its own accord to the Sea.</P> <P>The other is to turn it into <I>Main-Po</I> at <I>Stellata,</I> for, as at other times it hath done, it will carry it to the Sea happily.</P> <P>As to what concerns the making choice of the fir$t way, that which $eemeth to per$wade us to it is, that we therein do nothing that is new, in that it is but re$tored to the place whence it was removed in the year 1522. in the time of Pope <I>Adrian,</I> by an agreement made in way of contract, between <I>Alfon$o,</I> Duke of <I>Ferrara,</I> and the <I>Bologne$i</I>; and that it was diverted for rea$ons, that are either out of date, or el$e have been too long time deferred.</P> <P>In like manner the facility wherewith it may be effected, let- ting it run into the divided P<I>o,</I> whereby it will be turned to <I>Fer- rara,</I> or el$e carrying it by <I>Torre del Fondo,</I> to the mouth of <I>Ma$i,</I> and from thence thorow the Trench made by the <I>Ferrare$i,</I> along by <I>Panaro,</I> where al$o finding an ample Bed, and high and thick Banks, that will $erve at other times for it, and for the wa- ters of P<I>o,</I> there may a great expence be $pared.</P> <P>That what ever its Fall be, it would maintain the $ame, not having other Rivers, which with their Floods can hinder it; and that running confined between good Banks, without doubt it would not leave <I>la Motto</I> by the way; but e$pecially, that it would be $ufficient if it came to <I>Codigoro,</I> where being a$$i$ted by the Ebbing and Flowing of the Sea, it would run no hazard of having its Chanel filled up from thence downwards.</P> <P>That there might thence many benefits be derived to the City, by means of the Running Waters, and al$o no mean Navigation might be expected.</P> <P>On the contrary it is objected, That it is not convenient to <foot>think</foot> <p n=>104</p> think of returning this Torrent into the divided P<I>o,</I> by rea$on of the peril that would thence redound to this City.</P> <P>And that going by <I>Torre del Fondo,</I> through <I>Sanmartina</I> to the Mouth <I>de Ma$i</I> by the Chappel of <I>Vigarano</I> unto the Sea, it is by this way 70. miles; nor is the Fall greater than 26. 5. 6. Feet, $o that it would come to fall but 4. inches & an half, or thereabouts in a mile; whereas the common opinion of the skilfull (to the end that the Torrents may not depo$e their $and that they bring with them in Land-Floods) requireth the twenty fourth part of the hundredth part of their whole length, which in our ca$e, accounting according to the mea$ure of the$e places, is 16. inches <marg>* The inch of the$e places is $omewhat bigger than ours.</marg> a ^{*} mile; whereupon the $inking of the Mud and Sand would mo$t certainly follow, and $o an immen$e heightning of the Line of the Bottom, and con$equently a nece$$ity of rai$ing the Banks, the impo$$ibility of maintaining them, the danger of breaches and decayes, things very prejudicial to the <I>I$lets</I> of this City, and of <I>San Giorgio,</I> the ob$truction of the Drains, which from the Tower of <I>Tienne</I> downwards, fall into the $aid Chanel; to wit, tho$e of the Sluices of <I>Goro,</I> and the Drains, of the Meadows of <I>Ferrara</I>: And moreover, the damages that would ari$e unto the $aid <I>I$let</I> of S. <I>Giorgio,</I> and the Valleys of <I>Comachio,</I> by the wa- ters that $hould enter into the <I>Goro</I> or Dam of the Mills of <I>Belri- guardo,</I> thorow the Trenches of <I>Quadrea,</I> which cannot be $topt, becau$e they belong to the Duke of <I>Modena,</I> who hath right of diverting the waters of that place at his plea$ure to the work of turning Mills.</P> <P>The greater part of which Objections, others pretend to prove frivolous, by $aying, that its running there till at the la$t it was turned another way, is a $ign that it had made $uch an elevation of the Line, of its Bed as it required; denying that it needeth $o great a declivity as is mentioned above; and that for the fu- ture it would ri$e no more.</P> <P>That the $aid Dra ns and Ditches did empty into the $ame, whil$t P<I>o</I> was there; $o that they mu$t needs be more able to do $o when onely <I>Reno</I> runs that way.</P> <P>That there would no Breaches follow, or if they did, they would be onely of the water of <I>Reno,</I> which in few hours might be taken away (in tho$e parts they call damming up of Breaches, and mending the Bank, <I>taking away the Breaches</I>) and its a que- $tion whether they would procure more inconvenience than bene- fit, for that its Mud and Sand might in many places, by filling them up, occa$ion a $ea$onable improvement.</P> <P>Now omitting to di$cour$e of the $olidity of the rea$ons on the one$ide, or on the other, I will produce tho$e that move me to $u$pend my allowance of this de$ign.</P> <foot>The</foot> <p n=>105</p> <P>The fir$t is, that although I dare not $ub$cribe to the opinion of tho$e that require 16. inches Declivity in a mile to <I>Reno,</I> to prevent its depo$ing of Mud; yet would I not be the Author that $hould make a trial of it with $o much hazard, for having to $a- tisfie my $elf in $ome particulars cau$ed a Level to be taken of the Rivers <I>L'amone, Senio,</I> and <I>Santerno,</I> by <I>Bernardino Aleotti,</I> we found that they have more Declivity by much than Arti$ts re- quire, as al$o the <I>Reno</I> hath from <I>la Botta de Ghi$lieri</I> to the Chappel of <I>Vigarano,</I> for in the $pace of four miles its Bottom- Line falleth five feet and five inches. So that I hold it greater prudence to depend upon that example, than to go contrary to a common opinion, e$pecially $ince, that the effects cau$ed by <I>Reno</I> it $elf do confirm me in the $ame, for when it was for$aken by the P<I>o,</I> after a few years, either becau$e it had choaked up its Chanel with Sand, or becau$e its too long journey did increa$e it, it al$o naturally turned a$ide, and took the way of the $aid P<I>o</I> towards <I>Stellata.</I> Nay, in tho$e very years that it did run that way, it only began (as relations $ay) to make Breaches, an evi- dent $ign that it doth depo$e Sand, and rai$e its Bed; which a- greeth with the te$timony of $ome that were examined in the <I>Vi$itation</I> of the Publique Notary, who found great benefit by having Running Water, and $ome kind of pa$$age for Boats, and yet neverthele$s affirm that it for want of Running Water had made too high Stoppages and Shelfes of Sand; $o that if it $hould be re$tored to the Cour$e that it for$ook, I much fear that after a $hort time, if not $uddenly, it would leave it a- again.</P> <P>The $econd I take from the ob$ervation of what happened to <I>Panaro,</I> when with $o great applau$e of the <I>Ferare$i,</I> it was brought by Cardinal <I>Serra</I> into the $aid Chanel of <I>Volana</I>; for that notwith$tanding that it had Running Waters in much grea- ter abundance than <I>Reno</I>; yet in the time that it continued in that Chanel it rai$ed its Bed well neer five feet, as is to be $een below the Sluice made by Cardinal <I>Capponi</I> to his new Chanel; yea, the $aid Cardinal <I>Serra</I> who de$ired that this his under taking $hould appear to have been of no danger nor damage, was con- $trained at its Overflowings, to give it Vent into <I>Sanmartina,</I> that it might not break in upon, and prejudice the City; which dan- ger I $hould more fear from <I>Reno,</I> in regard it carrieth a greater abundance of Water and Sand</P> <P>Thirdly, I am much troubled (in the uncertainty of the $uc- ce$s of the affair) at the great expence thereto required; For in regard I do not approve of letting it in, neer to the Fortre$$e, for many re$pects, and carrying it by <I>la Torre del Fondo</I> to the Month <I>de Ma$t,</I> it will take up eight miles of double Banks, a <foot>Pppp thing</foot> <p n=>106</p> thing not ea$ie to be procured, by rea$on that the Grounds lie under Water; but from the Mouth <I>de Ma$i</I> unto <I>Codigoro,</I> it would al$o be nece$$ary to make new Scowrings of the Chanel; to the end, that the Water approaching (by wearing and carry- ing away the Earth on both $hores, might make a Bed $ufficient for its Body, the depth made for <I>Panaro</I> not $erving the turn, as I conceive; and if it $hould $uffice, when could the people of <I>Ferrara</I> hope to be re-imbur$ed and $atisfied for the charge thereof?</P> <P>Fourthly, it $erves as an Argument with me, to $ee that the very individual per$ons concerned in the Remotion or Diver$ion of the $aid Torrent, namely, the <I>Bologne$i</I> do not incline unto it, and that the whole City of <I>Ferrara,</I> even tho$e very per$ons who at pre$ent receive damage by it, cannot indure to hear thereof. The rea$on that induceth the$e la$t named to be $o aver$e thereto, is, either becau$e that this undertaking will render the introducti- on of the Water of <I>Main-Po</I> more difficult; or becau$e they fear the danger thereof; The others decline the Project, either for that they know that <I>Reno</I> cannot long continue in that Cour$e, or becau$e they fear that it is too much expo$ed to tho$e mens re- vengeful Cutting of it who do not de$ire it $hould; and if a man have any other wayes, he ought, in my opinion, to forbear that, which to $uch as $tand in need of its Removal, is le$$e $ati$- factory, and to $uch as oppo$e it, more prejudicial.</P> <P>To conclude, I exceedingly honour the judgment of Cardinal <I>Capponi,</I> who having to his Natural Ability and Prudence added a particular Study, Ob$ervation, and Experience of the$e Wa- ters for the $pace of three years together, doth not think that <I>Reno</I> can go by <I>Volana</I>; to which agreeth the opinion of Car- dinal S. <I>Marcello,</I> Legate of this City, of whom, for his exqui- $ite under$tanding, we ought to make great account. But if e- ver this $hould be re$olved on, it would be materially nece$$ary to unite the Quick and Running Waters of the little Chanel of <I>Cento,</I> of the Chanel <I>Navilio,</I> of <I>Guazzaloca,</I> and at its very beginning tho$e of <I>Dardagna,</I> which at pre$ent, is one of the Springs or Heads of <I>Panaro,</I> that $o they might a$$i$t it in carry- ing its Sand, and the matter of its Muddine$s into the Sea; and then there would not fail to be a greater evacuation and $cowr- ing; but withall the Proprietors in the I$let of <I>San Giorgio</I> and of <I>Ferrara</I> mu$t prepare them$elves to indure the inconveniences of Purlings or Sewings of the Water from the River thorow the Boggy Ground thereabouts.</P> <P>I $hould more ea$ily incline therefore to carry it into <I>Main-Po</I> at <I>Stellata,</I> for the Rea$ons that Cardinal <I>Capponi</I> mo$t ingeni- ou$ly enumerates in a $hort, but well-grounded Tract of his: not <foot>becau$e</foot> <p n=>107</p> becau$e that indeed it would not both by Purlings and by Brea- ches occa$ion $ome inconvenience; e$pecially, in the beginning: but becau$e I hold this for the incomodities of it, to be a far le$s evil than any of the re$t; and becau$e that by this means there is no occa$ion given to them of <I>Ferrara,</I> to explain that they are deprived of the hope of ever $eeing the P<I>o</I> again under the Walls of their City: To whom, where it may be done, it is but rea$on that $atisfaction $hould be given.</P> <P>It is certain that P<I>o</I> was placed by Nature in the mid$t of this great Valley made by the <I>Appennine</I> Hills, and by the Alps, to carry, as the Ma$ter-Drain to the Sea, that is the grand receptacle of all Waters; tho$e particular $treams which de$cend from them.</P> <P>That the <I>Reno</I> by all Geographers, <I>Strabo, Pliuy, Solimas, Mella,</I> and others is enumerated among the Rivers that fall into the $aid P<I>o.</I></P> <P>That although P<I>o</I> $hould of it $elf change its cour$e, yet would <I>Reno</I> go to look it out, if the works erected by humane ind u$try did not ob$truct its pa$$age; $o that it neither is, nor ought to $eem $trange, if one for the greater common good $hould turn it into the $ame.</P> <P>Now at <I>Stellata</I> it may go $everal waies into P<I>o,</I> as appeareth by the levels that were taken by my Order; of all which I $hould be$t like the turning of it to <I>la Botta de' Ghi$lieri,</I> carrying it above <I>Bondeno</I> to the Church of <I>Gambarone,</I> or a little higher or lower, as $hall be judged lea$t prejudicial, when it cometh to the execution, and this for two principal rea$ons: The one becau$e that then it will run along by the confines of the Church P tri- mony, without $eparating <I>Ferrara</I> from the re$t of it; The other is, Becau$e the Line is $horter, and con$equently the fall greater; for that in a $pace of ten miles and one third, it falleth twenty $ix feet, more by much than is required by Arti$ts; and would go by places where it could do but little hurt, notwith$tanding that the per$ons interre$$ed $tudy to amplifie it incredibly.</P> <P>On the contrary, there are but onely two objections that are worthy to be examined; One, That the Drains and Ditches of S. <I>Bianca,</I> of the Chanel of <I>Cento,</I> and of <I>Burana,</I> and all tho$e others that enter into P<I>o,</I> do hinder this diver$ion of <I>Reno,</I> by the encrea$ing of the waters in the P<I>o.</I> The other is that P<I>o</I> ri$ing about the Tran$om of the <I>Pila$ter</I>-Sluice, very near 20 feet, the <I>Reno</I> would have no fall into the $ame; whereupon it would ri$e to a terrible height, at which it would not be po$$ible to make, or keep the Banks made, $o that it would break out and drown the Meadowes, and cau$e mi$chiefs, and damages un$peakable and irreparable; as is evident by the experiment made upon <foot>Pppp 2 <I>Pana-</I></foot> <p n=>108</p> <I>Panaro,</I> which being confined between Banks, that it might go into P<I>o,</I> this not being neither in its greate$t excre$cen$e, it broke out into the territories of <I>Final,</I> and of <I>Ferrara.</I> And though that might be done, it would thereupon en$ue, that there being let into the Chanel of P<I>o,</I> 2800, $quare feet of water (for $o much we account tho$e of <I>Reno</I> and <I>Panaro,</I> taken together in their greate$t heights) the $uperficies of it would ri$e at lea$t four feet, in$omuch that either it would be requi$ite to rai$e its Banks all the way unto the Sea, to the $ame height, which the trea$ures of the <I>Indies</I> would not $uffice to effect; or el$e there would be a nece$- $ity of enduring exce$$ive Breaches. To the$e two Heads are the Arguments reduced, which are largely amplified again$t our opi- nion; and I $hall an$wer fir$t to the la$t, as mo$t material.</P> <P>I $ay therefore, that there are three ca$es to be con$idered: Fir$t, P<I>o</I> high, and <I>Reno</I> low. Secondly, <I>Reno</I> high, and P<I>o</I> low. Thirdly, <I>Reno</I> and P<I>o</I> both high together.</P> <P>As to the fir$t and $econd, there is no difficulty in them; for if P<I>o</I> $hall not be at its greate$t height, <I>Reno</I> $hall ever have a fall into it, and there $hall need no humane Artifice about the Banks: And if <I>Reno</I> $hall be low, P<I>o</I> $hall regurgitate and flow up into the Chanel of it; and al$o from thence no inconvenience $hall follow. The third remains, from which there are expected ma- ny mi$chiefs; but it is a mo$t undoubted truth, that the excre$cen- cies of <I>Reno,</I> as coming from the adjacent <I>Appennines</I> and Rains, are to continue but $even, or eight hours at mo$t, and $o would never, or very rarely happen to be at the $ame time with tho$e of P<I>o,</I> cau$ed by the melting of the $nowes of the Alps, at lea$t 400. miles di$tance from thence. But becau$e it $ometimes may hap- pen, I reply, that when it cometh to pa$s, <I>Reno</I> $hall not go into P<I>o,</I> but it $hall have allowed it one or two Vents; namely, into the Chanel of <I>Ferrara,</I> as it hath ever had; and into <I>Sanmartina,</I> where it runneth at pre$ent, and wherewith there is no doubt, but that the per$ons concerned will be well plea$ed, it being a great benefit to them, to have the water over-flow their grounds once every four or five years, in$tead of $eeing it anoy them continu- ally. Yea, the Vent may be regulated, re$erving for it the Cha- nel in which <I>Reno</I> at pre$ent runneth; and in$tead of turning it by a Dam at <I>la Betta de Chi$lieri,</I> perhaps, to turn it by help of $trong Sluices, that may upon all occa$ions be opened and $hut. And for my part, I do not que$tion but that the Proprietors them$elves in <I>Sanmartina</I> would make a Chanel for it; which receiving, and confining it in the time of the Vents, might carry the Sand into the P<I>o</I> of <I>Primaro:</I> Nor need there thence be fear- ed any $toppage by Mud and Sand, $ince that it is $uppo$ed that there will but very $eldom be any nece$$ity of u$ing it; $o that <foot>time</foot> <p n=>109</p> time would be allowed, upon occa$ion, to $cowr and clean$e it.</P> <P>And in this manner all tho$e Prodigies vani$h that are rai$ed with $o much fear from the enterance of the Water of <I>Reno</I> $welled into P<I>o,</I> when it is high, to which there needeth no other an$wer; yet neverthele$$e we do not take that quantity of Wa- ter, that is carried by <I>Reno,</I> and by <I>Panaro,</I> to be $o great as is affir- med: For that P. D. <I>Benedetto Ca$telli</I> hath no le$$e accutely than accurately ob$erved the mea$ures of this kind, noting that the breadth and depth of a River is not enough to re$olve the que$tion truly, but that there is re$pect to be had to the velocity of the Waters, and the term of time, things hitherto not con$i- dered by the Skilful in the$e affairs; and therefore they are not able to $ay what quantity of Waters the $aid Rivers carry, nor to conclude of the ri$ings that will follow thereupon. Nay, it is mo$t certain, that if all the Rivers that fall into <I>Po,</I> which are above thirty, $hould ri$e at the rate that the$e compute <I>Reno</I> to do, an hundred feet of Banks would not $uffice, and yet they have far fewer: So that this confirmes the Rule of R. P. D. <I>Bene- detto,</I> namely, that the proportion of the height of the Water of <I>Reno</I> in <I>Reno</I> to the height of the Water of <I>Reno</I> in P<I>o,</I> is compounded of the proportion of the breadth of the Chanel of <I>Po</I> to that of <I>Reno,</I> and of the velocity of the Water of <I>Reno</I> in <I>Po</I> to the velccity of the Water of <I>Reno</I> in <I>Reno</I>; a manife$t argument that there cannot in it, by this new augmentation of Waters follow any alteration that nece$$itates the rai$ing of its Banks, as appeareth by the example of <I>Panaro,</I> which hath been $o far from $welling P<I>o,</I> that it hath rather a$$waged it, for it hath carried away many Shelfs and many I$lets that had grown in its Bed, for want of Waters $ufficient to bear away the matter of Land-floods in $o broad a Chanel; and as is learnt by the trial made by us in <I>Panaro</I> with the Water of <I>Burana</I>; for erecting in the River $tanding marks, and $hutting the $aid Sluice, we could $ee no $en$ible abatement, nor much le$s after we had opened it $en$ible increa$ment; by which we judge that the $ame is to $uc- ceed to P<I>o,</I> by letting in of <I>Reno, Burana</I> having greater pro- portion to <I>Panaro</I> than <I>Reno</I> to P<I>o,</I> con$idering the $tate of tho$e Rivers in which the Ob$ervation was made. So that there is no longer any occa$ion for tho$e great rai$ings of Banks, and the danger of the ruptures as well of <I>Reno</I> as of P<I>o</I> do vani$h, as al- $o the fear le$t that the Sluices which empty into P<I>o</I> $hould re- ceive ob$truction: which if they $hould, yet it would be over in a few hours. And as to the Breaches of <I>Panaro</I> which happened in 1623. I know not why, $eeing that it is confe$$ed that the P<I>o</I> was not, at that time, at its height, one $hould rather charge it <foot>with</foot> <p n=>110</p> with the crime, than quit it thereof. The truth is, that the Bank was not made of proof, $ince that the $ame now continu- eth whole and good, and <I>Panaro</I> doth not break out; nay, there was, when it brake more than a foot and half of its Banks above the Water, and to $pare; but it broke thorow by a Moles wor- king, or by the hole of a Water-Rat, or $ome $uch vermine; and by occa$ion of the badne$s of the $aid Banks, as I finde by the te$timony of $ome witne$$es examined by my command, that I might know the truth thereof. Nor can I here forbear to $ay, that it would be better, if in $uch matters men were more candid and $incere. But to $ecure our $elves neverthele$$e, to the ut- mo$t of our power, from $uch like Breaches which may happen at the fir$t, by rea$on of the newne$$e of the Banks, I pre$uppo$e that from P<I>o</I> unto the place whence <I>Reno</I> is cut, there ought to be a high and thick Fence made with its Banks, $o that there would be no cau$e to fear any what$oever acce$$ions of Water, although that concurrence of three Rivers, which was by $ome more ingeniou$ly aggravated than faithfully $tated by that which was $aid above were true; to whom I think not my $elf bound to make any farther reply, neither to tho$e who $ay that <I>Po</I> will a$cend upwards into <I>Reno,</I> $ince that the$e are the $ame per$ons who would introduce a $mall branch of the $aid P<I>o</I> into the Chanel of <I>Ferrara,</I> that $o it may conveigh to the Sea, not <I>Reno</I> onely, but al$o all the other Brooks of which we complained; and becau$e that withal it is impo$$ible, that a River $o capacious as <I>Po</I> $hould be incommoded by a Torrent, that, as I may $ay, hath no proportion to it.</P> <P>I come now to the bu$ine$$e of the Ditches and Draines; and as to the Conveyance of <I>Burana,</I> it hath heretofore been deba- ted to turn it into <I>Main-Po,</I> $o that in this ca$e it will receive no harm, and though it were not removed, yet would it by a Trench under ground pur$ue the cour$e that it now holdeth, and al$o would be able to di$-imbogue again into the $aid new Chanel of <I>Reno,</I> which conforming to the $uperficies of the Water of <I>Po,</I> would continue at a lower level than that which <I>Panara</I> had when it came to <I>Ferrara,</I> into which <I>Burana</I> did neverthele$$e empty it $elf for $ome time.</P> <P>The Conveyance or Drain of <I>Santa Bianca,</I> and the little Chanel of <I>Cento</I> may al$o empty them$elves by two $ubterranean Trenches, without any prejudice where they run at pre$ent, or without any more works of that nature, they may be turned into the $aid new Chanel, although with $omewhat more of incon- venience; and withall, the Chanel of <I>Ferrara,</I> left dry, would be a $ufficient receptacle for any other Sewer or Drain what$oe- ver, that $hould remain there.</P> <foot>All</foot> <p n=>111</p> <P>All which Operations might be brought to perfection with 150. thou$and Crowns, well and faithfully laid out; which $umm the <I>Bologne$i</I> will not be unwilling to provide; be$ides that tho$e <I>Ferrare$i</I> ought to contribute to it, who $hall partake of the benefit.</P> <P>Let me be permitted in this place to propo$e a thing which I have thought of, and which peradventure might occa$ion two benefits at once, although it be not wholly new. It was in the time of <I>Pope Paul</I> V. propounded by one <I>Cre$cenzio</I> an Ingi- neer, to cut the <I>Main-Po,</I> above <I>le Papozze</I>; and having made a $ufficient evacuation to derive the water thereof into the P<I>o</I> of <I>Adriano,</I> and $o to procure it to be Navigable, which was not at that time effected, either by rea$on of the oppo$itions of tho$e, who$e po$$e$$ions were to be cut thorow, or by rea$on of the great $um of money that was nece$$ary for the effecting of it: But in viewing tho$e Rivers, we have ob$erved, that the $edge cutting might ea$ily be made below <I>le Papozze,</I> in digging thorow the Bank called <I>Santa Maria,</I> & drawing a Trench of the bigne$s that skilful Arti$ts $hall judge meet unto the P<I>o</I> ^{*} of <I>Ariano,</I> below the <marg>* Of <I>Adriān<*>.</I></marg> <I>Secche</I> of the $aid S. <I>Maria</I>; which as being a work of not above 160. Perches in length, would be fini$hed with onely 12000. Crowns.</P> <P>Fir$t; it is to be believed, that the waters running that way, would not fail to open that Mouth into the Sea, which at pre- $ent is almo$t choakt up by the Shelf of Sand, which the new Mouth of <I>Ponto Virro</I> hath brought thither; and that it would again bring into u$e the Port <I>Goro,</I> and its Navigation.</P> <P>And haply experience might teach us, that the $uperficies of P<I>o</I> might come to fall by this a$$wagement of Water, $o that the acce$$ion of <I>Reno</I> would que$tionle$s make no ri$ing in it: Whereupon, if it $hould $o fall out, tho$e Princes would have no rea$on to complain; who $eem to que$tion, le$t by this new acce$$ion of water into P<I>o,</I> the Sluices might be endangered. Which I thought not fit to omit to repre$ent to your Lord$hip; not, that I propo$e it to you as a thing ab$olutely certain, but that you might, if you $o plea$ed, lay it before per$ons who$e judge- ments are approved in the$e affairs.</P> <P>I return now from where I degre$t, and affirm it as indubita- ble, that <I>Reno</I> neither can, nor ought to continue longer where it at this day is; and that it cannot go into any other place but that, whither <I>Cardinal Capponi</I> de$igned to carry it, and which at pre$ent plea$eth me better than any other; or into <I>Volana,</I> whence it was taken away; the vigilance of Men being able to obviate part of tho$e mi$chiefs, which it may do there.</P> <P>But from its Removal, be$ides the alleviation of the harm <foot>which</foot> <p n=>112</p> which by it $elf is cau$ed, there would al$o re$ult the diminution of that which is occa$ioned by the other Brooks, to the right hand of the <I>Po</I> of <I>Argenta</I>; fora$much as the $aid <I>Po</I> wanting all the water of <I>Reno,</I> it would of nece$$ity come to ebb in $uch man- ner, that the Valleys would have a greater Fall into the $ame, and con$equently it would take in, and $wallow greater abun- dance of water; and by this means the Ditches and Draines of the Up-Lands would likewi$e more ea$ily Fall into them; e$- pecially if the $couring of <I>Zenzalino</I> were brought to perfection, by which the waters of <I>Marrara</I> would fall into <I>Marmorta</I>: And if al$o that of <I>Ba$tia</I> were enlarged, and fini$hed, by which there might enter as much water into the $aid P<I>o</I> of <I>Argenta,</I> as is taken from it by the removal of <I>Reno</I>; although that by that meanes the water of the Valleys would a$$wage double: Nor would the people of <I>Argenta,</I> the I$les of S. <I>Giorgio,</I> and <I>Comacchio</I> have any cau$e to complain; for that there would not be given to them more water than was taken away: Nay $ometimes whereas they had Muddy waters, they would have clear; nor need they to fear any ri$ing: And furthermore, by this means a very great quan- tity of ground would be re$tored to culture; For the effecting of all which, the $umm of 50. thou$and Crowns would go very far, and would $erve the turn at pre$ent touching tho$e Brooks, car- rying them a little farther in the mean time, to fill up the greater cavities of the Valleys, that we might not enter upon a va$ter and harder work, that would bring with it the difficulties of other operations, and $o would hinder the benefit which the$e people expect from the paternal charity of His Holine$s.</P> <foot>TO</foot> <p n=>113</p> <head>TO The Right Honourable, MONSIGNORE D. Ferrante Ce$arini.</head> <P>My Treati$e of the MENSURATION of RUN- NING WATERS, Right Honourable, and mo$t Noble Sir, hath not a greater Preroga- tive than its having been the production of the command of Pope <I>Vrban</I> VIII. when His Ho- line$s was plea$ed to enjoyn me to go with <I>Mon$ignore Cor$ini,</I> in the Vi$itation that was impo$ed upon him in the year 1625. of the Waters of <I>Ferrara, Bologna, Romagna,</I> and <I>Romagnola</I>; for that, on that occa$ion applying my whole Study to my $ervice and duty, I publi$hed in that Treati$e $ome particulars till then not rightly under$tood and con$idered (that I knew) by any one; although they be in them- $elves mo$t important, and of extraordinary con$equence. Yet I mu$t render thanks to Your Lord$hip for the honour you have done to that my Tract; but wi$h withal, that your E$teem of it may not prejudice the univer$al E$teem that the World hath of Your Honours mo$t refined judgement.</P> <P>As to that Point which I touch upon in the Conclu$ion, name- ly, That the con$ideration of the Velocity of Running Water $up- plyeth the con$ideration of the ^{*} Length omitted in the common <marg>* Larghezza, but mi$printed.</marg> way of mea$uring Running Waters; Your Lord$hip having com- manded me that in favour of <I>Practi$e,</I> and for the perfect di$co- very of the di$order that commonly happeneth now adayes in the di$tribution of the Waters of Fountains, I $hould demon- $trate that the knowledge of the Velocity $erveth for the finding of the Length: I have thought fit to $atisfie your Command by relating a Fable; which, if I do not deceive my $elf, will make out to us the truth thereof; in$omuch that the re$t of my Treati$e $hall thereby al$o become more manife$t and intelligible, even to <foot>Qqqq tho$e</foot> <p n=>432</p> tho$e who finde therein $ome kinde of ob$curity.</P> <P>In the dayes of yore, before that the admirable Art of Wea- ving was in u$e, there was found in <I>Per$ia</I> a va$tand unvaluable Trea$ure, which con$i$ted in an huge multitude of pieces of Er- me$in, or Damask, I know not whether; which, as I take it, amounted to near two thou$and pieces; which were of $uch a nature, that though their Breadth and Thickne$s were finite and determinate, as they u$e to be at this day; yet neverthele$s, their Length was in a certain $en$e infinite, for that tho$e two thou$and pieces, day and night without cea$ing, i$$ued out with their ends at $uch a rate, that of each piece there i$$ued 100. Ells a day, from a deep and dark Cave, con$ecrated by the Super$tition of tho$e people, to the fabulous <I>Arachne.</I> In tho$e innocent and early times (I take it to have been, in that $o much applauded and de$ired Golden age) it was left to the liberty of any one, to cut off of tho$e pieces what quantity they plea$ed without any diffi- culty: But that felicity decaying and degenerating, which was altogether ignorant of <I>Meum</I> and <I>Tuum</I>; terms certainly mo$t pernicious, the Original of all evils, and cau$e of all di$cords; there were by tho$e people $trong and vigilant Guards placed upon the Cave, who re$olved to make merchandize of the Stuffes; and in this manner they began to $et a price upon that ine$tima- ble Trea$ure, $elling the propriety in tho$e pieces to divers Mer- chants; to $ome they $old a right in one, to $ome in two, and to $ome in more. But that which was the wor$t of all, There was found out by the in$atiable avarice of the$e men crafty inventions to deceive the Merchants al$o; who came to buy the afore$aid commodity, and to make them$elves Ma$ters, $ome of one $ome of two, and $ome of more ends of tho$e pieces of $tuff; and in particular, there were certain ingenuous Machines placed in the more $ecret places of the Cave, with which at the plea$ure of the Guards, they did retard the velocity of tho$e Stuffs, in their i$$uing out of the Cave; in$omuch, that he who ought to have had 100. Ells of Stuff in a day, had not above 50, and he who $hould have had 400, enjoyed the benefit of 50. onely; and $o all the re$t were defrauded of their Rights, the $urplu$age being $old, appropriated, and $hared at the will of the corrupt Officers: So that the bu$ine$s was without all order or ju$tice, in$omuch that the Godde$s <I>Arachne</I> being di$plea$ed at tho$e people, deprived every one of their benefit, and with a dreadful Earthquake for ever clo$ing the mouth of the Cave, in puni$hment of $o much impiety and malice: Nor did it avail them to excu$e them$elves, by $aying that they allowed the Buyer the Breadth and Thick- ne$s bargained for; and that of the Length, which was infinite, <foot>there</foot> <p n=>115</p> there could no account be kept: For the wi$e and prudent Prie$t of the Sacred <I>Grotto</I> an$wered, That the deceit lay in the length, which they were defrauded of, in that the velocity of the ftuffe was retarded, as it i$$ued out of the Cave: and although the total length of the Piece was infinite, for that it never cea- $ed coming forth, and $o was not to be computed; yet never- thele$s its length con$idered, part by part, as it came out of the Cave, and was bargained for, continued $till finite, and might be one while greater, and another while le$$er, according as the Piece was con$tituted in greater or le$$er velocity; and he added withall, that exact Ju$tice required, that when they $old a piece of $tuff, and the propriety or dominion therein, they ought not only to have a$certained the breadth and thickne$$e of the Piece, but al$o to have determined the length, determining its ve- locity.</P> <P>The $ame di$order and confu$ion, that was repre$ented in the Fable, doth come to pa$$e in the Hi$tory of the Di$tribution of the Waters of Conduits and Fountains, $eeing that they are $old and bought, having regard only to the two Dimen$ions, I mean of Breadth and Height of the Mouth that di$chargeth the Wa- ter; and to remedy $uch an inconvenience, it is nece$$ary to de- termine the length in the velocity; for never $hall we be able to make a gue$$e at the quantity of the Body of Running Water, with the two Dimen$ions only of Breadth and Height, without Length.</P> <P>And to the end, that the whole bu$ine$s may be reduced to a mo$t ea$ie practice, by which the waters of Aqueducts may be bought and $old ju$tly, and with mea$ures alwayes ex- act and con$tant.</P> <P>Fir$t, the quantity of the Water ought diligently to be exa- mined, which the whole principal ^{*} Pipe di$chargeth in a time certain, as for in$tance, in an hour, in half an hour, or in a le$$e interval of time, (for knowing which I have a mo$t exact and ea$ie Rule) and finding that the whole principal pipe di$char- geth <I>v. g.</I> a thou$and Tuns of Water in the $pace of one or more hours, in $elling of this water, it ought not to be uttered by the ordinary and fal$e mea$ure, but the di$tribution is to be made with agreement to give and maintain to the buyer ten or twenty, or a greater number of Tuns, as the bargain $hall be made, in the $pace of an hour, or of $ome other $et and deter- minate time. And here I adde, that if I were to undertake to make $uch an adju$tment, I would make u$e of a way to divide and mea$ure the time with $uch accuratene$$e, that the $pace of an hour $hould be divided into four, $ix, or eight thou$and parts <foot>Qqqq 2 with-</foot> <p n=>116</p> without the lea$t errour; which Rule was taught me by my Ma$ter <I>Sign. Galilæo Galilæi,</I> Chief Philo$opher to the mo$t Se- rene <I>Grand Duke</I> of <I>Tu$cany.</I> And this way will $erve ea$ily and admirably to our purpo$e and occa$ion; $o that we $hall thereby be able to know how many Quarts of Water an A- queduct will di$charge in a given time of hours, moneths, or years. And in this manner we may con$titute a Cock that $hall di$charge a certain and determinate quantity of water in a time given.</P> <P>And becau$e daily experience $hews us, that the Springs of A- queducts do not maintain them alwayes equally high, and full of Water, but that $ometimes they increa$e, and $ometimes de- crea$e, which accident might po$$ibly procure $ome difficulty in our di$tribution: Therefore, to the end that all manner of $cru- ple may be removed, I conceive that it would be convenient to provide a Ci$tern, according to the occa$ion, into which there might alwayes fall one certain quantity of water, which $hould not be greater than that which the principal pipe di$chargeth in times of drought, when the Springs are bare of water, that $o in this Ci$tern the water might alwayes keep at one con$tant height. Then to the Ci$tern $o prepared we are to fa$ten the Cocks of particular per$ons, to whom the Water is $old by the Reverend Apo$tolique Chamber, according to what hath been ob$erved before; and that quantity of Water which remaineth over and above, is to be di$charged into another Ci$tern, in which the Cocks of the Waters for publick $ervices, and of tho$e which people buy upon particular occa$ions are to be placed. And when the bu$ine$$e $hall have been brought to this pa$$e, there will likewi$e a remedy be found to the $o many di$orders that continually happen; of which, for brevity $ake, I will in$tance in but four only, which concern both publique and private bene- fit, as being, in my judgment, the mo$t enormous and intole- rable.</P> <P>The fir$t inconvenience is, that in the common way of mea$u- ring, di$pen$ing, and $elling the Waters of Aqueducts, it is not under$tood, neither by the Buyer nor Seller, what the quantity truly is that is bought and $old; nor could I ever meet with any either Engineer or Architect, or Arti$t, or other that was able to decypher to me, what one, or two, or ten inches of water was. But by our above declared Rule, for di$pen$ing the Waters of Aqueducts we may very ea$ily know the true quantity of Water that is bought or $old, as that it is $o many Tuns an hour, $o ma- ny a day, $o many in a year, &c.</P> <P>The $econd di$order that happeneth, at pre$ent, in the di$tri- <foot>buting</foot> <p n=>117</p> bution of Aqueducts is, that as the bu$ine$$e is now governed, it lieth in the power of a $ordid Ma$on to take unju$tly from one, and give unde$ervedly to another more or le$$e Water than be- longeth to them of right: And I have $een it done, of my own experience. But in our way of mea$uring and di$tri- buting Waters, there can no fraud be committed; and put- ting the ca$e that they $hould be committed, its an ea$ie mat- ter to know it, and amend it, by repairing to the Tribunal appointed.</P> <P>Thirdly, it happens very often, (and we have examples there- of both antient and modern) that in di$pen$ing the Water after the common and vulgar way; there is $ometimes more Water di$- pended than there is in the Regi$ter, in which there will be regi- $tred, as they $ay, two hundred inches (for example) and there will be di$pen$ed two hundred and fifty inches, or more. Which pa$$age happened in the time of <I>Nerva</I> the Emperour, as <I>Giulio Frontino</I> writes, in his 2. Book, <I>De Aquaductibus Vrbis Romæ,</I> where he ob$erveth that they had <I>in Commentariis 12755. Qui- naries</I> of Water; and found that they di$pen$ed 14018. <I>Qui- naries.</I> And the like Errour hath continued, and is in u$e al$o modernly until our times. But if our Rule $hall be ob$erved, we $hall incur no $uch di$order, nay there will alwayes be given to every one his $hare, according to the holy end of exact ju$tice, which <I>dat unicuique quod $uum e$t.</I> As on the contrary, it is manife$t, that His Divine Maje$ty hateth and abominateth <I>Pon- dus & pondus, Men$ura & men$ura,</I> as the Holy Gho$t $peak- eth by the mouth of <I>Solomon</I> in the <I>Proverbs, Chap. 20. Pondus & Pondus, Men$ura & Men$ura, utrumque abominabile e$t apud Deum.</I> And therefore who is it that $eeth not that the way of dividing and mea$uring of Waters, commonly u$ed, is expre$ly again$t the Law of God. Since that thereby the $ame mea$ure is made $ometimes greater, and $ometimes le$$er; A di$order $o enormous and execrable, that I $hall take the boldne$s to $ay, that for this $ole re$pect it ought to be condemned and prohibited like- wi$e by human Law, which $hould Enact that in this bu$ine$s there $hould be imployed either this our Rule, or $ome other that is more exqui$ite and practicable, whereby the mea$ure might keep one con$tant and determinate tenor, as we make it, and not, as it is now, to make <I>Pondus & Pondus, Men$ur a & Men$ura.</I></P> <P>And this is all that I had to offer to Your mo$t Illu$trious Lord$hip, in obedience to your commands, re$erving to my $elf the giving of a more exact account of this my invention, when the occa$ion $hall offer, of reducing to practice $o holy, ju$t, and <foot>nece$$ary</foot> <p n=>118</p> nece$$ary a reformation of the Mea$ure of Running Waters and of Aqueducts in particular: which Rule may al$o be of great benefit in the divi$ion of the greater Waters to over-flow Grounds, and for other u$es: I humbly bow,</P> <P><I>Your Most Devoted,</I> and <I>Mo$t Obliged Servant,</I></P> <P>D. Benedetto Ca$telli, <I>Abb. Ca$<*>.</I></P> <head>FINIS.</head> <fig> <pb> <head>A TABLE</head> <head>Of the mo$t ob$ervable matters in this Treati$e of the MENSURATION of RUNNING WATERS.</head> <table> <row><col>A</col><col></col></row> <row><col>Abatements <I>of a River in different and unequal Diver$ions, is alwaies equal, which is proved with</I> 100. Syphons.</col><col><I>Page</I> 75</col></row> <row><col>Arno <I>River when it ri$eth upon a Land-Flood near the Sea one third of a Brace, it ri$eth about</I> Pi$a 6. <I>or 7. Braces.</I></col><col>82</col></row> <row><col>B</col><col></col></row> <row><col><I>Banks near to the Sea lower, than far from thence. Corollary</I> XIV.</col><col>16</col></row> <row><col>Brent <I>River diverted from the Lake o</I>f Venice, <I>and its effects.</I></col><col>64</col></row> <row><col>Brent <I>$uppo$ed in$ufficient to remedy the inconveniences of the Lake, and the fal$ity of that $uppo$ition.</I></col><col>67</col></row> <row><col>Brent, <I>and its benefits in the Lake.</I></col><col>70</col></row> <row><col><I>Its Depo$ition of Sand in the Lake, bow great it is.</I></col><col>78, 79</col></row> <row><col><I>Bridges over Rivers, and how they are to be made. Appendix</I> VIII.</col><col>20</col></row> <row><col>Burana <I>River, its ri$ing, and falling in</I> Panaro.</col><col>110</col></row> <row><col>C</col><col></col></row> <row><col>Ca$telli <I>applyed him$elf to this Study by Order of</I> Urban VIII.</col><col>2</col></row> <row><col>Chanel of Navigation <I>in the Valleys of</I> Bologna, <I>and its inconveniences.</I></col><col>99</col></row> <row><col><I>Carried into the</I> Po <I>of</I> Ferrara, <I>and its benefits</I></col><col>ibid.</col></row> <row><col>Ciampoli <I>alover of the$e Ob$ervations of Waters.</I></col><col>3</col></row> <row><col>D</col><col></col></row> <row><col><I>Difficulty of this bu$ine$s of Mea$uring Waters.</I></col><col>2</col></row> <row><col><I>Di$orders that happen in the di$tribution of the Waters of Aqueducts, and their re- medies.</I></col><col>113</col></row> <row><col><I>Di$tribution of the Waters of Fountains, and Aqueducts. Appendix</I> X.</col><col>22</col></row> <row><col><I>Di$tribution of Water to over-flow Grounds. Appendix</I> XI.</col><col>23, 69, 70</col></row> <row><col><I>Diver$ion of</I> Reno <I>and other Brooks of</I> Romagna, <I>advi$ed by</I> P. Spernazzati <I>to what end it was.</I></col><col>100</col></row> <row><col><I>Drains and Ditches, the benefit they receive by cutting away the Weeds and Reeds. Appendix</I> IX.</col><col>21</col></row> <row><col><I>Drains and Sewers ob$tructed, in the Diver$ion of</I> Reno <I>into</I> Main Po, <I>and a remedy for the $ame.</I></col><col>110</col></row> <row><col>E</col><col></col></row> <row><col><I>Engineers unver$'d in the matters of Waters.</I></col><col>2</col></row> <row><col><I>Erour found in the common way of Mea$uring Running Waters.</I></col><col>68, 69</col></row> <row><col><I>Errour in deriving the Water of</I> Acqua Paola. <I>Appendix</I> II.</col><col>17, 18</col></row> <foot><I>Eriour</I></foot> <pb> <row><col><I>Errour of</I> Bartolotti.</col><col>86, 87</col></row> <row><col><I>Errours of Engineers in the Derivation of Chenels. Corollary</I> XII.</col><col>12</col></row> <row><col><I>Errour of Engineers in Mea$uring of</I> Reno <I>in</I> Po. <I>Appendix</I> III.</col><col>ibid.</col></row> <row><col><I>Errour of other Engineers, contrary to the precedent. Appendix</I> IV.</col><col>Ibid.</col></row> <row><col><I>Errour of</I> Giovanni Fontana <I>in Mea$uring Waters, Corollary</I> XI.</col><col>9</col></row> <row><col><I>Errour of</I> Giulio Frontino <I>in Mea$uring the Waters of Aqueducts. Appen- dix</I> I.</col><col>17</col></row> <row><col><I>Errours committed in cutting the Bank at</I> Bondeno, <I>in the $wellings of</I> Po: <I>Corollary</I> XIII.</col><col>81</col></row> <row><col>F</col><col></col></row> <row><col><I>Fenns</I> Pontine, <I>Drained by Pope</I> Sixtus Quintus, <I>with va$t expence.</I></col><col>92</col></row> <row><col><I>The ruine and mi$carriage thereof.</I></col><col>93</col></row> <row><col><I>Tardity of the principal Chanel that Drains them, cau$e of the Drowning.</I></col><col>ibid.</col></row> <row><col><I>They are ob$tructed by the Fi$hing-Wears, which $uell the River.</I></col><col>94</col></row> <row><col><I>Waters of</I> Fiume Si$to, <I>which flow in great abundance into the</I> Evacuator <I>of the $aid Fenns.</I></col><col>94, 95</col></row> <row><col><I>Remedies to the di$orders of tho$e Fenns.</I></col><col>95, 96</col></row> <row><col>Fontana Giovanni, <I>his errours in Mea$uring Waters. Corollary</I> XI.</col><col>9</col></row> <row><col>Fiume Morto, <I>whether it ought to fall into the Sea, or into</I> Serchio,</col><col>79</col></row> <row><col><I>Let into</I> Serchio <I>and its inconveniences.</I></col><col>79, 80</col></row> <row><col><I>The dangerous ri$ing of its Waters, when to be expected.</I></col><col>81</col></row> <row><col><I>Its inconveniences when it is higher in level than</I> Serchio, <I>and why it ri$eth mo$t On the Sea-coa$ts, at $uch time as the Winds make the Sea to $uell.</I></col><col>83</col></row> <row><col>G</col><col></col></row> <row><col>Galilæo Galilæi. <I>hoxourably mentioned.</I></col><col><I>Page</I> 2, 28</col></row> <row><col><I>His Rule for mea$uring the time.</I></col><col>49</col></row> <row><col>H</col><col></col></row> <row><col><I>Height,</I> vide <I>Quick</I></col><col></col></row> <row><col><I>Heights different, made by the $ame $tream of a Brock or Torrent, according to the divers Velocities in the entrance of the River. Corollary</I> I.</col><col>6</col></row> <row><col><I>Heights different, made by the Torrent in the River, according to the different heights of the River. Corollary</I> II.</col><col>ibid.</col></row> <row><col>K</col><col></col></row> <row><col><I>Knowledge of Motion how much it importeth.</I></col><col>1</col></row> <row><col>L</col><col></col></row> <row><col><I>t</I></col><col></col></row> <row><col><I>Lake of</I> Perugia, <I>and, he Ob$ervation made on it. Appendix</I> XII.</col><col>42</col></row> <row><col><I>Lake of</I> Thra$imenus <I>and Con$iderations upon it, a Letter written to</I> Sig. Galilæo Galilæi.</col><col>28</col></row> <row><col><I>Lake of</I> Venice, <I>and Con$iderations upon it.</I></col><col>63, 73</col></row> <row><col><I>Low Waters which let the bottom of it be di$covered.</I></col><col>64</col></row> <row><col><I>The $toppage and choaking of the Ports, a main cau$e of the di$orders of the Lake, and the grand remedy to tho$e di$orders what it is.</I></col><col>66</col></row> <row><col><I>Lakes and Metrs along the Sea-coa$ts, and the cau$es thereof.</I></col><col>65</col></row> <row><col><I>Length of Waters, how it is to be Mea$ured.</I></col><col>70</col></row> <row><col>M</col><col></col></row> <row><col><I>Mea$ure and Di$tributions of Waters. Appendix</I> V.</col><col>18</col></row> <foot><I>Mea$ure</I></foot> <pb> <row><col><I>Mea$ure of Rivers that fall into others difficult. Coroll.</I> X:</col><col>9</col></row> <row><col><I>Mea$ure of the Running Water of a Chanel of an height known by a</I> Regulator <I>of a Mea- $ure given, in a time a$$igned. Propo$ition</I> I. <I>Problem</I> I.</col><col>50</col></row> <row><col><I>Mea$ure of the Water of any River, of any greatne$s, in a time given. Propo$ition</I> V. <I>Problem</I> III.</col><col>60</col></row> <row><col><I>Mea$ure that $hewes how much Water a River di$chargeth in a time given.</I></col><col>48</col></row> <row><col><I>Mole-holes,</I></col><col></col></row> <row><col><I>Motion the principal $ubject of Philo$ophy.</I></col><col>1</col></row> <row><col><I>Mud.</I> Vide <I>Sand.</I></col><col></col></row> <row><col>N</col><col></col></row> <row><col><I>Navigation from</I> Bologna <I>to</I> Ferrara, <I>is become impo$$ible, till $uch time as</I> Reno <I>be diverted.</I></col><col>101</col></row> <row><col><I>Navigation in the Lake of</I> Venice <I>endangered, and how restored.</I></col><col>65, 70</col></row> <row><col>P</col><col></col></row> <row><col><I>Perpendicularity of the Banks of the River, to the upper $uperficies of it.</I></col><col>37</col></row> <row><col><I>Perpendicularity of the Banks to the bottom.</I></col><col>37</col></row> <row><col><I>Perugia.</I> Vide <I>Lake.</I></col><col></col></row> <row><col><I>Pontine.</I> Vide <I>Fenns.</I></col><col></col></row> <row><col><I>Ports of</I> Venice, Malamocco, Bondolo, <I>and</I> Chiozza, <I>choaked up for want of Water in the Lake.</I></col><col>65</col></row> <row><col><I>Proportions of unequal Sections of equal Velocity, and of equal Sections of unequal Velo- city. Axiome</I> IV. <I>and</I> V.</col><col>38</col></row> <row><col><I>Proportions of equal and unequal quantities of Water, which pa$s by the Sections of dif- ferent Rivers. Propo$ition</I> II.</col><col>39</col></row> <row><col><I>Proportions of unequal Sections that in equal times di$charge equal quantities of Water. Propo$ition</I> III.</col><col>41</col></row> <row><col><I>Proportion wherewith one River falling into another, varieth in height. Propo- $ition</I> IV.</col><col>44</col></row> <row><col><I>Proportion of the Water di$charged by a River in the time of Flood, to the Water di$charged in an equal time by the $aid River, before or after the Flood. Propo$ition</I> V.</col><col>44</col></row> <row><col><I>Proportion of the Heights made by two equal Brooks or Streams falling into the $ame River. Propo$ition</I> VI.</col><col>45</col></row> <row><col><I>Proportion of the Water which a River di$chargeth encrea$ing in Quick-height by the ad- dition of new Water, to that which it di$chargeth after the encrea$e is made. Propo- $ition</I> IV. <I>Theor.</I> II.</col><col>54</col></row> <row><col><I>Proportion of a River when high, to it $elf when low. Coroll.</I> I.</col><col>55</col></row> <row><col>Q</col><col></col></row> <row><col><I>Quantity of Running Waters is never certain, if with the Vulgar way of Mea$uring them, their Velocities be not con$idered.</I></col><col>32</col></row> <row><col><I>Quantities of Waters which are di$charged by a River, an$wer in equality to the Velocities and times in which they are di$charged. Axiome</I> I, II, III.</col><col>38</col></row> <row><col>Quick-Height <I>of a River, what it is. Definition</I> V.</col><col>48</col></row> <row><col>R</col><col></col></row> <row><col><I>Rea$on of the Proverb,</I> Take heed of the $till Waters. <I>Coroll.</I> VI.</col><col>7</col></row> <row><col><I>Rea$ons of</I> Mon$ignore Cor$ini <I>again$t the diver$ion of</I> Reno <I>into the</I> Po <I>of</I> Volano.</col><col>105</col></row> <row><col><I>Rea$ons of</I> Cardinal Capponi <I>and</I> Mon$ig. Cor$ini, <I>for the turning of</I> Reno <I>into Main</I> Po.</col><col>106</col></row> <foot>Rrrr <I>Rea-</I></foot> <pb> <row><col><I>Two objections on the contrary, and an$wers to them.</I></col><col>104 <I>&</I> 105</col></row> <row><col><I>What ought to be the proportion of the Heights of</I> Reno <I>in</I> Reno, <I>and of</I> Reno <I>in</I> Po.</col><col>110</col></row> <row><col><I>Regulator what it is. Definition</I> IV.</col><col>48</col></row> <row><col><I>Relation of the Waters of</I> Bologna <I>and</I> Ferrara, <I>by</I> Mon$ignore Cor$ini</col><col>100</col></row> <row><col>Reno <I>in the Valleys, and its bad effects.</I></col><col>100, 101</col></row> <row><col><I>Two wayes to divert it.</I></col><col>103</col></row> <row><col><I>The facility and utility of tho$e wayes.</I></col><col>Ibid.</col></row> <row><col><I>The difficulties objected.</I></col><col>104</col></row> <row><col><I>Reply to</I> Bartolotti <I>touching the dangers of turning</I> Fiume Morto <I>into</I> Serchio.</col><col>83</col></row> <row><col><I>Retardment of the cour$e of a River cau$ed by its Banks. Appendix</I> VII.</col><col>19</col></row> <row><col><I>Ri$ings made by Flood-Gates but $mall. Appendix</I> XIII.</col><col>26</col></row> <row><col><I>Rivers that are $hallow $well much upon $mall $howers, $uch as are deep ri$e but little upon great Floods. Corollary</I> III.</col><col>6</col></row> <row><col><I>Rivers the higher they are, the $wifter.</I></col><col>Ibid.</col></row> <row><col><I>Rivers the higher they are, thele$$e they encrea$e upon Floods.</I></col><col>49</col></row> <row><col><I>Rivers when they are to have equal and when like Velocity.</I></col><col>Ibid.</col></row> <row><col><I>Rivers in falling into the Sea, form a Shelf of Sand called</I> Cavallo.</col><col>65</col></row> <row><col><I>Five Rivers to be diverted from the Lake of</I> Venice, <I>and the inconveniences that would en$ue thereupon.</I></col><col>74, 75</col></row> <row><col><I>A River of Quick-height, and Velocity in its Regulator being given, if the Height be redoubled by new Water, it redoubleth al$o in Velocity. Propo$ition</I> II. <I>The- orem</I> I.</col><col>51</col></row> <row><col><I>Keepeth the proportion of the heights, to the Velocities. Corollary</I></col><col>52</col></row> <row><col>S</col><col></col></row> <row><col><I>Sand and Mud that entereth into the Lake of</I> Venice, <I>and the way to examine it.</I></col><col>76</col></row> <row><col><I>Seas agitated and driven by the Winds $top up the Ports.</I></col><col>64, 65</col></row> <row><col><I>Sections of a River what they are. Definition</I> I.</col><col>37</col></row> <row><col><I>Sections equally $wift what they are. Definition</I> II.</col><col>Ibid.</col></row> <row><col><I>Sections of a River being given, to conceive others equal to them, of different breadth, height and Velocity. Petition.</I></col><col>38</col></row> <row><col><I>Sections of the $ame River, and their Proportions to their Velocities. Coroll.</I> I.</col><col>42</col></row> <row><col><I>Sections of a River di$charge in any what$oever place of the $aid River, equal quantities of Water in equal times. Propo$ition</I> I.</col><col>39</col></row> <row><col>Sile <I>River what mi$chiefes it threatneth, diverted from the Lake.</I></col><col>74</col></row> <row><col><I>Spirtings of Waters grow bigger the higher they go. Coroll.</I> XVI.</col><col>16</col></row> <row><col><I>Sreams of Rivers how they encrea$e and vary. Coroll.</I> I.</col><col>6</col></row> <row><col><I>Streams retarded, and the effects thereof. Coroll.</I> IX.</col><col>8</col></row> <row><col>T</col><col></col></row> <row><col><I>Table of the Heights, Additions, and Quantities of Waters, and its u$e.</I></col><col>56</col></row> <row><col><I>Thra$imenus.</I> Vide <I>Lake.</I></col><col></col></row> <row><col><I>Time how its mea$ured in the$e Operations of the Waters.</I></col><col>49</col></row> <row><col><I>Torrents encrea$e at the encrea$ing of a River, though they carry no more Water than before: Coroll.</I> IV.</col><col>6</col></row> <row><col><I>Torrents when they depo$e and carry away the Sand. Coroll.</I> V.</col><col>7</col></row> <row><col><I>Torrents and their effects in a River.</I></col><col>6, 7</col></row> <row><col><I>Torrents that fall into the Valleys, or into</I> Po <I>of</I> Volano, <I>and their mi$chiefs prevent- ed, by the diverting of</I> Reno <I>into</I> Main Po.</col><col>100</col></row> <row><col><I>Tyber and the cau$es of its inundations. Coroll.</I> VIII.</col><col>8</col></row> <foot><I>Val-</I></foot> <pb> <row><col>V</col><col></col></row> <row><col><I>Valleys of</I> Bologna <I>and</I> Ferrara, <I>their inundations and di$orders, whence they pro- ceed.</I></col><col>97</col></row> <row><col><I>Velocity of the Water $hewn by $everal Examples.</I></col><col>3</col></row> <row><col><I>Its proportion to the Mea$ure.</I></col><col>5</col></row> <row><col><I>Velocities equal, what they are.</I></col><col>47</col></row> <row><col><I>Velocities like, what they are.</I></col><col>47, 48</col></row> <row><col><I>Velocities of Water known, how they help us in finding the Lengths.</I></col><col>113</col></row> <row><col><I>A Fable to explain the truth thereof.</I></col><col>Ibid.</col></row> <row><col><I>Venice.</I> Vide <I>Lake.</I></col><col></col></row> <row><col><I>V$e of the</I> Regulator <I>in mea$uring great Rivers. Con$ideration I.</I></col><col>60</col></row> <row><col>W</col><col></col></row> <row><col><I>Waters falling, why they di$groß. Coroll.</I> XVI.</col><col>16</col></row> <row><col><I>Waters, how the Length of them is Mea$ured.</I></col><col>70</col></row> <row><col><I>Waters that are imployed to flow Grounds, how they are to be di$tributed.</I></col><col>19, 53, 54</col></row> <row><col><I>Waters to be carryed in Pipes, to $erve Aquaducts and Conduits, how they are to be Mea- $ured.</I></col><col>115, 116</col></row> <row><col><I>Way to know the ri$ing of Lakes by Raines.</I></col><col>28</col></row> <row><col><I>Way of the Vulgar to Mea$ure the Waters of Rivers.</I></col><col>68</col></row> <row><col><I>Wind Gun, and Tortable Fountain of</I> Vincenzo Vincenti <I>of</I> Urbin.</col><col>11</col></row> <row><col><I>Windes contrary, retard, and make Rivers encrea$e. Coroll.</I> VII.</col><col>8</col></row> </table> <head>The END of the TABLE of the Second Part of the Fir$t TOME.</head> <pb> <head>MATHEMATICAL DISCOURSES AND DEMONSTRATIONS, TOVCHING Two <I>NEW SCIENCES</I>; pertaining to THE MECHANICKS AND LOCAL MOTION:</head> <head>BY <I>GALILÆVS GALILÆVS LYNCEVS,</I> Chiefe <I>Phylo$opher</I> and <I>Mathematitian</I> to the mo$t Serene <I>GRAND DVKE</I> of <I>TVSCANY.</I> WITH <I>AN APPENDIX OF THE</I> Centre of Gravity Of $ome <I>SOLIDS.</I></head> <head>Engli$hed from the Originall <I>Latine</I> and <I>Italian, By THOMAS SALUSBURY, E$q</I>;</head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, <I>Anno Dom. MDCLXV.</I></head> <pb> <head>MATHEMATICAL Collections and Tran$lations: THE SECOND TOME: IN TWO PARTS.</head> <head><I>THE FIRST PART,</I></head> <head>Containing,</head> <P><I>I.</I> GALILEUS GALILEUS His <I>MATHEMATI- CAL Di$cour$es</I> and <I>Demon$trations,</I> touching two <I>NEW SCIENCES,</I> pertaining to the <I>MECHA- NICKS</I> and <I>LOCAL MOTIONS:</I> With an <I>Appendix</I> of the <I>CENTRE of GRAVITY</I> of $ome <I>SOLIDS.</I></P> <P><I>II.</I> GALILEUS His <I>MECHANICKS:</I> with $ome Additionall <I>Pieces.</I></P> <P><I>III.</I> RHENATUS DES CARTES His <I>MECHA- NICKS,</I> Tran$lated from the FRENCH <I>Manu$cript.</I></P> <P><I>IV.</I> ARCHIMEDES His Tract <I>De Insidentibus Humido,</I> or of the <I>NATATION</I> of <I>BODIES:</I> With the Notes and Demon$trations of NICHOLAUS TARTALEA, and FEDERICUS COMMANDINUS.</P> <P><I>V.</I> GALILEUS His <I>Di$cour$e</I> of <I>NATATION.</I></P> <P><I>VI.</I> NICOLAUS TARTALEA, His <I>Inventions</I> for <I>Diving</I> un- der <I>Water, Rai$ing</I> of <I>Ships</I> $unk, &c.</P> <head><I>By THOMAS SALUSBURY, E$q</I>;</head> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, <I>Anno Dom. MD CLXV.</I></head> <p n=>1</p> <head>GALILEUS, HIS DIALOGUES OF MOTION.</head> <head>The Fir$t Dialogue.</head> <head><I>INTERLOCUTORS,</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <P>SALVIATUS.</P> <P>The frequent relort (Gentlemen) to <marg><I>A De$cription of the Ar$enal of</I> Venice.</marg> your Famous Ar$enal of <I>Venice,</I> pre$en- teth, in my thinking, to your Speculative <marg><I>It is a large field for Wits to Philo- $ophate in.</I></marg> Wits, a large field to Philo$ophate in: and more particularly, as to that part which is called the <I>Mechanicks:</I> in re- gard that there all kinds of Engines, and Machines are continually put in u$e, by a huge number of Artificers of all $orts; among$t whom, as well through the ob$ervations of their Prede- ce$$ors, as tho$e, which through their own care they continually are making, it's probable, that there are $ome very learned, and bravely di$cours'd Men.</P> <P>SAGR. Sir, you are not therein mi$taken: and I my $elf, out of <foot>B 2</foot> <p n=>2</p> a natural Curio$itie, do frequentlie for my Recreation vi$it that place, and confer with the$e per$ons; which for a certain prehe- <marg>* Proti.</marg> minence that they have above the re$t we call ^{*} <I>Over$eers</I>: who$e di$cour$e hath oft helped me in the inve$tigation of not only won- derful, but ab$truce, and incredible Effects: and indeed I have been at a lo$$e $ometimes, and de$paired to penetrate how that could po$$ibly come to pa$$e, which far from all expectation my $en$es demon$trated to be true; and yet that which not long $ince that good Old man told us, is a $aying and propo$ition, though com- <marg><I>The Opinion of Common Artificers are often fal$e.</I></marg> mon enough, yet in my opinion wholly vain, as are many others, often in the mouths of unskilful per$ons; introduced by them, as I $uppo$e, to $hew that they under$tand how to $peak $omething about that, of which neverthele$$e they are incapable.</P> <P>SALV. It may be Sir, you $peak of that la$t propo$ition which he affirmed, when we de$ired to under$tand, why they made <marg><I>Great Ships apter than others to break their Keels in Launching, accor- ding to $ome.</I></marg> $o much greater provi$ion of $upporters, and other provi$ions, and reinforcements about that Galea$$e, which was to be launcht than is made about le$$er Ve$$els, and he an$wered us, that they did $o to avoid the peril of breaking its Keel, through the mighty weight of its va$t bulk, an inconvenience to which le$$er $hips are not subject.</P> <P>SAGR. I do intend the $ame, and chiefly that la$t conclu$ion, which he added to his others, and which I alwaies e$teemed a vain conceit of the Vulgar, namely, That in the$e and other Machines we mu$t not argue from the le$$e to the greater, becau$e many Mechanical Inventions take in little, which hold not in great. But being that all the Rea$ons of the Mechanicks, have their founda- tions from Geometry; in which I $ee not that greatne$$e and $malne$$e make Circles, Triangles, Cilinders, Cones, or any other $olid Figures $ubject to different pa$$ions: when the great Ma- chine is conformed in all its members to the proportions of the le$$e that is u$eful, and fit for exerci$e to which it is de$igned; I cannot $ee why it al$o $hould not be exempt from the unlucky, $ini$ter, and de$tructive accidents that may befall it.</P> <P>SALV The $aying of the Vulgar is ab$olutely vain, and $o fal$e, that its contrary may be affirmed with equal truth, $aying, <marg><I>Many Machines may be made more exact in great than in little.</I></marg> That many Machines may be made more perfect in great than lit- tle: As for in$tance, a Clock that $hews and $trikes the Houres, may be made more exact in one certain $ize, than in another le$$e. With better ground is that $ame conclu$ion u$urped by other more intelligent per$ons, who refer the cau$e of $uch effects in the$e great Machines different from what is collected from the pure, and ab$tracted Demon$trations of Geometry, to the imperfection of the matter, which is $ubject to many alterations, and defects. But here, I know not whether I may without contracting $ome <foot>$u$pition</foot> <p n=>3</p> $u$pition of Arrogance $ay, that thither al$o doth the recour$e to the defects of the matter (able to blemi$h the perfecte$t Mathe- matical Demon$trations) $uffice to excu$e the di$obedience of <marg><I>Great Material Machines, al- though framed In the $ame proportion as others of the $ame Matter that are le$$er, are le$$e $trong and able to re$i$t external Im- petu$s's than the le$$er.</I></marg> Machines in concrete, to the $ame ab$tracted and Ideal: yet not- with$tanding I will $peak it, affirming, That ab$tracting all imper- fections from the Matter, and $uppo$ing it mo$t perfect, and unal- terable, and from all accidental mutation exempt, yet neverthe- le$$e its only being Material, cau$eth, that the greater Machine, made of the $ame matter, and with the $ame proportions, as the le$$er; $hall an$wer in all other conditions to the le$$er in exact Symetry, except in $trength, and re$i$tance again$t violent inva$i- ons: but the greater it is, $o much in proportion $hall it be wea- ker. And becau$e I $uppo$e the Matter to be unalterable, that is alwaies the $ame, it is manife$t, that one may produce Demon$tra- tions of it, no le$$e $imply and purely Mathematical, then of eter- nal, and nece$$ary Affections: Therefore, <I>Sagredus,</I> Revoke the opinion which you, and, it may be, all the re$t hold, that have $tu- died the Mechanicks; that Machines, and Frames compo$ed of the $ame Matter, with punctual ob$ervation of the $elf $ame proporti- on between their parts, ought to be equally, or to $ay better, pro- portionally di$po$ed to Re$i$t; and to yield to External injuries and a$$aults: For if it may be Geometrically demon$trated, that the greater are alwaies in proportion le$s able to re$i$t, than the le$$e; $o that in fine there is not only in all Machines & Fabricks Arti$icial, but Natural al$o, a term nece$$arily a$cribed, beyond which neither Art, nor Nature may pa$$e; may pa$$e, I $ay, al- waies ob$erving the $ame proportions with the Identity of the Matter.</P> <P>SAGR. I already feel my Brains to turn round, and my Mind, (like a Cloud unwillingly opened by the Lightning,) I perceive to be $urprized with a tran$cient, and unu$ual Light, which from affar off twinkleth, and $uddenly a$toni$heth me; and with ab- $truce, $trange, and indige$ted imaginations. And from what hath been $poken, it $eems to follow, that, it is a thing impo$$ible to frame two Fabricks of the $ame Matter, alike, and unequal, and between them$elves in proportion equally able to Re$i$t; and were it to be done, yet it would be impo$$ible to find two only Launces of the $ame wood, alike between them$elves in $trength, <marg><I>A Wooden Launce fixed in a Wall at Right-Angles, and reduced to $uch a length and thick- ne$$e as that it may endure, but made a hairs breadth big- ger, breaketh with its own weight, is $ingly one and no more.</I></marg> and toughne$$e, but unequal in bigne$$e.</P> <P>SALV. So it is Sir; and the better to a$$ure you that we con- cur in opinion, I $ay, that if we take a Launce of wood of $uch a length and thickne$$e, that being fixed fa$t <I>(v. g.)</I> in a Wall at Right Angles, that is parallel to the Horizon, it is reduced to the utmo$t length, that it will hold at, $o that lengthened never- $o-little more, it would break, being over-burthened with its own <foot>B 2 weight,</foot> <p n=>4</p> weight, there could not be another $uch-a-one in the World: So that if its length (for example) were Centuple to its thickne$$e, there cannot be found another Launce of the $ame Matter, that being in length Centuple to its thickne$$e, $hall be able to $u$tain it $elf preci$ely, as that did, and no more: for all that are bigger $hall break, and the le$$er $hall be able, be$ides their own, to $u$tain $ome additional weight. And this that I $ay of the <I>State of bear- ing it $elf,</I> I would have under$tood to be $poken of every other Con$titution, and thus if one Tran$ome bear or $u$tain the force often Tran$omes equal to it, $uch another Beam cannot bear the weight of ten that are equal to it. Now be plea$ed, Sir, and you <marg><I>Truth upon a little Courting, throweth off her Vail, and $hews her Secrets maked.</I></marg> <I>Simplicius</I> to ob$erve, how true Conclu$ions, though at the fir$t $ight they $eem improbable, yet never $o little glanced at, do depo$e the Vailes which ob$cure them, and make a voluntary $hew of their $ecrets nakedly, and $imply. Who $ees not, that a Hor$e falling <marg><I>Great Animals receive more harm by a fall than le$- $er.</I></marg> from a height of three or four yards, will break his bones, but a Dog falling $o many yards, or a Cat eight or ten, will receive no hurt; nor likewi$e a Gra$hopper from a Tower, nor an Ant thrown from the Orbe of the Moon? Little Children e$cape all harm in their falls, whereas per$ons grown up break either their $hins or faces. And as le$$er Animals are in proportion more robu$tious, and $trong than greater, $o the le$$er Plants better $upport them- $elves: and I already believe, that both of you think, that an Oake two hundred foot high could not $upport its branches $pread like <marg><I>Nature could not have made of mea- ner Hor$es bigger, and have retained the $ame $trength, but by altering their Symetry.</I></marg> one of an indifferent $ize; and that Nature could not have made an Hor$e as big as twenty Hor$es, nor a Giant ten times as tall as a Man, unle$$e $he did it either miraculou$ly, or el$e by much alte- ring the proportion of the Members, and particularly of the Bones, enlarging them very much above the Symetry of common Bones. To $uppo$e likewi$e, that in Artificial Machines, the greater and le$$er are with equal facility made, and pre$erved, is a manife$t Er- rour: and thus for in$tance, $mall Spires, Pillars, and other $olid figures may be $afely moved, laid along, and reared upright, with- out danger of breaking them; but the very great upon every $ini- $ter accident fall in pieces, and for no other rea$on but their own weight. And here it is nece$$ary that I relate an accident, worthy of notice, as are all tho$e events that occur unexpectedly, e$pecial- ly when the means u$ed to prevent an inconvenience, proveth in <marg><I>A great Marble Pillar broken by its own weight, and why.</I></marg> fine the mo$t potent cau$e of the di$order. There was a very great Pillar of Marble laid along, and two Rowlers were put under the $ame neer to the ends of it; it came into the mind of a certain In- gineer $ome time after, that it would be expedient, the better to $ecure it from breaking in the mid$t through its own weight, to put under it in that part yet another Rowler: the coun$el $eemed generally very $ea$onable, but the $ucce$$e demon$trated it to be <foot>wholly</foot> <p n=>5</p> wholly contrary: for many moneths had not pa$t, before the Pil- lar crackt, and broke in the middle ju$t upon the new Rowler.</P> <P>SIMP. This was an accident truly $trange, and indeed <I>preter $pem,</I> e$pecially if it were derived from the addition of new $up- port in the middle.</P> <P>SALV. From that doubtle$s it did proceed; and the known cau$e of the Effect removeth the wonder: for the two pieces of the Pillar being taken from off the Rowlers, one of tho$e bearers on which one end of the Column had re$ted, was by length of time rotten, and $unk away; and that in the mid$t continuing $ound, and $trong, occa$ioned that half the Column lay hollow in the air without any $upport at the end; $o that its own unweildy weight, made it do that, which it would not have done, if it had re$ted only upon the two fir$t Bearers, for as they had $hrunk away it would have fol- lowed. And here none can think that this would have faln out in a little Column, though of the $ame $tone, and of a length an$we- rable to its thickne$$e, in the very $ame proportion as the thick- ne$s, and length of the great Pillar.</P> <P>SAGR. I am now a$$ured of the effect, but do not yet compre- hend the cau$e, how in the augmentation of Matter, the Re$i$tance and Strength ought not al$o to multiply at the $ame rate. And I admire at it $o much the more, in regard I $ee, on the contrary, in other ca$es the $trength of Re$i$tance again$t Fraction to encrea$e much more than the enlargement of the matter encrea$eth. For if (for example) there be two Nailes fa$tned in a Wall, the one twice asthick as the other, that $hall bear a weight not only double to this, but triple, and quadruple.</P> <P>SALV. You may $ay octuple, and not be wide of the truth: <marg><I>A Naile double in thickne$$e to another being fa$t- ned in a Wall, $u- $tains a Weight octuple to that of the le$$er.</I></marg> nor is this effect contrary to the former, though in appearance it $eemeth $o different.</P> <P>SAGR. Therefore <I>Salviatus,</I> explain unto us the$e Riddles, and level us the$e Rocks, if you can do it: for indeed I gue$$e this mat- ter of Re$i$tance to be a field repleni$hed with rare, and u$eful Con- templations, and if you be content that this be the $ubject of our this-daies di$cour$e, it will be to me, and I believe to <I>Simplicius,</I> very acceptable.</P> <P>SALV. I cannot refu$e to $erve you, $ince my Memory $erveth <marg><I>By Accademick here, as in his Dialogues of the Sy$teme,</I> Galile- us <I>meaneth him- $elf.</I></marg> me, in minding me of that which I formerly learnt of our <I>Accade- mick,</I> who hath made many Speculations on this $ubject, and all conformable (as his manner is) to Geometrical Demon$tration: in$omuch that, not without rea$on, this of his may be called a <I>New Science</I>; for though $ome of the Conclu$ions have been ob$erved <marg>Ari$totle <I>the fir$t Ob$erver of Me- chanical Conclu$i- ons, but they nei- ther not the mo$t curious nor $olidly demon$trated.</I></marg> by others, and in the fir$t place by <I>Ari$totle,</I> yet neverthele$$e are they not any of the mo$t curious, or (which more importeth) proved by nece$$ary Demon$trations deduced from their primary, <foot>and</foot> <p n=>6</p> and indubitable fundamentals. And becau$e, as I $ay, I de$ire de- mon$tratively to a$$ure you, and not with only probable di$cour- $es to per$wade you; pre$uppo$ing, that you have $o much know- ledge of the Mechanical Conclu$ions, by others heretofore funda- mentally handled, as $ufficeth for our purpo$e; it is requi$ite, that before we proceed any further, we con$ider what effect that is which opperates in the Fraction of a Beam of Wood, or other Solid, who$e parts are firmly connected; becau$e this is the fir$t <I>Notion,</I> where- on the fir$t and $imple principle dependeth, which as familiarly known, we may take for granted. For the clearer explanation whereof; let us take the Cilinder, or Pri$me, <I>A. B.</I> of Wood, or other $olid and coherent matter, fa$tned above in <I>A,</I> and hanging perpendicular; to which, at the other end <I>B,</I> let there hang the Weight <I>C</I>: It is manife$t, that how great $oever the Tenacity and coherence of the parts of the $aid Solid to one another be, $o it be not infinite, it may be overcome by the Force of the drawing Weight C: who$e Gravity I $uppo$e may be encrea$ed as much <fig> as we plea$e; by the encrea$e whereof the $aid Solid in fine $hall break, like as if it had been a Cord. And, as in a Cord, we under- $tand its re$i$tance to proceed from the mul- titude of the $trings or threads in the Hemp that compo$e it, $o in Wood we $ee its veins, and grain di$tended lengthwaies, that render it far more re$i$ting again$t Fraction, then any Rope would be, of the $ame thickne$$e: but in a Cylinder of $tone or Metal the Tenacity of its parts, (which yet $eemeth greater) de- pendeth on another kind of Cement, than of $trings, or grains, and yet they al$o being drawn with equivalent force, break.</P> <P>SIMP. If the thing $ucceed as you $ay, I under$tand well enough, that the thread or grain of the Wood which is as long as the $aid Wood may make it $trong and able to Re$i$t a great vio- lence done to it to break it: But a Cord compo$ed of $trings of Hemp, no longer than two, or three foot a piece, how can it be $o $trong when it is $pun out, it may be, to a hundred times that length? Now I would farther under$tand your opinion concern- ing the Connection of the parts of Metals, Stones, and other mat- ters deprived of $uch Ligatures, which neverthele$$e, if I be not deceived, are yet more tenacious.</P> <P>SALV. We mu$t be nece$$itated to digre$$e into new Specu- lations, and not much to our purpo$e, if we $hould re$olve tho$e difficulties you $tart.</P> <foot>SAGR.</foot> <p n=>7</p> <P>SAGR. But if Digre$$ions may lead us to the knowledge of new Truths, what prejudice is it to us, that are not obliged to a $trict and conci$e method, but that make our Congre$$ions only for our diverti$ement to digre$$e $ometimes, le$t we let $lip tho$e Notions, which perhaps the offered occa$ion being pa$t, may never meet with another opportunity of remembrance? Nay, who knows not, that many times curio$ity may thereby di$cover hints of more worth, than the primarily intended Conclu$ions? Therefore I entreat you to give $atisfaction to <I>Simplicius,</I> and my $elf al$o, no le$$e curious than he, and de$irous to under$tand what that Cement is, that holdeth the parts of tho$e Solids $o tenaciou$ly conjoyned, which yet neverthele$$e in conclu$ion are di$$oluble: a knowledge which furthermore is nece$$ary for the under$tanding of the coherence of the parts of tho$e very ligaments, whereof $ome Solids are compo$ed.</P> <P>SALV. Well, $ince it is your plea$ure, I will herein $erve you. <marg><I>What that Cement is that Connecteth the parts of Solids.</I></marg> And the fir$t difficulty is, how the threads of a Cord or Rope an hundred foot long $hould $o clo$ely connect together (none of them exceeding two or three foot) that it requireth a great violence to break them. But tell me, <I>Simplicius,</I> cannot you hold one $ingle $tring of Hemp $o fa$t between your fingers by one <marg><I>How a Rope or Cord re$i$teth Fra- ction.</I></marg> end, that I pulling by the other end $hould break it $ooner than get it from you? Que$tionle$$e you might: when then, tho$e threads are not only at the end, but al$o in every part of their length, held fa$t with much $trength by him that gra$peth them, is it not apparent, that it is a much harder matter to pluck them from him that holds them, then to break them? Now in the Cord, <marg><I>In breaking a Rope the parts are not $eparated, but bro- kon.</I></marg> the $ame act of twi$ting, binds the threads mutually within one another, in $uch $ort, that pulling the Cord with great force, the threads of it break in$under, but $eparate and part not from one another; as is plainly $een by viewing the $hort ends of the $aid threads in the broken place, that are not a $pan long; as they would be, if the divi$ion of the Cord had been made by the $ole $eperating of them in drawing the Cord, and not by breaking them.</P> <P>SAGR. For confirmation of this, let me add, that the Cord is $ometimes $een to break, not by pulling it length-waies, but by over-twi$ting it: an argument, in my judgment, concluding that the threads are $o enterchangeably compre$t by one another, that tho$e compre$$ings permit not the compre$$ed to $lip $o very little, as is requi$ite to lengthen it out that it wind about the Cord, which in the twining breaketh, and con$equently in $ome $inall mea$ure $wels in thickne$$e.</P> <P>SALV. You $ay very well; but con$ider by the way, how one truth draweth on another. That thread, which griped between the <foot>fingers</foot> <p n=>8</p> fingers, did not yield to follow him that would have forceably drawn it from between them, re$i$ted, becau$e it was $tayed by a double compre$$ion, $ince the upper finger pre$t no le$$e again$t the nether, than it pre$$ed again$t that. And there is no que$tion, that if of the$e two pre$$ures, one alone might be retained, there would remain half of that Re$i$tance, which depended conjunctive- ly on them both: but becau$e you cannot with removing, <I>v.g.</I> the upper finger take away its pre$$ion, without taking away the other part al$o; it will be nece$$ary by $ome new Artifice to retain one of them, and to find a way that the $ame thread may compre$$e it $elf again$t the finger or other $olid body upon which it is put; and this is done by winding the $ame thread about the Solid. For the better under$tanding whereof, I will briefly give it you in Figure; and let <I>A B</I> and C<I>D</I> be two Cilinders, and between them let there be di$tended the thread <I>E F,</I> which for greater plainne$$e I will repre$ent to be a $mall Cord: there is no doubt but that the two Cylinders being pre$$ed hard one again$t the other, the Cord <I>E F</I> pulled by the end <I>F</I> will Re$i$t no $mal force before it will $lip from between the two Solids compre$$ing it: but if we remove one of them, though the Cord <fig> continue touching the other, yet $hall it not by $uch contact be hindered from $lipping away. But if holding it fa$t, though but gently in the point A, towards the top of the Cylinder, we wind, or belay it about the $ame $pirally in A F L O T R, and pull it by the end R: it is manife$t, that it will begin to pre$$e the Cylinder, and if the windings and wreathes be many, it $hall in its effectual drawing alwaies pre$$e it $o much the $trai- ter about the Cylinder: and by multiplying the wreathes if you make the contact longer, and con$equently more invincible, the more difficult $till $hall it be to withdraw the Cord, and make it yield to the force that pulls it. Now who $eeth not, that the $ame Re$i$tance is in the threads, which with many thou$and $uch twinings $pin the thick Cord? Yea, the $tre$$e of $uch twi$ting bindeth with $uch Tenacity, that a few Ru$hes, and of no great length, ($o that the wreaths and windings are but few where- with they entertwine) make very $trong bands, called, as I take it, <marg>* Fu$ta.</marg> ^{*} Thum-ropes.</P> <P>SAGR. Your Di$cour$e hath removed the wonder out of my mind at two effects, whereof I did not well under$tand the rea- $on; One was to $ee, how two, or at the mo$t three twines of the <foot>Rope</foot> <p n=>9</p> Rope about the Axis of a Crane did not only hold it, that be- ing drawn by the immen$e force of the weight, which it held, it $lipt nor $hrunk not; but that moreover turning the Crane about, the $aid Axis with the $ole touch of the Rope which begirteth it, did in the after-turnings, draw and rai$e up va$t $tones, whil$t the $trength of a little Boy $ufficed to hold and $tay the other end of the $ame Cord. The other is at a plain, but cunning, In$trument found out by a young Kin$man of mine, by which with a Cord he could let him$elf down from a window without much gauling the palmes of his hands, as to his great $mart not long before he had done. For <marg><I>An Hand-Pully or In$trument in- vented by an ama- rous per$on to let him$elf down from any great height with a Cord with- out gauling his hands.</I></marg> the better under$tanding whereof, rake this Scheame: About $uch a Cylinder of Wood as A B, two Inches thick, and $ix or eight Inches long, he cut a hollow notch $pirally, for one turn and a <fig> half and no more, and of widene$$e fit for the Cord he would u$e; which he made to enter through the notch at the end A, and to come out at the other B, incircling after- wards the Cylinder in a barrel or $ocket of Wood, or rather Tin, but divided length- waies, and made with Cla$pes or Hinges to open and $hut at plea$ure: and then gra$p- ing and holding the $aid Barrel or Ca$e with both his hands, the rope being made fa$t above, he hung by his arms; and $uch was the compre$$ion of the Cord between the moving Socket and the Cylinder, that at plea$ure griping his hands clo$er he could $tay him$elf without de$cending, and $lacking his hold a little, he could let him$elf down as he plea$ed.</P> <P>SALV. Aningenious invention verily, and for a full explanati- on of its nature, me-thinks I di$cover, as it were by a $hadow, the light of $ome other additional di$coveries: but I will not at this time deviate any more from my purpo$e upon this particular: and the rather in regard you are de$irous to hear my opinion of the Re$i$tance of other Bodies again$t Fraction, who$e texture is not <marg><I>Why $uch Bodies re$i$t Fraction that are not connected with Fibrous fila- ments.</I></marg> with threads, and fibrous $trings, as is that of Ropes, and mo$t kinds of Wood: but the connection of their parts $eem to de- pend on other Cau$es; which in my judgment may be reduced to two heads; one is the much talked-of Repugnance that Nature hath again$t the admi$$ion of Vacuity: for another (this of Va- cuity not $ufficing) there mu$t be introduced $ome glue, vi$cous matter, or Cement, that tenaciou$ly connecteth the Corpu$cles of which the $aid Body is compacted.</P> <P>I will fir$t $peak of <I>Vacuity,</I> $hewing by plain experiments, <foot>C what</foot> <p n=>10</p> <marg><I>The fir$t Cau$e of the Cohorence of Bodies is their Re- pugnance to Vacu- ity.</I></marg> what and how great its virtue is. And fir$t of all the $eeing at plea$ure two flat pieces of either Marble, Metal, or Gla$$e, exqui- $itely planed, $lickt, and poli$hed, that being laid upon one the other, without any difficulty $lide along upon each other, if drawn <marg><I>This is proved by the Coherence of two poli$hed Mar- bles.</I></marg> $idewaies, (a certain argument that no glue connects them,) but that going about to $eperate them, keeping them equidi$tant, there is found $uch repugnance, that the uppermo$t will be lif- ted up, and will draw the other after it, and keep it perperually rai$ed, though it be pretty thick, and heavy, evidently proveth to us, how much Nature abhorreth to admit, though for a $hort mo- ment of time, the void $pace, that would be between them, till the concour$e of the parts of the Circum-Ambient Air $hould have po$$e$t, and repleated it. We $ee likewi$e, that if tho$e two Plates be not exactly poli$hed, and con$equently their contact not every where exqui$ite; in going about to $eparate them gently, there will be found no Renitence more than that of their meer weight, but in the $udden rai$ing, the nether Stone will ri$e, and in$tantly fall down again, following the upper only for that very $mall time which $erveth for the expan$ion of that little Air which interpo- $eth betwixt the Plates, that did not every where touch, and for the ingre$$ion of the other circumfu$ed. The like Re$i$tance, which $o $en$ibly di$covers it $elf betwixt the two Plates, cannot be doubted to re$ide al$o between the parts of a Solid, and that it en- tereth into their connection, at lea$t in part, and as their Concomi- tant Cau$e.</P> <P>SAGR. Hold, I pray you, and permit me to impart unto you a particular Con$ideration, ju$t now come into my Mind, and this it is; That $eeing how the lower Plate followeth the upper, and is by a $peedy motion rai$ed, we are thereby a$certained that (con- trary to the $aying of many Philo$ophers, and perchance of <I>Ari- $totle</I> him$elf) the Motion in <I>Vacuity</I> would not be In$tantaneous; for $hould in be $uch, the propo$ed Plates without the lea$t repug- nance would Seperate; $ince the $elf $ame in$tant of time would $uffice for their $eparation, and for the concour$e of the Ambient Air to repleat that <I>Vacuity,</I> which might remain between them. By the Inferiour Plates following the Superiour therefore may be gathered, that in the <I>Vacuity</I> the Motion would not be In$tanta- <marg><I>Vacuity partly the cau$e of the Cohe- rence between the parts of Solids.</I></marg> neous. And al$o it may be inferred, that even betwixt tho$e Plates there re$teth $ome <I>Vacuity,</I> at lea$t for $ome very $hort time; that is, for $o long as the Ambient Air is moving whil$t it concurreth to replete the <I>Vacuum:</I> for if there did no <I>Vacuity</I> remain, there would be no need either of the Concour$e, or Motion of the Am- bient We mu$t therefore $ay that <I>Vacuity</I> $ometimes is admit- ted, though by Violence or again$t Nature, (albeit it is my opi- nion, that nothing is contrary to Nature, but that which is im- <foot>po$$ible</foot> <p n=>11</p> po$$ible, which again never is.) But here $tarts up another diffi- culty, and it is, That though Experience a$$ures me of the truth of the Conclu$ion, yet my Judgment is not thorowly $atisfied of the Cau$e, to which $uch an effect may be a$cribed. For as much as the effect of the Seperation of the two Plates, is in time before the Vacuity which $hould $ucceed by con$equence upon the Separa- tion. And becau$e, in my opinion, the Cau$e ought, if not in <marg><I>Of a Po$itive Ef- fect the Cau$e is Po$itive.</I></marg> Time, at lea$t in Nature, to precede the Effect: and that of a Po- $itive Effect, the Cau$e ought al$o to be Po$itive; I cannot con- ceive, how the Cau$e of the Adhe$ion of the two Plates, and of their Repugnance to Separation, (Effects that are already in Act) $hould be a$$igned to Vacuity, which yet is not, but $hould follow. And of things that are not in being, there can be no Ope- <marg><I>Non-entity is at- tended with Non- operation.</I></marg> ration; according to the infallible Maxime of Philo$ophy.</P> <P>SIMP. But $ince you grant <I>Ari$totle</I> this Axiome, I do not think you will deny another that is mo$t excellent, and true; to <marg><I>Nature doth not attempt Impo$$ibi- lities.</I></marg> wit, That Nature doth not attempt Impo$$ibilities: Upon which Axiom I think the Solution of our doubt depends: becau$e there- fore a void $pace is of it $elf impo$$ible, Nature forbids the doing that, in con$equence of which Vacuity would nece$$arily $ucceed; and $uch an act is the $eparation of the two Plates.</P> <P>SAGR. Now, (admitting this which <I>Simplicius</I> alledgeth is a $ufficient Solution of my Doubt) in per$uance of the di$cour$e with which I began, it $eemeth to me, that this $ame Repugnance to Vacuity $hould be a $ufficient Cement in the parts of a Solid of Stone, Metal, or what other $ub$tance is more firmly conjoyned, and aver$e to Divi$ion. For if a $ingle Effect, hath but one $ole Cau$e, as I under$tand, and think; or if many be a$$igned, they are reducible to one alone: why $hould not this of Vacuity, which certainly is one, be $ufficient to an$wer all Re$i$tances?</P> <P>SALV. I will not at this time enter upon this conte$t, whether Vacuity, without other Cement, be in it $elf alone $ufficient to keep together the $eparable parts of firm Bodies; but yet this I $ay, that the Rea$on of the Vacuity, which is of force, and con- oluding in the two Plates, $ufficeth not of it $elf alone for the firm connection of the parts of a $olid Cylinder of Marble, or Metal, the which forced with great violence, pulling them $treight out, in fine, divide and $eparate. And in ca$e I have found a way to di$tingui$h this already-known Re$i$tance dependent on Va- ouity, from all others what$oever that may concur with it in $trengthening the Connection, and make you $ee how that it alone is not neer $ufficient for $uch an Effect, would not you grant that it would be nece$$ary to introduce $ome other? Help him out, <I>Sim- plicius,</I> for he $tands $tudying what to an$wer.</P> <P>SIMP. The Su$pen$ion of <I>Sagredus</I> mu$t needs be upon ano- <foot>C 2 ther</foot> <p n=>12</p> ther account, there being no place left for doubting of $o clear, and nece$$ary a Con$equence.</P> <P>SAGR. You Divine <I>Simplicius,</I> I was thinking if a Million of Gold <I>per annum,</I> coming from <I>Spaine,</I> not being $ufficient to pay the Army, whether it was nece$$ary to make any other provi$ion than of Money to pay the Souldiers. But proceed, <I>Salviatus,</I> and $uppo$ing that I admit of your Con$equence, $hew us how to $e- parate the opperation of Vacuity from the other, that mea$uring it we may $ee how it's in$ufficient for the Effect of which we $peak.</P> <P>SALV. Your Genius hath prompted you. Well, I will tell you the way to part the Virtue of Vacuity from the re$t, and then how to mea$ure it. And to $ever it, we will take a continuate matter, <marg><I>How to mea$ure the Virtue of Va- cuity in Solids di- $tinct from other convenient Cau$es of their Coherence. Water a Continu- ate Matter, and void of all other a- ver$ion to $eparati- on, $ave that of Va- cuity.</I></marg> who$e parts are de$titute of all other Re$i$tance to Separation, $ave only that of Vacuity, $uch as Water at large hath been demon- $trated to be in a certain Tractate of our <I>Accademick.</I> So that when ever a Cylinder of Water is $o di$po$ed, that being drawn we find a Re$i$tance again$t the $eparation of its parts, this mu$t be acknowledged to proceed from no other cau$e, but from re- pugnance to Vacuity. But to make $uch an experiment, I have imagined a device, which with the help of a $mall Diagram, may be better expre$t than by my bare words. Let this Figure C A B D be the Profile of a Cylinder of Metal, or of Gla$s, which mu$t be made hollow within, but turned exactly round; into who$e Concave mu$t enter a Cylinder of Wood, exqui$itely fitted to touch every where, who$e Profile is noted by E G H F, which Cylinder may be thru$t up- <fig> wards, and downwards: and this I would have bored in the middle, $o that there may a rod of Iron pa$s thorow, hooked in the end K, and the other end I, $hall grow thicker in fa$hion of a Cone, or Top; and let the hole made for the $ame thorow the Cylinder of Wood be al$o cut hollow in the upper part, like a Conical Superficies, and exactly fitted to receive the Conick end I, of the Iron I K, as oft as it is drawn down by the part K. Then I put the Cylinder of Wood E H into the Concave Cylinder A D, and would not have it come to touch the upper- mo$t Superficies of the $aid hollow Cylinder, but that it $tay two or three fingers breadth from it: and I would have that $pace filled with Water; which $hould be put therein, holding the Ve$$el with the mouth C D up- wards; and thereupon pre$s down the Stopper E H, holding the Conical part I $omewhat di$tant from the hollow that was made <foot>for</foot> <p n=>13</p> for it in the Wood, to leave way for the Air to go out, which in thru$ting down the Stopper will i$$ue out by the hole of the Wood, which therefore $hould be made a little wider than the thickne$s of the Hook of Iron I K. The Air being let out, and the Iron pull'd back, which clo$e $toppeth the wood with its Conick part I, then turn the ve$$el with its mouth downwards, and fa$ten to the hook K a Bucket that may receive into it $and, or other weigh- ty matter, and you may hang $o much weight thereat, that at length the Superiour $urface of the Stopper E F will $eparate and for$ake the inferiour part of the Water; to which nothing el$e held it con- nected but the Repugnance again$t Vacuity: afterwards weighing the Stopper with the Iron, the Bucket, and all that was in it, you will have the quantity of the Force of the Vacuity. And if affixing to a Cylinder of Marble, or Chri$tal, as thick as the Cylinder of Water, $uch a weight, that together with the proper weight of the Marble or Chri$tal it $elf, equalleth the gravity of all tho$e fore- named things, a Rupture follow thereupon; we may without doubt affirm, that the only rea$on of Vacuity holdeth the parts of Marble and Chri$tal conjoyned: but not $ufficing; and $eeing that to break it there mu$t be added four times as much weight, it mu$t be confe$$ed, that the Re$i$tance of Vacuity is one part of $ive, and that the other Re$i$tance is quadruple to that of Vacuity.</P> <P>SIMP. It cannot be denied, but that the Invention is Ingen- ous: but I hold it to be $ubject to many difficulties, which makes me que$tion it; for who $hall a$$ure us, that the Air cannot pene- trate between the Gla$s, and the Stopper, though it be clo$e $topt with Flax, or other pliant matter? And al$o it's a Que$tion, whe- ther Wax or Turpentine will $erve to make the Cone I, $top the hole clo$e: Again, Why may not the parts of the Water with- draw and rarefie them$elves? Why may not the Air, or Exhalati- ons, or other more $ubtil Sub$tances penetrate through the Poro$i- ties of the Wood, or Gla$s it $elf?</P> <P>SALV. <I>Simplicius</I> is very nimble at rai$ing doubts, and, in part, helping us to re$olve them, as to the Penetration of the Air through the Wood, or between the Wood and Gla$s. But I moreover ob$erve, that we may at the $ame time $ecure our $elves, and with- all acquire new Notions, if the fore-named doubts take place; for if the Water be by Nature, howbeit with violence, capable of ex- tention, as it falleth out in Air, you $hall $ee the Stopper to de- $cend: and if in the upper part of the Gla$s we make a $mall pro- minent Bo$s, as this V; in ca$e any Air, or other more Tenuous or Spirituous Matter $hould penetrate thorow the Sub$tance, or Poro$i- ty of the Gla$s, or Wood, it would be $een to reunite (the water giving place) in the eminence V: which things not being percei- ved, we re$t a$$ured that the Experiment was made with due <foot>caution:</foot> <p n=>14</p> caution: and $ee that the Water is not capable o$ exten$ion, nor the Gla$s permeable by any matter, though never $o $ubtil.</P> <P>SAGR. And I, by means of the$e Di$cour$es have found the Cau$e of an Effect, that hath for a long time puzled my mind <marg><I>The Nature of the attraction of Wa- ter by Pumps.</I></marg> with wonder, and kept it in Ignorance. I have heretofore ob- $erved a Ci$tern, wherein, for the drawing thence of Water, there was made a Pump, by $ome one that thought, perhaps, (but in vain) to be thereby able to draw, with le$s labour, the $ame, or greater quantity of Water, than with the ordinary Buckets; and this Pump had its Sucker and Value on high, $o that the Water was made to a$cend by Attraction, and not by Impul$e, as do the Pumps that work below. This, whil$t there is any Water in the Ci$tern to $uch a determinate height, will draw it plentifully; but when the Water ebbeth below a certain Mark, the Pump will work no more. I conceited, the fir$t time that I ob$erved this ac- cident, that the Engine ____ had been $poyled, and looking for the Workman, that he might amend it; he told me, that there was no defect at all, other than what was in the Water, which being fallen too low, permitted not it $elf to be rai$ed to $uch a height; <marg><I>Water rai$ed or at- tracted by a Pump ri$eth not above eleven yards.</I></marg> and farther $aid, that neither Pump, or other Machine, that rai$eth the water by Attraction, was po$$ibly able to make it ri$e a hair more than eighteen Braces, and be the Pumps wide or narrow, this is the utmo$t limited mea$ure of their height. And I have hitherto been $o dull of apprehen$ion, that though I knew that a Rope, a Stick, and a Rod of Iron might be $o and $o lengthened, that at la$t, holding it up on high in the Air, its own weight would break it, yet I never remembred, that the $ame would much more ea$ily happen in a Rope, or Thread of Water. And what other is that which is attracted in the Pump than a Cylinder of Water, which having its contraction above, prolonged more and more, in the end arriveth to that term, beyond which being drawn, it breaketh by its foregoing over-weight, ju$t as if it was a Rope.</P> <P>SALV. It is even $o as you $ay; and becau$e the $aid height of eighteen Braces is the prefixed term of the Elevation, to which any quantity of Water, be it (that is to $ay, be the Pump) broad, narrow, or even, $o narrow as to the thickne$s of a $traw, can $u- $tain it $elf; when ever we weigh the water contained in eighteen Braces of Pipe, be it broad or narrow, we have the value of Re$i- $tance of Vacuity in Cylinders of what$oever $olid matter, of the thickne$s of the propo$ed Pipes. And $ince I have $aid $o much, <marg><I>To what length Cy- linders or Ropes of any Matter may be prolonged, be- yond which being charged they break by their own weight</I></marg> we will $hew, that a man may ea$ily find in all Metals, Stones, Tim- bers, Gla$$es, <I>&c.</I> How far one may lengthen out Cylinders, $trings, or rods of any thickne$s, beyond which, being oppre$t with their own weight, they can no longer hold, but break in pieces. Take for example a Bra$s wyer of any certain thickne$s, and length, <foot>and</foot> <p n=>15</p> and fixing one of its ends on high, add gradually more and more weight to the other, till at la$t it break, and let the greate$t weight that it can bear be <I>v. gr.</I> fifty pounds. It is manife$t that fifty pound of Bra$s more than its own weight, which let us $uppo$e, for example, to be one eighth of an Ounce, drawn out into a Wyer of the like thickne$s, would be the greate$t length of the Wyer that could bear it $elf. Then mea$ure how long the Wyer was which brake, and let it be for in$tance a y ard; and becau$e it weighed one eighth of an Ounce; and poi$ed, or bore it $elf, and fifty pounds more; which are Four Thou$and Eight Hundred eighths of Ounces; we $ay, that all Wyers of Bra$s, whatever thickne$s they be of, can hold, at the length of Four Thou$and Eight Hundred and one yards, and no more: and $o, a Bra$s Wyer being able to hold to the length of 4801 yards; the Re$i$tance it findeth dependent on Vacuity, in re$pect of the remainder, is as much as is equivalent to the weight of a Rope of Water eighteen Braces long, and of the $ame thickne$s with the $aid Bra$s Wyer: and finding Bra$s to be <I>v. gr.</I> nine times heavier than Water, in any Wyer of Bra$s, the Re$i$tance again$t Fraction dependent on the rea$on of Vacuity, importeth as much as two Braces of the $ame Wyer weigheth. And thus arguing, and operating, we may find the length of the Wyers, or Threads of all Solid Matters re- duced to the utmo$t length that they can $ub$i$t of, and al$o what part Vacuity hath in their Re$i$tance.</P> <P>SAGR. It re$teth now, that you declare to us wherein con$i$ts the remainder of that Tenacity, that is, what that Glue or Reni- tence is, which connecteth together the parts of a Solid, be$ides that which is derived from Vacuity; becau$e I cannot imagine what that Cement is, that cannot be burnt, or con$umed in a ve- ry hot Furnace in two, three, or four Moneths, nor ten, nor an hun- dred; and yet Gold, Silver, and Gla$s, $tanding $o long Liqui$i- ed, when it is taken out, its parts return, upon cooling, to reunite, and conjoyn, as before. And again, becau$e the $ame difficulty which I meet within the Connection of the parts of the Gla$s, I find al$o in the parts of the Cement, that is, what thing that $hould be which maketh them cleave $o clo$s together.</P> <P>SALV. I told you but even now, that your Genius prompted you: I am al$o in the $ame $trait: and al$o whereas I have in gene- ral told you, how that Repugnance again$t Vacuity is unque$ti- onably that which permits not, nnle$s with great violence, the $e- paration of the two Plates, and moreover of the two great pieces of the Pillar of Marble, or Bra$s, I cannot $ee why it $hould not al$o take place, and be likewi$e the Cau$e of the Coherence of the le$- $er parts, and even of the very lea$t and la$t, of the $ame Matters: <marg><I>There is but one $ole Cau$e of one $ole Effect.</I></marg> and being that of one $ole Effect, there is but one only true, and <foot>mo$t</foot> <p n=>16</p> mo$t potent Cau$e; if I can find no other Cement, why may I not try whether this of Vacuity, which I have already found, may be $ufficient?</P> <P>SIMP. But when you have already demon$trated the Re$i- $tance of the great Vacuity in the $eparation of the two great parts of a Solid to be very $mall in compari$on of that which con- necteth, and con$olidates the little Particles, or Atomes, why will you not $till hold, for certain, that this is extreamly differing from that?</P> <P>SALV. To this <I>Sagredus</I> an$wereth, That every particular Souldier is $till paid with money collected by the general Impo$i- tions of Shillings and Pence, although a Million of Gold $ufficeth not to pay the whole Army. And who knows, but that other ex- ceeding $mall Vacuities may operate among$t tho$e $mall Atomes, (even like as that was of the $elf-$ame money) wherewith all the parts are connected? I will tell you what I have $ometimes fancied: and I give it you, not as an unque$tionable Truth, but as a kind of Conjecture very undige$ted, $ubmitting it to exacter con- $iderations: Pick out of it what plea$eth you, and judge of the re$t <marg><I>Mo$t $mall Va- cuities di$$emina- ted and interpo$ed between the $mall Corpu$cles of So- lids the probable cau$e of the con$i- $tence or connecti- on of tho$e Corpu$- cles to one another,</I></marg> as you think fit. Con$idering $ometimes how the Fire, penetra- ting and in$inuating between the $mall Atomes of this or that Me- tal, which were before $o clo$ely con$olidated, in the end $epa- rates, and di$unites them; and how, the Fire being gone, they re- turn with the $ame Tenacity as before to Con$olidation, without dimini$hing in quantity, (at all in Gold, and very little in other Metals,) though they continue a long time melted; I have thought that that might happen, by rea$on the extream $mall parts of the Fire, penetrating through the narrow pores of the Metal (through which the lea$t parts of Air, or of many other Fluids, could not for their clo$ene$s perforate) by repleating the $mall interpo$ing Vacuities might free the minute parts of the $ame from the vio- lence, wherewith the $aid Vacuities attract them one to another, prohibiting their $eparation: and thus becoming able to move freely, their Ma$s might become fluid, and continue $uch, as long as the $mall parts of the Fire $hould abide betwixt them: and that tho$e departing, and leaving the former Vacuities, their wonted attractions might return, and con$equently the Cohe$ion of the parts. And, as to the Allegation made by <I>Simplicius,</I> it may, in my opinion, be thus re$olved; That although $uch Vacuities $hould be very $mall, and con$equently each of them ea$ie to be over- come, yet neverthele$s their innumerable multitude innumerably <marg><I>Innumerable A- tomes of Water in- $inuating into Ca- bles draw and rai$e an immen$e weight</I></marg> (if it be proper $o to $peak) multiplieth the Re$i$tances: and we have an evident proof what, and how great is the Force that re$ul- teth from the conjunction of an immen$e number of very weak Moments, in $eeing a Weight of many thou$ands of pounds, held <foot>by</foot> <p n=>17</p> by mighty Cables, to yield, and $uffer it $elf at la$t to be over- come by the a$$ault of the innumerable Atomes of Water; which, either carryed by the South-wind, or el$e by being di$tended into very thin Mi$ts that move to and fro in the Air, in$inuate them- $elves between $tring and $tring of the Hemp of the harde$t twi- $ted Cables; nor can the immen$e force of the pendent Weight prohibit their enterance; $o that perforating the $trict pa$$ages be- tween the Pores, they $well the Ropes, and by con$equence $hor- ten them, whereupon that huge Ma$s is forcibly rai$ed.</P> <P>SAGR. There's no doubt but that $o long as a Re$i$tance is not <marg><I>Any finite Re$i- $tance is $uperable by any the lea$t Force, multiplied.</I></marg> infinite, it may by a multitude of mo$t minute Forces be over- come; in$omuch that a competent number even of Ants would be able to carry to $hore a whole $hips lading of Corn: for Sen$e giveth us quotidian examples, that an Ant carrieth a $ingle grain with ea$e; and its cleer, that in the Ship there are not infinite grains, but that they are compri$ed in a certain number; and if you take another number four or $ix times bigger than that, and take al$o another of Ants equal to it, and $et them to work, they $hall carry the Corn, and the Ship al$o. It is true indeed, that it will be needful that the number be great, as al$o in my judgment that of the <I>Vacuities,</I> which hold together the $inall parts of the Mettal.</P> <P>SALV. But though they were required to be infinite, do you think it impo$$ible?</P> <P>SAGR. Not if the Mettal were of an infinite ma$$e; other- wi$e ----</P> <P>SALV. Otherwi$e what? Go to, feeing we are faln upon Paradoxes, let us $ee if we can any way demon$trate, how that in a continuate finite exten$ion, it is not impo$$ible to finde infi- nite <I>Vacuities:</I> and then, if we gain nothing el$e, yet at lea$t we <marg>Ari$totles <I>admi- rable Problem of two Concentrick Circles that turn round, and its true re$olution.</I></marg> $hall finde a $olution of that mo$t admirable Problem propound- ed by <I>Ari$totle</I> among$t tho$e which he him$elf calleth admirable, I mean among$t his <I>Mechanical Que$tions</I>; and the Solution may haply be no le$$e plain and concluding, than that which he him$elf brings thereupon, and different al$o from that which Learned <marg>Mon$ig. Gueva ra <I>honourably men- tioned.</I></marg> <I>Mon$ig. di Guevara</I> very acutely di$cu$$eth. But it is fir$t requi$ite to declare a Propo$ition not toucht by others, on which the $olution of the que$tion dependeth, which afterwards, if I deceive not my $elf, will draw along with it other new and admirable Notions; for under$tanding whereof the more exactly, we will give it you in a Scheme: We $uppo$e, therefore an equilateral, and equian- gled Poligon of any number of Sides at plea$ure, de$cribed about this Center G; and in this example let it be a Hexagon A B C D E F; like to which, and concentrick with the $ame mu$t be di$tributed another le$$er, which we mark H I K L M N; <foot>D and</foot> <p n=>18</p> and let one Side of the greater A B be prolonged indeterminately towards S, and of the le$$e the corre$pondent Side H I is to be produced in like manner towards the $ame part, repre$enting the Line H T, parallel to A S; and let another pa$$e by the Center equidi$tant from the former, namely G V. This done, we $uppo$e the greater Poligon to turn about upon the Line A S, carrying with it the other le$$er Poligon. It is manife$t, that the point B, the term of the Side A B, $tanding $till, whil$t the Revolution begins, the angle A ri$eth, and the point C de$cendeth, de$cribing the arch C Q; $o that the Side B C is applyed to the line B Q, equal to it $elf: but in $uch conver$ion the angle I of the le$$er Poligon ri$eth above the Line I T. for that I B is oblique upon A S: nor will the point I fall upon the parallel I T, before the point C come to Q: and by that time I $hall be de$cended unto O after it had de$cribed the Arch I O, without the Line H T: and at the $ame time the Side I K $hall have pa$s'd to O P. But the Cen- ter G $hall have gone all this time out of the Line G V, on which it $hal not fall, until it $hall fir$t have de$cribed the Arch G C. Having made this fir$t $tep, the greater Poligon $hall be tran$po$ed to re$t with the Side B C upon the Line B Q; the Side I K of the le$$er upon the Line O P, having skipt all the Line I O without touching <fig> it; and the Center G $hall be removed to C, making its whole cour$e without the Parallel G V: And in fine all the Figure $hall be remitted into a Po$ition like the fir$t; $o that the Revolution being continued, and coming to the $econd $tep, the Side of the greater Poligon D C $hall remove to Q X; K L of the le$$er (ha- ving fir$t skipt the Arch P Y) $hall fall upon Y Z, and the Center proceeding evermore without G V $hall fall on it in R, after the great skip C R. And in the la$t place, having fini$hed an entire Conver$ion, the greater Poligon will have impre$$ed upon A S, $ix <foot>Lines</foot> <p n=>19</p> Lines equal to its Perimeter without any interpo$itions or skips: the le$$er Poligon likewi$e $hall have traced $ix Lines equal to its Perimeter, but di$continued by the interpo$ition of five Arches, under which are the Chords, parts of the parallel H T not toucht by the Poligon: And la$tly, the Center G never hath toucht the Parallel G V except in $ix points. From hence you may compre- hend, how that the Space pa$$ed by the le$$er Poligon, is almo$t equal to that pa$$ed by the greater, that is the Line H T is almo$t equal to A S, then which it is le$$er only the quantity of one of the$e Arches, taking the Line H T, together with all its Arches. Now, this which I have declared and explained to you in the exam- ple of the$e Hexagons, I would have you under$tand to hold true in all other Poligons, of what number of Sides $oever they be, $o that they be like Concentrick, and Conjoyned; and that at the Conver$ion of the greater, the other, how much $oever le$$er, be $uppo$ed to revolve therewith: that is, you mu$t under$tand, I $ay, that the Lines by them pa$$ed are very near equal, computing in- to the Space pa$t by the le$$er, the Intervals under the little Ar- ches not toucht by any part of the Perimeter of the $aid le$$er Po- ligon. Let therefore the greater Poligon, of a thou$and Sides, pa$s round, and mea$ure out a continued Line equal to its Perimeter; and in the $ame time the le$s pa$$eth a Line almo$t as long, but compounded of a thou$and Particles equal to its thou$and Sides, but di$continued with the interpo$ition of a thou$and void Spaces: for $uch may we call them, in relation to the thou$and little Lines toucht by the Sides of the Poligon. And what hath been $poken hitherto admits of no doubt or $cruple. But tell me, in ca$e that about a Center, as $uppo$e the point A, (in the former Scheme) we $hould de$cribe two Circles concentrick, and united together; and that from the points C and B of their Semi-Diameters, there be drawn the Tangents C E, and B F, and by the Center A the Pa- rallel A D; $uppo$ing the greater Circle to be turned upon the Line B F, (drawn equal to its Circumference, as likewi$e the other two C E, and A D;) when it hath compleated one Revolution, what $hall the le$$er Circle, and Center have done? The Center $hall doubtle$s have run over, and touched the whole Line A D, and the le$s Circumference $hall with its touches have mea$ured all C E, doing the $ame as did the Poligons above; and different only in this, that the Line H T was not touched in all its Parts by the Perimeter of the le$$er Poligon, but there were as many parts left untoucht with the interpo$ition of $alts, or skipped $paces; as were the$e parts touched by the Sides: but here in the Circles, the Circumference of the le$$er Circle, never $eparates from the Line C E, $o as to leave any of its parts untou cht; nor is the parts touching of the Circumference, le$s than the part toucht of the <foot>D 2 Right-</foot> <p n=>20</p> Right-line. Now how is it po$$ible that the le$$er Circle $hould without skips run a Line $o much bigger than its Circumfe- rence?</P> <P>SAGR. I was con$idering whether one might not $ay, that like as the Center of the Circle trailed alone upon A D toucht, it all being yet but one $ole Point; $o likewi$e might the Points of the le$$er Circumference, drawn by the revolution of the greater, go gliding along $ome $mall part of the Line C E.</P> <P><I>S</I>ALV. This cannot be, for two rea$ons; fir$t, becau$e there is no rea$on why $ome of the touches like to C $hould go gliding along $ome part of the Line C E, more than others: and though there $hould; $uch touches being (becau$e they are points) in$i- nite, the glidings along upon C E would be infinite; and $o being, they would make an infinite Line, but the Line C E is finite. The other rea$on is, that the greater Circle, in its Revolution continu- ally changing contact, the le$$er Circle mu$t of nece$$ity do the like; there being no other Point but B, by which a Right Line can be drawn to the Center A, and pa$$ing through C; $o that the greater Circumference changing Contact, the le$s doth change it al$o; nor doth any Point of the le$s touch more than one Point of its Right-Line C E: be$ides, that al$o in the conver$ion of the Po- ligons, no Point of the Perimeter of the le$s falls on more than one Point of the Line, which was by the $aid Perimeter traced, as may be ea$ily under$tood, con$idering the Line I K is parallel to B C, whereupon, till ju$t that B C fall on B R, I K continueth elevated above I P, and toucheth it not before B C is on the very Point of uniting with B Q, and then all in the $ame in$tant I K uniteth with O P, and afterwards immediately ri$eth above it again.</P> <P>SAGR. The bu$ine$s is really very intricate, nor can I think on any Solution of it, therefore do you explain it to us as far as you judge needful.</P> <P>SALV. I $hould, for the evincing hereof, have recour$e to the con$ideration of the fore-de$cribed Poligons, the effect of which is intelligible and already comprehended, and would $ay, that like as in the Poligons of an hundred thou$and Sides, the Line pa$$ed and mea$ured by the Perimeter of the greater, that is by its hundred thou$and Sides continually di$tended, is not con$iderably bigger than that mea$ured by the hundred thou$and Sides of the le$s, but with the interpo$ition of an hundred thou$and void $paces interve- ning; fo I would $ay in the Circles (which are Poligons of innu- merable Sides) that the Line mea$ured by the infinite Sides of the great Circle, lying continued one with another, to be equalled in length by the Line traced by the infinite Sides of the le$s, but by the$e including the interpo$ition of the like number of intervening Spaces: and like as the Sides are not quantitative, but yet infinite <foot>in</foot> <p n=>21</p> in number, $o the interpo$ing Vacuitics are not quantitative, but infinite in number; that is, tho$e are infinite Points all filled, and the$e are infinite points, part filled, and part empty. And here I would have you note, that re$olving, and dividing a Line into quan- titative parts, and con$equently of a finite number, it is not po$$ible to di$po$e them into a greater extention than that which they po$- $e$t whil$t they were continued, and connected, without the inter- po$ition of a like number of void Spaces; but imagining it to be re$olved into parts not quantitative, namely, into its infinite indivi- $ibles, we may conceive it produced to immen$ity without the in- terpo$ition of quantitative void $paces, but yet of infinite indivi$i- ble Vacuities. And this which is $poken of $imple lines, $hould al$o be under$tood of Superficies, and Solid Bodies, con$idering that they are compo$ed of infinite Atomes not non-quantitative; if we would divide them into certain quantitative parts, there's no que$tion, but that we cannot di$po$e them into Spaces more ample than the Solid before occupied, unle$s with the interpo$ition of a certain number of quantitative void Spaces; void, I $ay, at lea$t of the matter of the Solid: but if we $hould propo$e the highe$t, and ultimate re$olution made into the fir$t, non-quantitative, but infinite fir$t compoun- ding parts, we may be able to conceive $uch compounding parts extended unto an immen$e Space without the interpo$ition of quantitative void Spaces; but only of infinite non-quantitative Va- cuities: and in this manner a man may draw out, <I>v. gr.</I> a little Ball of Gold into a very va$t expan$ion without admitting any quan- titative void Spaces; yet neverthele$s we may admit the Gold to be compounded of infinite induci$$ible ones.</P> <P>SIMP. Me thinks that in this point you go the way of tho$e di$- $eminated Vacuities of a certain <I>Ancient Philo$opher</I> ------</P> <P>SALV. But you add not: [<I>who denied Divine Providence:)</I> as on $uch another occa$ion, $ufficiently be$ides his purpo$e, a cer- tain Antagoni$t of our <I>Accademick</I> did $ubjoyn.</P> <P>SIMP. I $ee very well, and not without indignation, the malice of $uch contradictors; but I $hall forbear the$e Cen$ures, not only upon the $core of Good-Manners, but becau$e I know how di$a- greeing $uch Tenets are to the well-tempered, and well-di$po$ed mind of a per$on, $o Religious and Pious, yea, Orthodox and Ho- ly, as you, Sir. But returning to my purpo$e; I find many $cruples to ari$e in my mind about your la$t Di$cour$e, which I know not how to re$olve. And this pre$ents its $elf for one, that if the Cir- cumferences of two Circles are equall to the two Right Lines C E, and B F, this taken continually, and that, with the interpo$i- tion of infinite void Points; how can A D, de$cribed by the Center, which is but one $ole Point, be $aid to be equal to the $ame, it con- taining infinite of them? Again, that $ame compo$ing the Line of <foot>Points,</foot> <p n=>22</p> Points, the divi$ible of indivi$ibles, the quantitative of non-quan- titative, is a rock very hard, in my judgment, to pa$s over: And the very admitting of Vacuity, $o thorowly confuted by <I>Ari$totle,</I> no le$s puzleth me than tho$e difficulties them$elves.</P> <P>SALV. There be, indeed, the$e and other difficulties; but re- member, that we are among$t Infinites, and Indivi$ibles: tho$e in- comprehen$ible by our finite under$tanding for their Grandure; and the$e for their minutene$s: neverthele$s we $ee that Humane Di$cour$e will not be beat off from ruminating upon them, in which regard, I al$o a$$uming $ome liberty, will produce $ome of my conceits, if not nece$$arily concluding, yet for novelty $ake, which is ever the me$$enger of $ome wonder: but perhaps the car- rying you $o far out of your way begun, may $eem to you imper- tinent, and con$equently little plea$ing.</P> <P>SAGR. Pray you let us enjoy the benefit, and priviledge, of free $peaking which is allowed to the living, and among$t friends; e$pe- cially, in things arbitrary, and not nece$$ary; different from Di$cour$e with dead Books, which $tart us a thou$and doubts, and re$olve not one of them. Make us therefore partakers of tho$e Con$iderations, which the cour$e of our Conferences $ugge$t unto you; for we want no time, $eeing we are di$engaged from urgent bu$ine$$es, to continue and di$cu$$e the other things mentioned; and particular- ly, the doubts, hinted by <I>Simplicius,</I> mu$t by no means e$cape us.</P> <P>SAIV. It $hall be $o, $ince it plea$eth you: and beginning at the fir$t, which was, how it's po$$ible to imagine that a $ingle Point is equal to a Line; in regard I can do no more for the pre$ent, I will attempt to $atisfie, or, at lea$t, qualifie one improbability with another like it, or greater; as $ome times a Wonder is $wallowed up in a Miracle. And this $hall be by $hewing you two equal Su- perficies, and at the $ame time two Bodies, likewi$e equal, and placed upon tho$e Superficies as their Ba$es; and that go (both the$e and tho$e) continually and equally dimini$hing in the $elf- <marg><I>The equal Super- ficies of two Solids continually $ub- $tracting from them both equal parts, are reduced, the one into the Circumference <*>f a Circle, and theo- ther into a Point.</I></marg> $ame time, and that in their remainders re$t alwaies equal between them$elves, and (la$tly) that, as well Super$icies, as Solids, deter- mine their perpetual precedent equalities, one of the Solids with one of the Superficies in a very long Line; and the other Solid with the other Superficies in a $ingle Point: that is, the latter in one Point alone, the other in infinite.</P> <P><I>S</I>AGR. An admirable propo$al, really, yet let us hear you ex- plain and demon$trate it.</P> <P>SALV. It is nece$$ary to give you it in Figure, becau$e the proof is purely Geometrical. Therefore $uppo$e the Semicircle A F B, and its Center to be C, and about it de$cribe the Rectangle A D E B, and from the Center unto the Points D and E let there be drawn the Lines C D, and C E; Then drawing the Semi-Dia- <foot>meter</foot> <p n=>23</p> meter C F, perpendicular to one of the two Lines A B, or D E and immoveable; we $uppo$e all this Figure to turn round about that Perpendicular: It is manife$t, that there will be de$cribed by the Parallelogram A D E B, a Cylinder; by the Semi-circle A F B, an Hemi-Sphære; and by the Triangle C D E a Cone. This pre- $uppo$ed, I would have you imagine the Hemi$phære to be taken away, leaving behind the Cone, and that which $hall remain of the Cylinder; which for the Figure, which it $hall retain like to a Di$h, we will hereafter call a Di$h: touching which, and the Cone, we will $ir$t demon$trate that they are equal; and next a Plain being drawn parallel to the Circle, which is the foot or Ba$e of the Di$h, who$e Diameter is the Line D E, and its Center F; we will demon$trate, that $hould the $aid Plain pa$s, <I>v. gr.</I> by the Line G H, cutting the Di$h in the points G I, and O N; and the Cone in the points H and L; it would cut the part of the Cone C H L, equal alwaies to the part of the Di$h, who$e Profile is repre$ented to us by the Triangles G A I, and B O N: and more- over we will prove the Ba$e al$o of the $ame Cone, (that is the Circle, who$e Diameter is H L) to be equal to that circular Su- perficies, which is Ba$e of the part of the Di$h; which is, as we may $ay, a Rimme as broad as G I; (note here by the way what Mathematical Definitions are: they be an impo$ition of names, or, we may $ay, abreviations of $peech, ordain'd and introduced to prevent the trouble and pains, which you and I meet with, at pre- $ent, in that we have not agreed together to call <I>v. gr.</I> this Super- ficies a circular Rimme, and that very $harp Solid of the Di$h a round Razor:) now how$oever you plea$e to call them, it $ufficeth you to know, that the Plain produced to any di$tance at plea$ure, $o that it be parallel to the Ba$e, <I>viz.</I> to the Circle who$e Diame- ter D E cuts alwaies the two Solids, namely, the part of the Cone C H L, and the upper part of the Di$h equal to one another: and likewi$e the two Superficies, Ba$is of the $aid Solids, <I>viz.</I> the $aid Rimme, and the Circle H L, equal al$o to one another. Whence followeth the forementioned Wonder; namely, that if we $hould $uppo$e the cutting-plain to be $ucce$$ively rai$ed towards the <fig> Line A B, the parts of the Solid cut are alwaies equall, as al$o the Superficies, that are their Ba$es, are evermore equal; and, in fine, rai$ing the $aid Plain higher and higher, the two Solids (ever equal) as al$o their Ba$es, (Su- perficies ever equal) $hall one couple of them terminate in a Cir- cumference of a Circle, and the other couple in one $ole point; <foot>for</foot> <p n=>24</p> for $uch are the upper Verge or Rim of the Di$h, and the Vertex of the Cone. Now whil$t that in the diminution of the two So- lids, they till the very la$t maintain their equality to one another, it is, in my thoughts, proper to $ay, that the highe$t and ultimate terms of $uch Diminutions are equal, and not one infinitely bigger than the other. It $eemeth therefore, that the Circumference of an im- men$e Circle may be $aid to be equal to one $ingle point; and this that befalls in Solids, holdeth likewi$e in the Superficies their Ba$es; that they al$o in the common Diminution con$erving al- waies equality, in fine, determine at the in$tant of their ultimate Diminution the one, (that is, that of the Di$h) in their Circum- ference of a Circle, the other (to wit, that of the Cone) in one $ole point. And why may not the$e be called equal, if they be the la$t remainders, and foot$teps left by equal Magnitudes? And note again, that were $uch Ve$$els capable of the immen$e Cœle$tial Hemi$pheres: both their upper Rims, and the points of the contai- ned Cones (keeping evermore equally to one another) would fi- nally determine, tho$e, in Circumferences equal to tho$e of the greate$t Circles of the Cœle$tial Orbes, and the$e in $implo points. Whence, according to that which $uch Speculations per$wade us to, all Circumferences of Circles, how unequal $oever, may be $aid to be equal to one another, and each of them equal to one $ole point.</P> <P>SAGR. The Speculation is, in my e$teem, $o quaint and curi- ous, that, for my part, though I could, yet would I not oppo$e it, for I take it for a piece of Sacriledge to deface $o fine a Structure, by $purning at it with any pedantick contradiction; yet for our en- tire $ati<*>faction, give us the proof (which you $ay is Geometrical) of the equality alwaies retained between tho$e Solids, and tho$e their Ba$es, which I think mu$t needs be very $ubtil, the philo$o- phical Contemplation being $o nice, which depends on the $aid Conclu$ion.</P> <P>SALV. The Demon$tration is but $hort, and ea$ie. Let us keep to the former Figure, in which the Angle I P C being a Right An- gle, the Square of the Semi-Diameter I C is equal to the two Squares of the Sides I P, and P C. But the Semi-Diameter I C, is equal to A C, and this to G P; and C P is equal to P H; therefore the Square of the Line G P is equal to the two Squares of I P, and P H, and the Quadruple to the Quadruples; that is, the Quadrate of the Diameter G N is equal to the two Quadrates I O, and H L: and becau$e Circles are to each other, as the Squares of their Dia- meters; the Circle who$e Diameter is G N, $hall be equall to the two Circles who$e Diameters are I O, and H L; and taking away the Common Circle, who$e Diameter is I O; the re$idue of the Circle G N $hall be equal to the Circle, who$e Diameter is H L. <foot>And</foot> <p n=>25</p> And this is as to the fir$t part: Now as for the other part, we will, for the pre$ent, omit its Demon$tration, as well becau$e that if you would $ee it, you $hall find it in the twelfth Propo$ition of the Se- <marg>Lucas Valerius, <I>the other</I> Archi- chimedes <I>of our Age, hath written admirably,</I> De Centro Gravita- tis Solidorum.</marg> cond Book <I>De centro Gravitatis Solidorum,</I> publi$hed by <I>Signeur Lucas Valerius,</I> the new <I>Archimedes</I> of our Age; who upon ano- ther occa$ion hath made u$e of it; as becau$e in our ca$e it $uffi- ceth to have $een, how the Superficies, already explained, are ever- more equal; and that alwaies dimini$hing equally, they in the end determine, one in a $ingle point, and the other in the Circumfe- rence of a Circle, be it never-$omuch bigger, for in this lyeth our Wonder.</P> <P>SAGR. The Demon$tration is as ingenious, as the reflection grounded upon it is admirable. Now let us hear $omewhat about the other Doubt $ugge$ted by <I>Simplicius,</I> if you have any particu- lars worth note to hint thereupon, but I $hould incline to think it impo$$ible to be, in regard it is a Controver$ie that hath been $o canva$$ed.</P> <P>SALV. You $hall have $ome of my particular thoughts thereon; fir$t repeating what but even now I told you, namely, that Infini- ty alone, as al$o Indivi$ibility, are things incompre hen$ible to us: now think how they will be conjoyned together: and yet if you would compound the Line of indivi$ible points, you mu$t make them infinite; and thus it will be requi$ite to apprehend in the $ame in$tant both Infinite, and Indivi$ible. The things that ar $e- veral times have come into my mind, on this occa$ion, are many; part whereof, and the more con$iderable, it may be, I cannot upon $uch a $udden remember; but it may happen, that in the $equal of the Di$cour$e, coming to put que$tions and doubts to you, and particularly to <I>Simplicius,</I> they may, on the other $ide, re-mind me of that, which without $uch excitement would have lain dor- mant in my Fancy: and therefore, with my wonted freedom, per- mit me that I produce any wild conjectures, for $uch may we fitly call them in compari$on of $upernatural Doctrines, the only true and certain determiners of our Controver$ies, and unerring guides in our ob$cure, and dubious paths, or rather Laberinths.</P> <P>Among$t the fir$t In$tances that are wont to be produced <marg>Continuum <I>com- pounded of Indivi- $ibles.</I></marg> again$t tho$e that compound <I>Continuum</I> of Indivi$ibles, this is u$u- ally one; That an Indivi$ible, added to another Indivi$ible, produ- ceth not a thing divi$ible; for if that were $o, it would follow, that even the Indivi$ibles were divi$ible: for if two Indivi$ibles, as for example, two Points conjoyned, $hould make a Quantity that $hould be a divi$ible Line, much more $uch $hould one be that is compounded of three, five, $even, or others, that are odd num- bers; the which Lines, being to be cut in two equal parts, render divi$ible that Indivi$ible which was placed in the middle. In this <foot>E and</foot> <p n=>26</p> and other Objections of this kind you may $atisfie the propo$er of them, telling him, that neither two Indivi$ibles, nor ten, nor an hundred, no, nor a thou$and can compound a Magnitude divi$ible, and quantitative, but being infinite they may.</P> <P>SIMP. Here already ri$eth a doubt, which I think unre$olvable; and it is, that we being certain to find Lines one bigger than ano- ther, although both contain infinite Points, we mu$t of nece$$ity confe$s, that we have found in the $ame Species a thing bigger than infinite; becau$e the Infinity of the Points of the greater Line, $hall exceed the Infinity of the Points of the le$$er. Now this a$$igning of an Infinite bigger than an Infinite is, in my opinion, a conceit that can never by any means be apprehended.</P> <P>SALV. The$e are $ome of tho$e difficulties, which re$ult from the Di$cour$es that our finite Judgments make about Infinites, gi- ving them tho$e attributes which we give to things finite and ter- minate; which I think is inconvenient; for I judge that the$e terms of Majority, Minority, and Equality $ute not with Infinites, of which we cannot $ay that one is greater, or le$s, or equal to ano- ther: for proof of which there cometh to my mind a Di$cour$e, which, the better to explain, I will propound by way of Interroga- tories to <I>Simplicius</I> that $tarted the que$tion.</P> <P>I $uppo$e that you very well under$tand which are Square Num- bers, and which not Square.</P> <P>SIMP. I know very well, that the Square Number is that which proceeds from the multiplication of another Number into it $elf; and $o four, and nine, are Square Numbers, that ari$ing from two, and this from three multiplied into them$elves.</P> <P>SALV. Very well; And you know al$o, that as the Products are called Squares: the Produ$ors, that is, tho$e that are multiplied, are called Sides, or Roots; and the others, which proceed not from Numbers multiplied into them$elves, are not Squares. So that if I $hould $ay, all Numbers comprehending the Square, and the not Square Numbers, are more than the Square alone, I $hould $peak a mo$t unque$tionable truth: Is it not $o?</P> <P>SIMP. It cannot be denied.</P> <P>SALV. Farther que$tioning, if I ask you how many are the Numbers Square, you can an$wer me truly, that they be as many, as are their propper Roots; $ince every Square hath its Root, and every Root its Square, nor hath any Square more than one $ole Root, or any Root more than one $ole Square.</P> <P>SIMP. True.</P> <marg><I>An Infinite Num- ber, as it contains infinite Square and Cupe Roots, $o it conta neth infi- nite Square and Cube Numbers.</I></marg> <P>SALV. But if I $hall demand how many Roots there be, you cannot deny but that they be as many as all Numbers, $ince there is no Number that is not the Root of $ome Square: And this be- ing granted, it is requi$ite to affirm, that Square Numbers are as <foot>many</foot> <p n=>27</p> many as their Roots, and Roots are all Numbers: and yet in the beginning we $aid, that all Numbers are far more than all Squares, the greater part not being Squares: and yet neverthele$s the num- ber of the Squares goeth dimini$hing alwaies with greater propor- tion, by how much the greater number it ri$eth to; for in an hun- dred there are ten Squares, which is as much as to $ay, the tenth part are Squares: in ten thou$and only the hundredth part are Squares: in a Million only the thou$andth, and yet in an Infinite Number, if we are able to comprehend it, we may $ay the Squares are as many, as all Numbers put together.</P> <P>SAGR. What is to be re$olved then on this occa$ion?</P> <P>SALV. I $ee no other deci$ion that it may admit, but to $ay, that all Numbers are infinite, Squares are infinite, their Roots are infinite; and that neither is the multitude of Squares le$s than all Numbers, nor this greater than that: and in conclu$ion, that the Attributes of Equality, Majority, and Minority, have no place in Infinites, but only in terminate quantities. And therefore when <I>Simplicius</I> propoundeth to me many unequal <I>L</I>ines, and demand- eth of me, how it can be, that in the greater there are no more Points than in the le$s: I an$wer him, That there are neither more, nor le$s, nor ju$t $o many; but in each of them infinite. Or if I had an$wered him, that the Points in one, are as many as there are Square Numbers; in another bigger, as many as all Numbers; in a le$s, as many as the Cubick Numbers, might not I have given $a- tisfaction, by a$$igning more to one, than to another, and yet to every one infinite? And thus much as to the fir$t difficulty.</P> <P>SAGR. Hold, I pray you, and give me leave to add unto what hath been $poken hitherto, a thought which I ju$t now light on, and it is this, that granting what hath been $aid, me-thinks, that not on- ly it's improper to $ay, one Infinite is greater than another Infinite, but al$o, that it's greater than a Finite; for if an Infinite Number were greater, <I>v. gr.</I> than a Million; it would thereupon follow, that pa$$ing from the Million to others, and $o to others continual- ly greater, one $hould pa$s on towards Infinity; which is not $o: but on the contrary, to how much the greater Numbers we go, $o much the more we depart from Infinite Number; becau$e in Num- bers, the greater you take, $o much the rarer and rarer alwaies are Square Numbers contained in them; but in Infinite Number the Squares can be no le$s than all Numbers, as but ju$t now was con- cluded: therefore the going towards Numbers alwaies greater, and greater, is a departing farther from Infinite Number.</P> <P>SALV. And $o by your ingenious Di$cour$e we may conclude, that the Attributes of Greater, Le$$er, or Equal, have no place, not only among$t Infinites; but al$o betwixt Infinites, and Fi- nites.</P> <foot>E 2 I</foot> <p n=>28</p> <P>I pa$s now to another Con$ideration; and it is, that in regard that the Line, and every continued quantity are divideable conti- nually into divi$ibles, I $ee not how we can avoid granting that the compo$ition is of infinite Indivi$ibles: becau$e a divi$ion and $ub- divi$ion that may be pro$ecuted perpetually $uppo$eth that the parts are infinite; for otherwi$e the $ubdivi$ion would be termina- ble: and the parts being Infinite, it followeth of con$equence that they be non-quantitative; for infinite quantitative parts make an infinite exten$ion: and thus we have a <I>Continuum</I> compoun- ded of infinite Indivi$ibles.</P> <P>SIMP. But if we may continually pro$ecute the divi$ion in quantitative parts, what need have we, for $uch re$pect, to intro- duce the non-quantitative?</P> <P>SALV. The very po$$ibility of perpetually pro$ecuting the di- vi$ion in quantitative parts induceth the nece$$ity of the compo$iti- on of infinite non-quantitative. Therefore, coming clo$er to you, I demand you to tell me re$olutely, whether the quantitative parts in <I>Continuum</I> be in your judgment finite or infinite?</P> <P>SIMP. I reply, that they are both Infinite, and Finite; Infinite in Power, and Finite in Act. Infinite in Power, that is, before the Divi$ion; but Finite in Act, that is, after they are divided: for the parts are not actually under$tood to be in the whole, till it is di- vided, or at lea$t marked; otherwi$e we $ay that they are in Power.</P> <P>SALV. So that a Line <I>v. gr.</I> twenty foot long, is not $aid to contain twenty Lines of one foot a piece, actually, but only after it is divided into twenty equal parts: but is till then $aid to contain them only in power. Now be it as you plea$e; and tell me whe- ther, when the actual Divi$ion of $uch parts is made, that fir$t whole encrea$eth or dimini$heth, or el$e continueth of the $ame bigne$s?</P> <P>SIMP. It neither encrea$eth, nor dimini$heth.</P> <P>SALV. So I think al$o. Therefore the quantitative parts in <I>Con- tinuum</I> quantity, be they in Act, or be they in Power, make not its quantity bigger or le$$er: but it is very plain that the$e quantita- tive parts, actually contained in their whole, if they be infinite, make it an infinite Magnitude; therefore quantitative parts, though infinite only in power, cannot be contained, but only in an infinite Magnitude: therefore in a finite Magnitude infinite quan- titative parts can be contained neither in Act, nor Power.</P> <P>SAGR. How then can it be true, that the <I>Continuum</I> may be ince$$antly divided into parts $till capable of new divi$ions?</P> <P>SALV. It $eems that that di$tinction of Power, and Act, makes that fea$ible one way, which another way would be impo$$ible. But I will $ee to adju$t the$e matters by making another account: <foot>And</foot> <p n=>29</p> And to the Que$tion, which was put, Whether the quantitative parts in a terminated <I>Continuum</I> be finite or infinite; I will an$wer directly contrary to that which <I>Simplicius</I> replied, namely, that they be neither finite, nor infinite.</P> <P>SIMP. I $hould never have found $uch an an$wer, not imagi- ning that there was any mean term between finite and infinite; $o that the divi$ion or di$tinction which makes a thing to be either Finite, or Infinite, is imperfect and deficient.</P> <P>SALV. In my opinion it is; and $peaking of ^{*} Di$crete Quan- <marg><I>Quantitative parts in Di$crete Quan- tity are neither fi- nite nor infinite, but an$werable to every given Num- ber.</I></marg> tities, me thinks that there is a third mean term between Finite and Infinite, which is that which an$wereth to every a$$igned Number: So that being demanded in our pre$ent ca$e, Whether the quanti- tative parts in <I>Continuum</I> be Finite, or Infinite, the mo$t congru- ous reply is to $ay, that they are neither Finite, nor Infinite, but $o many, as that they <I>An$wer</I> to any number a$$igned: the which to do, it is nece$$ary that they be not comprehended in a limited Number, for then they would not an$wer to a greater: nor, again, is it nece$$ary, that they be infinite, for no a$$igned Number is infi- nite. And thus at the plea$ure of the Demander, a Line being propounded, we may be able to a$$ign in it an hundred quantita- tive parts, or a thou$and, or an hundred thou$and, according to the number which he be$t likes; $o that it be not divided into in- finite. I grant therefore to the Philo$ophers, that <I>Continuum</I> con- taineth as many quantitative parts as they plea$e, and grant them that it containeth the $ame either in Act, or in Power, which they be$t like: but this I add again, that in like manner, as in a Line of ten yards, there are contained ten Lines of one yard a piece, and thirty Lines of a foot a piece, and three hundred and $ixty Lines of an inch a piece, $o it contains infinite Points; denominate them in Act, or in Power, as you will: and I remit my $elf in this matter to your opinion and judgment, <I>Simplicius.</I></P> <P>SIMP. I cannot but commend your Di$cour$e: but am great- ly afraid, that this parity of the Points, being contained in the like manner as the quantitative parts, will not agree with ab$olute ex- actne$s; nor $hall it be $o ea$ie a matter for you to divide the gi- ven Line into infinite Points, as for tho$e Philo$ophers to divide it into ten yards, or thirty feet, nay, I hold it wholly impo$$ible to effect $uch a divi$ion: $o that this will be one of tho$e Powers that are never reduced to Act.</P> <P>SALV. The trouble, pains, and long time without which a thing is not fea$ible, render it not impo$$ible; for I think al$o, that you cannot $o ea$ily effect a divi$ion to be made of a Line into a thou$and parts; and much le$s being to divide it into 937, or $ome other great Prime Number. But if I di$patch this, which you, it may be, judge an impo$$ible divi$ion, in as $hort a time, as another <foot>would</foot> <p n=>30</p> would require to divide it into forty, you will be content more willingly to admit of it in our future Di$cour$e?</P> <P>SIMP. I am plea$ed with your way of arguing, as you now do mix it with $ome plea$antne$s: and to your que$tion I reply, that the facility would $eem more than $ufficient, if the re$olving it into Points were but as ea$ie, as to divide it into a thou$and parts.</P> <P>SALV. Here I will tell you a thing, which haply will make you wonder in this matter of going about, or being able to re$olve the Line into its Infinites, keeping that order which others ob$erve in dividing it into forty, $ixty, or an hundred parts; namely, by di- viding it fir$t into two, then into four: in which order he that $hould think to find its infinite Points would gro$ly delude him$elf; for by that progre$$ion, though continued to eternity, he $hould never arrive to the divi$ion of all its quantitative parts: yea, he is in that way $o far from being able to arrive at the intended term of Indivi$ibility, that he rather goeth farther from it; and whil$t he thinks by continuing the divi$ion, and multiplying the multi- tudes of the parts, to approach to Infinite, I am of opinion, that he more and more removes from it: and my rea$on is this; In the Di$cour$e, we had even now, we concluded, that, in an infinite Number, there was, of nece$$ity, as many Square, or Cube Num- bers, as there were Numbers; $ince that tho$e and the$e were as ma- ny as their Roots, and Roots comprehend all Numbers: Next we did $ee, that the greater the Numbers were that were taken, the $eldomer are their Squares to be found in them, and $eldomer yet their Cubes: Therefore it is manife$t, that the greater the Number is to which you pa$s, the farther you remove from Infinite Num- ber: from whence it followeth, that turning backwards, ($eeing that $uch a progre$$ion more removes us from the de$ired term) if <marg><I>The Unite of all Numbers may mo$t properly be $aid to be Infinite.</I></marg> any number may be $aid to be infinite it is the Unite: and, indeed, there are in it tho$e conditions, and nece$$ary qualities of the Infi- nite Number, I mean, of containing in it as many Squares as Cubes, and as Numbers.</P> <P>SIMP. I do not apprehend very well, how this bu$ine$s $hould be under$tood.</P> <P>SALV. The thing hath no difficulty at all in it, for the Unite is a Square, a Cube, a Squared Square, and all other Powers; nor is there any particular what$oever e$$ential to the Square, or to the Cube, which doth not agree with the Unite; as <I>v. gr.</I> one proper- ty of two Square-numbers is to have between them a Number mean-proportional; take any Square number for one of the terms, and the Unite for the other, and you $hall likewi$e ever find be- tween them a Number Mean-proportional. Let the two Square Numbers be 9 and 4, you $ee that between 9 and 1 the Mean- proportional is 3, and between 4 and 1 the Mean-proportional <foot>is</foot> <p n=>31</p> is 2, and between the two Squares 9 and 4, 6 is the Mean. The property of Cubes is to have nece$$arily between them two Num- bers Mean-proportional. Suppo$e 8, and 27, the Means between them are 12 and 18; and between the Unite and 8 the Means are 2 and 4; betwixt the Unite and 27 there are 3, and 9. We therefore conclude, <I>That there is no other Infinite Number but the Vnite.</I> And the$e be $ome of tho$e Wonders, that $urmount the comprehen$ion of our Imagination, and that advertize us how ex- ceedingly they err, who di$cour$e about Infinites with tho$e very Attributes, that are u$ed about Finites; the Natures of which have no congruity with each other. In which affair I will not conceal from you an admirable accident, that I met with $ome time $ince, explaining the va$t difference, yea, repugnance and contrariety of Nature, that a terminate quantity would incur by changing or pa$- $ing into Infinite. We a$$ign this Right Line A B, of any length at plea$ure, and any point in the $ame, as C being taken, dividing it into two unequal parts: I $ay, that many couples Lines, (hold- ing the $ame proportion between them$elves as have the parts A C, and B C,) departing from the terms A and B to meet with one another; the points of their Inter$ection $hall all fall in the Circumference of one and the $ame Circle: as for example, A L and B L departing [or <I>being drawn</I>] from the Points A and B, and having between them$elves the $ame proportion, as have the parts A C and B C, and concurring in the point L: and the $ame pro- portion being between two others A K, and B K, concurring in K, al$o others as A I, and B I; A H, and B H; A G, and B G; A F, and B F; A E, and B E: I $ay, that the points of their Inter$ecti- on L, K, I, H, G, F, E, do all fall in the Circumference of one and the $ame Semi-circle: $o that we $hould imagine the point C to mve conti- <fig> nuallyafter $uch a $ort, that the Lines produced from it to the fix- ed terms A and B retain alwaies the $ame propor- tion that is be- tween the fir$t parts A C and C B, that point C $hall decribe the Circumference of a Circle, as we $hall $hew you pre$ently. And the Circle in $uch $ort de$cribed $hall be alwaies greater and greater $ucce$$ively, according as the point C is taken nearer to the middle point which is O; and the Circle $hall be le$$er which $hall be de$cribed from a point nearer to the extremity B, in$omuch, that from the <foot>infinite</foot> <p n=>32</p> infinite Points which may be taken in the Line O B, there may be de$cribed Circles (moving them in $uch $ort as above is pre$cri- bed) of any Magnitude; le$$er than the Pupil of the eye of a Flea, and bigger than the Equinoctial of the <I>Primum Mobile.</I> Now, if rai$ing any of the Points comprehended betwixt the terms O and B, from every one we may de$cribe Circles, and va$t ones from the Points nearer to O; then if we rai$e the Point O it $elf, and continue to move it in $uch $ort as afore$aid, that is, that the Lines drawn from it to the terms A and B keep the $ame proporti- on as have the fir$t Lines A O, and O B, what Line $hall be de$cri- bed? There would be de$cribed the Circumference of a Circle, but of a Circle bigger than the bigge$t of all Circles, therefore of a Circle that is infinite: but it doth al$o de$cribe a Right Line, and perpendicular upon A B, erected from the Point O, and produced <I>in infinitum</I> without ever turning to reunite its la$t term with the fir$t, as the others did; for the limited motion of the Point C, after it had de$igned the upper Semi-circle C H E, continued to de- $cribe the Lower E M C, reuniting its extream terms in the point C: But the Point O being moved to de$ign (as all the other Points of the Line A B, for the Points taken in the other part O A $hall de$ign their Circles, and tho$e Points neare$t to O the greate$t) its Circle; to make it the bigge$t of all, and con$e- quently infinite, it can never return any more to its fir$t term, and <marg><I>The difference be- twixt a finite and infinite Circle.</I></marg> in a word de$igneth an Infinite Right-Line for the Circumference of its Infinite Circle. Con$ider now, what difference there is be- tween a finite Circle, and an infinite; $eeing that this in $uch man- ner changeth its being that it wholly lo$eth both its being, and power of being; for we have already well comprehended, that there cannot be a$$igned an infinite Circle; by which we may con$equently know that there can be no infinite Sphære, or other Body, or figured Superficies. Now what $hall we $ay to this Meta- morpho$is in pa$$ing from Finite to Infinite? And why $hould we find greater repugnance, whil$t $eeking Infinity in Numbers, we <marg><I>Vnity participates of Infinity.</I></marg> come to conclude it to be in the Unite? And whil$t that breaking a Solid into many pieces, and pur$uing to reduce it into very $mall powder, it were re$olved into its infinite Atomes, admitting no far- ther divi$ion, why may we not $ay that it is returned into one $ole <I>Continuum,</I> but perhaps fluid, as the Water, or Quick$ilver, or other Metall melted? And do we not $ee Stones liquified into Gla$s, and Gla$s it $elf with much Fire to become more fluid than Water?</P> <P>SAGR. Should we therefore think Fluids to be $o called, be- cau$e they are re$olved into their fir$t, infinite, indivi$ible com- pounding parts?</P> <P>SALV. I know not how to find a better an$wer to re$olve cer- <foot>tain</foot> <p n=>33</p> tain $en$ible appearances, among$t which this is one: When I take a hard Body, be it either Stone, or Metal, and with a Hammer, or very fine File, endeavour to divide it, as much as is po$$ible, into its mo$t minute and impalpable powder; it is very clear, that its lea$t Atomes, albeit for their $malne$s they are imperceptible, one by one, to our $ight and touch; yet are they quantitative, figured, and numerable: and it happens in them, that being accumulated to- gether, they continue in heap; and being laid hollow, or with a pit in the mid$t, the hollowne$s or pit remains, the parts heaped about it not returning to fill it up; and being $tirr'd, or $haken, they $uddenly $ettle $o $oon as their external mover leaves them, And the like effects are $een in all the Aggregates of $mall Bodies, bigger, and bigger, and of any kind of Figure, although Sphærical; as we $ee in heaps of Pea$e, Wheat, Bird $hot, and other matters. But if we try to find the like accidents in Water, you will meet with none of them; but, being rai$ed, it in$tantly returns to a level, if it be not by a ve$$el, or $ome other external $tay upheld; being made hollow, it pre$ently diffu$eth to fill up the Cavity; and be- ing long moved, it continually undulates, and $preads its waves very far. From this, I think, we may very rationally infer, that the minute <marg><I>Fluid Bodies are $uch, for that they are re$olved into their fir$t Indivi$i- ble Atomes.</I></marg> parts of Water, into which it $eemeth to be re$olved, ($ince it hath le$s con$i$tence than any the fine$t powder, yea, hath no con$i- $tence at all) are va$tly differing from Atomes quantitative and divi$ible; nor know I how to find any other difference therein than that of being indivi$ible. Methinks, al$o, that its mo$t exqui- $ite tran$parency, affords us $ufficient grounds to conjecture there- of; for if we take the mo$t diaphanous Chri$tal that is, and begin to break, and pound it to powder, when it is in powder it lo$eth its tran$parency, and $o much the more, the $maller it is pounded; but yet Water which is ground to the highe$t degree, hath al$o the highe$t degree of Diaphaneity Gold and Silver, reduced by <I>Aqua- fortis</I> into a $maller Powder than any File can make, yet they con- tinue powder, and become not fluid; nor do they liquifie till the Indivi$ibles of the Fire, or of the Sun-beams di$$olve them, as, I be- lieve, into their fir$t and highe$t infinite and indivi$ible compoun- ding parts.</P> <P>SAGR. This which you have hinted of the Light I have many times ob$erved with admiration: I have $een, I $ay, a burning- Gla$s, of a foot Diameter, liquifie or melt lead in an in$tant; whence I came to be of opinion, that if the Gla$$es were very big, and very polite, and of Parabolical Figure, they would no le$s melt every other Metal in a very $hort time; $eeing that that, not very big, nor very clear, and of a Sphærical Concave, with $uch force melted Lead, and burnt every combu$tible matter: effects, that make the wonders, reported of the Burning-gla$$es of <I>Archimedes,</I> credible to me.</P> <foot>F SALV.</foot> <p n=>34</p> <marg>Archimedes <I>his Burning — Gla$$es admirable.</I></marg> <P>SALV. Touching the Effects of the Gla$$es, invented by <I>Ar- chimedes,</I> all the Miracles, that $everal Writers record of them, are to me rendred credible by the reading of <I>Archimedes</I> his own Books, which I have with infinite amazement peru$ed and $tudied: and if any doubts had been left me; that which la$t of all Father <marg>Buonaventura Cavalieri, <I>the Je- $uate, a famous Mathematician, and his Book en- titled,</I> Lo Spec- chio U$torio.</marg> <I>Buonaventura Cavalieri</I> hath publi$hed, touching <I>Lo Specehio V$torio,</I> (or the Burning gla$s) and which I have read with ad- miration, is $ufficient to re$olve them all.</P> <P>SAGR. I have al$o $een that Tract, and peru$ed it with much delight and wonder; and becau$e I formerly had knowledge of the Author, I was confirmed in the opinion which I had conceived of him, that he was like to prove one of the principal Mathemati- cians of our Age. But returning to the admirable effects of the Sun- Beams in melting of Metals, are we to believe that $uch, and $o violent an operation is without Motion, or el$e that it is with Motion, but extream $wift?</P> <marg><I>Burnings are per- formed with a mo$t $wift Motion.</I></marg> <P>SALV. We $ee other burnings, and meltings to be performed with Motion, and with a mo$t $wift Motion. Ob$erve the ope- rations of Lightnings, of Powder in Mines, and in Petards, and, in $um, how by quickning the flame of Coles, mixt with gro$s and impure vapours, by Bellows, encrea$eth its force in the melting of Metals: $o that I cannot $ee how the Action of Light, albeit mo$t pure, can be without Motion, and that al$o ve- ry $wift.</P> <P><I>S</I>AGR. But what and how great ought we to judge this Velo- city of the Light? Is it haply <I>In$tantaneous,</I> and done in a moment, or, as the re$t of Motions, performed in Time? May we not by Experiment be a$$ured what it is?</P> <P>SIMP. Quotidian experience $hews the expan$ion of Light to be <I>In$tantaneous</I>; in that beholding a Cannon, let off at a great di$tance, the fla$h of the fire, without interpo$ition of time, is tran$- mitted to our eye, but $o is not the Report to our ear untill a con- $iderable time after.</P> <P>SAGR. True, but, I pray you, what doth this obvious experi- ment evince; but only this, that the Report is longer in arriving at our Ear, than the Fla$h at our Eye; but it a$$ures me not, that the tran$mi$$ion of the Light is therefore <I>In$tantaneous</I> rather than in Time, but only mo$t $wift. Nor doth $uch an ob$ervation con- clude more than that other, of $uch who $ay, that as $oon as the Sun cometh to the Horizon, its Light arriveth at our eye: for who $hall a$$ure me, that its beams arrive not at the $aid term, afore they reach our $ight?</P> <P>SALV. The inconcludency of the$e, and other ob$ervations of the like Nature, made me once think of $ome other way, whereby we may without errour be a$certained whether the illumination, <foot>that</foot> <p n=>35</p> that is, whether the expan$ion of the Light were really <I>In$tantane- ous</I>; $eeing that the very $wift Motion of Sound, a$$ureth us, that that of Light cannot but be extream $wift. And the experiment I <marg><I>The Velocity of Light, how to find by Experiment whether it be In- $tantaneosu or not.</I></marg> hit upon, was this; I would have two per$ons take each of them a Light, which, by holding it in a Lanthorn, or other coverture, they may cover, and di$cover at plea$ure by interpo$ing their hand to the fight of each other; and, that placing them$elvs again$t one another, $ome few paces di$tance, they may practice the $peedy di$covery, and occultation of their Lights from the $ight of each other: So that when one $eeth the others Light, he immediatly di$clo$e his: which corre$pondence, after $ome Re$pon$es mutually made, will become $o exactly In$tantaneous, that, without $en$ible variation, at the di$covery of the one, the other $hall at the $ame time ap- pear to the $ight of him that di$clos'd the fir$t. Having adju$ted this practice at this $mall di$tance, let us place the two per$ons with two $uch Lights at two or three miles di$tance; and by night re- newing the $ame experiment; Let them inten$ely ob$erve if the Re$pon$es of the di$clo$ures, and occultations do follow the $ame tenour which they did near hand: for if they keep the $ame pro- portion, it may be with certainty enough concluded, that the ex- pan$ion of Light is In$tantaneous; but if it $hould require time in a di$tance of three miles, which importeth $ix for the going of one, and return of the other, the $tay would be $ufficiently ob$er- vable. And if this Experiment be made at greater di$tances, namely, at eight or ten miles, we may make u$e of the <I>Tele$cope,</I> the Ob$ervators accommodating each of them one at the places, where by night the Lights are to be ob$erved; which though not very big, and $o not vi$ible, at that great di$tance, to the eye at large; (though ea$ie to be di$clo$ed, and hid) by help of the <I>Tele$copes</I> before admitted, and fixed they may commodiou$ly be di$cerned.</P> <P>SAGR. The Invention $eems to me no le$s certain than ingenu- ous; but tell us what upon experimenting it you concluded.</P> <P><I>S</I>ALV. Really, I have not tryed it, $ave only at a $mall di$tance, namely, le$s than a Mile: whereby I could come to no certainty whether the apparence of the oppo$ite Light was truly In$tantane- ous; But if not In$tantaneous, yet it was of exceeding great Velo- city, and I may $ay Momentary: and for the pre$ent, I would re- $emble it to that Motion which we $ee a fla$h of Lightning make in the Clouds ten or more Miles off: of which Light we di$tin- gui$h the beginning, and, I may fay, the $ource and ri$e of it, in a particular place in tho$e Clouds; but yet its wide expan$ion imme- diatly $ucceeds among$t tho$e adjacent: which to me $eems an ar- gument that it is $ome $mall time in doing; becau$e had the illu- mination been made all at once, and not by degrees, it feems to <foot>F 2 me</foot> <p n=>36</p> me that we could not have di$tingui$hed its original, or rather the Center of its flake, and extream Dilatations. But into what Oceans do we by degrees engage our $elves? Among$t <I>Vacuities, Infinites, Indivi$ibles,</I> and <I>Instantaneous Motions</I>; $o that we $hall not be able by a thou$and Di$cour$es to recover the Shore?</P> <P>SAGR. They are things, indeed, very di$proportionate to our under$tanding. Behold Infinite, $ought among$t Numbers, $eemeth to determine in the Unite: From Indivi$ibles ari$eth things that are continually divi$ible: Vacuity $eems only to re$ide indivi$ibly mixt with Repletion: and, in brief, the$e things $o change the nature of tho$e under$tood by us, that even the Circumference of a Circle becometh an Infinite Right-Line; which, if I well re- member, is that Propo$ition which you, <I>Salviatus,</I> are to mani- fe$t by Geometrical Demon$tration. Therefore, if you think fit, it would be well, without any more digre$$ions, to make it out to us.</P> <P>SALV. I am ready to $erve you in demon$trating the en$uing Problem for your fuller information.</P> <head>PROPOSITION.</head> <P><I>A Right-Line being given, divided, according to any proportion, into unequal parts, to de$cribe a Circle, to the Circumference of which, at any point of the $ame, two Right-Lines being produced from the terms of the given Right Line, they may retain the $ame pro- portion that the parts of the $aid Line given have to one another, $o that tho$e be Homologous which de- part $rom the $ame terms.</I></P> <P>Let the given Right-Line be AB, unequally divided ac- cording to any proportion in the point C; it is required to de$cribe a Circle at any point of who$e Circumference two Right Lines, produced from the terms A and B, concurring, have the $ame proportion to each other, that A C, hath to B C, $o that tho$e be Homologous which depart from the $ame term. Upon the Center C, at the di$tance of the le$$er part C B, let a Circle be $uppo$ed to be de$cribed, to the Circumference of which from the point A the Right-line A D is made a Tangent, and indetermi- nately prolonged towards E: and let the Contact be in D, and draw a Line from C to D, which $hall be perpendicular to A E; and let B E be perpendicular to B A, which produced, $hall inter- <foot>$ect</foot> <p n=>37</p> $ect A E, the Angle A being acute: Let the inter$ection be in E, from whence let fall a Perpendicular to A E, which produced, will meet with A B infinitely prolonged in F. I $ay, fir$t, that the Right-lines F E, and F C are equal: $o that drawing the Line E C, we $hall, in the <fig> two Triangles D E C, B E C, have the two Sides of the one, D E, and C E, equal to the two Sides of the other B E, and E C; the two Sides, D E, and E B, being Tangents to the Circle D B, and the Ba$es D C, and C B, are likewi$e equal: wherefore the two Angles D E C, and B E C, $hall be equal. And becau$e the Angle B C E wanteth of being a Right- Angle, as much as the Angle B E C; and the Angle C E F, to make it a Right-Angle, wants the Angle C E D, tho$e Supple- ments being equal, the Angles F C E, and F E C $hall be equal, and $o con$equently the Sides F E, and F C; wherefore making the point F a Center, and at the di$tance F E, de$cribing a Circle, it $hall pa$s by the point C. De$cribe it, and let it be C E G. I $ay, that this is the Circle required, by any point of the Circumfe- rence of which, any two Lines that $hall inter$ect, departing from the terms A and B, $hall be in proportion to each other, as are the two parts A C, and B C, which be$ore did concur in the point C. This is manife$t in the two that concur or inter$ect in the point E, that is A E, and B E; the Angle E of the Triangle A E B being divided in the mid$t by C E; $o that as A C is to C B, $o is A E to B E. The $ame we prove in the two A G, and B G, determined in the point G. Therefore being (by the Similitude of the Tri- angles A F E, and E F B) that as A F is to E F, $o is E F to F B; that is, as A F is to F C, $o is C F to F B: So by Divi$ion; as A C is to C F, (that is, to F G) $o is C B to B F; and the whole A B is to the whole B G, as the part C B to the part B F: and by Com- po$ition; as A G is to G B, $o is C F to F B; that is, as E F to F B, that is, as A E to E B, and A C to C B: Which was to be de- mon$trated. Again, let any other Point be taken in the Circum- ference, as H; in which the two Lines A H and B H concur. I $ay, in like manner as before, that as A C is to C B, $o is A H to B H. Continue H B untill it inter$ect the Circumference in I, and draw <foot>a</foot> <p n=>38</p> a Line joyning I to F. And becau$e it hath been proved already that as A B is to B G, $o is C B to B F, the Rectangle A B F $hall be equall to the Rectangle C B G, that is I B H: and therefore, as A B is to B H, $o is I B to B F, and the Angles at B are equal: Therefore A H is to H B, as I F, that is E F, to F B, and as A E to E B.</P> <P>I $ay moreover, that it is impo$$ible, that the Lines, which have this $ame proportion, departing from the terms A and B, $hould meet in any point, either within or without the $aid Circle: For- a$much as if it be po$$ible that two Lines $hould concur in the point L, placed without; let them be A L, and B L; and continue L B to the Circumference in M, and conjoyn M to F. If therefore A L is to B L, as A C to B C, that is, as M F to F B, we $hall have two Triangles A L B, and M F B, which about the two Angles A L B and M F B have their Sides proportional, their upper Angles in the point B equal, and the two remaining Angles F M B and L A B le$s than Right Angles (for that the Right-angle at the point M hath for its Ba$e the whole Diameter C G, and not the $ole part B F, and the other at the point A is acute by rea$on the Line A L Homologous to A C, is greater than B L Homologous to B C) Therefore the Triangles A B L, and M B F are like: and therefore as A B is to B L, $o is M B to B F; Wherefore the Rectangle A B F $hall be equall to the Rectangle M B L. But the Rectangle A B F hath been demon$trated to be equal to that of C B G: Therefore the Rectangle M B L is equal to the Rectangle C B G, which is impo$$ible: Therefore the Concour$e of the Lines cannot fall without the Circle. And in like manner it may be de- mon$trated that it cannot fall within; Therefore all the Concour- $es fall in the Circumference it $elf.</P> <P>But it is time that we return to give $atisfaction to the Intreaty of <I>Simplicius,</I> $hewing him that the re$olving the Line into its in- finite Points is not only not impo$$ible, but that it hath in it no more difficulty than to di$tingui$h its quantitative parts; pre$up- po$ing one thing (notwith$tanding) which I think, <I>Simplicius,</I> you will not deny me, and that is this; that you will not require me to $ever the Points one from another, and $hew you them one by one di$tinctly upon this paper: for I my $elfe $hould be content, if without enjoyning to pull the four or $ix parts of a Line from one another, you $hould but $hew me its divi$ions marked, or at mo$t inclined to Angles, framing them into a Square, or a Hexa- gon; therefore I per$wade my $elf, that for the pre$ent you will grant them then $ufficiently, and actually di$tingui$hed.</P> <P>SIMP. I $hall indeed.</P> <P>SALV. Now if the inclining of a Line to Angles, framing <marg><I>How infinite points are a$$igned in a finite Line.</I></marg> therewith $ometimes a Square $ometimes an Octagon, $ometimes <foot>a</foot> <p n=>39</p> a Poligon of Forty, of an <I>H</I>undred, of a Thou$and Angles be a mutation $ufficient to reduce into Act tho$e four, eight, forty, hundred, or thou$and parts, which were, as you $ay, Potentially in the $aid Line at fir$t: if I make thereof a Poligon of infinite Sides, namely, when I bend it into the Circumference of a Circle, may not I, with the like leave, $ay, that I have reduced tho$e infi- nite parts into Act, which you before, whil$t it was $traight, $aid were Potentially contained in it? Nor may $uch a Re$olution be denied to be made into its Infinite Points, as well as that of its four parts in forming thereof a Square, or into its thou$and parts in forming thereof a Mill-angular Figure; by rea$on that there wants not in it any of the Conditions found in the Poligon of a thou- $and, or of an hundred thou$and Sides. This applied or layed to a Right-Line covereth it, touching it with one of its Sides, that is, with one of its hundred thou$andth parts; the Circle, which is a Poligon of infinite Sides, toucheth the $aid Right-line with one of its Sides, that is one $ingle Point divers from all its Colaterals, and therefore divided, and di$tinct from them, no le$s than a Side of the Poligon from its Conterminals. And as the Poligon turned round upon a Plane de$cribes, with the con$equent tacts of its Sides, a Right-line equal to its Perimeter: $o the Circle, rowled upon $uch a Plane, de$cribes or $tamps upon it, by its infinite $ucce$$ive Contacts, a Right-line, equall to its own Circumference. I know not at pre$ent, <I>Simplicius,</I> whether or no the Peripateticks, (to whom I grant, as a thing mo$t certain, that <I>Continuum</I> may be di- vided into parts alwaies divi$ible, $o that continuing the divi$ion and $ubdivi$ion there can be no end thereof) will be content to yield to me, that none of tho$e divi$ions are the ultimate, as in- deed they be not, $ince that there alwaies remains another; but that only to be the la$t, which re$olves it into infinite Indivi$ibles; to which I yield we can never attain, dividing and $ubdividing it $ucce$$ively into a greater, and greater multitude of parts: but making u$e of the way which I propound to di$tingui$h and re- $olve all the infinite parts at one only draught, (an Artifice which ought not to be denied me) I could per$wade my $elf they would $atisfie them$elves, and admit this compo$ition of <I>Continu-</I> <marg>Continuum <I>com- pounded of Indivi- $ibles.</I></marg> <I>um</I> to con$i$t of Atomes ab$olutely indivi$ible: And e$pecially, this one path being more current than any other to extricate us out of very intricate Laberinths; $uch as are, (be$ides that alrea- dy touched of the Coherence of the parts of Solids) the concei- ving the bu$ine$s of Rarefaction and Conden$ation, without running into the inconvenience of being forced to admit forth of void Spaces or Vacuities; and for this a Penetration of Bodies: in- conveniences, which both, in my opinion, may ea$ily be avoided, by granting the fore$aid Compo$ition of Indivi$ibles.</P> <foot>SIMP.</foot> <p n=>40</p> <P>SIMP. I know not what the Peripateticks would $ay, in regard that the Con$iderations you have propo$ed would be, for the mo$t part, new unto them, and as $uch, it is requi$ite that they be exa- mined: and it may be, that they would find you an$wers, and powerful Solutions, to unty the$e knots, which I, by rea$on of the want of time and ingenuity proportionate, cannot for the pre$ent re$olve. Therefore, $u$pending this particular for this time, I would gladly under$tand how the introduction of the$e Indivi$i- bles facilitateth the knowledge of Conden$ation, and Rarefa- ction, avoiding at the $ame time a <I>Vacuum,</I> and the Penetration of Bodies.</P> <P>SAGR. I al$o much long to under$tand the $ame, it being to my Capacity $o ob$cure: with this <I>provi$o,</I> that I be not couzen- ed of hearing (as <I>Simplicius</I> $aid but even now) the Rea$ons of <I>Ari$totle</I> in confutation of a <I>Vacuum,</I> and con$equently the Solu- tions which you bring, as ought to be done, whil$t that you ad- mit what he denieth.</P> <P>SALV. I will do both the one and the other. And as to the fir$t it's nece$$ary, that like as in favour of Rarefaction, we make u$e of the Line de$cribed by the le$$er Circle bigger than its own Cir- cumference, whil$t it was moved at the Revolution of the greater; $o, for the under$tanding of Conden$ation, we $hall $hew, how that, at the conver$ion made by the le$$er Circle, the greater de$cribeth a Right-line le$s than its Circumference; for the clearer explicati- on of which, let us $et before us the con$ideration of that which befalls in the Poligons. In a de$cription like to that other; $up- po$e two Hexagons about the common Center L, which let be A B C, and H I K, with the Parallel-lines H O M, and A B C, up- on which they are to make their Revolutions; and the Angle I, of the le$$er Poligon, re$ting at a $tay, turn the $aid Poligon till $uch time as I K fall upon the Parallel, in which motion the point K $hall de$cribe the Arch K M, and the Side K I, $hall unite with the part I M; while this is in doing, you mu$t ob$erve what the Side C B of the greater Poligon will do. And becau$e the Revolution is made upon the Point I, the Line I B with its term B $hall de- $cribe, turning backward the Arch B b, below the Parallel c A, $o that when the Side K I $hall fall upon the Line M I, the Line B C $hall fall upon the Line b c, advancing forwards only $o much as is the Line B c, and retiring back the part $ubtended by the Arch B b, which falls upon the Line B A, and intending to continue af- ter the $ame manner the Revolution of the le$$er Poligon, this will de$cribe, and pa$s upon its Parallel, a Line equal to its Perimeter; but the greater $hall pa$s a Line le$s than its Perimeter, the quan- tity of $o many of the lines <I>B</I> b as it hath Sides, wanting one; and that $ame line $hall be very near equal to that de$cribed by <foot>the</foot> <p n=>41</p> the le$$er Poligon, exceeding it only the quantity of b B. Here then, without the lea$t repugnance the cau$e is $een, why the grea- ter Poligon pa$$eth or moveth not (being carried by the le$s) with its Sides a greater Line than that pa$$ed by the le$s; that is, becau$e that one part of each of them falleth upon its next coter- minal and precedent.</P> <P>But if we $hould con$ider the two Circles about the Center A, re$ting upon their Parallels, the le$$er touching his in the point B, and the greater his in the <fig> point C; here, in begin- ning to make the Revolu- tion of the le$s, it $hall not occur as before, that the point B re$t for $ome time immoveable, $o that the Line B C giving back, carry with it the point C, as it befell in the Poligons, which re$ting fixed in the point I till that the Side K I falling upon the Line I M, the Line I B carried back B, the term of the Side C B, as far as b, by which means the Side B C fell on b c, $uper-po$ing or re$ting the part B b upon the Line B A, and advancing forwards only the part <I>B</I> c, equal to I M, that is to one Side of the le$$er Poligon: by which $uperpo$i- tions, which are the exce$$es of the greater Sides above the le$s, the advancements which remain equal to the Sides of the le$$er Poli- gon come to compo$e in the whole Revolution the Right-line equal to that traced, and mea$ured by the le$$er Poligon. But <marg><I>A Circle is a Poli- gon of infinite in- divi$ible quantita- tive Sides.</I></marg> now, I $ay, that if we would apply this $ame di$cour$e to the ef- fect of the Circles, it will be requi$ite to confe$s, that whereas the Sides of what$oever Poligon are comprehended by $ome Number, the Sides of the Circle are infinite; tho$e are quantitative and di- vi$ible, the$e non-quantitative and Indivi$ible: the terms of the Sides of a Poligon in the Revolution $tand $till for $ome time, that is, each $uch part of the time of an entire Conver$ion, as it is of the whole Perimeter: in the Circles likewi$e the $tay o$ the terms <marg><I>An In$tant or Mo- ment of quantita- tive Time, is the $ame as a Point of a quantitative Line.</I></marg> of its infinite Sides are momentary, for a Moment is $uch part of a limited Time, as a Point is of a Line, which containeth infinite of them; the regre$$ions made by the Sides of the greater Poligon, are not of the whole Side, but only of its exce$s above the Side of the <foot>G le$$er</foot> <p n=>42</p> le$$er, getting forwards as much $pace as the $aid le$$er Side: in Circles, the Point, or Side C in the in$tantaneous re$t of B recedeth as much as is its exce$s above the Side B, advancing forward as much as the quantity of the $ame B: And in $hort, the infinite indivi$ible Sides of the greater Circle with their infinite indivi$ible Regre$$ions, made in the infinite in$tantaneous $taies of the infi- nite terms of the infinite Sides of the le$$er Circle, and with their infinite Progre$$es, equal to the infinite Sides of the $aid le$$er Circle, they compo$e and mea$ure a Line equall to that de$cribed by the le$$er Circle, containing in it $elf infinite $uperpo$itious non-quantitative, which make a Con$tipation and Conden$ation without any penctration of quantitative parts: which cannot be contrived to be done in the Line divided into quantitative parts, as is the Perimeter of any Poligon, which being di$tended in a Right-line at length, cannot be reduced to a le$$er length, unle$s the Sides fall upon and Penetrate one the other. This Con$tipati- on of parts non-quantitative, but infinite without Penetration of quantitative parts, and the former Di$traction above declared of <marg><I>Rarefaction is the di$traction of infi- nite Indivi$ibles by the interpo$ition of infinite indivi$i- ble Vaeuities.</I></marg> infinite Indivi$ibles by the interpo$ition of indivi$ible Vacui- ties, I believe, is the mo$t that can be $aid for the Conden$ation and Rarefaction of Bodies, without being driven to introduce Pe- netration of Bodies, or quantitative Void Spaces. If there be any thing therein that plea$eth you, make u$e of it, if not, account it <marg><I>Conden$ation, ac- cording to the ope- ration of the Au- thor, proceeds from the Con$tipation of quantitative and indivi$ible parts.</I></marg> vain, and my di$cour$e al$o; and $eek out $ome other explanation that may better $atisfie your Judgment. Only the$e two words by the way, let us remember that we are among$t Infinites, and In- divi$ibles.</P> <P>SAGR. That the Conceit is ingenious, and to my eares wholly new, and $trange, I freely confe$s, but whether or no Nature pro- ceed in this order, I know not how to re$olve; Truth is, that till $uch time as I hear $omething that may better $atisfie me, that I may not $tand $ilent, I will adhere to this. But haply <I>Simplicius</I> may have $omwhat, which I have not yet met with, to explicate the explication, which is produced by Philo$ophers in $o ab$truce a matter; for, indeed, what I have hitherto read about Conden$a- tion, is to me $o den$e, and that of Rarefaction $o $ubtill, that my weak $ight neither penetrates the one, nor comprehends the other.</P> <P>SIMP. I am full of confu$ion, and find great Rubbs in the one path, and in the other, and more particularly in this new one: for according to this Rule, an Ounce of Gold might be rarefied and drawn forth into a Ma$s bigger than the whole Earth, and the whole Earth conden$ed and reduced into a le$s Ma$s than a Nut; which I neither believe, nor think that you your $elf do believe: and the Con$iderations and Demon$trations by you hitherto de- <foot>livered,</foot> <p n=>43</p> livered, as they are things Mathematical, ab$tract and $eparate from Sen$ible Matter, I believe, that when they come to be apply- ed to Matters Phy$ical and Natural, they will not exactly comply with the$e Rules.</P> <P>SALV. It is not in my power, nor, as I believe, do you de$ire, that I $hould make that vi$ible which is invi$ible; but as to $uch things as may be comprehended by our Sen$es, in regard that you <marg><I>Gold in the gilding of Silver is drawn forth and di$gro$- $ed immen$ly.</I></marg> have in$tanced in Gold, do we not $ee an immen$e exten$ion to be made of its parts? I know not whether you may have $een the Method that Wyer-drawers ob$erve in di$gro$$ing Gold Wyer: which in reality is not Gold, $ave only in the Superficies, for the internal $ub$tance is Silver; and the way of di$gro$$ing it is this. They take a Cylinder, or, if you will, Ingot of Silver, about half a yard long, and about three or four Inches thick, and this they <marg>* Or Thumb- breadths.</marg> gild or over-lay with Leaves of beaten Gold, which, you know, is $o thin that the Wind will blow it to and again, and of the$e Leaves they lay on eight or ten, and no more. So $oon as it is gilded, they begin to draw it forth with extraordinary force, ma- king it to pa$s thorow the hole of the Drawing Iron, and then reiterate this forceable di$gro$sment again and again thorow holes $ucce$$ively narrower, $o that, after $everal of the$e di$gro$ments, they bring it to the $malne$s of the hair of a womans head, if not $maller, and yet it $till continueth gilded in its Superficies or out- $ide: Now I leave you to con$ider to what a finene$s and di$ten$i- on the $ub$tance of the Gold is brought.</P> <P>SIMP. I do not $ee how it can be inferred from this Experi- ment, that there may be a di$gro$ment of the matter of the Gold $ufficient to effect tho$e wonders which you $peak of: Fir$t, For that the fir$t gilding was with ten Leaves of Gold, which make a con$iderable thickne$s: Secondly, howbeit in the exten$ion and di$gro$ment that Silver encrea$eth in length, it yet withall dimi- ni$heth $o much in thickne$s, that compen$ating the one dimen$i- on with the other, the Superficies doth not $o enlarge, as that for overlaying the Silver with Gold, the $aid Gold need to be reduced to a greater thinne$s than that of its fir$t Leaves.</P> <P>SALV. You much deceive your $elf, <I>Simplicius,</I> for the en- crea$e of the Superficies is Subduple to the exten$ion in length, as I could Geometrically demon$trate to you.</P> <P>SAGR. I be$eech you, both in the behalf of my $elf, and of <I>Simplicius,</I> to favour us with that Demon$tration, if $o be you think that we can comprehend it.</P> <P>SALV. I will $ee whether I can, thus upon the $udden, recall it to mind. It is already manife$t, that that $ame fir$t gro$s Cylin- der of Silver, and the Wyer di$tended to $o great a length are two equal Cylinders, in regard that they are the $ame Silver; $o that <foot>G 2 if</foot> <p n=>44</p> if I $hall $hew you what proportion the Superficies of equall Cy- linders have to one another, we $hall obtain our de$ire. I $ay there- fore, that</P> <head>PROPOSITION.</head> <P><I>The Superficies of Equal Cylinders, their Ba$es being $ub$tracted, are to one another in $ubduple proportion of their lengths.</I></P> <P>Take two equall Cylinders, the heights of which let be A B, and C D: and let the Line E be a Mean-proportional between them. I $ay, the Superficies of the Cylinder A B, the Ba$es $ub$tracted, hath the $ame proportion to the Superficies of the Cylinder C D, the Ba$es in like manner $ub$tracted, as the Line A B hath to the Line E, which is $ubduple of the proportion of A B to C D. Cut the part of the Cylinder A B in F, and let the height A F be equal to C D: And becau$e the Ba$es of equal Cy- linders an$wer Reciprocally to their heights, the Circle, Ba$e of the Cylinder C D, to the Circle, Ba$e of the <fig> Cylinder A B, $hall be as the height B A to D C: And becau$e Circles are to one ano- ther as the Squares of their Diameters, the $aid Squares $hall have the $ame proportion, that B A hath to C D: But as B A, is to C D, $o is the Square B A to the Square of E: Therefore tho$e four Squares are Pro- portionals: And therefore their Sides $hall be Proportionals. And as the Line A B is to E, $o is the Diameter of the Circle C to the Diameter of the Circle A: But as are the Diameters, $o are the Circumferences; and as are the Circumferences, $o likewi$e are the Superficies of Cylin- ders equal in Height. Therefore as the Line A B is to E, $o is the Superficies of the Cylinder C D to the Superficies of the Cylinder A F. Becau$e therefore the height A F to the height A B, is as the Superficies A F to the Superficies A B: And as is the height A B to the Line E, $o is the Superficies C D to the Superficies A F: Therefore by Perturbation of Proportion as the height A F is to E, $o is the Superficies C D to the Superficies A B: And, by Con- ver$ion, as the Superficies of the Cylinder A B is to the Superficies of the Cylinder C D, $o is the Line E to the Line A F; that is, to the Line C D: or as A B to E: Which is in $ubduple proportion of A B to C D: Which is that which was to be proved.</P> <foot>Now</foot> <p n=>45</p> <P>Now if we apply this, that hath been demon$trated, to our purpo$e; pre$uppo$ing that that $ame Cylinder of Silver, that was gilded whil$t it was no more than half a yard long, and four or five Inches thick, being di$gro$$ed to the $inene$s of an hair, is prolon- ged unto the exten$ion of twenty thou$and yards (for its length would be much greater) we $hall find its Superficies augmented to two hundred times its former greatne$s: and con$equently, tho$e Leaves of Gold, which were laid on ten in number, being di$ten- ded on a Superficies two hundred times bigger, a$$ure us that the Gold which covereth the Superficies of the $o many yards of Wyer is left of no greater thickne$s than the twentieth part of a Leaf of ordinary Beaten-Gold. Con$ider, now, how great its thinne$s is, and whether it is po$$ible to imagine it done without an immen$e di- $tention of its parts: and whether this $eem to you an Experi- ment, that tendeth likewi$e towards a compo$ition of infinite In- divi$ibles in Phy$ical Matters: Howbeit there want not other more $trong and nece$$ary proofs of the $ame.</P> <P>SAGR. The Demon$tration $eemeth to me $o ingenuous, that although it $hould not be of force enough to prove that fir$t intent for which it was produced, (and yet, in my opinion, it plainly makes it out) yet neverthele$s that $hort $pace of time was well $pent which hath been employed in hearing of it.</P> <P>SALV. In regard I $ee, that you are $o well plea$ed with the$e Geometrical Demon$trations, which bring with them certain pro. fit, I will give you the fellow to this, which $atisfieth to a very cu- rious Que$tion. In the former we have that which hapneth in Cylinders that are equall, but of different heights or lengths: it will be convenient, that you al$o hear that which occurreth in Cy- linders equal in Superficies, but unequal in heights; my meaning alwaies is, in tho$e Superficies only that encompa$s them about, that is, not comprehending the two Ba$es $uperiour and inferiour. I $ay, therefore, that</P> <head>PROPOSITION.</head> <P><I>Upon Cylinders, the Superficies of which the Ba$es be- ing $ub$tracted are equal, have the $ame proportion to one another as their heights Reciprocally taken.</I></P> <P>Let the Superficies of the two Cylinders A E and C F be equall; but the height of this C D greater than the height of the other A B. I $ay, that the Cylinder A E hath the $ame proportion to the Cylinder C F, that the height C D hath to A B. Becau$e therefore the Superficies C F is equall to the <foot>Superficies</foot> <p n=>46</p> $uperficies A E, the Cylinder C F $hall be le$$e than A E: For if they were equal, its Superficies, by the la$t Propo$ition would be greater than the Superficies A E, and <fig> much the more, if the $aid Cylinder C F were greater than A E. Let the Cylinder I D be $uppo$ed equal to A E: There- fore, by the precedent Propo$ition, the Superficies of the Cylinder I D $hall be to the Superficies A E, as the height I F to the Mean-proportional betwixt I F & A B. But the Superficies A E being by Suppo$ition equal to C F and I D, ha- ving the $ame proportion to C F that the height I F hath to C D: Therefore C D is the Mean-Proportional between I F and A B. Moreover, the Cylinder I D being equal to the Cylinder A E, they $hall both have the $ame proporti- on to the Cylinder C F: But I D is to C F, as the height I F is to C D: Therefore the Cylinder A E $hall have the $ame proportion to the Cylinder C F, that the line I F hath to C D; that is, that C D hath to A B: Which was to be demon$trated.</P> <marg><I>Of Corn-$acks with a Board at the Bottom, made of the $ame Stuffe, but different in height, which are the more capa- cious.</I></marg> <P>From hence is collected the Cau$e of an Accident, which the Vulgar do not hearken to without admiration; and it is, how it is po$$ible that the $ame piece of ^{*}Cloth, being longer one way than another, if a Sack be made thereof to hold Corn, as the u$ual manner is, with a Board at the bottom, will hold more, making u$e of the le$$er breadth of the Cloth, for the height of the Sack, <marg>* Or Sacking.</marg> and with the other encompa$$ing the Board at the bottom, than if it be made up the other way: As if for Example, the Cloth were one way $ix foot, and the other way twelve, it will hold more, when with the length of twelve one encompa$$eth the Board at the bottom, the Sack being $ix foot high, than if it encompa$$ed a bottom of $ix foot, having twelve for its height. Now, by what hath been demon$trated, there is added to the Knowledge in ge- neral that it holds more that way than this, the Specifick, and particular Knowledge of how much it holdeth more: which is, That it will hold more in proportion as it is lower, and le$$er, as it is higher. And thus in the mea$ures afore taken, the Cloth be- ing twice as long as broad, when it is $ewed the length-ways it will hold but half $o much, as it will do the other way. And likewi$e <marg>* Bugnola, any Ve$$el made of Rushes or Wick- er.</marg> having a Mat to make a ^{*} Frale or Basket twenty five foot long, and $uppo$e $even broad; made up the long-way it will hold but onely $even of tho$e mea$ures, whereof the other way it will hold five and twenty.</P> <foot>SAGR.</foot> <p n=>47</p> <P><I>S</I>AGR. And thus to our particular content we continually di$- cover new Notions of great Curio$ity, and not unaccompanyed with Utility. But in the particular glanced at but even now, I really believe, that among$t $uch as are altogether void of the knowledge of Geometry, there would not be found one in twen- ty, but at the fir$t da$h would not be mi$taken, and wonder that tho$e Bodies that are contained within equal Superficies, $hould not likewi$e be in every re$pect equal; like as they run in- to the $ame errour, $peaking of the Superficies, when for deter- mining, as it frequently falls out, of the amplene$$e of $everal Cities, they think they have obtained their de$ire $o $oon as they know the $pace of their Circuits, not knowing that one Circuit may be equal to another, and yet the place conteined by this much larger than the place of that: which befalleth not onely in irregular Superficies, but in the regular; among$t which tho$e of more Sides are alwayes more capacious than tho$e of fewer; $o that in fine, the Circle, as being a Poligon of infinite Sides, is more capacious than all other Poligons of equal Perimeter; of which I remember, that I with particular delight $aw the Demon- $tration on a time when I $tudied the Sphere of <I>Sacrobo$co,</I> with a very learned Commentary upon the $ame.</P> <P>SALV. It is mo$t certain; and I having likewi$e light upon that very place, it gave me occa$ion to inve$tigate, how it may with one $ole Demon$tration be concluded, that the Circle is greater than all the re$t of regular I$operemitral Figures, and of others, tho$e of more Sides bigger than tho$e of fewer.</P> <P>SAGR. And I that take great plea$ure in certain $elect and no- wi$e-trivial Demon$trations, entreat you with all importunity to make me a partaker therein.</P> <P>SALV. I $hall di$patch the $ame in few words, demon$trating the following Theorem, namely;</P> <foot>PRO-</foot> <p n=>48</p> <head>PROPOSITION.</head> <P><I>The Circle is a Mean-Proportional betwixt any two Regular Homogeneal Poligons, one of which is cir- cum$cribed about it, and the other I$operimetral to it: Moreover, it being le$$e than all the circum$cri- bed, it is, on the contrary, bigger than all the I$operi- metral. And, again of the circum$cribed, tho$e that have more angles are le$$er than tho$e that have fewer; and on the other $ide of the I$operimetral, tho$e of more angles are bigger.</I></P> <P>Of the two like Poligons A and B, let A be circum$cribed about the Circle A, and let the other B, be I$operime- tral to the $aid Circle: I $ay, that the Circle is the Mean- proportional betwixt them. For that (having drawn the Semidi- ameter A C) the Circle being equal to that Right-angled Trian- gle, of who$e Sides including the Right angle, the one is equal <fig> to the Semidiameter A C, and the other to the Circumference: And likewi$e the Poligon A being equal to the right angled Tri- angle, that about the right angle hath one of its Sides equal to the $aid right line A C, and the other to the Perimeter of the $aid Poligon: It is manife$t, that the circum$cribed Poligon hath the $ame proportion to the Circle, that its Perimeter hath to the Cir- cumference of the $aid Circle; that is, to the Perimeter of the Poligon B, which is $uppo$ed equal to the $aid Circumference: But the Poligon A hath a proportion to the Poligon B, double to that of its Perimeter, to the Perimeter of B (they being like Fi- gures:) Therefore the Circle A is the Mean-proportional be- tween the two Poligons A and B. And the Poligon A being bigger than the Circle A, it is manife$t that the $aid Circle A is bigger than the Poligon B, its I$operimetral, and con$e- quently the greate$t of all Regular Poligons that are I$operimetral <foot>to</foot> <p n=>49</p> to it. As to the other particular, that is to prove, that of the Poligons circum$cribed about the $ame Circle, that of fewer Sides is bigger than that of more Sides; but that, on the contrary, of the I$operimetral Poligons, that of more Sides is bigger than that of fewer Sides, we will thus demon$trate. In the Circle who$e Center is O, and Semidiameter O A, let there be a Tangent A D, and in it let it be $uppo$ed, for example, that A D is the half of the Side of the Pentagon circum$cribed, and A C the half of the Side of the Heptagon, and draw the right lines O G C, and O F D; and on the Center O, at the di$tance O C, draw the Arch E C I: And becau$e the Triangle D O C is greater than the Sector E O C, and the Sector C O I greater than the Triangle C O A; the Triangle D O C $hall have greater proportion to the Triangle C O A, than the Sector E O C, to the Secant C O I, that is, than the Secant F O G to the Secant G O A. And, by Compo$ition, Permutation of Proportion, the Triangle D O A $hall have greater proportion to the Secant F O A, than the Tri- angle C O A to the Secant G O A: And ten Triangles D O A $hall have greater proportion to ten Secants F O A, than four- teen Triangles C O A to fourteen Sectors G O A: That is the circum$cribed Pentagon $hall have greater proportion to the Cir- cle, than hath the Heptagon: And therefore the Pentagon $hall be greater than the Heptagon. Let us now $uppo$e an Hep- tagon and a Pentagon I$operimetral to the $ame Circle. I $ay, that the Heptagon is bigger than the Pentagon. For that the $aid Cir- cle being the Mean proportional between the Pentagon circum- $cribed and the Pentagon its I$operimetral, and likewi$e the Mean between the Circum$cribed and I$operimetral Heptagon: It ha- ving been proved that the Circum$cribed Pentagon is greater then the Circum$cribed Heptagon, the $aid Pentagon $hall have greater proportion to the Circle, than the Heptagon: that is, the Circle $hall have greater proportion to its I$operimetral Pentagon, than to its I$operimetral Heptagon: Therefore the Pentagon is le$$er than the I$operimetral Heptagon. Which was to be demon- $trated</P> <P>SAGR. A mo$t ingenious Demon$tration, and very acute. But whither are we run to ingulph our $elves in Geometry, when as we were about to con$ider the Difficulties propo$ed by <I>Simpli- cius,</I> which indeed are very con$iderable, and in particular, that of Conden$ation, is in my opinion, very ab$truce.</P> <P>SALV. If Conden$ation and Rarefaction are oppo$ite Motions, where there is $een an immen$e Rarefaction, one cannot deny an extraordinary Conden$ation: but immen$e Rarefactions, and, which encrea$eth the wonder, almo$t Momentary, we $ee every day: for what a boundle$$e Rarefaction is that of a little quan- <foot>H tity</foot> <p n=>50</p> <marg><I>Rarefaction im- min$e is that of a little Gunpow- der into a va$t ma$s of Fire.</I></marg> tity of Gunpowder re$olved into a va$t ma$$e of Fire? And what, beyond this, the (I could almo$t $ay) indeterminate Expan$ion of its Light? And if that Fire and this Light $hould reunite toge- ther, which yet is no impo$$ibility, in regard, that at the fir$t they lay in that little room, what a Conden$ation would this be? If you $tudy for them, you will find hundreds of $uch Rarefacti- ons, which are much more readily ob$erved, than Conden$ati- ons: for Den$e matters are more tractable, and $ubject to our Sen$es. For we can ea$ily order Wood at plea$ure, and we $ee it re$olved into Fire, and into Light, but we do not in the $ame manner $ee the Fire and the Light Conden$e to the making of Wood: We $ee Fruits, Flowers, and many other $olid matters re$olved in a great mea$ure into Odors, but we do not after the $ame manner $ee the odoriferous Atomes concurre to the con$titu- tion of the Oderate Solids; but where Sen$ible Ob$ervation is wanting, we are to $upply it with Rea$on, which will $uffice to make us apprehen$ive, no le$$e of the Motion to the Rarefaction and re$olution of Solids, than, to the Conden$ation of rare and mo$t tenuous Sub$tances. Moreover, we que$tion how to effect the Conden$ation and Rarefaction of the Bodies which may be rarefied and conden$ed, $tudying in what manner it may be done without introducing of a <I>Vacuum,</I> and Penetration of Bodies; which doth not hinder, but that in Nature there may be matters which admit no $uch accidents, and con$equently do not allow roome for tho$e things which you phra$e inconvenient and im- po$$ible. And la$tly, <I>Simplicius,</I> I have on the the $core of $atis- fying you, and tho$e Philo$ophers that hold with you, taken $ome pains in con$idering how Conden$ation and Rarefaction may be under$tood to be performed without admitting Penetra- tion of Bodies, and introducing the Void Spaces called Vacuities, Effects which you deny and abhorre: for if you would but grant them, I would no longer $o re$olutely contradict you. There- fore either admit the$e Inconveniences, or accept of my Spe- culations, or el$e finde out others more conducing to the purpo$e.</P> <P>SAGR. As to the denying of Penetration, I am wholly of opi- nion with the Peripatetick Philo$ophers; as to that of a <I>Vacuum,</I> I would $ee the Demon$tration of <I>Ari$totle</I> thorowly examined, wherewith he oppo$eth the $ame, and what you, <I>Salviatus,</I> will an$wer to it. <I>Simplicius</I> $hall do me the favour punctually to recite the proof of the Philo$opher; and you, <I>Salviatus,</I> to an- $wer it.</P> <P>SIMP. <I>Ari$totle,</I> as neer as I can remember, breaks out again$t certain of the Ancients, who introduced Vacuity, as nece$$ary to Motion, $aying, that this without that could not be effected; <foot>to</foot> <p n=>51</p> to this <I>Ari$totle</I> making oppo$ition, demon$trateth, that on the contrary, the effecting of Motion (as we $ee) de$troyeth the Po$iti- on of <I>Vacuum</I>; and his method therein is this. He maketh two Suppo$itions, one is touching Moveables different in Gravity moved in the $ame <I>Medium:</I> the other is concerning the $ame Moveable moved in $everal <I>Medium's.</I> As to the fir$t, he $uppo- $eth that Moveables different in Gravity, move in the $ame <I>Medium</I> with unequal Velocities, which bear to each other the $ame proportion as their Gravities: $o that, for example, a Move- able ten times heavier than another, moveth ten times more $wift- ly. In the other Po$ition he a$$umes, that the Velocity of the $ame Moveable in different <I>Medium's</I> are in Reciprocal to that of the thickne$$e or Den$ity of the $aid <I>Medium's</I>: $o that $uppo- $ing <I>v. gr.</I> that the Cra$$itude of the Water was ten times as great as that of the Air, he will have the Velocity in the Air to be ten times more than the Velocity in the Water. And from this $e- cond A$$umption he draweth his Demon$tration in this manner. Becau$e the tenuity of <I>Vacuum</I> infinitely $urpa$$eth the corpu- lence, though never $o $ubtil, of any whatever Replete <I>Medi- um,</I> every Moveable that in the Replete <I>Medium</I> moveth a cer- tain $pace in a certain time, in a <I>Vacuum</I> would pa$$e the $ame in an in$tant: But to make a Motion in an in$tant is impo$$ible: Therefore to introduce Vacuity in the accompt of Motion is im- po$$ible.</P> <P>SALV. The Argument one may $ee to be <I>ad hominem,</I> that is, <marg>Ari$totle's <I>Argu- ment again$t a</I> Vacuum <I>is</I> ad hominem.</marg> again$t tho$e who would make a <I>Vacuum</I> nece$$ary to Motion; but if I $hall admit of the Argument as concludent, granting withal, that in Vacuity there would be no Motion; yet the Po$i- tion of Vacuity taken ab$olutely, and not in relation to Motion, is not thereby overthrown. But to tell you what tho$e Ancients, peradventure, might an$wer, that $o we may the better di$cover how far the Demon$tration of <I>Ari$totle</I> holds good, methinks that one might oppo$e his A$$umptions, denying them both. And as to the fir$t: I greatly doubt that <I>Ari$totle</I> never experimented how true it is, that two $tones, one ten times heavier than the o- ther, let fall in the $ame in$tant from an height, <I>v. gr.</I> of an hun- dred yards, were $o different in their Velocity, that upon the arrival of the greater to the ground, the other was found not to have de$cended $o much as ten yards.</P> <P>SIMP. Why, it may be $een by his own words, that he confe$- $eth he had made the Experiment, for he $aith, [<I>We $ee the more grave</I>] now that <I>Seeing</I> implieth that he had tried the Experi- ment.</P> <P>SAGR. But I, <I>Simplicius,</I> that have made proof thereof, do a$- $ure you, that a Cannon bullet that weigheth one hundred, rwo <foot>H 2 hun-</foot> <p n=>52</p> hundred, and more pounds, will not one Palme anticipate the ar- rival of a Musket-bullet to the ground, that weigheth but half a pound, falling likewi$e from an height of two hundred yards.</P> <P><I>S</I>ALV. But without any other Experiments, we may by $hort and nece$$ary Demon$trations cleerly prove, that it is not true that a Moveable more grave moveth more $wiftly than another le$$e grave, confining our meaning $till to Moveables of the $ame Mat- ter; and, in $hort, to tho$e of which <I>Ari$totle</I> $peaketh. For tell me, <I>Simplicius</I> whether you admit, that to every cadent grave Body there belongeth by nature one determinate Velocity; $o as that it cannot be encrea$ed or dimini$hed in it without u$ing vi- olence to it, or impo$ing $ome impediment upon it?</P> <P>SIMP. It cannot be doubted, but that the $ame Moveable in the $ame <I>Medium</I> hath one e$tabli$hed and by-nature-determinate Velocity, which cannot be increa$ed, unle$$e with new <I>Impetus</I> conferred on it, or dimini$hed, $ave onely by $ome impediment that retards it.</P> <P>SALV. If therefore we had two Moveables, the natural Velo- cities of which were unequal, it is manife$t, that if we joyned the $lower with the $wifter, this would be in part retarded by the $lower, and that in part accelerated by the other more $wift. Do not you concur with me in this opinion?</P> <P>SIMP. I think that it ought undoubtedly $o to $ucceed.</P> <P>SALV. But if this be $o, and, it be likewi$e true that a great Stone moveth with ($uppo$e) eight degrees of Velocity, and a le$- $er with fewer, then joyning them both together, the compound of them will move with a Velocity le$$e than eight Degrees: But the two Stones joyned together make one Stone greater than that before, which moved with eight degrees of Velocity: There- fore this greater Stone moveth le$$e $wiftly than the le$$er, which is contrary to your Suppo$ition. You $ee therefore, that from the $uppo$ing that the more grave Moveable moveth more $wiftly than the le$$e grave, I prove unto you that the more grave mo- veth le$$e $wiftly.</P> <P>SIMP. I find my $elf at a lo$$e, for the truth is, that the le$- $er Stone being joyned to the greater, weight is added unto it, and weight being added to it, I cannot $ee why there $hould not Ve- locity be added to it, or at lea$t why it $hould be dimini$hed in it.</P> <P>SALV. Here you run into another errour, <I>Simplicius,</I> for it is not true, that that $ame le$$er Stone encrea$eth the weight of the greater.</P> <P>SIMP. Oh wonderful! this quite $urpa$$eth my apprehen$ion.</P> <P>SALV. Not at all, if you will but $tay till I have di$covered to you the Equivokes, of which you are in doubt: Therefore <foot>you</foot> <p n=>53</p> you mu$t know that it is nece$$ary to di$tingui$h betwixt grave Bodies $et on Moving, and the $ame con$tituted in Re$t; a Stone put into the Ballance not onely acquireth greater weight, by lay- ing another Stone upon it, but al$o the addition of, a Flake of Hemp will make it weigh more by tho$e $ix or ten ounces that the Hemp $hall weigh; but if you $hould freely let fall the Stone tied to the Hemp from an high place, do you think that in the Motion the Hemp weigheth down the Stone, $o as to accelerate its Motion; or el$e do you believe that it will retard it, $u$tain- ing it in part? We indeed feel our $houlders laden, $o long as we will oppo$e the Motion that the weight would make which lyeth upon our backs; but if we $hould de$cend with the $ame Velocity wherewith that $ame grave Body would naturally de$cend, in what manner will you that it pre$$e or bear upon us? Do not you $ee that this would be a wounding one with a Lance that runneth before you, with as much or more $peed than you pur$ue him. You may conclude therefore that in the free and natural fall, the le$$er Stone doth not bear upon the greater, and con$equently doth not encrea$e their weight, as it doth in Re$t.</P> <P>SIMP. But what if the greater was put upon the le$$er?</P> <P>SALV. It would encrea$e their weight, in ca$e its Motion were more $wift; but it hath been already concluded, that in ca$e the le$$er $hould be more $low it would in part retard the Velocity of the greater, $o that there Compound would move le$$e $wiftly; being greater than the other, which is contrary to your A$$umpti- on: Let us conclude therefore, that great Moveables, and like- wi$e little, being of the $ame Specifical Gravity, move with like Velocity.</P> <P>SIMP. Your di$cour$e really is full of ingenuity, yet methinks it is hard to conceive that a drop of Bird-$hot, $hould move as $wiftly as a Canon-bullet.</P> <P>SALV. You may $ay a grain of Sand as fa$t as a Mill-$tone. I would not have you, <I>Simplicius,</I> to do as $ome others are wont to do, and diverting the di$cour$e from the principal de$ign, fa- $ten upon $ome one $aying of mine that may want an hairs-breadth of the truth, and under this hair hide a defect of another man as big as the Cable of a Ship. <I>Aristotle</I> $aith, a Ball of Iron of an hundred pounds weight falling, from an height of an hundred yards, commeth to the ground before that one of one pound is de$cended one $ole yard: I $ay, that they arrive at the earth both in the $ame time: You find, that the bigger anticipates the le$$er two Inches, that is to $ay, that when the great one falls to the ground, the o- ther is di$tant from it two inches: you go about to hide under the$e two inches the ninety nine yards of <I>Ari$totle,</I> and $peaking onely to my $mall errour, pa$$e over in $ilence the other great one. <foot><I>Ari-</I></foot> <p n=>54</p> <I>Ari$totlee</I> affirmeth, that Moveables of different Gravities in the $ame <I>Medium</I> move (as far as concerneth Gravity) with Veloci- ties proportionate to their Weights; and exemplifieth it by Moveables, wherein one may di$cover the pure and ab$olute effect of Weight, omitting the other Con$iderations, as well of Figures, as of the minute Motions; which things receive great alteration from the <I>Medium,</I> which altereth the $imple effect of the $ole Gravity; wherefore we $ee Gold, that is heavier than any other matter, being reduced into a very thin Leaf, to go flying to and again through the Air, the like do Stones beaten to very $mall Powder. But if you would maintain the Univer$al Propo$ition, it is requi$ite that you $hew the proportion of the Velocities to be ob$erved in all grave Bodies, and that a Stone of twenty pounds moveth ten times $wifter than one of two: which, I tell you, is fal$e, and that falling from an height of fifty or an hundred yards, they come to the ground in the $ame in$tant.</P> <P>SIMP. Perhaps in very great heights of Thou$ands of yards that would happen, which is not $een to occur in the$e le$$er heights.</P> <P>SALV. If this was the Meaning of <I>Ari$totle,</I> you have in- volved him in another Errour, which will be found a Lie; for there being no $uch perpendicular altitudes found on the Earth, its a clear ca$e, that <I>Ari$totle</I> was not able to have made an Experi- ment thereof; and yet would per$wade us that he had, whil$t he $aith, that the $aid effect is <I>$een.</I></P> <P>SIMP. <I>Ari$totle</I> indeed makes no u$e of this Principle, but of that other, which I believe is not obnoxious to the$e doubts.</P> <P>SALV. Why that al$o is no le$$e fal$e than this; and I admire that you do not of your $elf perceive the fallacy, and di$cern, that $hould it be true, that the $ame Moveable in <I>Medium's</I> of dif- ferent Subtilty and Rarity, and, in a word, of different Ce$$ion, $uch, for example, as are Water and Air, move with a greater Velocity in the Air than in the Water, according to the propor- tion of the Airs Rarity to the Rarity of the Water, it would follow that every Moveable that de$cendeth in the Air would de$cend al$o in the Water: Which is $o fal$e, that very many Bodies de$cend in the Air, that in the Water do not onely not de$cend, but al$o ri$e upwards.</P> <P>SIMP I do not under$tand the nece$$ity of your Con$equence: and I will $ay farther, that <I>Ari$totle</I> $peaketh of tho$e Grave- bodies that de$cend in the one <I>Medium</I> and in the other, and not of tho$e that de$cend in the Air and a$cend in the Water.</P> <P>SALV. You produce for the Phil$opher $uch Pleas as he, with- out all doubt, would never alledge, for that they aggravate the fir$t mi$take. Therefore tell me, if the Cra$situde of the Water, <foot>or</foot> <p n=>55</p> or whatever it be that retardeth the Motion, hath any proporti- on to the Cra$$itude of the Air that le$$e retards it; and if it have; do you a$$ign it us, at plea$ure.</P> <P>SIMP. It hath $uch a proportion, and we will $uppo$e it to be decuple; and that therefore the Velocity of a Grave Body, that de$cends in both the Elements, $hall be ten times $lower in the Wa- ter than in the Air.</P> <P>SALV. I will take one of tho$e Grave-Bodies that de$cend in the Air, but not in the Water; as for in$tance, a Ball of Wood, and de$ire that you will a$$ign it what Velocity you plea$e, whil$t it de$cends through the Air.</P> <P>SIMP. Suppo$e we, that it move with twenty degrees of Velo- city.</P> <P>SALV. Very well: And it is manife$t, that that Velocity to $ome other le$$er, may have the $ame proportion, that the Cra$$i- tude of the Water hath to that of the Air; and that this $hall be the Velocity of the two only degrees: $o that exactly to an hair, and in direct conformity to the A$$umption of <I>Ari$totle,</I> it $hould be concluded, That the Ball of Wood, which in the Air, ten times more yielding, moveth de$cending with twenty degrees of Veloci- ty, in the Water $hould de$cend with two, and not return from the bottom to flote a-top, as it doth: unle$s you will $ay, that the a$cending of the Wood to the top is the $ame in the Water, as its $inking to the bottom with two degrees of Velocity; which I do not believe. But $eeing that the Ball of Wood de$cends not to the bottom, I rather think that you will grant me, that $ome other Ball, of other matter different from Wood, might be found that de$cends in the Water with two degrees of Velocity.</P> <P>SIMP. Que$tionle$$e there might; but it mu$t be of a matter con$iderably more grave than Wood.</P> <P>SALV. This is that which I de$ired to know. But this $econd Ball, which in the Water de$cendeth with two degrees of Velocity, with what Velocity will it de$cend in the Air? It is requi$ite (if you will maintain <I>Ari$totles</I> Rule) that you an$wer that it will move with twenty degrees: But you your $elf have a$$igned twen- ty degrees of Velocity to the Ball of Wood; Therefore this, and the other that is much more grave, will move thorow the Air with equall Velocity. Now how doth the Philo$opher reconcile this Conclu$ion with that other of his, that the Moveables of different Gravity, move in the $ame <I>Medium</I> with different Velocities, and $o different as are their Gravities? But, without any deep $tudies, how comes it to pa$s that you have not ob$erved very frequent, and very palpable Accidents, and not con$idered two Bodies, that in the Water will move one an hundred times more $wiftly than the other, but that again in the Air that $wifter one will not out-go the <foot>other</foot> <p n=>56</p> other, one $ole Cente$m? As for example, an Egge of Marble will de$cend in the Water an hundred times fa$ter than one of an Hen, when as in the Air, at the height of twenty Yards it will not anti- cipate it four Inches: and, in a word, $uch a certain Grave Body will $ink to the bottom in three hours in ten fathom Water, that in the Air will pa$s the $ame $pace in one or two pul$es, and $uch another (as for in$tance a Ball of Lead) will pa$s that number of fathoms with ea$e in le$s than double the time. And here I $ee plainly, <I>Simplicius,</I> that you find, that herein there is no place left for any di$tinction, or reply. Conclude we therefore, that that $ame Argument concludeth nothing again$t <I>Vacuum</I>; and if it $hould, it would only overthrow Spaces con$iderably great, which neither I, nor, as I take it, tho$e <I>Ancients</I> did $uppo$e to be natu- rally allowed, though, perhaps, with violence they may be effe- cted, as, me thinks, one may collect from $everal Experiments, which it would be two tedious to go about at pre$ent to produce.</P> <P>SAGR. Seeing that <I>Simplicius</I> is $ilent, I will take leave to $ay $omething. In regard you have with $ufficient plainne$$e demon- $trated, that it is not true, That Moveables unequally grave move in the $ame <I>Medium</I> with Velocities proportionate to their Gravities, but with equal: de$iring to be under$tood to $peak of Bodies of the $ame Matter, or of the $ame Specifick Gravity, but not (as I con- ceive) of Gravities different <I>in Spetie,</I> (for I do not think that you intend to prove unto us, that a Ball of Cork moveth with like Velocity to one of Lead;) and having moreover very manife$tly demon$trated, that it is not true, That the $ame Moveable in <I>Me- diums</I> of different Re$i$tances retain in their Velocities and Tardi- ties the $ame proportion as have their Re$i$tances: to me it would be a very plea$ing thing to hear, what tho$e be which are ob$erved as well in the one ca$e as in the other.</P> <P>SALV. The Que$tions are ingenuous, and I have many times thought of them: I will relate unto you the Contemplations made upon them, and what at length I did from thence infer. After I had a$$ured my $elf that it was not true, That the $ame Moveable in <I>Medium's</I> of different Re$i$tance ob$erveth in its Velocity the proportion of the Ce$$ion of tho$e <I>Media</I>; nor yet, again, That in the $ame <I>Medium</I> Moveables of different Gravity retain in their Velocities the proportion of tho$e Gravities ($peaking alwaies of Gravitles different <I>in $pecie</I>) I began to put both the$e Accidents together, ob$erving that which befell the Moveables different in Gravity put into <I>Mediums</I> of different Re$i$tance, and I perceived that the inequality of the Velocities were found to be alwaies greater in the more re$i$ting <I>Medium's,</I> than in the more yielding; and that with $uch a diver$ity, that of two Moveables that, de- $cending thorow the Air, differ very little in Velocity of Motion, <foot>one</foot> <p n=>57</p> one will, in the Water, move ten times fa$ter than the other; yea: that $uch, as in the Air do $wiftly de$cend, in the Water not only will not de$cend, but will be wholly deprived of Motion, and, which is yet more, will move upwards: for one $hall $ome- times find $ome kind of Wood, or $ome knot, or root of the $ame, that in the Water will lye $till, when as in the Air it will $wiftly de$cend.</P> <P>SAGR. I have many times $et my $elf with an extream patience to $ee if I could reduce a Ball of Wax, (which of it $elf doth not go to the bottom) by adding to it grains of $and, to $uch a degree of Gravity like to the Water, as to make it $tand $till in the mid$t of that Element; but I could never, by all the care I u$ed, $ucceed in my attempt; $o that I cannot tell, whether any Solid matter may be found $o naturally alike in Gravity to Wa- ter, as that being put into any place of the $ame, it can re$t or lye $till.</P> <P>SALV. In this, as well as in a thou$and other actions, many Animals are more ingenuous than we. And, in this ca$e, Fi$hes <marg><I>Fi$hes equilibrate them$elves admi- rably in the Water.</I></marg> would have been able to have given you $ome light, being in this affair $o skilful, that at their plea$ure they ^{*} equilibrate them$elves, <marg>* Or poi$e.</marg> not only with one kind of Water, but with $uch, as, either of their own nature, or by means of $ome $upervenient muddine$s, or for their $altne$s (which maketh a great alteration) are very diffe- rent; equilibrate them$elves, I $ay, $o exactly, that without $tir- ring in the lea$t they lye $till in every place: and this, in my opi- nion, they do, by making u$e of the In$trument given them by Na- ture to that end, <I>$cilicet,</I> of that Bladder which they have in their Bodies, which by a very narrow neck an$wereth to their mouth; and by that they either, when they would $tand $till, $end forth part of the Air that is contained in the $aid Bladders, or, $wimming to the top they draw in more, making them$elves by that art one while more, another while le$s heavy than the Water, and at their plea$ures equilibrating them$elves to the $ame.</P> <P>SAGR I deceived $ome of my Friends with another device; for I had made my boa$t unto them, that I would reduce that Ball of Wax to an exact <I>equilibrium</I> with the Water, and having put $ome $alt Water in the bottom of the Ve$$el, and a-top of that $ome fre$h, I $hewed them the Ball, which in the mid$t of the Water $tood $till, and being thru$t to the bottom, or to the top, $taid nei- ther in this nor that $cituation, but returned to the mid$t.</P> <P>SALV. This $ame Experiment is not void of utility; for Phy$i- <marg><I>A Ball of Wax prepared to make the Experiment of the different Gra- vities of Waters.</I></marg> cians, in particular, treating of $undry qualities of Waters, and among$t other things, principally of the more or le$s Gravity or Levity of this or that: by $uch a Ball, in $uch manner poi$ed and adju$ted that it may re$t ambiguous, if I may $o $ay, between <foot>I a$cending</foot> <p n=>58</p> a$cending and de$cending in a Water, upon the lea$t difference of weight between two Waters, if that Ball $hall de$cend in the one; in the other, that is more grave, it $hall a$cend. And the Experiment is $o exact, that the addition of but only two grains of Salt, put into $ix pounds of Water, $hall make that Ball to a$cend from the bottom to the $urface, which was but a little be- <marg><I>Water bath no Re$i$tance to Di- vi$ion.</I></marg> fore de$cended thither. And moreover, I will tell you this in con- firmation of the exactne$s of this Experiment, and withall for a clear proof of the Non-re$i$tance of Water to divi$ion, that not only the ingravitating it with the mixture of $ome matter heavier than it, maketh that $o notable difference, but the warming or cooling of it a little produceth the $ame effect, and with $o $ubtil an operation, that the infu$ing four diops of other Water, a lit- tle warmer, or a little colder, than the $ix pounds, $hall cau$e the Ball to ri$e or $ink in the $ame; to $ink in it upon the infu$ion of the warm, and to ri$e at the infu$ion of the cold. Now $ee how much tho$e Philo$ophers are deceived, who would introduce in Water vi$co$ity, or other conjunction of parts which make it to re$i$t Divi$ion or Penetration.</P> <P>SAGR. I have $een many Convincing Di$cour$es touching <marg>* The Tract cited in this place is that which we di$po$e fir$t in Order, in the fir$t part of this Tome,</marg> this Argument in a ^{*} Treati$e of our <I>Accademick</I>; yet never the le$s there is re$ting in me a $trong $cruple, which I know not how to remove: For if nothing of Tenacity, or Coherence re$ides among$t the parts of Water, how can it bear it $elf up in rea$onable big and high Tumours; in particular, upon the leaves of Cole-worts without di$per$ing or levelling?</P> <P>SALV. Although it be true, that he who is Ma$ter of a true Conclu$ion, may re$olve all Objections that can be brought again$t it, yet will not I arrogate to my $elf the power $o to do; nor ought my in$ufficiency becloud the $plendour of Truth. Fir$t, therefore, I confe$s that I know not how it com<*>h to pa$s, that tho$e Globes of Water $u$tain them$elves at $uch an height and bigne$s, albeit I certainly know that it doth not proceed from any <marg><I>Water formed into great drops upon the Leaves of Col- worts, how they con$i$t.</I></marg> internal Tenacity that is between its parts; $o that it remaineth neœ$$ary, that the Cau$e of that Effect do re$ide without. That it is not Internal, be$ides tho$e Experiments already $hewn you, I can prove by another mo$t convincing one. If the parts of that Wa- ter, which con$erveth it $elf in a Globe or Tumour whil$t it is en- compa$$ed by the Air, had an internal Cau$e for $o doing, they would much better $u$tain them$elves being environed by a <I>Medi- um,</I> in which they had le$s propen$ion to de$cend, than they have in the Ambient Air: But every Fluid Body more grave than the Air would be $uch a <I>Medium</I>; as, for in$tance, Wine: And there- fore, infu$ing Wine about that Globe of Water, it might rai$e it $elf on every $ide, and yet the parts of the Water, conglutinated <foot>by</foot> <p n=>59</p> by the internal Vi$co$ity, never di$$olve: But it doth not happen $o; nay, no $ooner doth the circumfu$ed liquor approach thereto, but, without $taying till it ri$e much about it, the little globes of Water will di$$olve and become flat, re$ting under the Wine, if it was red. The Cau$e therefore of this Effect is External, and per- haps in the Ambient Air: and, indeed, one may ob$erve a great di$$ention between the Air and Water; which I have ob$erved in another Experiment; and this it is: If I fill a ^{*} Ball of Chri$tal, <marg>* Or bottle.</marg> that hath a mouth as narrow as the hollow of a $traw, with water, and when it is thus full, turn it with its mouth downwards, yet will not the Water, although very heavy, and prone to de$cend tho- row the Air, nor the Air, as much di$po$ed on the other hand, as being very light, to a$cend thorow the Waters, yet will they not (I $ay) agree that that $hould de$cend, i$$uing out at the mouth, and this a$cend, entering in at the $ame: but they both continue aver$e and contumacious. Again, on the contrary, if I pre$ent to that mouth a ve$$el of red Wine, which is almo$t in$en$ibly le$s grave than Water, we $hall $ee it in an in$tant gently to a$cend by red $treams thorow the Water, and the Water with like Tardity to de$cend through the Wine, without ever mixing with each other, till that in the end, the Ball will be full of Wine, and the Water Will all $ink unto the bottom of the Ve$$el underneath. Now what are we to $ay, or what are we to infer, but a di$agreement between the Water and Air, occult to me, but perhaps -----</P> <P>SIMP. I can $carce refrain my laughter to $ee the great Anti- pathy that <I>Salviatus</I> hath to Antipathy, $o that he will not $o much as name it, and yet it is $o accommodate to re$olve the doubt.</P> <P>SALV. Now let this, for the $ake of <I>Simplicius</I> be the $oluti- on of our $cruple; and leaving the Digre$$ion, let us return to our purpo$e. Seeing that the difference of Velocity in Moveables of divers Gravities is found to be more and more, as the <I>Mediums</I> are more and more Re$i$ting: And withall, that in a <I>Medium</I> of Quick$ilver, Gold doth not only go to the bottom more $wiftly than Lead, but it alone de$cends in it, and all other Metals and Stones move upwards therein, and flote thereon; whereas between Balls of Gold, Lead, Bra$s, Porphiry, or other grave matters, the in- equality of motion in the Air $hall be almo$t wholly in$en$ible, for it is certain, that a Ball of Gold in the end of the de$cent of an <marg><I>Re$i$tance of the</I> Medium <I>remo- ved, all Matters, though of different Gravities would move with like Velocity.</I></marg> hundred yards $hall not out-$trip one of Bra$s four Inches: $eeing this, I $ay, I have thought, that if we wholly took away the Re$i$tance of the <I>Medium,</I> all Matters would de$cend with equall Velocity.</P> <P>SIMP. This is a bold $peech, <I>Salviatus,</I> I $hall never believe that in <I>Vacuity</I> it $elf, if $o be one $hould allow Motion in it, a lock of Wooll would move as $wiftly as a piece of Lead.</P> <foot>I 2 SALV.</foot> <p n=>60</p> <P>SALV. Fair and $oftly, <I>Simplicius,</I> your $cruple is not $o ab- $truce, nor I $o incautelous, that you $hould need to think that I was not advi$ed of it, and that con$equently I have not found a re- ply to it. Therefore, for my explanation, and your information, hearken to what I $hall $ay. We are upon the examination of what would befall Moveables exceeding different in weight in a <I>Medium,</I> in ca$e it $hould have no Re$i$tance, $o that all the diffe- rence of Velocity that is found between the $aid Moveables ought to be referred to the $ole inequality of Weight. And becau$e on- ly a Space altogether void of Air, and of every other, though te- nuous and yielding Body, would be apt $en$ibly to $hew us what we $eek, $ince we want $uch a Space, let us $ucce$$ively ob$erve that which happeneth in the more $ubtill and le$$e re$i$ting <I>Mediums,</I> in compari$on of that which we $ee to happen in others le$$e $ubtill and more re$i$ting: for if we $hould really find the Moveables different in Gravity to differ le$$e and le$$e in Velocity, according as the <I>Mediums</I> are found more and more yielding; and that, finally, although extreamly unequal in weight, in a <I>Medium</I> more tenuous than any other, though not void, the difference of Velo- city di$covers it $elf to be very $mall, and almo$t unob$ervable, I conceive that we may, and that upon very probable conjecture, believe, that in a <I>Vacuum</I> their Velocities would be exactly equal. Therefore let us con$ider that which hapneth in the Air; wherein to have a Figure of an uniform Superficies, and very light Matter, I will that we take a blown Bladder, in which the included Air will weigh little or nothing in a <I>Medium</I> of the Air it $elf, becau$e it can make but very $mall Compre$$ion therein, $o that the Gravi- ty is only that little of the $aid film, which would not be the thou- $andth part of the weight of a lump of Lead of the bigne$s of the $aid Bladder when blown. The$e, <I>Simplicius,</I> being let fall from the height of four or $ix yards, how great a $pace, do you judge, that the Lead would anticipate the Bladder in its de$cent? A$$ure your $elf that would not move thrice, no nor twice as fa$t, although even now you would have had it to have been a thou- $and times more $wift.</P> <P>SIMP. It is po$$ible that at the beginning of the Motion, that is, in the fir$t five or $ix yards this might happen that you $ay; but in the progre$$e, and in a long continuation I believe, that the Lead would leave it behind, not only $ix, but al$o eight and ten parts of twelve.</P> <P>SALV. And I al$o believe the $ame: and make no que$tion, but that in very great di$tances the Lead will have pa$$ed an hun- dred miles of <I>way,</I> ere the Bladder will have pa$$ed $o much as one. But this, <I>Simplicius,</I> which you propound, as an effect contrary to my A$$ertion, is that which mo$t e$pecially confirmeth it. It is (I <foot>once</foot> <p n=>61</p> once more tell you) my intent to declare, That the difference of Gravity is in no wi$e the cau$e of the divers velocities of Movea- bles of different Gravity, but that the $ame dependeth on exteri- our accidents, & in particular, on the Re$i$tance of the <I>Medium,</I> $o that, this being removed, all Moveables move with the $ame de- grees of Velocity. And this I chiefly deduce from that which but now you your $elf did admit, and which is very true, namely, that of two Moveables, very different in weight, the Velocities more and more differ, according as the ^{*} Spaces are greater and greater that <marg>* Or Waies.</marg> they pa$$e: an Effect which would not follow, if it did depend on the different Gravities: for they being alwaies the $ame, the pro- portion betwixt the Spaces would likewi$e alwaies continue the $ame, which proportion we $ee $till $ucce$$ively to encrea$e in the continuance of the Motion; for that the heavie$t Moveable in the de$cent of one yard will not anticipate the lighte$t the tenth part of that Space or Way, but in the fall of twelve yards will out-go it a third part, in that of an hundred will out$trip it 90/100.</P> <P>SIMP. Very well: But following you $tep by $tep, if the dif- ference of weight in Moveables of different Gravities cannot cau$e the difference of proportion in their Velocities, for that the Gravities do not alter; neither then can the <I>Medium,</I> which is $uppo$ed alwaies to continue the $ame, cau$e any alteration in the proportion of the Velocities.</P> <P>SALV. You wittily bring an in$tance again$t my Po$ition, that <marg><I>The Velocity of Grave Bodies de- $cending Natural- ly to the Center do go continually en- crea$ing till that by the encrea$e of the Re$i$tance of the</I> Medium <I>it becometh uniform.</I></marg> it is very nece$$ary to remove. I $ay therefore, that a Grave Body hath, by Nature, an intrin$ick Principle of moving towards the Common Center of heavy things, that is to that of our Terre$trial Globe, with a Motion continually accelerated, and accelerated alwaies equally, <I>$cilicet,</I> that in equal times there are made equal ^{*} additions of new Moments, and degrees of Velocities: and this ought to be under$tood to hold true at all times when all acciden- <marg>* Or aqui$ts.</marg> tal and external impediments are removed; among$t which there is one that we cannot obviate, that is the Impediment of the <I>Me- dium,</I> which is Repleat, when as it $hould be opened and latterally moved by the falling Moveable, to which tran$ver$e Motion the <I>Medium,</I> though fluid, yielding and tranquile, oppo$eth it $elf with a Re$i$tance one while le$$er, and another while greater and greater, according as it is more $lowly or ha$tily to open to give pa$$age to the Moveable, which, becau$e, as I have $aid, it goeth of its own nature continually accelerating, it cometh of con$e- quence to encounter continually greater Re$i$tance in the <I>Medi- um,</I> and therefore Retardment, and diminution in the acqui$t of new degrees of Velocity; $o that in the end, the Velocity arriveth to that $wiftne$$e, and the Re$i$tance of the <I>Medium,</I> to that $trength, that ballancing each other, they take away all further <foot>Acceleration,</foot> <p n=>62</p> Acceleration, and reduce the Moveable to an Equable and Uni- form Motion, in which it afterwards continually abides. There is therefore in the <I>Medium</I> augmentation of Re$i$tance, not becau$e it changeth its E$$ence, but becau$e the Velocity altereth where- with it ought to open, and laterally move, to give pa$$age to the falling Body, which goeth continually accelerating. Now the ob$erving, that the Re$i$tance of the Air to the $mall Moment or <I>Impetus</I> of the Bladder is very great, and to the great weight of the Lead is very $mall, makes me hold for certain, that if one $hould wholly remove it, by adding to the Bladder great a$$i$tance, and but very little to the Lead, their Velocities would equalize each other. Taking this Principle therefore for granted, That in the <I>Medium</I> wherein, either by rea$on of Vacuity, or otherwi$e, there were no Re$i$tance that might abate the Velocity of the Motion, $o that of all Moveables the Velocities were alike, we might con- <marg><I>To find the Pro- portions of the Ve- locities of different Moveables in the $ame, and in diffe- rent</I> Mediums.</marg> gruou$ly enough a$$ign the proportions of the Velocities of like and unlike Moveables, in the $ame and in different, Replear, and therefore Re$i$ting <I>Medium's.</I> And this we might effect by $tudy- ing how much the Gravity of the <I>Medium</I> abateth from the Gra- vity of the Moveable, which Gravity is the In$trument wherewith the Moveable makes its Way, repelling the parts of the <I>Medium</I> on each Side: an operation that doth not occur in void <I>Mediums</I>; and therefore there is no difference to be expected from the di- ver$e Gravity: and becau$e it is manife$t, that the <I>Medium</I> abateth from the Gravity of the Body by it contained, as much as is the weight of $uch another ma$s of its own Matter, if the Velocities of the Moveables that in a non-re$i$ting <I>Medium</I> would be (as hath been $uppo$ed) equal, $hould dimini$h in that proportion, we $hould have what we de$ired. As for example; $uppo$ing that Lead be ten thou$and times more grave than Air, but Ebony a thou$and times only; of the Velocities of the$e two Matters, which ab$olutely taken, that is, all Re$i$tance being removed, would be equal, the Air $ub$tracts from the ten thou$and degrees of the Lead one, and from the thou$and degrees of the Ebony likewi$e abateth one, or, if you will, of its ten thou$and, ten. If there- fore the Lead and the Ebony $hall de$cend thorow the Air from any height, which, the retardment of the Air removed, they would have pa$$ed in the $ame time, the Air will abate from the ten thou$and degrees of the Leads Velocity one, but from the ten thou$and degrees of Ebony's Velocity it will abate ten: which is as much as to $ay, that dividing that Altitude, from which tho$e Moveables departed into ten thou$and parts, the Lead will arrive at the Earth, the Ebony being left behind, ten, nay, nine of tho$e $ame ten thou$and parts. And what el$e is this, but that a Ball of Lead, falling from a Tower two hundred yards high, to find how <foot>much</foot> <p n=>63</p> much it will anticipate one of Ebony of le$$e than four Inches? The Ebony weigheth a thou$and times more than the Air, but that Bladder $o blown, weigheth only four times $o much; the Air therefore from the intrin$ick and natural Velocity of the Ebony $ubducteth one degree of a thou$and, but from that, which al$o in the Bladder would ab$olutely have been the $ame, the Air $ub- ducts one part of four: $o that by that time the Ball of Ebony falling from the Tower, $hall come to the ground, the Bladder $hall have pa$$ed but three quarters of that height. Lead is twelve times heavier than Water, but Ivory only twice as heavy; the Water therefore, from their ab$olute Velocities which would be equal, $hall abate in the Lead the twelfth part, but in the Ivory the half: when therefore, in the Water, the Lead $hall have de- $cended eleven fathom, the Ivory $hall have de$cended $ix. And, arguing by this Rule, I believe, that we $hall find the Experiment much more exactly agree with this $ame Computation, than with that of <I>Ari$totle.</I> By the like method we might find the Veloci- ties of the $ame Moveable in different fluid <I>Mediums,</I> not compa- ring the different Re$i$tances of the <I>Mediums,</I> but con$idering the exce$$es of the Gravity of the Moveable over and above the Gra- vities of the <I>Mediums: v. gr.</I> ^{*} Tin is a thou$and times heavier than <marg>* Or Pewter.</marg> Air, and ten times heavier than Water; therefore dividing the ab- $olute Velocity of the Tin into a thou$and degrees, it $hall move in the Air, (which deducteth from it the thou$andth part,) with nine hundred ninety nine, but in the Water with nine hundred only; being that the Water abateth the tenth part of its Gravity, and the Air the thou$andth part. Take a Solid $omewhat heavier than Water, as for in$tance, the Wood called Oake, a Ball of which weighing, as we will $uppo$e, a thou$and drams, a like quantity of Water will weigh nine hundred and fifty, but $o much Air will weigh but two drams,: it is manife$t, that $uppo$ing that its ab$o- lute Velocity were of a thou$and degrees, in Air there would re- main nine hundred ninety eight, but in the Water only fifty; be- cau$e that the Water of the thou$and degrees of Gravity taketh away nine hundred and fifty, and leaves fifty only; that Solid there- fore would move well-near twenty times as fa$t in the Air as Wa- ter; like as the exce$$e of its Gravity above that of the Water is the twentieth part of its own. And here I de$ire that we may con- $ider, that no matters, having a power to move downwards in the Water, but $uch as are more grave in Species than it; and con$e- quently many hundreds of times, more grave than the Air, in $eeking what the proportions of their Velocities are in the Air and Water, we may, without any con$iderable errour, make account that the Air doth not deduct any thing of moment from the ab$o- lute Gravity, and con$equently, from the ab$olute Velocity of $uch <foot>mat-</foot> <p n=>66</p> matters: $o that having ea$ily found the exce$$e of their Gravi- ty above the Gravity of the Water, we may $ay that their Velo- city in the Air, to their Velocity in the Water hath the $ame propor- tion, that their total Gravity hath to the exce$$e of this above the Gravity of the Water. For example, a Ball of Ivory weigh- eth twenty ounces, a like quantity of Water weigheth $eventeen ounces: therefore the Velocity of the Ivory in Air, to its Velocity in Water is very neer as twenty to three.</P> <P>SAGR. I have made a great acqui$t in a bu$ine$$e of it $elf cu- rious, and in which, but without any benefit, I have many times wearied my-thoughts: nor would there any thing be wanting for the putting the$e Speculations in practice, $ave onely the way how one $hould come to know of what Gravity the Air, is in com- pari$on to the Water, and con$equently to other heavy matters.</P> <P>SIMP. But in ca$e one $hould finde, that the Air in$tead of Gravity had Levity, what ought one to $ay of the foregoing di$- cour$es, otherwi$e very ingenuous?</P> <P>SALV. It would be nece$$ary to confe$$e that they were truly Aerial, Light, and Vain. But will you que$tion whether the Air be heavy, having the expre$$e <I>Text</I> of <I>Ari$totle</I> that affirmeth it, $aying, That all the Elements have Gravity, even the Air it $elf; <marg>* Or <I>Boracho</I>; a bottle made of a Goat skin, u$ed to hold wine and other Liquids.</marg> a $igne of which ($ubjoyns he) we have in that a ^{*} Bladder blown, weigheth heavier than un$well'd.</P> <P>SIMP. That a <I>Boracho,</I> or Bladder blown, weigheth more, might proceed, as I could $uppo$e, not from the Gravity that is in the Air, but in the many gro$$e Vapours intermixed with it in the$e our lower Regions; by means whereof I might $ay, that the Gravity of the Bladder, or <I>Boracho</I> encrea$eth.</P> <P>SALV. I would not have you $ay it, and much le$$e that you $hould make <I>Aristotle</I> $peak it, for he treating of the Elements, and de$iring to per$wade me that the Element of Air is grave, making me to $ee it by an Experement: if in comming to the proof he $hould $ay: Take a Bladder, and fill it with gro$$e Vapours; and ob$erve that its weight will encrea$e; I would tell him that it would weigh yet more if one $hould fill it with bran; but would afterwards adde; that tho$e Experiments prove, that bran, and gro$$e Vapours are grave: but as to the Element of Air, I $hould be left in the $ame doubt as before. The Experiment of <I>Ari$totle</I> therefore is good, and the Propo$ition true. But I will not $ay $o much, for a certain other rea$on taken expre$ly out of a Philo$o- pher who$e name I do not remember, but am $ure that I have read it, who argueth the Air to be more grave than light, becau$e it more ea$ily carrieth grave Bodies downwards, than the light up- wards.</P> <P>SAGR. Good i- faith. By this rea$on then, the Air $hall be <foot>much</foot> <p n=>65</p> much heavier than the Water, $ince, that all Bodies are carried more ea$ily downwards thorow the Air than thorow the Water, and all light Bodies more ea$ily upwards in this than in that: nay, infinite matters a$cend in the Water, that in the Air de$cend. But be the Gravity of the Bladder, <I>Simplicius,</I> either by rea$on of the gro$$e Vapours, or pure Air, this nothing concerns our pur- po$e, for we $eek that which happeneth to Moveables that move in this our Vaporous Region. Therefore, returning to that which more concerneth me, I would for a full and ab$olute informati- on in the pre$ent bu$ine$$e, not onely be a$$ured that the Air is grave, as I hold for certain, but I would, if it be po$$ible, know what its Gravity is. Therefore, <I>Salviatus,</I> if you have wherewith to $atisfie me in this al$o, I entreat you to favour me with the $ame.</P> <P>SALV. That there re$ideth in the Air po$itive Gravity, and <marg><I>The Air hath Po- $itive Gravity.</I></marg> not, as $ome have thought, Levity, which haply is in no Mat- ter to be found, the Experiment of the Blown-Bladder, alledged by <I>Ari$totle,</I> affordeth us a $ufficiently-convincing Argument; for if the quality of ab$olute and po$itive Levity were in the Air, then the Air being multiplied and compre$$ed, the Levity would encrea$e, and con$equently the propen$ion of going upwards: but Experience $hews the contrary. As to the other demand, that <marg><I>How that Gravity may be computed.</I></marg> is, of the Method how to inve$tigate its Gravity, I have tried to do it in this manner: I have taken a pretty bigge Gla$$e ^{*} Bottle, <marg>* <I>Un Fia$co,</I> tho$e long-neckt gla$$e bottles in which we have our <I>Florence</I> Wine brought to us.</marg> with its neck bended, and a Finger-$tall of Leather fa$t about it, having in the top of the $aid Finger-$tall in$erted and fa- $tened a Valve of Leather, by which with a Siringe I have made pa$$e into the Bottle by force a great quantity of Air, of which, becau$e it admits of great Conden$ation, it may take in two or three other Bottles-ful over and above that which is naturally con- tained therein. Then I have in an exact Ballance very preci$ely weighed that Bottle with the Air compre$$ed within it, adju$ting the weight with $mall Sands. Afterwards, the Valve being opened, and the Air let out, that was violently conteined in the Ve$$el, I have put it again into the Scales, and finding it notably aleviated, I have by degrees taken $o much Sand from the other Scale, keep- ing it by it $elf, that the Ballance hath at la$t $tood <I>in Equilibrio</I> with the remaining counter-poi$e, that is with the Bottle. And here there is no que$tion, but that the weight of the re$erved Sand is that of the Air that was forceably driven into the Bottle, and which is at la$t gone out thence. But this Experiment hitherto a$- $ureth me of no more but this, that the Air violently deteined in the Ve$$el, weigheth as much as the re$erved Sand, but how much the Air re$olutely and determinately weigheth in re$pect of the Water, or other grave matter, I do not as yet know, nor can <foot>K I</foot> <p n=>66</p> I tell, unle$$e I mea$ure the quantity of the Air compre$$ed: and for the di$covering of this a Rule is nece$$ary, which I have found may be performed two manner of wayes, one of which is to take $uch another Bottle or Flask as the former, and in like manner bended, with a Finger-$tall of Leather, the end of which may clo$ely imbrace the Volve of the other, and let it be very fa$t tied about it. It's requi$ite, that this $econd Bottle be bored in the bottom, $o that as by that hole we may thru$t in a Wier, wherewith we may, at plea$ure, open the $aid Volve, to let out the $uperfluous Air of the other Ve$$el, after it hath been weighed: but this $econd Bottle ought to be full of Water. All being pre- pared in the manner afore$aid, and with the Wier opening the Volve, the Air i$$uing out with impetuo$ity, and pa$$ing into the Ve$$el of Water, $hall drive it out by the hole at the Bottom: and it is manife$t, that the quantity of Water which $hall be thru$t out, is equal to the Ma$$e and quantity of Air that $hall have i$$ued from th'other Ve$$el: that Water therefore being kept, and returning to weigh the Ve$$el lightned of the Air com- pre$$ed (which I $uppo$e to have been weighed likewi$e fir$t with the $aid forced Air) and the $uperfluous $and being laid by, as I directed before; it is manife$t, that this is the ju$t weight of $o much Air in ma$$e, as is the ma$$e of the expul$ed and re$erved Water; which we are to weigh, and $ee how many times its weight $hall contain the weight of the re$erved $and: and we may without errour affirme, that the Water is $o many times heavier than Air; which $hall not be ten times, as it $eemeth <I>Ari$totle</I> held, but very neer four hundred, as the $aid Experiment $heweth.</P> <P>The other way is more expeditious, and it may be done with one Ve$$el onely, that is with the fir$t accomodated after the man- ner before directed, into which I will not that any other Air be put, more than that which naturally is found therein; but I will, that we inject Water without $uffering any Air to come out, which being forced to yield to the $upervenient Water mu$t of nece$$ity be compre$$ed: having gotten in, therefore, as much Water as is po$$ible, (but yet without great violence one cannot get in three quarters of what the Bottle will hold) put it into the Scales, and very carefully weigh it: which done, holding the Ve$$el with the neck upwards, open the Volve, letting out the Air, of which there will preci$ely i$$ue forth $o much as there is Water in the Bottle. The Air being gone out, put the Ve$$el again into the Scales, which by the departure of the Air will be found lightened, and abating from the oppo$ite Scale the $uperfluous weight, it $hall give us the weight of as much Air as there is Water in the Bottle.</P> <P>SIMP. The Contrivances you found out cannot but be con- <foot>fe$$ed</foot> <p n=>67</p> fe$$ed to be witty and very ingenuous, but whil$t, me thinks, they fully $atisfie my under$tanding, they another way occa$ion in me much Confu$ion, for it being undoubtedly true that the Ele- ments in their proper Region are neither heavy nor light, I can- not comprehend, how and which way that portion of Air, which $eemeth to have weighed <I>v. gr.</I> four drams of $and, $hould af- terwards have that $ame Gravity in the Air, in which the $and is contained that weigheth again$t it: and therefore me thinks that the Experiment ought not to be practiced in the Element of Air, but in a <I>Medium</I> in which the Air it $elf might exerci$e its quality of Gravitation, if it really be owner thereof.</P> <P>SALV. Certainly the Objection of <I>Simplicius</I> is very acute, and therefore its nece$$ary, either that it be unan$werable, or that the Solution be no le$$e acute. That that Air, which compre$- $ed, appeared to weigh as much as that $and, left at liberty in its Element is no longer to weigh any thing as the Sand doth, is a thing manife$t: and therefore for making of $uch an Experiment, its requi$ite to choo$e a place and <I>Medium</I> wherein the Air as well as the Sand might weigh: for, as hath $everal times been $aid, the <I>Medium</I> $ub$tracts from the Weight of every Matter that is im- merged therein, $o much, as $uch another quantity of the $aid <I>Medium,</I> as is that of the ma$$e immer$ed, weigheth: $o that the Air depriveth the Air of all its Gravity. The operation, there- <marg><I>The Air compre$- $ed and violently pent up, weigheth in a</I> Vacuum; <I>and how its weight is to be e$timated.</I></marg> fore, to the end it were made exactly, ought to be tried in a <I>Va- cuum,</I> wherein every grave Body would exerci$e its Moment without any diminution. In ca$e therefore, <I>Simplicius,</I> that we $hould weigh a portion of Air in a <I>Vacuum,</I> would you then be convinced and a$$ured of the bu$ine$$e?</P> <P>SIMP. Verily I $hould: but this is to defire, or enjoyn that which is impo$$ible.</P> <P>SALV. And therefore the obligation mu$t needs be great that you owe to me, when ever I $hall for your $ake effect an impo$$ibi- lity: but I will not $ell you that which I have already given you: for we, in the foregoing Experiment, weigh the Air in a <I>Vacuum,</I> and not in the Air, or in any other Replete <I>Medium.</I> That from the Ma$s, <I>Simplicius,</I> that in the fluid <I>Medium</I> is immerged certain Gravity is $ub$tracted by the $aid <I>Medium,</I> this commeth to pa$s by rea$on that it re$i$teth its being opened, driven back, and in a word commoved; a $ign of which is its pronene$s to return in$tant- ly to fill the Space up again, that the immer$ed ma$s occupied in it, as $oon as ever it departeth thence; for if it $uffered not by that immer$ion, it would not operate again$t the $ame. Now tell me, when you have in the Air the Bottle before filled with the $ame Air naturally contained therein, what divi$ion, repul$e, or, in $hort, what mutation doth the external ambient Air receive from the $e- <foot>K 2 cond</foot> <p n=>68</p> cond Air that was newly infu$ed with force into the Ve$$el? Doth it enlarge the Bottle, whereupon the Ambient ought the more to retire it $elf to make room for it? Certainly no: And therefore we may $ay, that the $econd Air is not immer$ed in the Ambient, not occupying any Space therein; but is as if it was in a <I>Vacuum,</I> nay more, is really con$tituted in it, and is placed in Vacuities that were not repleted by the former un-conden$ed Air. And, really, I know not how to di$cern any difference between the two Con$ti tutions of Inclo$ed and <I>Ambient,</I> whil$t in this the <I>Ambient</I> doth no-ways pre$s the Inclo$ed, and in that the Inclo$ed doth not re- repul$e the <I>Ambient</I>: and $uch is the placing of any matter in a <I>Vacuum,</I> and the $econd Air compre$sed in the Flask. The weight therefore that is found in that $ame conden$ed Air, is the $ame that it would have, were it freely di$tended in a <I>Vacuum.</I> Tis true in- deed, that the weight of the Sand that weigheth again$t it, as ha- ving been in the open Air, would in a <I>Vacuum</I> have been a little more than ju$t $o heavy; and therefore it is nece$$ary to $ay, that the weighed Air is in reality $omewhat le$$e heavy than the Sand that counterpoi$eth it, that is, $o much, by how much the like quantity of Air would weigh in a <I>Vacuum.</I></P> <P>SIMP. I had thought that there was $omething to have been wi$hed for in the Experiments before produced; but now I am thorowly $atisfied.</P> <marg><I>The difference, though very great, of the Gravity of Moveables hath no part in differer- cing their Veloci- ties.</I></marg> <P>SALV. The things by me hitherto alledged, and in particular, this, That the difference of Gravity, although exceeding great, hath no part in diver$ifying the Velocities of Moveables, $o that, notwith$tanding any thing depending on that, they would all move with equal Celerity, is $o new, and at the fir$t apprehen$i- on $o remote from probability, that, were there not a way to de- lucidate it, and make it as clear as the Sun, it would be better to pa$$e it over in $ilence, than to divulge it: therefore $eeing that I have let it e$cape from me, its fit that I omit neither Expe- riment nor Rea$on that may corroborate it.</P> <P>SAGR. Not onely this, but many other al$o of your A$$erti- ons are $o remote from the Opinions and Doctrines commonly received, that $ending them abroad, you would $tir up a great number of Antagoni$ts: in regard, that the innate Di$po$ition of Men doth not $ee with good eyes, when others in their Studies di$cover Truths or Fallacies, that were not di$covered by them- $elves: and with the title of Innovators of Doctrines, little plea- $ing to the ears of many, they $tudy to cut tho$e knots which they cannot untie, and with $ub-terranean Mines to blow up tho$e Structures, which have been with the ordinary Tools by patient Architects erected: but with us here, who are far from any $uch thoughts, your Experiments and Arguments are <foot>$uf-</foot> <p n=>69</p> $ufficient to give full $atisfaction: yet neverthele$$e, if $o be you have other more palpable Experiments, and more convincing Rea$ons we would very gladly hear them.</P> <P>SALV. The Experiment made with two Moveables, as different in weight as may be, by letting them de$cend from a place on high, thereby to $ee whether their Velocity be equal, meets with $ome difficulty: for if the height $hall be great, the <I>Medium,</I> which is to be opened and laterally repelled by the <I>Impetus</I> of the cadent Body, $hall be of much greater prejudice to the $mall Mo- ment of the light Moveable, than to the violence of the heavy one; whereupon in a long way the light one will be left behind: and in a little altitude it might be doubted whether there were really any difference, or if there were, whether it would be $en$ible. Therefore I have oft been thinking to reiterate the de- $cent $o many times from $mall heights, and to accumulate toge- ther $o many of tho$e minute differences of time, as might inter- cede between the arrival or fall of the heavy Body to the ground, and the arrival of the light one, which $o conjoyned, would make a time not onely ob$ervable, but ob$ervable with much facility Moreover, that I might help my $elf with Motions as $low as po$- $ible may be, in which the Re$i$tance of the <I>Medium</I> operates le$$e in altering the effect that dependeth on $imple Gravity, I have had thoughts to cau$e the Moveable to de$cend upon a de- clining Plane, not much rai$ed above the Plane of the Horizon; for upon this, no le$$e than in perpendicularity, we may di$cover that which is done by Grave Bodies different in weight: and pro- ceeding farther, I have de$ired to free my $elf from any what$o- ever impediment, that might ari$e from the Contact of the $aid Moveables upon the $aid declining Plane: and la$tly, I have ta- ken two Balls, one of Lead, and one of Cork, that above an hun- dred times more grave than this, and have fa$tened them to two $mall threads, each equally four or five yards long, tyed on high: and having removed a$wel the one as the other Ball from the $tate of Perpendicularity, I have let them both go in the $ame Moment, and they de$cending by the Circumferences of Circles de$cribed by the equal Strings their Semidiameters, and having pa$$ed beyond the Perpendicular, they afterwards by the $ame way returned back, and reiterating the$e Vibrations, and re- turns of them$elves neer an hundred times, they have $hewn ve- ry $en$ibly, that the grave <I>Pendulum</I> moveth $o exactly under the time of the light one, that it doth not in an hundred, no nor in a thou$and Vibrations, anticipate the time of one $mall moment, but that they keep an equal pa$$e in their Recur$ions. They al$o $hew the Operation of the <I>Medium,</I> which conferring $ome im- pediment on the Motion, doth much more dimini$h the Vibrati- <foot>ons</foot> <p n=>70</p> ons of the Cork, than that of the Lead: not that it maketh them more or le$$e frequent, nay, when the Arches pa$$ed by the Cork were not of above five or $ix degrees, and tho$e of the Lead fif- ty, they did pa$s them under the $ame times.</P> <P>SIMP. If this be $o, how is it then that the Velocity of the Lead is not greater than that of the Cork? that pa$$ing a jour- ney of $ixty degrees, in the time that this pa$seth hardly $ix?</P> <P><I>S</I>ALV. But what would you $ay, <I>Simplicius,</I> in ca$e they $hould both di$patch their Recur$ions in the $ame time, when the Cork being removed thirty degrees from the Perpendicular, $hould pa$s an arch of $ixty, and the Lead removed from the $ame middle point onely two degrees, $hould run an arch of four? would not then the Cork be $o much more $wift than the Lead? and yet Experience $hews that $o it happeneth: therefore ob$erve, The <I>Pendulum</I> of Lead being carried <I>v. gr.</I> fifty degrees from the Perpendicular, and thence let go, $wingeth, and pa$$ing beyond the Perpendicular, neer fifty more degrees, de$cribeth an arch of well neer an hundred degrees; and returning of its $elf back again, it de$cribeth another arch, not much le$$e than the former, and continuing its Vibrations, after a great number of them, it finally returneth to Re$t: Each of tho$e Vibrations are made un- der equal times a$wel tho$e of ninety degrees, as tho$e of fifty, twenty, ten, or four; $o that by con$equence, the Velocity of the Moveable doth $ucce$$ively langui$h and abate, in regard, that under equal times it doth $ucce$$ively pa$$e arches continually le$$er and le$$er. The like, yea the $elf $ame effect is performed by the Cork, hanging by a $tring of the like length, $ave that in a le$$e number of Vibracions it returneth to Re$t, as being le$s apt, by means of its Levity, to overcome the ob$tacle of the Air: and yet neverthele$s all the Vibrations, both great and $mall, are made under times equal to one another, and equal al$o to the times of the times of the Vibrations of the Lead. Whereupon it is true, that if whil$t the Lead pa$$eth an arch of fifty degrees, the Cork pa$seth one but of ten, the Cork is then more $low than the Lead: but it will al$o happen on the other $ide, that the Cork pa$seth the arch of fifty degrees, when the Lead pa$seth but that of ten or $ix; and $o in $everal times the Lead $hall be $wifter onewhile, and the Cork another while: but if the $ame Moveables $hall al$o under the $ame equal times, pa$s arches that are equal, one may then very $afely $ay, that their Velocities are equal.</P> <P>SIMP. This di$cour$e $eems to me concluding, and not con- cluding, and I finde in my thoughts $uch a Confu$ion, ari$ing from the one-while $wift, another-while $low, another-while ex- treme $low motion of both the one and other Moveable; as that <foot>it</foot> <p n=>71</p> it permits me not to di$cern clearly, whether it be true, That their Velocities are alwaies equal.</P> <P>SAGR. Give me leave, I pray you, <I>Salviatus,</I> to interpo$e two words. And tell me, <I>Simplicius,</I> whether you admit, that it may be $aid with ab$olute verity that the Velocities of the Cork and of the Lead are equal, in ca$e, that both of them departing at the $ame moment from Re$t, and moving by the $ame declivities, they $hould alwaies pa$$e equal Spaces in equal times?</P> <P>SIMP. This admits of no doubt, nor can it be contradicted.</P> <P>SAGR. It hapneth now in the Pendulums that each of them pa$$eth now $ixty degrees, now fifty, now thirty, now ten, now eight, four, and two; and when each of them pa$$eth the Arch of $ixty degrees they pa$$e it in the $ame time; in the Arch of fifty the $ame time is $pent by both the one and the other Moveable; $o in the Arch of thirty, of ten, and of the re$t: and therefore it is con- cluded, that the Velocity of the Lead in the Arch of $ixty degrees, is equal to the Velocity of the Cork in the $ame Arch of $ixty de- grees: and that the Velocities in the Arch of fifty, are likewi$e equal to one the other, and $o in the re$t. But it is not $aid, that the Velocity that is exerci$ed in the Arch of $ixty is equal to the Ve- locity that is exerci$ed in the Arch of fifty, nor this to that of the Arch of thirty. But the Velocities are alwaies le$$er, in the le$$er Arches. And this is collected from our $en$ibly $eeing the $ame Moveable con$ume as much time in pa$$ing the great Arch of $ixty degrees, as in pa$$ing the le$$er of fifty, or the lea$t of ten: and, in a word, in their being all pa$$ed alwaies under equal times. It is true therefore, that both the Lead and the Cork $ucce$$ively retard the Motion, according to the Diminution of the Arches, but yet do not alter their harmony in keeping the equality of Velocity in all the $ame Arches by them pa$$ed. I de$ired to $ay thus much, more to try whether I have rightly apprehended the Conceit of <I>Salvia- tus,</I> than out of any nece$$ity that I thought <I>Simplicius</I> to $tand in of a more plain Explanation than that of <I>Salviatus,</I> which is, as in all other things, extreamly clear, and $uch, that, it being fre- quent with him to re$olve Que$tions, in appearance not only ob- $cure, but repugnant to Nature, and to the Truth, with Rea$ons, or Ob$ervations, or Experiments very trite and familiar to every one, it hath (as I have under$tood from divers) given occa$ion to one of the mo$t e$teemed Profe$$ors of our Age to put the le$$e e$teem upon his Novelties, holding them to have as much of Sor- didne$$e, for that they depend on over low and popular Funda- mentals: as if the mo$t admirable and mo$t-to-be-prized Proper- ty of the Demon$trative Sciences, were not to $pring and ari$e from Principles known, under$tood, and granted by every one. But let us, for all that, continue to banquet our $elves with this diet <foot>that</foot> <p n=>72</p> that is $o light of dige$tion; and $uppo$ing that <I>Simplicius</I> is fully $atisfied in under$tanding and admitting, That the intern Gravity of different Moveables hath no $hare in differencing their Veloci- ties, $o that all of them, for ought that dependeth on that, would move with the $ame Velocities; tell us, <I>Salviatus,</I> in what you place the $en$ible and apparent inequalities of Motion; and an- $wer to that In$tance that <I>Simplicius</I> produceth, and which I like- wi$e confirm, I mean, of $eeing a Cannon Bullet move more $wift- ly than a drop of Bird-$hot, for the difference of Velocity $hall be but $mall, in re$pect of that which I object again$t you of Movea- bles of the $ame matter, of which $ome of the greater will de$cend in a <I>Medium,</I> in le$$e than one beat of the Pul$e, that $pace, that others which are le$$er will not pa$$e in an hour, nor in four, nor in twenty; $uch are pebbles and minute gravel-$tones, e$pecially, that $mall $and which muddieth the Water; in which <I>Medium</I> they will not de$cend in many hours $o much as two fathoms, which Stones, and tho$e of no great bigne$$e, do pa$$e in one beat of the Pul$e.</P> <P>SALV. That which the <I>Medium</I> operates, in retarding Movea- bles, the more according as they are compared to one another, le$s grave <I>in $pecie,</I> hath been already declared, $hewing that it pro- ceeds from the $ub$traction of weight. But how one and the $ame <I>Medium</I> can with $o great difference dimini$h the Velocity in Moveables that differ only in Magnitude, although they are of the $ame Matter, and of the $ame Figure, requireth for its expli- cation a more $ubtil di$cour$e, than that which $ufficeth for under- $tanding how the more dilated Figure of the Moveable, or the Motion of the <I>Medium</I> that is made contrary to the Moveable, re- <marg><I>The greater or le$s Scabro$ity and Po- ro$ity of the Super- ficies of Movea- bles, a probable cau$e of their grea- ter or le$$er Retar- dation.</I></marg> tardeth the Velocity of the $aid Moveable. I reduce the cau$e of the $aid Problem to the Scabro$ity, and Poro$ity, that is common- ly, and, for the mo$t part, nece$$arily found in the Superficies of Solid Bodies, the which Scabro$ities, in their Motion, go repul$ing and commoving the Air, or other Ambient <I>Medium</I>: of which we have an evident te$timony, in that we hear the Bodies, though made as round as is po$$ible for them to be, to hum whil$t they pa$$e ve- ry $wiftly thorow the Air; and they are not only heard to hum, but to whir and whi$tle, if there be but in them $ome more than ordi- nary cavity or prominency. We $ee al$o, that in turning round every rotund Solid maketh a little wind: And what need more? Do we not hear a notable whirring, and in a very $harp Accent, made by a Top, while it turneth round on the ground with great Celerity? The $hrilne$s of which whizzing groweth flatter accor- ding as the Velocity of the <I>Vertigo</I> doth by degrees more and more $lacken: a nece$$ary Argument likewi$e of the commotion and percu$$ion of the Air by tho$e (though very $mall) Scabro$i- <foot>ties</foot> <p n=>73</p> ties of their Superficies. It is not to be doubted, but that the$e in the de$cent of Moveables, grating upon, and repul$ing the fluid Am- bient, procure retardment in the Velocity, and $o much the greater, by how much the Superficies $hall be greater, as is that of le$$er Solids compared to bigger.</P> <P>SIMP. Stay, I pray you, for here I begin to be at a lo$$e: for though I under$tand and admit, that the Confrication of the <I>Medi- um</I> with the Superficies of the Moveable retardeth the Motion, and that it more retardeth it where <I>(ceteris paribus)</I> the Superficies is greater, yet do I not comprehend upon what ground you call the Superficies of le$$er Solids greater: & farthermore if, as you affirm, the greater Superficies ought to cau$e greater retardment, the greater Solids ought to be the $lower, which is not $o: but this Objection may ea$ily be removed, by $aying, that although the greater hath a greater Superficies, it hath al$o a greater Gravity, upon which the impediment of the greater Superficies hath not $o much more prevalent influence, than the impediment of the le$$er Superficies hath upon the le$$er Gravity, as that the Velocity of the greater Solid $hould become the le$$er. And therefore I $ee no rea$on why one $hould alter the equality of the Velocities, whil$t, that looking how much the Moving Gravity dimini$heth, the faculty of the Re- tarding Superficies doth dimini$h at the $ame rate.</P> <P>SALV. I will re$olve all that which you object in one word. Therefore, <I>Simplicius,</I> you will without controver$ie admit, that when, of two equal Moveables of the $ame Matter, and alike in Fi- gure (which undoubtedly would move with equal $wiftne$$e) as well the Gravity, as the Superficies of one of them dimini$heth, (yet $till retaining the $imilitude of Figure) the Velocity like- wi$e, for the $ame rea$on, would not be dimini$hed in that which was le$$ened.</P> <P>SIMP. Really, I think, that it ought $o to follow as you $ay, granting the pre$ent Doctrine with a <I>$alvo</I> $till to our Doctrine, which teacheth, that the greater or le$$er Gravity hath no operati- on in accelerating or retarding Motion.</P> <P>SALV. And this I confirm; and grant you likewi$e your Po- $ition, from whence, in my opinion, may be inferred, That in ca$e the Gravity dimini$heth more than the Superficies, there may be introduced in the Moveable, in that manner dimini$hed, $ome re- tardment of Motion, and that greater and greater, by how much in proportion, the diminution of the Weight was greater than the di- minution of the Superficies</P> <P>SIMP. I make not the lea$t que$tion of it.</P> <marg><I>Solids cannot be dimini$hed at the $ame rate in Super- ficies as in Weight, retaining the $imi- litude of the Fi- gures.</I></marg> <P>SALV. Now know, <I>Simplicius,</I> that in Solids one cannot di- mini$h the Superficies $o much as the Weight keeping the $imili- tude of Figure. For it being manife$t, that in dimini$hing of grave <foot>L Solids,</foot> <p n=>74</p> Solids, the Weight le$$eneth as much as the Bulk, when ever the Bulk happens to be dimini$hed more than the Superficies, (care being had to retain the $imilitude of Figure) the Gravity likewi$e would come to be more dimini$hed than the Superficies. But <I>Geo- metry</I> teacheth us, that there is much greater proportion between the Bulk and the Bulk in like Solids, than between their Superfi- cies. Which for your better under$tanding, I $hall explain in $ome particular ca$e. Therefore fancy to your $elf, for example, a Dye, one of the Sides of which is <I>v. gr.</I> two Inches long, $o that one of its Surfaces $hall be four Square Inches, and all $ix, that is, all its Superficies twenty four Square Inches. Then $uppo$e the $ame Dye at three $awings cut into eight $mall Dice, the Side of every one of which will be one Inch, and one of its Surfaces an Inch Square, and its whole Superficies $ix Square Inches, of which the whole Dye contained twenty four in its Superficial content. Now, you $ee, that the Superficial content of the little Dye is the fourth part of the Superficial content of the great one, (for $ix is the fourth part of twenty four) but the Solid content of the $aid Dye is only the eighth part: therefore the Bulk, and con$equently the Weight, doth much more dimini$h than the Superficies. And if you $ubdivide the little Dye into eight others, we $hall have for the whole Superficial content of one of the$e, one and an half Square Inches, which is the $ixteenth part of the Superficies of the fir$t Dye; but its Bulk, or Ma$s, is only the $ixty fourth part of that. You $ee therefore, how that in only the$e two divi$ions the Bulks decrea$e four times fa$ter than their Superficies: and if we $hould pro$ecute the Subdivi$ion, untill that we had reduced the fir$t So- lid into a $mall powder, we $hould find the Gravity of the minute Atomes to be le$$ened an hundred and an hundred times more than their Superficies. And this which I have exemplified in Cubes, hapneth in all like Solids, the Bulks of which are in Se$- quialter proportion of their Superficies. You $ee, therefore, in how much greater proportion the Impediment of the Contact of the Superficies of the Moveable with the <I>Medium</I> encrea$eth in $mall Moveables, than in greater: and if we $hould add, that the Sca- bro$ities in the very $mall Superficies of the minute Atomes are not happily le$$er than tho$e of the Superficies of greater Solids, that are diligently poli$hed, ob$erve how fluid, and void of all Re- $i$tance being opened, the <I>Medium</I> is required to be, when it is to give pa$$age to $o feeble a Virtue. And therefore take notice, <I>Sim- plicius,</I> that I did not equivocate, when even now I $aid, That the Superficies of le$$er Solids is greater, in compari$on of that of bigger.</P> <P>SIMP. I am wholly $atisfied: and I verily believe, that if I were to begin my Studies again, I $hould follow the Coun$el of <I>Plato,</I> <foot>and</foot> <p n=>75</p> and enter my $elf fir$t in the Mathematicks, which I $ee to proceed very $crupulou$ly, and refu$e to admit any thing for certain, $ave that which they nece$$arily demon$trate.</P> <P>SAGR. I have taken great delight in this Di$cour$e; but, be- fore we pa$$e any further, I would be glad to be $atisfied in one particular, which newly came into my thoughts, when but ju$t now you $aid, that Like-Solids are in Se$quialter proportion to their Superficies for I have $een, and under$tood the Propo$ition <marg><I>Solids are to each other in Se$quial- ter proportion to their Superficies.</I></marg> with its Demon$tration, in which it is proved, That the Superficies of Like-Solids are in duplicate proportion of their Sides; and ano- ther that proveth the $ame Solids to be in triple proportion of the $ame Sides; but the proportion of Solids to their Superficies, I do not remember that I ever $o much as heard it mentioned.</P> <P>SALV. You your $elf have an$wered and declared the doubt. For that which is triple of a thing of which another is double, doth it not come to be Se$quialter of this double? Yes doubtle$$e. Now, if Superficies are in double proportion of the Lines, of which the Solids are in triple proportion, may not we $ay, That the Solids are in Se$quialter proportion of their Superficies?</P> <P>SAGR. I under$tand you very well. And although other par- ticulars, pertaining to the matter of which we have treated, do re- main for me to ask, yet if we $hould thus run from one Digre$$ion to another, it will be late before we $hould come to the Que$tions principally intended, which concern the diver$ities of the Acci- dents of the Re$i$tances of Solids again$t Fraction; and therefore, if you $o plea$e, we may return to the fir$t Theme, which we pro- po$ed in the beginning.</P> <P>SALV. You $ay very well; but the $o many, and $o different things that have been examined, have $toln $o much of our time, that there is but little of it left in this day to $pend in our other principal Argument, which is full of Geometrical Demon$trati- ons that are to be con$idered with attention: $o that I $hould think it were better to adjourn our meeting till to morrow, as well for this which I have told you, as al$o becau$e I might bring with me $ome Papers, on which I have, in order, $et down the Theorems and Problems, in which are propo$ed and demon$trated the different Pa$$ions of this Subject, which, it may be, would not otherwi$e with requi$ite Method come into my mind.</P> <P>SAGR. I very gladly comply with your advice, and $o much the more willingly, in regard that, for a Conclu$ion of this daies Con- ference, I $hall have time to hear you re$olve $ome doubts that I find in my mind concerning the Point la$t handled. Of which one is, Whether we are to hold, that the Impediment of the <I>Medium</I> may be $ufficient to a$$ign bounds to the Acceleration of Bodies of very grave Matter, that are of great Bulk, and of a Spherical Figure: <foot>L 2 and</foot> <p n=>76</p> and I in$tance in the Spherical Figure, that I might take that which is contained under the lea$t Superficies, and therefore le$$e $ubject to Retardment. Another $hall be, touching the Vibrations of Pen- dulums, and this hath many heads: One $hall be, Whether all, both Great, Mean, and Little, are made really and preci$ely under equal Times: And another, What is the proportion of the Times of Moveables, $u$pended at unequal $trings, of the Times of their Vibrations I mean.</P> <P>SALV. The Que$tions are ingenious, and, like as it is incident to all Truths, I $uppo$e, that, which ever of them we handle, it will draw after it $o many other Truths, and curious Con$equences, that I cannot tell whether the remainder of this day may $uffice for the di$cu$$ing of them all.</P> <P>SAGR. If they $hall be but as delightful as the precedent, it would be more grateful for me to employ as many daies, not to $ay, hours, as it is unto night, and I believe that <I>Simplicius</I> will not be cloy'd with $uch Argumentations as the$e.</P> <P>SIMP. No certainly: and e$pecially, when the Que$tions trea- ted of are Phy$ical, touching which we read not the Opinions or Di$cour$es of other Philo$ophers.</P> <marg><I>Any Body, of any Figure, Greatne$s, and Gravity, is checked by the Re- nitence of the</I> Me- dium, <I>though ne- ver $o tenuous, in $uch $ort, that the Motion continuing, it is reduced to equability.</I></marg> <P>SALV. I come therefore to the fir$t, affirming without any hæ$itation, that there is not a Sphere $o big, nor of Matter $o grave, but that the Renitence of the <I>Medium,</I> though very tenuous, checks its Acceleration, and in the continuation of the Motion reduceth it to Equability, of which we may draw a very clear Argument from Experience it $elf. For if any falling Moveable were able in its continuation of Motion to attain any degree of Velocity, no Velocity that $hould be conferred upon it, could be $o great but that it would depo$e it, and free it $elf of it by help of the Impe- diment of the <I>Medium.</I> And thus, a Cannon-bullet, that had de $cended through the Air, <I>v. gr.</I> four yards, and had, for example, acquired ten degrees of Velocity, and that with the$e $hould enter into the Water, in ca$e the Impediment of the Water were not able to prohibit $uch a certain <I>Impetus</I> in the Ball, it would en- crea$e it, or at lea$t would continue it unto the bottom; which is not ob$erved to en$ue: nay, the Water, although it were but a few fathoms in depth, would impede and debilitate it in $uch a man- ner, that it will make but a $mall impre$$ion in the bottom of the River or Lake. It is therefore manife$t, that that Velocity, of which the Water had ability to deprive it in a very $hort way, would never be permitted to be acquired by it, though in a depth of a thou$and Fathoms. And why $hould it be permitted to gain it in a thou$and, to be taken from it again in four? What need we more? Do we not $ee the immen$e <I>Impetus</I> of the Ball, $hot from the Cannon it $elf, to be in $uch a manner flatted by the interpo- <foot>$ition</foot> <p n=>77</p> $ition of a few Fathom of Water, that without any harm to the Ship, it but very hardly reacheth to make a dent in it? The Air al- $o, though very yielding, doth neverthele$$e repre$$e the Velocity of the falling Moveable, although it be very heavy, as we may by $uch like Experiments collect; for if from the top of a very high Tower we $hould di$charge a Mu$quet downwards, this will make a le$$er impre$$ion on the ground, than if we $hould di$charge the Mu$quet at the height of four or $ix yards above the Plane: an evident $ign, that the <I>Impetus,</I> wherewith the Bullet i$$ueth from the Gun, di$charged on the top of the Tower, doth gradually di- mini$h in de$cending thorow the Air: therefore the de$cending from any what$oever great height will not $uffice to make it ac- quire that <I>Impetus,</I> of which the Re$i$tance of the Air deprived it, when it had in any manner been conferred upon it. The batte- ry likewi$e that the force of a Bullet, $hot from a Culverin, $hall make in a Wall at the di$tance of twenty Paces, would not, I be- lieve, be $o great, if the Bullet was $hot perpendicularly from any immen$e Altitude. I believe, therefore, that there is a Bound or term belonging to the Acceleration of every Natural Moveable that departs from Re$t, and that the Impediment of the <I>Medium</I> in the end reduceth it to ^{*} Equality, in which it afterwards alwaies <marg>* Or Equability.</marg> continueth.</P> <P>SAGR. The Experiments are really, in my opinion, much to the purpo$e: nor doth any thing remain, unle$$e the Adver$ary $hould fortifie him$elf, by denying, that they will hold true in great and ponderous Ma$$es, and that a Cannon-bullet coming from the Concave of the Moon, or from the upper Region of the Air, would make a greater percu$$ion than coming from the Cannon.</P> <P>SALV. There is no que$tion, but that many things may be objected, and that they may not be all $alved by Experiments; ne- verthele$$e in this contradiction, me thinks, there is $omething that may fall under con$ideration; <I>$cilicet,</I> that it is very probable, <marg><I>A Grave Body, falling from an Altitude, acqui- reth $o much</I> Im- petus <I>at its arri- val to the ground, as in all probabili- ty, would $uffice to recarry it to the $ame height from whence it fell.</I></marg> that the Grave Body, falling from an Altitude, acquireth $o much <I>Impetus,</I> at its arrival to the ground, as would $uffice to return it to that height, as is plainly $een in a <I>Pendulum</I> rea$onable weighty, that being removed fifty or $ixty degrees from the Perpendicular, gaineth that Velocity and Virtue which exactly $ufficeth to force it to the like Recur$ion, that little abated, which is taken from it by the Impediment of the Air. To con$titute, therefore, the Cannon- bullet in $uch an Altitude as may $uffice for the acqui$t of an <I>Impe- tus,</I> as great as that which the Fire giveth it in its i$$uing from the Piece, it would $uffice to $hoot it upwards perpendicularly with the $aid Cannon, and then ob$erving, whether in its fall it maketh an impre$$ion equal to that of the percu$$ion made near at hand in its i$$uing forth; but, indeed, I believe, that it would not be any <foot>whit</foot> <p n=>78</p> whit near $o forcible. And therefore I hold that the Velocity, which the Bullet hath near to its going out of the Piece, would be one of tho$e that the Impediment of the Air would never $uffer it to acquire, whil$t it $hould with a natural Motion de$cend, leaving the $tate of Re$t, from any great height. I come now to the other Que$tions belonging to <I>Pendulums,</I> matters which to many would $eem very frivolous, and more e$pecially to tho$e Philo$ophers that are continually bu$ied in the more profound Que$tions of Natural Philo$ophy: yet, notwith$tanding, will not I contemn them, being encouraged by the Example of <I>Ari$totle</I> him$elf, in whom I admire this above all things; that he hath not, as one may $ay, omitted any matter that any waies merited con$ideration, which he hath not $poken of: and now upon the Que$tions you propounded, I think I can tell you a certain conceit of mine upon $ome Problems con- cerning Mu$ick, a noble Subject, of which $o many famous men, and <I>Ari$totle</I> him$elf, have written; and touching it, he con$ide- reth many curious Problems: $o that if I likewi$e $hall from $o fa- miliar and $en$ible Experiments, draw Rea$ons of admirable acci- dents on the Argument of Sounds, I may hope that my di$cour$es will be accepted by you.</P> <P>SAGR Not only accepted, but by me, in particular, mo$t pa$- $ionately de$ired, in regard that I taking a great delight in all Mu- $ical In$truments, and being rea$onably well in$tructed concerning Con$onances, have alwaies been ignorant and perplexed with endeavouring to know, whence it cometh that one $hould more plea$e and delight me than another; and that $ome not only pro- cure me no delight, but highly di$plea$e me: the trite Ptoblem al- $o of the two Chords $et to an Uni$on, one of which moveth and actually $oundeth at the touching of the other, I al$o am unre$ol- ved in: nor am I very clearly informed concerning the Forms of Con$onances, and other particularities.</P> <P><I>S</I>ALV. We will $ee, if from the$e our <I>Peudulums</I> one may ga- ther any $atisfaction in all the$e Doubts. And as to the fir$t Que- $tion, that is, Whether the $ame <I>Pendulum</I> doth really and punctu- ally perform all its Vibrations, great, le$$er, and lea$t, under Times preci$ely equal; I refer my $elf to that which I have heretofore learnt from our <I>Academian,</I> who plainly demon$trateth, that the <marg><I>Moveables de$cen- ding along the Chords, that are Subten$es to any Arch of a Circle, pa$$e as well the greater as the le$- $er Chords in equal Times.</I></marg> Moveable that $hould de$cend along the Chords, that are Subten- $es to any Arch, would nece$$arily pa$$e them all in equal Times, as well the Subten$e under an hundred and eighty degrees, (that is, the whole Diameter) as the Subten$es of an hundred, $ixty, ten, two, or half a degree, or of four minutes: $till $uppo$ing that they all determine in the lowe$t Point touching the Horizontal Plane. Next as to the de$cendents by the Arches of the $ame Chords eli- vated above the Horizon, and that are not greater than a Qua- <foot>drant,</foot> <p n=>79</p> drant, that is, than ninety degrees, Experience likewi$e $hews, that <marg><I>Moveables and</I> Pendula <I>de$cend- ing along the Ar- ches of the $ame Chords, elivated as far as 90 deg. pa$s the $aid Arches in Times equal, but that are $horter than the tran$iti- ons along the Chords.</I></marg> they pa$$e all in Times equal, but yet $horter than the Times of the pa$$ages by the Chords: an effect which hath $o much of won- der in it, by how much at the fir$t apprehen$ion one would think the contrary ought to follow: For the terms of the beginning, and the end of the Motion being common, and the Right-Line be- ing the $horte$t, that can be comprehended between the $aid Terms, it $eemeth rea$onable, that the Motion made by it $hould be fini$hed in the $horte$t Time, which yet is not $o: but the $hor- te$t Time, and con$equently, the $wifte$t Motion, is that made by the Arch of which the $aid Right-Line is Chord. In the next <marg><I>The Times of the Vibrations of Mo- vables, hanging at alonger or $horter thread, are to one another in propor- tion $ubduple the lengths of the $trings, at which they hang.</I></marg> place, as to the Times of the Vibrations of Moveables, $u$pended by $trings of different lengths, tho$e Times are in Subduple pro- portion to the lengths of the $trings, or, if you will, the lengths are in duplicate proportion to the Times, that is, are as the Squares of the Times: $o that if, for example, the Time of a Vibration of one <I>Pendulum</I> is double to the Time of a Vibration of another, it followeth, that the length of the $tring of that is quadruple to the length of the $tring of this. And in the Time of one Vibration of that, another $hall then make three Vibrations, when the $tring of that $hall be nine times as long as the other. From whence doth follow, that the length of the $trings have to each other the $ame proportion, that the Squares of the Numbers of the Vibrations that are made in the $ame Times have.</P> <P>SAGR. Then, if I have rightly under$tood you, I may ea$ily <marg><I>To find the Length of any Rope, or $tring, at which a Moveable hang- eth, by the frequen- cy of its Vibrations</I></marg> know the length of a $tring, hanging at any never-$o-great height, although the $ublime term of the $u$pen$ion were invi$ible to me, and I only $aw the other lower extream. For if I $hall fa$ten a weight of $ufficient Gravity to the $aid $tring here below, and $et it on vibrating to and again, and a friend telling $ome of its Recur- $ions, and I at the $ame time tell the Recur$ions of another Movea- ble, $u$pended at a $tring that is preci$ely a yard long, by the Numbers of the Vibrations of the$e <I>Pendula,</I> made in the $ame Time, I will find the length of the $tring. As for example, $uppo$e that in the time that my friend hath counted twenty Recur$ions of the long $tring, I had told two hundred and forty of my $tring, that is one yard long: $quaring the two numbers twenty and two hundred and forty, which are 400, and 57600, I will $ay, that the long $tring containeth 57600 of tho$e Mea$ures, of which my $tring containeth 400. and becau$e the $tring is one $ole yard, I will divide 57600 by 400, and the quotient will be 144, and I will af- firm that $tring to be 144 yards long.</P> <P>SALV. Nor will you be mi$taken one Inch; and e$pecially, if you take a great Number of Vibrations.</P> <P>SAGR. You give me frequent occa$ion to admire the Riches, <foot>and</foot> <p n=>80</p> and withal the extraordinary bounty of Nature, whil'$t by things $o common, and, I might in a certain $ence $ay, vile, you go col- lecting of Notions very curious, new, and oftentimes, remote from all imagination. I have an hundred times con$idered the Vi- brations, in particular, of the Lamps in $ome Churches, hanging by very long ropes, when they have been unawares $tirred by any one: but the mo$t that I inferred from that $ame Ob$ervati- on, was the improbability of the Opinion of tho$e who hold, that $uch-like Motions are maintained and continued by the <I>Medi- um,</I> that is by the Air: for it $hould $eem to me, that the Air had a great judgment, and withal but little bu$ine$$e to $pend $o ma- ny hours time in vibrating an hanging Weight with $o much Regu- larity: but that I $hould have learnt, that that $ame Moveable, $u$pended at a $tring of an hundred yards long, being removed from Perpendicularity one while ninety degrees, and another while one degree onely, or half a degree, $hould $pend as much time in pa$$ing this little, as in pa$$ing that great Arch, certainly would never have come into my head, for I $till think, that it bordereth upon Impo$sibility. Now I am in expectation to hear that the$e petty Notions will a$sign me $uch Rea$ons of tho$e Mu$ical Pro- blems, as may, in part at lea$t, give me $atisfaction.</P> <marg><I>Every</I> Pendulum <I>hath the Time of its Vibration $o li- mited; that it is not po$$ible to make it move under any other Period.</I></marg> <P>SALV. Above all things, you are to know, that every <I>Pendu- lum</I> hath the Time of its Vibrations $o limited, and prefixed, that it is impo$$ible to make it move under any other Period, than that onely one, which is natural unto it. Let any one take the $tring in hand, to which the Weight is fa$tened, and trie all the wayes he can to encrea$e or decrea$e the frequency of its Vibrations, and he $hall finde it labour in vain: but we may, on the contrary, on a <I>Pendulum,</I> though grave and at re$t, by onely blowing up- on it, conferre a Motion, and a Motion con$iderably great, by reiterating the bla$ts, but under the Time that is properly be- longing to its Vibrations: for if at the fir$t bla$t we $hould have re- moved it from Perpendicularity half an Inch, adding a $econd, after that it being returned towards us, is ready to begin the $e- cond Vibration, we $hould conferre new Motion on it, and $o $ucce$$ively with other bla$ts, but given in Time, and not when the <I>Pendulum</I> is comming towards us (for $o we $hould impede; and not help the Motion) and $o continuing with many Impul- $es, we $hould confer upon it $uch an <I>Impetus,</I> that a greater force by much than that of a bla$t of our breath, will be required to $tay it.</P> <P>SAGR. I have, from my childhood, ob$erved, that one man a- lone, by means of the$e Impul$es, given in Time, hath been able to towl a very great Bell, and when it was to cea$e, I have $een four or $ix men more lay hold on the Bell-rope, and they have all <foot>been</foot> <p n=>81</p> been rai$ed from the ground: $o many together being unable to arre$t that <I>Impetus,</I> which one alone, with regular Pulls, had con- ferred upon the Bell.</P> <P>SALV. An example, that declareth my meaning with no le$$e <marg><I>The Chord of a Mu$ical In$tru- ment touched, mo- veth, and maketh the Chords $et to an Uni$on, Fifth and Eighth, with it to $ound; and why.</I></marg> propriety than this that I have premi$ed, doth $ute to render the rea$on of the admirable Problem of the Chord of the Lute or Viol, which moveth, and maketh not onely that really to $ound, which is tuned to the Uni$on, but that al$o which is $et to an Eighth and a Fifth. The Chord being toucht, its Vibrations begin, and continue all the Time that its Sound is heard to endure: the$e Vibrations make the Air neer adjacent to vibrate and tremble, who$e tremblings and quaverings di$tend them$elves a great way, and $trike upon all the Chords of the In$trument, and al$o of o- <marg><I>Sundry Problems touching Mu$ical Proportions, and their Solutions.</I></marg> thers neer unto it: the Chord that is $et to an Uni$on, with that which is toucht, being di$po$ed to make its Vibrations ^{*} in the $ame Time, beginneth at the fir$t impul$e to move a little, and <marg>* Or under.</marg> a $econd, a third, a twentieth, and many more, overtaking it, all in ju$t and Periodick Times, it receiveth at la$t, the $ame Tre- mulation, with that fir$t touched, and one may clearly $ee it go, dilating its Vibrations exactly according to the Pace of its Mo- ver. This Undulation that di$tendeth it $elf thorow the Air, mo- veth, and makes to vibrate, not onely the Chords, but likewi$e any other Body di$po$ed to trembling, and to vibrate in the very Time of the trembling Chord: $o that if we fix in the Sides of the In$trument $everal $mall pieces of Bri$tles, or of other flexible matters, you $hall $ee upon the $ounding of the Viol, now one, now another of tho$e Corpu$cles tremble, according as that Chord is toucht, who$e Vibrations return in the $ame Time: the others will not move at the $triking of this Chord, nor will that Bri$tle tremble at the $triking of another Chord. If with the Bow one $martly $trike the Ba$e-Chord of a Viol, and $et a drinking Gla$$e, thin and $mooth, neer unto it, if the Tone of the Chord be an Uni$on to the Tone of the Gla$$e, the Gla$$e $hall dance, and $en$ibly re-$ound. Again, the ample dilating of the Tremor or Undulation of the <I>Medium</I> about the Body re$ounding, is ap- parently $een in making the Gla$$e to $ound, by putting a little Water in it, and then chafing the brim or edge of it with the tip of the finger: for the included Water is ob$erved to undulate in a mo$t regular order: and the $ame effect will be yet more clearly $een, by $etting the foot of the Gla$$e in the bottom of a rea$o- nable large Ve$$el, in which there is Water as high almo$t as to the brim of the Gla$$e, for making it to $ound, as before, with the Confrication of the finger, we $hall $ee the trembling of the Water to diffu$e it $elf mo$t regularly, and with great Velocity, to a great di$tance round about the Gla$$e; and it hath many <foot>M times</foot> <p n=>82</p> times been my fortune, in making a rea$onable big Gla$$e, almo$t full of Water, to $ound as afore$aid, to $ee the Waves in the Water, at fir$t formed with an exact equality; and it hapning $ometimes, that the Tone of the Gla$$e ri$eth an Eighth higher, at the $ame in$tant, I have $een every one of the $aid Waves to divide them$elves in two: an accident that very clearly proveth the forme of the Octave to be the double.</P> <P>SAGR. The $ame hath al$o befaln me more than once, to my delight, and al$o benefit: for I $tood a long time perplexed a- bout the$e Forms of Con$onants, not conceiving, that the Rea- $on, commonly given thereof by the Authours that have hither- to written learnedly of Mu$ick, were $ufficiently convincing, they tell us, that the Diapa$on, that is the Eighth, is contained by the double, the Diapente, which we call the Fifth, by the Se$quialter: for a Chord being di$tended on the ^{*} Monochord, <marg>* An In$trument of but one $tring; called by <I>Mar- $ennus la Tromper- te Marine.</I></marg> $triking it all; and afterwards $triking but the half of it, by pla- cing a Bridge in the middle, one heareth an Eighth; and if the Bridge be placed at a third of the whole Chord, touching the whole, and then the two thirds, it $oundeth a Fifth; whereupon they infer, that the Eighth is contained between two and one, and the Fifth between three and two. This Rea$on, I $ay, $eemed to me not nece$$arily concluding for the a$$igning ju$tly the double and the Se$quialter, for the natural Forms of the Diapa$on and the Diapente. And that which moved me $o to think, was this. There are three ways, by which we may $harpen the Tone of a Chord: one is, by making it $horter, the other is by di$tending; or making it more ten$e; and the third is by making it thinner. If, retaining the $ame Tention and thickne$$e, we would hear an Eighth, it is nece$$ary to $horten it to one half, which is done by $triking it all, and then half. But if, retaining the $ame length and thickne$$e, we would have it ri$e to an Eighth, by $crewing it higher, it will not $uffice to $tretch it double as much, but we $hall need the quadruple, $o that, if before it was $tretched by a Weight of one pound, it will be needful to fa$ten four pound to it to $harpen it to an Eighth. And la$tly, if, keeping the $ame length and Tention, we would have a Chord, that by being $mal- ler, rendereth an Eighth, it will be nece$$ary, that it retain onely a fourth part of the thickne$$e of the other more Grave. And this which I $peak of the Eighth, that is, that its form taken from the Tention, or from the thickne$$e of the Chord, is in duplicate proportion to that which it receiveth from the length, is to be under$toood of all other Mu$ical Intervals: for that which the length giveth us in a Se$quialter proportion, <I>i. e.</I> by $triking it all, and then the two thirds, if you would have it proceed from the Tention, or from the di$gro$$ing, you mu$t double the Se$qui- <foot>alter</foot> <p n=>83</p> alter proportion, taking the double Se$quiquartan: and if the Grave Chord were $tretched by four pound weight, fa$ten to the Acute not $ix, but nine: and, as to the thickne$$e, make the Grave Chord thicker than the Acute, according to the proportion of nine to four, to have the Fifth. The$e being mo$t exact Experi- ments, I thought, that I $aw no rea$on, why the$e Sage Philo$o- phers $hould e$tabli$h the form of the Eighth to be rather the dou- ble, than quadruple; and the Form of the Fifth to be rather the Se$quialter, than the double Se$quiquartan. But becau$e the numbring of the Vibrations of a Chord, which in giving a $ound, are extreme frequent, is altogether impo$$ible, I $hould always have been in doubt, whether or no it were true, that the more Acute Chord of the Eighth, made in the $ame time, double the number of the Vibrations of the more Grave, if the Waves, which may be continued as long as you plea$e, by making the Gla$s to $ound and vibrate, had not $en$ibly $hewn me, that in the $elf $ame moment that ($ometimes) the Sound is heard to ri$e to an Eighth, there are $een to ari$e other Waves more minute, which with infinite $moothne$s cut in the middle each of tho$e fir$t.</P> <P>SALV. An excellent Ob$ervation for di$tingui$hing one by one the Undulations ari$ing from the Tremulation of the re- $ounding Body: which are tho$e that diffu$ing them$elves tho- row the Air, make the titillation upon the Drum of our Ear, that in our Soul becommeth a Sound: But whereas beholding and ob- $erving them in the Water, endure no longer than the confrica- tion of the finger la$teth, and al$o in that time they are not per- manent, but are continually made and di$$olved, would it not be an ingenious undertaking, if one could make, with much exqui$itene$$e, $uch, as would continue a long time; I mean Moneths and Years, $o as to give a man opportunity mea$ure, and with ea$e to number them?</P> <P>SAGR. I a$$ure you I $hould highly value $uch an Invention.</P> <P>SALV. The di$covery was accidental, and the Ob$ervation and applicative improvement of it onely were mine, and I hold it to be a Circum$tance of noble Contemplation, althongh a bu$i- ne$$e in its $elf $ufficiently homely. Scraping a Bra$$e Plate with an Iron Chizzel to fetch out $ome Spots, in moving the Chizzel to and again upon it pretty quick, I heard it (once or twice among$t many gratings) to Sibilate and $end forth a whi$tling noi$e, very $hrill and audible: and looking upon the Plate, I $aw a long row of $mall $treaks, parallel to one another, and di$tant from one another by mo$t equal Intervals: returning to my $craping again, I perceived by $everal trials, that in tho$e $crapings, and tho$e onely that whi$tled, the Chizzel left the $treaks upon the <foot>M 2 Plate:</foot> <p n=>84</p> Plate: but when the Scraping pa$$ed without any Sibilation, there was not $o much as the lea$t $ign of any $uch $treaks. Re- peating the Experiment $everal times afterwards, $craping now with greater, now with le$$e velocity, the Sibilation hapned to be of a Tone $ometimes acuter, $ometimes graver; and I ob$erved the marks made in the more acute $ounds to be clo$er together, and tho$e of the more grave farther a$under: and $ometimes al$o, according as the $elf $ame $crape was made towards the end, with greater velocity than at the beginning, the $ound was heard to grow $harper, and the $treaks were ob$erved to $tand thicker, but ever with extream neatne$$e, and marked with exact equidi- $tance: and farther-more, in the Sibilating $crapes; I felt the Chizzel to $hake or tremulate in my hand, and a certain chilne$$e to run along my arm; and in $hort, I $aw the $ame effected upon the Toole, which we u$e to ob$erve in whi$pering, and after- wards $peaking aloud, for $ending forth the breath without forming a $ound, we do not perceive any moving in the throat and mouth, in compari$on of that which we di$cern to be in the Wind-pipe and Throat of every one, in $ending forth the voice; and e$pecially in grave and loud Tones. I have likewi$e $ome- times among$t the Chords of the Viols, ob$erved two that were Uni$ons to the Sibilations made by $craping after the manner I told you, and that were mo$t different in Tone, from which two they preci$ely were di$tant a perfect Fifth, and then mea$uring the intervals of the $treaks of both the Scrapes, I $aw the di- $tance that conteined forty five $paces of the one, conteined thirty of the other: which, indeed, is the Form attributed to the Diapente. But here, before I proceed any farther, I will tell you, that of the three manners of rendring a Sound Acute, that which you refer to the $lenderne$$e or finene$$e of the Chord, may with more truth be a$cribed to the Weight. For the alteration ta- ken from the thickne$$e, an$wereth, when the Chords are of the $ame matter; and $o a Gut-$tring to make an Eighth, ought to be four times thicker than the other Gut-$tring; and one of Wier four times thicker than another of Wier. But if I would make an Eighth with one of Wier to one of Gut-$tring, I am not to make it four times thicker, but four times graver, $o that, as to thickne$$e, this of Wier $hall not be four times thicker, but quadruple in Gravity, for $ome times it $hall be more $mall than its re$pon- dent to the Acuter Eighth, that is of Gut-$tring. Hence it com- meth to pa$$e that, $tringing an In$trument with Chords of Gold, and another with Chords of Bra$$e, if they $hall be of the $ame length, thickne$$e, and Tention, Gold being almo$t twice as heavy, the Strings $hall prove about a Fifth more Grave. And here it is to be noted, that the Gravity of the Moveable more re- <foot>$i$teth</foot> <p n=>85</p> $i$teth the Velocity, than the thickne$$e doth; contrary to what others at the fir$t would think: for indeed, in appearance, its more rea$onable, that the Velocity $hould be retarded by the Re$i$tance of the <I>Medium</I> again$t Opening in a Moveable thick and light, than in one grave and $lender: and yet in this ca$e it happeneth quite contrary. But pur$uing our fir$t Intent, I $ay, That the ncere$t and immediate rea$ons of the Forms of Mu$ical Intervals, is neither the length of the Chord, nor the Tention, nor the thickne$$e, but the proportion of the numbers of the Vibrations, and Percu$$ions of the Undulations of the Air that beat upon the Drum of our Ear, which it $elf al$o doth tremulate under the $ame mea$ures of Time. Having e$tabli$hed this Point, we may, perhaps, a$$ign a very apt rea$on, whence it commeth, that of tho$e Sounds that are different in Tone, $ome Couples are re- ceived with great delight by our Sence, others with le$s, and others occa$ion in us a very great di$turbance; which is to $eek a rea$on of the Con$onances more or le$$e perfect, and of Di$lo- nances. The mole$tation and har$hne$$e of the$e proceeds, as I believe, from the di$cordant Pul$ations of two different Tones, which di$proportionally $trike the Drum of our Ear: and the Di$$onances $hall be extreme har$h, in ca$e the Times of the Vi- brations were incommen$urable. For one of which take that, when of two Chords $et to an Uni$on, one is $ounded, and $uch a part of another, as is the Side of the Square of its Diameter; a Di$$onance like to the ^{*} Tritone, or Semi-diapente. Con$onan- <marg>* Or a fal$e Fifth.</marg> ces, and with plea$ure received, $hall tho$e Couples of Sounds be, that $hall $trike in $ome order upon the Drum; which order requireth, fir$t, that the Pul$ations made in the $ame Time be commen$urable in number, to the end, the Cartillage of the Drum, may not $tand in the perpetual Torment of a double inflection of allowing and obeying the ever di$agreeing Percu$$ions. Therefore the fir$t and mo$t grateful Con$onance $hall be the Eighth, being, that for every $troke, that the Grave-$tring or Chord giveth upon the Drum, the Acute giveth, two; $o that both beat together in every $econd Vibration of the Acute Chord; and $o of the whole number of $trokes, the one half accord to $trike together, but the $trokes of the Chords that are Uni$ons, alwayes joyn both together, and therefore they are, as if they were of the $ame Chord, nor make they a Con$onance. The Fifth delighteth likewi$e, in regard, that for every two $troaks of the Grave Chord, the Acute giveth three: from whence it followeth, that numbering the Vibrations of the Acute Chord, the third part of that number will agree to beat together; that is, two Solitary ones interpo$e between every couple of Con$onances; and in the Di- ate$$eron there interpo$e three. In the $econd, that is in the <I>Se$-</I> <foot><I>quioctave</I></foot> <p n=>86</p> <I>quioctave</I> Tone for every nine Pul$ations, one onely $trikes in Con- $ort with the other of the Graver Chord; all the re$t are Di$cords, and received upon the Drum with regret, and are judged Di$$o- nances by the Ear.</P> <P>SIMP. I could wi$h this Di$cour$e were a little explained.</P> <P>SALV. Suppo$e this line A B the Space, and dilating of a Vi- bration of the Grave Chord; and the line C D that of the Acute Chord, which with the other giveth the Eighth: and let A B be divided in the mid$t in E. It is manife$t, that the Chords begin- ing to move at the terms A and C, by that time the Acute Vibra- tion $hall be come to the term D, the other <fig> $hall be di$tended onely to the half E, which not being the bound or term of the Motion, it $trikes not: but yet a $troak is made in D. The Vibrations afterwards returning from D to C, the other pa$$eth from E to B, where- upon the two Percu$$ions of B and C $trike both together upon the Drum: and $o con- tinuing to reiterate the like $ub$equent Vi- brations; one $hall $ee, that the union of the Percu$$ions of the Vibrations C D with tho$e of A B, happen al- ternately every other time: but the Pullations of the terms A B are alwayes accompanied with one of C D, and that alwayes the $ame: which is manife$t, for $uppo$ing that A and C $trike to- gether; in the time that A is pa$$ing to B, C goeth to D, and returneth back to C: $o that the $troaks at B and C are al$o together. But now let the two Vibrations A B and C D be tho$e that produce the Diapente, the times of which are in proportion Se$quialter, and divide A B of the Grave Chord, in three equal parts in E and O; And $uppo$e the Vibrations to begin at the $ame moment from the terms A and C: It is manife$t, that at the $troke that $hall be made in D, the Vibration of A B $hall have got no farther than O, the Drum therefore receiveth the Pul$a- tion D onely: again in the return from D to C, the other Vibra- tion pa$$eth from O to B, and returneth to O, making the Pul- $ation in B, which likewi$e is $olitary, and in Counter-time, (an accident to be con$idered:) for we having $uppo$ed the fir$t Pul$ations to be made at the $ame moment in the terms A and C, the $econd, which was onely by the term D, was made as long after as the time of the tran$ition C D, that is A O, imports; but that which followeth, made in B, is di$tant from the other one- ly $o much as is the time O B, which is the half: afterwards con- tinuing the Recur$ion from O to A, whil$t the other goeth from C to D, the two Pul$ations come to be made both at once in A and D. There afterwards follow other Periods like to the$e, that <foot>is,</foot> <p n=>87</p> is, with the interpo$ition of two $ingle and $olitary Pul$ations of the Acute Chord, and one of the Grave Chord, likewi$e $olita- ry, is interpo$ed between the two $olitary $trokes of the Acute. So that if we did but $uppo$e the Time divided into Moments, that is, into $mall equal Particles: $uppo$ing that in the two fir$t moments, I pa$$ed from the Concordant Pul$ations made in A and C to O and D, and that in D, I make a Percu$$ion: and that in the third and fourth moment I return from D to C, $triking in C, and that from O, I pa$t to B, and returned to O, $triking in B; and that la$tly in the fifth and $ixth moment from O and C, I pa$t to A and D $triking in both: we $hall have the Pul$ations di$tributed with $uch order upon the Drum, that $uppo$ing the Pul$ations of the two Chords in the $ame in$tant, it $hall two moments after receive a $olitary Percu$$ion, in the third moment anothor, $oli- tary likewi$e, in the fourth another $ingle one, and two moments after, that is, in the $ixth, two together; and here ends the Period, and, if I may $o $ay, Anomaly; which Period is oft-times afterwards replicated.</P> <P>SAGR. I can hold no longer, but mu$t needs expre$$e the con- tent I take in hearing rea$ons $o appo$itely a$$igned of effects that have $o long time held me in darkne$$e and blindne$$e. Now I know why the Uni$on differeth not at all from a $ingle Tone: I $ee why the Eighth is the principal Con$onance, but withal $o like to an Uni$on, that, as an Uni$on, it is taken and cojoyned with others: it re$embleth an Uni$on, for that whereas the Pul- $ations of Chords $et to an Uni$on, keep time in $triking, the$e of the Grave Chord in an Eighth alwayes keep time with tho$e of the Acute, and of the$e one interpo$eth alone, and in equal di$tances, and as, one may $ay, without any variety, whereupon that Con$onance is over $weet. But the Fifth, with tho$e its Counter-times, and with the interpo$ures of two $olitary Pul$a- tions of the Acute Chord, and one of the Grave Chord, between the Couples of Di$cordant Pul$ations, and tho$e three $olitary ones, with an interval of time, as great as the half of that which interpo$eth between each Couple, and the $olitary Percu$$ions of the Acute Chord, maketh $uch a Titillation and Tickling upon the Cartillage of the Drum of the Ear, that al- laying the Dulcity with a mixture of Acrimony, it $eemeth at one and the $ame time to ki$$e and bite.</P> <P>SALV. It is convenient, in regard I $ee, that you take $uch de- light in the$e Novelties, that I $hew you the way whereby the Eye al$o, and not the Ear alone, may recreate it $elf in beholding the $ame $ports that the Ear feeleth. Su$pend Balls of Lead or o- ther heavy matter on three $trings of different lengths, but in $uch proportion, that while the longer maketh two Vibrations, <foot>the</foot> <p n=>88</p> the $horter may make four, and the middle one three; which will happen, when the longe$t containeth $ixteen feet, or other mea$ures, of which the middle one containeth nine, and the $horte$t four: and removing them all together from Perpendi- cularity, and then letting them go, you $hall $ee a plea$ing In- termixtion of the $aid <I>Pendulums</I> with various encounters, but $uch, that, at every fourth Vibration of the longe$t, all the three will concurre in one and the $ame term together, and then again will depart from it, reiterating anew the $ame Period: the which commixture of Vibrations, is the $ame, that being made by the Chords, pre$ents to the Ear an Eighth, with a Fifth in the mid$t. And if you qualifie the length of other $trings in the like di$po- $ure, $o that their Vibrations an$wer to tho$e of other Mu$ical, but Con$onant Intervals, you $hall $ee other and other Inter- weavings, and alwaies $uch, that in determinate times, and after determinate numbers of Vibrations, all the $trings (be they three, or be they four) will agree to joyn in the $ame moment, in the term of their Recur$ions, and from thence to begin $uch another Period: but if the Vibrations of two or more $trings are either Incommen$urable, $o, that they never return harmoniou$ly to ter- minate determinate numbers of Vibrations, or though they be not Incommen$urable, yet if they return not till after a long time, and after a great number of Vibrations, then the $ight is con- founded in the di$orderly order of irregular Intermixtures, and the Ear with wearine$$e and regret receiveth the intemperate Im- pul$es of the Airs Tremulations, that without Order or Rule, $ucce$$ively beat upon its Drum.</P> <P>But whither, my Ma$ters, have we been tran$ported for $o many hours by various Problems, and unlook't for Di$cour$es? We have made it Night, and yet we have handled few or none of the points propounded; nay we have $o lo$t our way, that I $car$e remember our fir$t entrance, and that $mall Introduction, which we laid down, as the Hypothe$is and beginning of the fu- ture Demon$trations.</P> <P>SAGR It will be convenient, therefore, that we break up our Conference for this time, giving our Minds leave to compo$e them$elves in the Nights Repo$e, that we may to Morrow (if you plea$e $o far to favour us) rea$$ume the Di$cour$es de$ired, and chiefly intended.</P> <P>SALV. I $hall not fail to be here to Morrow at the u$ual hour, to $erve and enjoy you.</P> <head><I>The End of the Fir$t Dialogue.</I></head> <p n=>89</p> <head>GALILEUS, HIS DIALOGUES OF MOTION.</head> <head>The Second Dialogue.</head> <head><I>INTERLOCUTORS,</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <P>SAGREDUS.</P> <P><I>Simplicius,</I> and I, $taid expecting your com- ing, and we have been trying to recall to memory our la$t Con$ideration, which, as the Principle and Suppo$ition, on which you ground the Conclu$ions that you in- tended to Demon$trate to us, was that Re$i$tance, that all Bodies have to <I>Fracti- on,</I> depending on that Cement, that con- nects and glutinates the parts, $o, as that they do not $eparate and divide without a powerful attraction: and our enquiry hath been, what might be the Cau$e of that Coherence, which in $ome Solids is very vigorous; propounding that of <I>Vacuum</I> for the principal, which afterwards occa$ioned $o many Digre$$ions as held us the whole day, and far from the <foot>N matter</foot> <p n=>90</p> matter at fir$t propo$ed, which was the Contemplation of the Re- $i$tances of Solids to Fraction.</P> <P>SALV. I remember all that hath been $aid, and returning to our begun di$cour$e; What ever this Re$i$tance of Solids to brea- king by a violent attraction, is $uppo$ed to be, it is $ufficient, that it is to be found in them: which, though it be very great again$t the $trength of one that draweth them $treight out, it is ob$erved to be le$$e in forcing them tran$ver$ely, or $idewaies: and thus we $ee, for example, a rod of Steel, or Gla$$e to $u$tain the length-waies a weight of a thou$and pounds, which, fa$tned at Right-Angles in- to a Wall, will break if you hang upon it but only fifty. And of this $econd Re$i$tance we are to $peak, enquiring, according to what proportions it is found in Pri$mes, and Cylinders of like and unlike figure, length, and thickne$s, and, withal, of the $ame mat- ter. In which Speculation, I take for a known Principle, that which in the Mechanicks is demon$trated among$t the Pa$$ions of the Vectis, which we call the Leaver: namely, That in that u$e of the Leaver, the Force is to the Re$i$tance in Reciprocal proportion, as the Di$tances from the Fulciment to the $aid Force and the Re- $i$tance.</P> <P>SIMP. This <I>Ari$totle,</I> in his Mechanicks, demon$trated before any other man.</P> <P>SALV. I am content to grant him the precedency in time, but for the firmne$$e o$ Demon$tration, I think, that <I>Archimedes</I> ought to be preferred far before him, on one $ole Propo$ition of whom, by him demon$trated in his Book, <I>De Equiponderantium,</I> depend the Rea$ons, not only of the Leaver, but of the greater part of the other Mechanick In$truments.</P> <P>SAGR. But $ince that this Principle is the foundation of all that which you intend to demon$trate to us, it would be very re- qui$ite, that you produce us the proof of this $ame Suppo$ition, if it be not too long a work, giving us a full and perfect informati- on thereof.</P> <P>SALV. Though I am to do this, yet it will be better, that I lead you into the field of all our future Speculations, by an enterance $omewhat different from that of <I>Archimedes</I>; and that, $uppo- $ing no more, but only that equal Weights, put into a Ballance of equal Arms, make an <I>Equilibrium,</I> (a Principle likewi$e $uppo$ed by <I>Archimedes</I> him$elf.) I come, in the next place, to demon- $trate to you, that not only it is as true as the other, That unequal Weights make an <I>Equilibrium</I> in a Stiliard of Armes unequal, ac- cording to the proportion of tho$e Weights Reciprocally $u$pen- ded, but that it is one and the $ame thing to place equal Weights at equal di$tances, as to place unequal Weights at di$tances that are in Reciprocal Proportion to the Weights. Now for a plain <foot>Demon-</foot> <p n=>91</p> Demon$tration of what I $ay, de$cribe a Solid Pri$m or Cylinder A B, [<I>as in</I> Figure 1. <I>at the end of this Dialogue,</I>] $u$pended by its ends at the Line H I, and $u$tained by two Cords, H A, and I B. It is manife$t, that if I $u$pend the whole by the Cord C, placed in the middle of the Beam or Ballance H I, the Pri$m A B will be equilibrated, one half of its weight, being on one $ide, and the other half on the other $ide of the Point of Su$pen$ion C by the Princi- ple that we pre$uppo$ed. Now let the Pri$m be divided into un- equal parts by the Line D, and let the part D A be grea- ter, and D B le$$er; and to the end, that $uch divi$ion being made, the Parts of the Pri$m may re$t in the $ame $cituation and con$ti- tution, in re$pect of the Line H I, let us help it with a Cord E D, which, being fa$tened in the Point E, $u$taineth the parts A D, and D B: It is not to be doubted, but that there being no local muta- tion in the Pri$m, in re$pect of the Ballance H I, it $hall remain in the $ame $tate of Equilibration. But it will re$t in the $ame Con- $titution likewi$e, if the Part of the Pri$m, that is now $u$pended at the two extreams, or ends with Cords A H and D E, be hanged at one $ole Cord G L, placed in the mid$t: and likewi$e the other part D B, will not change $tate, if $u$pended by the middle, and $u$tained by the Cord F M. So that the Cords H A, E D, and I B being untied, and only the two Cords G L, and F M being left, the <I>Equilibrium</I> will $till remain, the Su$pen$ion being $till made at the Point C. Now, here let us confider, that we have two Grave Bodies A D, and D B, hanging at the terms G and F of a Beam G F, in which the <I>Equilibrium</I> is made at the Point C: in $uch manner, that the di$tance of the $u$pen$ion of the Weight A D from the Point C, is the Line C G, and the other part C F, is the di$tance at which the other Weight D B hangeth. It remaineth, therefore, only to be demon$trated, that tho$e Di$tances have the $ame proportion to one another, as the Weights them$elves have, but reciprocally taken: that is, that the di$tance G C is to the di- $tance C F, as the Pri$m D B to the Pri$m D A, which we prove thus. The Line G E being the half of E H, and E F the half of E I, all G F $hall be equall to all H I, and therefore equal to C I: and taking away the common part C F, the remainder G C $hall be equal to the remainder F I, that is, to F E: and C E taken in common, the two Lines G E and C F $hall be equal: and, there- fore, as G E, is to E F, $o is F C, to C G: but as G C is to E F, $o is the double to the double; that is H E to E I; that is, the Pri$m A D to the Pri$m D B. Therefore by Equality of proportion, and by Conver$ion, as the di$tance G C is to the di$tance C F, $o is the Weight B D to the Weight D A: which is that that I was to demon$trate. If you under$tand this, I believe that you will not $cruple to admit, that the two Pri$mes A D, and D B make an <foot>N 2 <I>Equili-</I></foot> <p n=>92</p> <I>Equilibrium</I> in th Point C, for the half of the whole Solid A B is on the right hand of the Su$pen$ion C, and the other half on the left; and that in this manner there are repre$ented two equal Weights, di$po$ed and di$tended at two equal di$tances. Again, that the two Pri$mes A D, and D B, being reduced into two Dice, or two Balls, or into any two other Figures, (provided that they keep the $ame Su$pen$ions G and F) do continue to make their <I>Equilibrium</I> in the Point C, I believe none can deny, for that it is mo$t manife$t, that Figures change not weight, where the $ame quantity of matter is retained. From which we may gather the general Conclu$ion, That two Weights, whatever they be, make an <I>Equilibrium</I> at Di$tances reciprocally an$wering to their Gra- vities. This Principle, therefore, being e$tabli$hed, before we pa$s any farther, I am to propo$e to Con$ideration, how the$e Forces, Re$i$tances, Moments, Figures, may be con$idered in Ab$tract, and $eparate from Matter, as al$o in Concrete and conjoyned with Matter; and in this manner tho$e Accidents that agree with Figures, con$idered as Immaterial, $hall receive certain Modifica- tions, when we $hall come to add Matter to them, and con$equent- ly Gravity. As for example, if we take a Leaver, as for in$tance B A [<I>as in</I> Fig. 2.] which, re$ting upon the Fulciment E, we ap- ply to rai$e the heavy Stone D: It is manife$t by the Principle de- mon$trated, that the Force placed at the end B, $hall $uffice to equal the Re$i$tance of the Weight D, if $o be, that its Moment have the $ame proportion to the Moment of the $aid D, that the Di$tance A C hath to the Di$tance C B: and this is true, if we confider no other Moments than tho$e of the $imple Force in B, and of the Re$i$tance in D, as if the $aid Leaver were immaterial, and void of Gravity. But if we bring to account the Gravity al$o of the In$trument or Leaver it $elf, which hapneth $ometimes to be of Wood, and $ometimes of Iron; it is manife$t, that the weight of the Leaver, being added to the Force in B, it will alter the pro- portion, which it will be requi$ite to deliver in other terms. And therefore before we pa$$e any farther, it is nece$$ary, that we di- $tingui$h between the$e two waies of Con$ideration, calling that a taking it ab$olutely, when we $uppo$e the In$trument to be taken in Ab$tract, that is, disjunct from the Gravity of its own Matter; but conjoyning the Matter, as al$o the Gravity, with $imple and ab$olute Figures, we will phra$e the Figures conjoyn'd with the Matter, Moment, or Force compounded.</P> <P>SAGR I mu$t of nece$$ity break the Re$olution I had taken, not to give occa$ion of digre$$ing, for I $hould not be able to $et my $elf to hear what remaines with attention, if a certain $cruple were not removed that cometh into my head; and it is this, That I gue$$e you make compari$on between the Force placed in B, and <foot>the</foot> <p n=>93</p> the total Gravity of the Stone D, of which Gravity me thinks, that one, and that, very probably, the greater part, re$teth upon the Plane of the Horizon: $o that----</P> <P>SALV. I have rightly apprehended you, $o that you need $ay no more, but only take notice, that I named not the total Gravity of the Stone, but $pake of the Moment that it hath, and exerci$eth at the Point A, the extream term of the Leaver B A, which is ever le$s than the entire weight of the Stone; and is variable according to the Figure of the Stone, and according as it hapneth to be more or le$$e elevated.</P> <P>SAGR. I am $atisfied in that particular, but I have one thing more to de$ire, namely, that for my perfect information, you would demon$trate to me the way, if there be one, how I may find what part of the total weight that is, which cometh to be born by the $ubjacent Plane, and what that which gravitates upon the Leaver at the extream A.</P> <P><I>S</I>ALV. Becau$e I can give you $atisfaction in few words, I will not fail to $erve you: therefore, de$cribing a $light Figure thereof, be plea$ed to $uppo$e, that the Weight, who$e Center of Gravity is A, [<I>as in</I> Fig. 3.] re$teth upon the Horizon with the term B, and at the other end is born up by the Leaver C G, on the Fulciment N, by a Power placed in G: and that from the Center A, and term C, Perpendiculars be let fall to the Horizon, A O, and C F. I $ay, That the Moment of the whole Weight $hall have to the Moment of the whole Power in G, a proportion compounded of the Di- $tance G N to the Di$tance N C, and of F B to B O. Now, as the Line F B is to B O, $o let N C be to X. And the whole Weight A being born by the two Powers placeed in B and C, the Power B is to C, as the di$tance F O to O B: and by Compo$ition, the two Powers B and C together, that is, the total Moment of the whole Weight A, is to the Power in C, as the Line F B is to the Line B O; that is, as N C to X: But the Moment of the Power in C is to the Moment of the Power in G, as the Di- $tance G N is to N C: Therefore, by Perturbation of proportion, the whole Weight A is to the Moment of the Power in G, as G N to X: But the proportion of G N to X is compounded of the pro- portion G N to N C, and of that of N C to X; that is, of F B to B O: Therefore the Weight A is to the Power that bears it up in G, in a proportion compounded of G N to N C, and of that of F B to B O: which is that that was to be demon$trated. Now re- turning to our fir$t intended Argument, all things hitherto decla- red being under$tood, it will not be hard to know the rea$on, whence it cometh to pa$$e that</P> <foot><I>A</I></foot> <p n=>94</p> <head>PROPOSITION I.</head> <P><I>A Solid Pri$m or Cylinder of Gla$$e, Steel, Wood, or other Frangible Matter, that being $u$pended length- waies, will $u$tain a very great Weight hanged Thereat, will, Sidewaies, (as we $aid even now) be broken in pieces by a far le$$er Weight, according as its length $hall exceed its thickne$s.</I></P> <P>Wherefore let us de$cribe the Solid Pri$m A B C D, fixed into a Wall by the Part A B, and in the other extream $uppo$e the Force of the Weight E; (alwaies under$tanding the Wall to be erect to the Horizon, and the Pri$m or Cylinder fa$tened in the Wall at Right-An- gles) it is manife$t, that being to break, it will be broken in the place B, where the Mortace in the Wall $erveth for Fulciment, and B C for the part of the Leaver in which lieth the force, and the thick- ne$$e of the Solid B A is the other part of the Leaver, in which lieth the Re$i$tance, which con$i$teth in the unfa$tening, or divi- ding, that is to be made of the part of the Solid B D, that is with- out the Wall from that which is within: and by what hath been declared, the Moment <fig> of the Force placed in C, is to the Moment of the Re$i$tance that lieth in the thickne$$e of the Pri$m, that is, in the Connection of the Ba$e B A, with the parts con- tiguous to it, as the length C B is to the half of B A: And therefore the ab$olute Re$i$tance again$t Fraction that is in the Pri$m B D, (which ab$olute Re$i- $tance is that which is made by drawing it downwards, for at that time the motion of the Mover is the $ame with that of the Body Moved) again$t the fracture to be made by help of the Leaver <foot>B C,</foot> <p n=>95</p> B C, is as the Length B C to the half of A B in the Pri$m, which in the Cylinder is the Semidiameter of its Ba$e. And this is our fir$t Propo$ition. And ob$erve, that what I have $aid ought to be un- der$tood, when the Confideration of the proper Weight of the So- lid B D is removed: which Solid I have taken as weighing nothing. But in ca$e we would bring its Gravity to account, conjoyning it with the Weight E, we ought to add to the Weight E the half of the Weight of the Solid B D: $o that the Weight B D being <I>v. gr.</I> two pounds, and the Weight of E ten pounds, we are to take the Weight E, as if it were eleven pounds.</P> <P>SIMP. And why not as if it were twelve?</P> <P>SALV. The Weight E, <I>Simplicius,</I> hanging at the term C, gravitates in re$pect of B C, with all its Moment of ten pounds, whereas if only B D were pendent, it would weigh with its whole Moment of two pounds; but, as you $ee, that Solid is di$tributed thorow all the length B C, uniformly, $o that its parts near to the extream B, gravitate le$$e than the more remote: $o that, in a word, compen$ating tho$e with the$e, the weight of the whole Pri$m is brought to operate under the Center of its Gravity, which an$we- reth to the middle of the Leaver B C: But a Weight hanging at the end C, hath a Moment double to that which it would have hanging at the middle: And therefore the half of the Weight of the Pri$m ought to be added to the Weight E, when we would u$e the Moment of both, as placed in the Term C.</P> <P>SIMP. I apprehend you very well, and, if I deceive not my $elf, me thinks, that the Power of both the Weights B D and E, $o placed, would have the $ame Moment, as if the whole Weight of B D, and the double of E were hanged in the mid$t of the Leaver B C.</P> <P>SALV. It is exactly $o, and you are to bear it in mind. Here we may immediatly under$tand</P> <head>PROPOSITION II.</head> <P><I>How, and with what proportion, a Ruler, or Pri$m, more broad than thick, re$i$teth Fraction, better if it be forced according to its breadth, than according to its thickne$$e.</I></P> <P>For under$tanding of which, let a Pri$m be $uppo$ed A D: [<I>as in</I> Fig. 4.] who$e breadth is A C, and its thickne$s much le$$er C B: It is demanded, why we would attempt to break it edge-waies, as in the fir$t Figure it will re$i$t the great Weight T, but placed flat-waies, as in the $econd Figure, it will not re$i$t <foot>X, le$-</foot> <p n=>96</p> X, le$$er than T: Which is manife$ted, $ince we under$tand the Fulciment, one while under the Line B C, and another while under C A, and the Di$tances of the Forces to be alike in both Ca$es, to wit, the length <I>B</I> D. But in the fir$t Ca$e, the Di$tance of the Re- $i$tance from the Fulciment, which is the half of the Line C A, is greater than the Di$tance in the other Ca$e, which is the half of B C: Therefore the Force of the Weight T, mu$t of nece$$ity be grea- ter than X, as much as the half of the breadth C A is greater than half the thichne$$e B C, the fir$t $erving for the Counter-Leaver of C A, and the $econd of C B to overcome the $ame Re$i$tance, that is the quantity of the <I>Fibres,</I> or $trings of the whole Ba$e A B. Conclude we therefore, that the $aid Pri$m or Ruler, which is broader than it is thick, re$i$teth, bresking more the edge-waies than the flat-waies, according to the Proportion of the breadth to the thickne$s.</P> <P>It is requi$ite that we begin in the next place</P> <head>PROPOSITION III.</head> <P><I>To find according to what proportion the encrea$e of the Moment of the proper Gravity is made in a Pri$m or Cylinder, in relation to the proper Re$i$tance again$t Fraction, whil$t that being parallel to the Horizon, it is made longer and longer: Which Mo- ment I find to encrea$e $ucce$sively in duplicate Pro- portion to that of the prolongation.</I></P> <P>For demon$tration whereof, de$cribe the Pri$m or Cylin- der A D, firmly fa$tned in the Wall at the end A, and let it be equidi$tant from the Horizon, and let the $ame be under$tood to be prolonged as far as E, adding thereto the part <I>B</I> E. It is manife$t, that the prolongation of the Leaver A <I>B</I> to C encrea$eth, by it $elf alone, that is taken ab$olutely, the Moment of the Force pre$$ing again$t the Re$i$tance of the Separation and Rupture to be made in A, according to the pro- portion of C A to <I>B</I> A: but, moreover, the Weight of the Solid affixed <I>B</I> E, encrea$eth the Moment of the pre$$ing Gravity of the Weight of the Solid A <I>B,</I> according to the Proportion of the Pri$m A E to the Pri$m A <I>B</I>; which proportion is the $ame as that of the length A C, to the length A <I>B</I>: Therefore it is clear <foot>that</foot> <p n=>97</p> that the two augmentations of the Lengths and of the Gravities being put together, the Moment compounded of both is in double <fig> proportion to ei- ther of them. We conclude there- fore, That the Mo- ments of the For- ces of Pri$mes and Cylinders of equal thickne$$e, but of unequal length, are to one another in duplicate proporti- on to that of their Lengths; that is, are as the Squares of their Lengths.</P> <P>We will $hew, in the $econd place, according to what proportion the Re$i$tance of Fraction in Pri$mes and Cylinders encrea$eth, when they continue of the $ame length, and encrea$e in thickne$s. And here I $ay, that</P> <head>PROPOSITION IV.</head> <P><I>In Pri$mes and Cylinders of equal length, but unequal thickne$s, the Re$i$tance again$t Fraction encrea$eth in a proportion iriple to the Diameters of their Thickne$$es, that is, of their Ba$es.</I></P> <P>Let the two Cylinders be the$e A and <I>B, [as in</I> Fig. 5.] who$e equal lengths are D G, and F H, the unequal <I>B</I>a$es the Circles, who$e Diameters are C D, and E F. I $ay, that the Re$i$tance of the Cylinder <I>B</I> is to the Re$i$tance of the Cylinder A again$t Fraction, in a proportion triple to that which the Diameter F E hath to the Diameter D C. For if we con$ider the ab$olute and $imple Re$i$tance that re$ides in the <I>B</I>a$es, that is, in the Circles E F, and D C to breaking, offering them vio- lence by pulling them end-waies, without all doubt, the Re$i$tance of the Cylinder <I>B,</I> is $o much greater than that of the Cylinder A, by how much the Circle E F is greater than C D; for the Fibres, Filaments, or tenacious parts, which hold together the Parts of the Solid, are $o many the more. <I>B</I>ut if we con$ider, that in offering <foot>O them</foot> <p n=>98</p> them violence tran$ver$ly we make u$e of two Leavers; of which the Parts or Di$tances, at which the Forces are applied are the Lines D G, and F H, the Fulciments are in the Points D and F; but the other Parts or Di$tances, at which the Re$i$tances are placed, are the Semidiameters of the Circles D C and E F, becau$e the Fila- ments di$per$ed thorow the whole Superficies of the Circles are as if they were all reduced into the Centers: con$idering, I $ay, tho$e Leavers, we would be under$tood to intend, that the Re$i$tance in the Center of the Ba$e E F again$t the Force of H, is $o much grea- ter than the Re$i$tance of the Ba$e C D, again$t the Force placed in G, (and the Forces in G and H are of equal Leavers D G, and F H) as the Semidiameter F E is greater than the Semidiameter D C, the Re$i$tance again$t Fraction, therefore, in the Cylinder B, encrea$eth above the Re$i$tance of the Cylinder A, according to both the proportions of the Circles E F and D C, and of their Semidiameters, or, if you will, Diameters: <I>B</I>ut the proportion of the Circles is double of that of the Diameters; Therefore the pro- portion of the Re$i$tances, which is compounded of them, is in triplicate proportion of the $aid Diameters: Which is that which I was to prove. <I>B</I>ut becau$e al$o the Cubes are in triplicate pro- portion to their Sides, we may likewi$e conclude, <I>That the Re$i- $tances of Cylinders of equal Length, are to one another as the Cubes of their Diameters.</I></P> <P>From that which we have Demon$trated we may likewi$e con- clude, that</P> <head>COROLARY.</head> <P><I>The Re$i$tances of Pri$ms, and Cylinders of equal length are in Se$quialter proportion to that of the $aid Cylinders.</I></P> <P>The which is manife$t, becau$e the Pri$ms and Cylinders, equal in height, are to one another, in the $ame proportion as their <I>B</I>a$es; that is, the double of the Sides or Diameters of the $aid <I>B</I>a$es: <I>B</I>ut the Re$i$tances (as hath been demon$trated) are in triplicate proportion to the $aid Sides or Diameters: Therefore the proportion of the Re$i$tances is Se$quialter to the proportion of the $aid Solids, and, con$equently, to the Weights of the $aid Solids.</P> <P>SIMP. It is convenient, that, before we proceed any farther, I be re$olved of a certain Doubt, and this it is, That I have not hi- therto heard propo$ed to Con$ideration another certain kind of Re$i$tance, that, in my opinion, is $ucce$$ively dimini$hed in So- lids, according as they are more and more prolonged, and not on- ly in u$ing them $idelongs, but al$o leng thwaies, in the $elf $ame <foot>manner</foot> <p n=>99</p> manner ju$t as we $ee a very long Cord to be much le$$e apt to $u$tain a great weight, than if it were $hort: $o that I believe, that a Ruler of Wood or Iron will $u$tain a much greater weight, if it $hall be $hort, than if it $hall be very long; under$tanding it al- waies to be u$ed lengthwaies, and not tran$ver$ly; and al$o its own weight being accounted for, which in the longer is greater.</P> <P>SALV. I fear, <I>Simplicius,</I> that in this Point you, with many others, are deceived, if $o be, that I have rightly apprehended your meaning, $o that you would $ay, that a Cord <I>v. gr.</I> forty yards long cannot $u$tain $o much, as if u$e were made but of one or two yards of the $ame Rope.</P> <P>SIMP. That is it, which I would have $aid, and as yet it $eemeth a very probable Propo$ition.</P> <P>SALV. But I hold it not only improbable, but fal$e: and think that I can very ea$ily reclaim you from your Errour. Therefore let us $uppo$e this Rope A B, [<I>as in</I> Fig. 6.] fa$tned on high by the end A, and by the other end let there hang the Weight C, by the force of which, the $aid Rope is to be broken. Do you a$$ign me the particular place, <I>Simplicius,</I> where the Rupture is to happen.</P> <P>SIMP. Let it be in the place D.</P> <P>SALV. I ask what is the cau$e why it $hould break in D.</P> <P>SIMP. The rea$on thereof is, becau$e the Rope was not $trong enough in that part, to $u$tain <I>v. gr.</I> an hundred pounds of weight, for $o much is the Rope D B with the Stone C.</P> <P>SALV. Therefore when ever $uch a Rope $hall come to be vio- lently $tretched by tho$e hundred pounds of weight, it $hall break in that place.</P> <P>SIMP So I think.</P> <P>SALV. But tell me now; if one did hang the $ame Weight, not at the end of the Rope <I>B,</I> but near to the point D, as for in$tance, in E, or el$e did tye the Rope not at the height A, but very near, and almo$t at the Point D it $elf, as in F, tell me, I $ay, whether the Point D would feel the $ame weight of an hundred pounds.</P> <P>SIMP. It would $o, $till joyning the piece of Rope E <I>B</I> to the Stone C.</P> <P>SALV. If then the Rope in the Point D commeth to be drawn by the $aid hundred pounds of weight, it will break by your con- ce$$ion. And yet F E, is a $mall piece of the length A <I>B</I>: why do you $ay then, that the long Rope is weaker than the $hort one? <I>B</I>e content, therefore, to $uffer your $elf to be reclaimed from an Errour, in which you have had many Companions, and tho$e in other things very knowing. And let us go on: and having demon- $trated, that Pri$ms and Cylinders encrea$e their Moments above <foot>O 2 their</foot> <p n=>100</p> their Re$i$tances, according to the Squares of their Lengths (alwaies provided, that they retain the $ame thickne$$e) and that likewi$e, the$e that are equally long, but different in thickne$$e, encrea$e their Re$i$tances according to the proportion of the Cubes of the Sides or Diameters of their Ba$es, we may enquire what befal- leth to tho$e Solids, being different in length and thickne$s, in which I ob$erve, that</P> <head>PROPOSITION V.</head> <P><I>Pri$ms and Cylinders, of different length and thickne$s, have their Re$i$tances again$t Fraction, in a propor- tion compounded of the proportion of the Cubes of the Diameters of their Ba$es, and of the proportion of their lengths reciprocally taken.</I></P> <P>Let the$e two A B C, and D E F, [<I>as in</I> Fig. 7.] be $uch Cy- linders. I $ay, the Re$i$tance of the Cylinder A C $hall be to the Re$i$tance of the Cylinder D F, in a proportion com- pounded of the proportion of the Cube of the Diameter A B, to the Gube of the Diameter D E, and of the proportion of the Length E F to the Length B C. Suppo$e E G equal to B C, and to the Lines A B, and D E, let C H be a third proportional, and I, a fourth; and as E F is to B C, $o let I be to S. And becau$e the Re$i$tance of the Cylinder A C is to the Re$i$tance of the Cylin- der D G, as the Cube A B to the Cube D E; that is, as the Line A B to the Line I: and the Re$i$tance of the Cylinder G D is to the Re$i$tance of the Cylinder D F, as the Length F E is to the Length E G; that is, as the Line I is to S: Therefore by Equali- ty of proportion, as the Re$i$tance of the Cylinder A C is to the Re$i$tance of the Cylinder D F, $o is the Line A B to S: But the Line A B is to S, in a proportion compounded of A B to I, and of I to S: Therefore the Re$i$tance of the Cylinder A C is to the Re- $i$tance of the Cylinder D F, in a proportion compounded of A B to I, that is, as the Cube of A B to the Cube of D E, and of the proportion of the Line I to S; that is, of the Length E F to the Length B C: Which was to be demon$trated.</P> <P>After the Propo$ition la$t demon$trated, we will con$ider what hapneth between like Cylinders and Pri$ms, of which we will de- mon$trate, how that</P> <foot>PRO-</foot> <p n=>101</p> <head>PROPOSITION VI.</head> <P><I>Of like Cylinders and Pri$ms the Moments compoun- ded, that is to $ay, re$ulting from their Gravities, and from their Lengths, which are, as it were, Lea- vers, have to one another a proportion Se$quialter to that which is between the Re$i$tances of their $ame Ba$es.</I></P> <P>For demon$tration of which let us de$cribe the two like Cy- linders A B, and C D, [<I>as in</I> Fig. 8.] I $ay, that the Mo- ment of the Cylinder A B, to overcome the Re$i$tance of its Ba$e B, hath to the Moment of C D, to overcome the Re$i$tance of its Ba$e C, a proportion Se$quialter to that which the $ame Re- $i$tance of the Ba$e B, hath to the Re$i$tance of the Ba$e D: And becau$e the Moments of the Solids A B, and C D, to over- come the Re$i$tances of their Ba$es B and D, are compounded of their Gravities, and of the Forces of their Leavers, and the Force of the Leaver A B is equal to the Force of the Leaver C D, and that becau$e the length A B hath the $ame proportion to the Semi- diameter of the Ba$e B, (by the $imilitude of the Cylinders) that the Length C D hath to the Semidiameter of the Ba$e D; it re- maineth, that the total Moment of the Cylinder A B, be to the total Moment of C D, as the $ole Gravity of the Cylinder A B is to the $ole Gravity of the Cylinder C D; that is, as the $aid Cy- linder A B is to the $aid C D: But the$e are in triplicate propor- tion to the Diameters of their Ba$es <I>B</I> and D; and the Re$i$tances of the $ame <I>B</I>a$es, being to one another as the $aid <I>B</I>a$es, they are con$equently in duplicate proportion to their $ame <I>B</I>a$es: There- fore the Moments of Cylinders are in Se$quialter proportion to the Re$i$tances of their <I>B</I>a$es.</P> <P>SIMP. This Propo$ition, indeed, is not only new, but unexpe- cted to me, and at fir$t $ight, very remote from the judgment that I $hould have conjecturally pa$t upon it: for in regard, that the$e Figures are in all other re$pects alike, I $hould have thought that their Moments likewi$e $hould have retained the $ame proportion towards their proper Re$i$tances.</P> <P>SAGR. This is the Demon$tration of that Propo$ition, that in the beginning of our Di$cour$es, I $aid, I thought------ I had $ome glimps of.</P> <P>SALV. That which now befalleth, <I>Simplicius,</I> hapned for $ome <foot>time</foot> <p n=>102</p> time to my $elf, believing, that the Re$i$tances of like Solids were alike, till that a certain, and that no very fixed or accurate Ob$er- vation $eemed to repre$ent unto me, that Solids do not contain an equal tenure in their Toughne$s, but that the bigger are le$$e apt to $uffer violent accidents, as lu$ty men are more damnified by their falls than little children; and, as in the begining we $aid, we $ee a great <I>B</I>eam or Column break to pieces falling from the $ame height, and not a $mall Pri$in or little Cylinder of Marble. This $ame Ob$ervation gave me the hint for finding of that which I am now about to demon$trate; a Quality truly admirable, for that among$t the infinite Solid-like Figures, there are not $o much as two, who$e Moments retain the $ame proportion towards their proper Re$i$tances.</P> <P>SIMP. Now you put me in mind of $omething in$erted by <I>Ari- $totle</I> among$t his Mechanical Que$tions, where he would give a Rea$on, whence it is, that <I>B</I>eams the longer they are, they are by $o much the more weak, and bend more and more, although the $hort ones be the $lendere$t, and the long ones thicke$t: and, if I well re- member, he reduceth the Rea$on to the $imple Leaver.</P> <P>SALV. It is very true, and becau$e the Solution $eemeth not wholly to remove the cau$e of doubting <I>Mon$ignore di Guevara,</I> who, the truth is, with his mo$t learned <I>Commentaries</I> hath highly enobled and illu$trated that Work, enlargeth him$elf with other accute Speculations for the obviating all difficulties, yet him$elf al$o remaining perplexed in this point, whether, the lengths and thickne$$es of $uch Solid Figures, encrea$ing with the $elf $ame proportion, they ought to retain the $ame tenure in their Tough- ne$$es and Re$i$tances again$t their breaking, and likewi$e again$t their bending. After I had long con$idered thereon, I have, in this manner found, that which I am about to tell you. And fir$t I will demon$trate that</P> <foot>PRO-</foot> <p n=>103</p> <head>PROPOSITION VII.</head> <P><I>Of like and heavy Pri$ms or Cylinders there is one only, and no more, that is reduced (being charged with its own weight) to the ultimate $tate between breaking and holding it $elf together: $othat every greater, as being unable to re$i$t its own weight, will break, and every le$$er re$i$teth $ome Force that is employed again$t it to break, it.</I></P> <P>Let the heavy Pri$m be A B [<I>as in</I> Fig 9.] reduced to the utmo$t length of its Con$i$tance, $o that being lengthned never $o little more it will break: I $ay, that this is the only one among$t all tho$e that are like unto it, (which yet are infinite) that is capable of being reduced to that dubious and tickli$h $tate; $o that every greater being oppre$$ed with its own weight will break, and every le$$er not, nay, will be able to re$i$t $ome additi- on of a new violence, over and above that of its own weight. Fir$t, take the Pri$m C E, like to, and greater than A B. I $ay, that this cannot con$i$t, but will break, being overcome by its own Gravity. Suppo$e the part C D as long as A B. And becau$e the Re$i$tance C D is to that of A B, as the Cube of the thickne$$e of C D to the Cube of the thickne$s of A B; that is, as the Pri$m C E to the Pri$m A B (being alike:) Therefore the Weight of C E is the greate$t that can be $u$tained at the length of the Pri$m C D: But the Length C E is greater: Therefore the Pri$m C E will break. But let F G be le$$et: it $hall be demon$trated like- wi$e ($uppo$ing F H equal to B A) that the Re$i$tance of F G is to that of A B, as the Pri$m F G is to the Pri$m A B, in ca$e that the Di$tance A B, that is F H, were equal to F G, but it is greater: Therefore the Moment of the Pri$m F G, placed in G, doth not $uffice to break the Pri$m F G.</P> <P>SAGR. A mo$t manife$t and brief Demon$tration, inferring the truth and nece$$ity of a Propo$ition that at fir$t $ight $eemeth far from probability. It would be requi$ite, therefore, to alter much the proportion betwixt the Length and Thickne$$e of the greater Pri$m by making it thicker or $horter, to the end it might be re- duced to that nice $tate of indifferency between holding and brea- king; and the Inve$tigation of that $ame State, as I think, would be no le$$e ingenuous.</P> <P>SALV. Nay, rather more, as it is al$o more laborious: and I am <foot>$ure</foot> <p n=>104</p> $ure I have $pent no $mall time to find it; and I will now impart it to you: Therefore</P> <head>PROP. VIII. PROBL. I.</head> <P><I>A Cylinder or Pri$m of the utmo$t length not to be bro- ken by its own weight, and al$o a greaver length, be- ing given, to find the thickne$$e of another Cylinder or Pri$m that under-given length is the only one, and bigge$t, that can re$i$t its own weight.</I></P> <P>Let the Cylinder B C [<I>as in</I> Fig. 10.] be the bigge$t that can re$i$t its own weight, and let D E be a Length greater than A C; it is required to find the Thickne$$e of the Cylin- der, that under the Length D E is the greate$t re$i$ting its own weight. Let I be a third proportional to the Lengths D E, and A C; and as D E is to I, $o let the Diameter F D be to the Dia- meter B A: and make the Cylinder F E. I $ay, that this is the big- ge$t, and only one among$t all that are like to it that re$i$teth its own weight. To the Lines D C and I let M be a third propor- tional, and O a fourth. And $uppo$e F G equal to A C. And be- cau$e the Diameter F D is to the Diameter A B, as the Line D E to I, and O is a fourth proportional to D E and I, the Cube of F D $hall be to the Cube of B A as D E is to O: But as the Cube of F D is to the Cube of B A, $o is the Re$i$tance of the Cylinder D G to the Re$i$tance of the Cylinder B C: Therefore the Re$i- $tance of the Cylinder D G is to that of the Cylinder B C, as the Line D F is to O. And becau$e the Moment of the Cylinder B C is equal to its Re$i$tance, if we $hew that the Moment of the Cylin- der F E is to the Moment of the Cylinder B C, as the Re$i$tance D F to the Re$i$tance B A; that is, as the Cube of F D to the Cube of B A; that is, as the Line D E to O, we $hall have our intent: that is, that the Moment of the Cylinder F E is equal to the Re$i- $tance placed in F D. The Moment of the Cylinder F E is to the Moment of the Cylinder D G, as the Square of D E is to the Square of A C; that is, as the Line D E to I: But the Moment of the Cylinder D G is to the Moment of the Cylinder B C, as the Square D F to the Square B A; that is, as the Square of D E to the Square of I; that is, as the Square of I to the Square of M; that is, as I to O: Therefore, by Equality of proportion, as the Mo- ment of the Cylinder F E is to the Moment of the Cylinder B C, $o is the Line D E to O; that is, the Cube D F to the Cube B A; that is, the Re$i$tance of the Ba$e D F to the Re$i$tance <foot>of</foot> <p n=>105</p> of the Ba$e B A: Which is that that was $ought.</P> <P>SAGR This, <I>Salviatus,</I> is a long Demon$tration, and very hard to be born in mind at the fir$t hearing, therefore I could wi$h, that you would plea$e to repeat it.</P> <P>SALV. I will do what you $hall command; but haply it would be better to produce one more conci$e and $hort: but then it will be requi$ite to de$cribe a Figure $omewhat different.</P> <P>SAGR. The favour will then be the greater: and be$tow upon me the draught of that already explained, that I may at my lea$ure con$ider it again.</P> <P>SALV. I will not fail to $erve you. Now, $uppo$e a Cylinder A, <marg><I>The la$t Problem performed another way.</I></marg> [<I>as in</I> Fig. 11.] the Diameter of who$e Ba$e let be the Line D C, and let this A be the greate$t that can $u$tain it $elf and not break, than which we will find a bigger, which likewi$e $hall be the big- ge$t al$o, and the only one that $u$taineth it $elf. Let us de$ire one like to the $aid A, and as long as the a$$igned Line, and let this be <I>v. gr.</I> E, the Diameter of who$e Ba$e let be K L; and to the two Lines D C, and K L let M N be a third proportional; which let be the Diameter of the Ba$e of the Cylinder X, in length equal to E. I $ay, that this X is that which we $eek. And becau$e the Re$i- $tance D C is to the Re$i$tance K L, as the Square D C to the <I>S</I>quare K L; that is, as the Square K L to the Square M N; that is, as the Cylinder E to the Cylinder X; that is, as the Moment E to the Moment X: But the Re$i$tance K L is to M N, as the Cube of K L is to the Cube of M N; that is, as the Cube B C to the Cube K L; that is, as the Cylinder A to the Cylinder E; that is, as the Moment A to the Moment E: Therefore, by Perturbation of proportion, as the Re$i$tance D C is to M N, $o is the Moment A to the Moment X: Therefore the Pri$m X, is in the $ame Con$ti- tution of Moment and Re$i$tance as the Pri$m A.</P> <P>But let us make the Problem more general, and let the Propo- $ition be this:</P> <P><I>The Cylinder</I> A C <I>being given, and its Moment to-</I> <marg><I>The la$t Propo$i- tion made more ge- neral.</I></marg> <I>wards its Re$i$tance being $uppo$ed at plea$ure, and any Length</I> D E <I>being a$signed, to find the Thick- ne$$e af the Cylinder who$e Length is</I> D E, <I>and who$e Moment towards its Re$i$tance retaineth the $ame proportion, that the Moment of the Cylinder</I> A C <I>doth to its Re$i$tance.</I></P> <foot>P Rea$$uming</foot> <p n=>106</p> <P>Rea$$uming the above $aid Figure and almo$t the $ame Me- thod, we will $ay: Becau$e the Moment of the Cylinder F E hath the $ame proportion to the Moment of the part D G, that the Square E D hath to the Square F G; that is that the Line D E hath to I: and becau$e the Moment of the Cylinder F G is to the Moment of the Cylinder A C, as the Square F D to the Square A B; that is, as the Square D E to the Square I; that is, as the Square I to the Square M; that is, as the Line I to O: Therefore, <I>ex æquali,</I> the Moment of the Cylinder F E hath the $ame proportion to the Moment of the Cylinder A C, that the Line D E hath to the Line O; that is, that the Cube D E hath to the Cube of I; that is, that the Cube of F D hath to the Cube of A B; that is, that the Re$i$tance of the Ba$e F D hath to the Re$i$tance of the Ba$e A B: Which was to be performed.</P> <P>Now, let it be ob$erved, that from the things hitherto demon$tra- ted, we may plainly gather, how Impo$$ible it is, not only for Art, but <marg>* Oares are u$ed in the Ships or Gallies of the Mediterrane, up- on which our Author was a Coa$ter.</marg> for Nature her $elf to encrea$e her Machines to an immen$e Va$t- ne$$e: $o that it would be impo$$ible by Art to build extraordina- ry va$t Ships, Palaces, or Temples, who$e ^{*} Oars, Sail-yards, Beams, Iron Bolts, and, in a word, their other parts $hould con$i$t or hold together: neither again could Nature make Trees of unmea$ura- <marg><I>Bones of Animals magnified beyond their ratural $ize, would not $ub$i$t, if it be required to retain the $ame proportion of thick- ne$s and hardne$s in them that is in tho$e of Natural Animals.</I></marg> ble greatne$$e, for that their Arms or Bows being oppre$$ed with their own weight would at la$t break: and likewi$e it would be impo$$ible for her to make $tructures of Bones for men, Hor$es, or other Animals, that might $ub$i$t, and proportionatly perform their Offices, when tho$e Animals $hould be augmented to im- men$e heights, unle$$e $he $hould take Matter much more hard and Refi$ting than that which $he commonly u$eth, or el$e $hould de- form tho$e Bones by augmenting them beyond their due Symetry, and making the Figure or $hape of the Animal to become mon- $trou$ly big: Which haply was hinted by my mo$t Witty Poet, where de$cribing an huge Giant, he $aith,</P> <P><I>Non $i puo compartir quanto $ia lungo,</I></P> <P><I>Si $mi$uratamente è tutto gro$$o.</I></P> <marg><I>Example of the Bone of an Animal enlarged to thrice the Natural pro- portion, how much thicker it ought to be to perform its office.</I></marg> <P>And for a $hort example of this that I $ay, [<I>as in</I> Fig. 12.] I have heretofore drawn the Figure of a Bone only trebled in Length, and augmented in Thickne$$e in $uch proportion, as that it may in its great Animal perform the office proportionate to that of the le$$er Bone in a $maller Animal, and the Figures are the$e: whereby you $ee what a di$proportionate Figure that of the aug- mented Bone becometh. Whence it is manife$t, that he that would in an huge Giant keep the proportions that the Members have in <foot>an</foot> <p n=>107</p> an ordinary Man, mu$t either find Matter much more hard and re- $i$ting to make Bone of, or el$e mu$t admit that its Strength is in proportion much more weak than in Men of middle Stature: other- wi$e, encrea$ing the Giant to an immea$urable height he would be oppre$$ed, and fall under his own weight. Whereas on the con- trary, in dimini$hing of Bodies we do not $ee the Strength and Forces to dimini$h in the $ame proportion, nay, in the le$$er the Robu$tiou$ne$$e encrea$eth with a great proportion. So that I believe, that a little Dog could carry on his back two or three Dogs equal to him$elf, but I do not think that an Hor$e could carry $o much as one $ingle Hor$e of his own $ize.</P> <P>SIMP. But if it be $o, I have great rea$on to doubt the Im- men$e bulks that we $ee in Fi$hes, for (if I rightly under$tand you) a Whale $hall be as big as ten Elephants, and yet they $u- $tain them$elves.</P> <P>SALV. Your doubt, <I>Simplicius,</I> prompts me with another Con- dition which I perceived not before, which is al$o able to make Giants and other very big Animals to con$i$t, and act them$elves no le$$e than $maller, and this will happen when not only Strength is added to the Bones and other Parts, who$e office it is to $u$tain their own and the $upervenient weight; but the $tructure of the Bones being left with the $ame proportions, the $ame Fabricks would ju$t in the $ame manner, yea, with much more ea$e, con- $i$t, when the Gravity of the matter of tho$e Bones, or that of the Fle$h, or whatever el$e is to re$t it $elf upon the Bones is dimini- $hed in that proportion: and of this $econd Artifice, Nature hath made u$e in the framing of Fi$hes, making their Bones, and Pulps, not only very light, but without any Gravity.</P> <P>SIMP. I $ee very well, <I>Salviatus,</I> whither your Di$cour$e ten- deth: you will $ay, that becau$e the Element of Water is the Ha- bitation of Fi$hes, which by its Corpulence, or, as others will, by its Gravity dimini$heth the weight of Bodies demerged in it, for that rea$on the Matter of Fi$hes, not weighing any thing, may be $u$tained without $urcharging their Bones: but this doth not $uf- fice, for although the re$t of the $ub$tance of the Fi$h weigh not, yet without doubt the matter of their Bones hath its weight: and who will $ay, that the Rib of a Whale that is as big as a Beam doth not weigh very much, and in Water $inketh to the Bot- tom? The$e therefore $hould not be able to $ub$i$t in $o va$t a Bulk.</P> <P>SALV. You argue very cunningly; and for an an$wer to your Doubt, tell me, whether you have ob$erved Fi$hes to $tand im- moveable under water at their plea$ures, and not to de$cend to- wards the Bottom, or rai$e them$elves towards the top without making $ome motion with their Fins?</P> <foot>P 2 SIMP.</foot> <p n=>108</p> <P>SIMP. This is a very manife$t Ob$ervation.</P> <marg><I>The Cau$e why Fi$hes do equili- brate them$elves in the Water.</I></marg> <P>SALV. This power therefore that the Fi$hes have to $tay them- $elves, as if they were immoveable in the mid$t of the Water, is a mo$t infallible argument, that the Compofition of their Corporeal Ma$$e equalleth the Specifick Gravity of the Water, $o that if there be found in them $ome parts that are more grave than the Water, it is nece$$arily requi$ite that they have others $o much le$$e grave, $o that the <I>Equilibrium</I> may be ballanced. If therefore the Bones be more grave, it is nece$$ary that the Pulps, or other Matters that are in them, be more light; and the$e will with their lightne$$e counterpoi$e and compen$ate the weight of the Bones. So that in Aquatick Animals the quite contrary hapneth to that which befals the Terre$trial, namely, that in the latter it is the of- fice of the Bones to $u$tain their own weight, and the weight of the Fle$h; and in the former, the <I>Fle$h [if one may $o call it]</I> <marg><I>Aquatick Animals greater than the Terre$trial, and for what Rea$on.</I></marg> beareth up its own weight, and that of the Bones. And therefore cea$e to wonder how there may be mo$t va$t Animals in the Wa- ter, but not on the Earth, that is, in the Air.</P> <P>SIMP. I am $atisfied, and moreover ob$erve, that the$e which we call Terre$trial Animals, ought with more rea$on to be called Aerial; becau$e in the Air they really live, and by the Air they are environ'd, and of the Air they breath.</P> <P>SAGR. The Di$cour$e of <I>Simplicius</I> plea$eth me, as al$o his Doubt and its Solution. And farthermore I comprehend very ea- $ily, that one of the$e huge Fi$hes being haul'd on $hore, could not perchance be able to $u$tain it $elf for any time; but that the Con- nections of the Bones being relaxed, its Ma$$e would be cru$h'd un- der its own weight.</P> <P>SALV. For the pre$ent, I encline to the $ame Opinion: nor am I far from thinking that the $ame would happen to that huge Ship, which floating in the Sea is not di$$olved by its weight, and the bur- den of its Lading and Artilery, but on dry ground, and environed with Air, it perhaps would fall in pieces. But let us pur$ue our bu- $ine$$e, and demon$trate, that</P> <foot>PROP.</foot> <p n=>109</p> <head>PROP. IX. PROBL. II.</head> <P><I>A Pri$me or Cylinder with its weight, and the great- e$t Weight $u$tained by it being given, to find the greate$t Length, beyond which being prolonged. it would break under its own Weight.</I></P> <P>Let there be given the Pri$me A C (<I>as in</I> Fig. 13.) with its weight, and likewi$e let the Weight D be given, the great- e$t that can be $u$tained by the extreme C: it is required to finde the greate$t Length unto which the $aid Pri$me may be pro- longed, without breaking. As the weight of the Pri$me A C is to the Compound of the weights A C, with the double of the Weight D, $o let the length C A be to C A H: between which let A G be a Mean-Proportional. I $ay that A G is the Length $ought. For the depre$$ing Moment of the Weight D in C, is equal to the Moment of the double weight D, if it be placed in the middle of A C, where is al$o the Center of the Moment of the Pri$me A C: The Moment, therefore, of the Re$i$tance of the Pri$me A C, which re$ides in A, is equivalent to the gravi- tation of the double of the Weight D with the weight A C, but hanged in the mid$t of A C. And becau$e it hath been made, that as the Moment of the $aid Weights $o $ituated, that is, of the double of D, with A C, is to the Moment of A C, $o is H A to A C, between which A G is a Mean Proportional: There- fore the Moment of D doubled with the Moment of A C, is to the Moment A C, as the Square G A to the Square A C: But the pre$$ing Moment of the Pri$me G A, is to the Moment of A C, as the Square G A to the Square A C: Therefore the Length A G is the greate$t that was $ought, namely, that unto which the Pri$me A G being prolonged, it would $u$tain it $elf, but beyond it would break.</P> <P>Hitherto we have con$idered the Moments and Re$i$tances of $olid Pri$mes and Cylinders, one end of which is $uppo$ed im- moveable, and to the other onely the Force of a pre$$ing weight is applyed, con$idering it by it $elf alone, or joyned with the Gravity of the $ame Solid, or el$e the $ole Gravity of the $aid Solid. Now I de$ire that we may $peak $omething of tho$e $ame Pri$mes or Cylinders, in ca$e they were $u$tained at both ends, or did re$t upon one $ole point taken between the ends. And fir$t, I $ay that,</P> <foot>PROP.</foot> <p n=>110</p> <head>PROPOSITION X.</head> <P><I>The Cylinder that being charged with its own Weight $hall be reduced to its greate$t Length, beyond which it would not $u$tain it $elf, whether it be born up in the middle by one $ole Fulciment, or el$e by two at the ends, may be double in length to that which $hould be fa$tned in the Wall, that is $u$tained at but one end.</I></P> <P>Which of it $elt is very obvious; for if we $hall $up- po$e of the Cylinder which I de$cribe A B C, its half A B to be the utmo$t Length that is able to be $u$tained, being fa$tened at the end B, it $hall be $u$tained in the $ame manner, if being laid upon the Fulciment G, it $hall be counterpoi$ed by its other half B C. And likewi$e, if of the Cy- linder D E F, the Length $hall be $uch that onely one half of it can be $u$tained, being fa$tened at the end D, and con$equent- ly the other E F, fixed at the end F; it is manife$t, that placing the Fulciments H and I under the ends D and F, every Moment of Force or of Weight that is added in E, will there make the Fracture.</P> <P>That which requireth a more $ubtil Speculation is, when $ub- $tracting from the proper Gravity of $uch Solids, it were pro- pounded to us</P> <head>PROP. XI. PROBL. III.</head> <P><I>To find whether that Force or weight, that being ap- plied to the middle of a Cylinder $u$tained at the ends, would $uffice to break it, could do the $ame, applied in any other place, neerer to one end than to the other.</I></P> <P>As for Example, whether we de$iring to break a Staffe and took it with the ends in our hands, and $etting our knee, to the mid$t of it, the $ame Force that $hould $uf- fice to break it in that manner, would al$o $uffice in ca$e the knee <foot>were</foot> <p n=>111</p> were $et, not in the mid$t, but neerer to one of the ends.</P> <P>SAGR. I think the Problem is toucht upon by <I>Ari$totle</I> in his <I>Mechanical Que$tions.</I></P> <P>SALV. The Que$tion of <I>Aristotle</I> is not preci$ely the $ame, for he $eeks no more, but to render a rea$on why le$$e labour is required to break the Staffe, holding the hands at the ends of it, that is, far di$tant from the Knee, than if we held them neerer: and he giveth a general Rea$on of the $ame, reducing the cau$e of it to the Leavers, which are longer when the Arms are ex- tended, gra$ping the ends. Our Que$tion addeth $omething more, $eeking whether, $etting the Knee in the mid$t, or in ano- ther place, but alwayes keeping the hands at the ends, the $ame Force $erveth in all $ituations.</P> <P>SAGR. At fir$t apprehen$ion it $hould $eem that it doth, for that the two Leavers retain in a certain fa$hion the $ame Moment, $eeing that as the one is $hortned, the other is lengthened.</P> <P>SALV. Now you $ee, how ea$ie it is to make Equivocations, and with what caution and circum$pection we are to walk, lea$t we run into them. This that you $ay, and which indeed at the fir$t $ight carrieth with it $o much of probability, is in the $trict- ne$$e of it $o fal$e, that whether the Knee, which is the Fulci- ment of the two Leavers, be placed or not placed in the mid$t, it maketh $uch alteration, that of that Force which would $uffice to make the Fracture in the mid$t, it being to be made in $ome other place, it will not $uffice to apply four times $o much, nor ten, nor an hundred, no nor a thou$and. Upon this we will make $ome general Con$ideration, and then we will come to the Specifick Determination of the Propo$ition, according to which, the Forces for making of Fractures gradually vary more in one point than in another.</P> <P>Let us fir$t de$igne this Truncheon A B to be broken in the mid$t upon the Fulciment C, and neer unto that let us de$igne it again, but under the Characters D E, to be broken on the Fulciment F, remote from the middle. Fir$t it is manife$t, that the Di$tances A C and C B being equal, the Force $hall be $ha- red equally in the ends B and A. Again, according as the Di- $tance D F groweth le$$e than the Di$tance A C, the Moment of the Force placed in D groweth le$$e than the Moment in A, that is placed at the Di$tance C A, and le$$eneth according to the proportion of the Line D F to A C; and con$equently, it is requi$ite to encrea$e it to equalize or exceed the Re$i$tance of F: But the Di$tance D F may dimini$h <I>in infinitum,</I> in relation to the Di$tance A C: Therefore it is requi$ite, that it be po$$ible for the Force to be applyed in D, to encrea$e <I>in infinitum,</I> that it may countervail the Re$i$tance in F. But, on the contrary, ac- <foot>cording</foot> <p n=>112</p> cording as the Di$tance F E encrea$eth above C B, it is requi$ite to dimini$h the Force in E, that it may compen$ate the Re$i- $tance in F: But the Di$tance F E in relation to C B, cannot en- crea$e <I>in infinitum,</I> by drawing the Fulciment F towards the end D, no nor yet to the double: Therefore, the Force in E, that it may compen$ate the Re$i$tance in F, $hall be alwayes more than half of the Force in B. We may comprehend, therefore, the ne- ce$$ity of augmenting the Moments of the Collected Forces in E and D infinitely to equalize or exceed the Re$i$tance placed in F, according as the Fulciment F $hall approach neerer and neerer to the end D.</P> <P><I>S</I>AGR. What will <I>Simplicius</I> $ay to this? Mu$t he not con- fe$$e the Virtue of Geometry to be a more powerful in$trument than all others, to $harpen the Wit, and di$po$e it to di$cour$e and $peculate well? and that <I>Plato</I> had great rea$on to de$ire that his Scholars $hould be well grounded in the Mathematicks? I have very well under$tood the nature of the Leaver, and how that its Length encrea$ing or decrea$ing, the Moment of the Force and of the Re$i$tance augmented or dimini$hed, and yet in the determination of the pre$ent Problem I deceived my $elf, and that not a little, but infinitely much.</P> <P>SIMP. The truth is, I begin to $ee that Logick, although it be a mo$t appo$ite In$trument to regulate our Di$cour$e, doth not attain, as to the prompting of the Mind with Invention, unto the acutene$$e of Geometry.</P> <P>SAGR. In my conceit, Logick giveth us to under$tand, whe- ther the Di$courfes and Demon$trations already made and found are concluding, but that it teacheth us how to finde concluding Di$cour$es and Demon$trations; the truth is, I do not believe: But it will be better, that <I>Salviatus</I> $hew us according to what pro- portion the Moments of the Forces do go increa$ing, to overcome the Re$i$tance of the $ame Piece of Wood, according to the $e- veral places of the Fracture.</P> <P>SALV. The proportion that you $eek, proceedeth after $uch a manner, that</P> <foot>PROP.</foot> <p n=>113</p> <head>PROPOSITION XII.</head> <P><I>If in the length of a Cylinder we $hall marke two places, upon which we would make the Fracture of the $aid Cylinder, the Re$i$tances of tho$e two places have the $ame proportion to each other, as have the Re- ctangles made by the Di$tances of tho$e places reciprocally taken.</I></P> <P>Let the two Forces (<I>as in</I> Fig. 16.) be A and B the lea$t, to break in C, and E and F likewi$e the lea$t, to break in D. I $ay the Forces A and B have the $ame proportion to the Forces E and F, that the Rectangle A D B hath to the Rectan- gle A C B. For the Forces A and B, have to the Forces E and F, a proportion compounded of the Forces A and B, to the Force B, of B to F, and of F to E and E: But as the Forces A and B are to the Force B, $o is the Length B A to A C; and as the Force B is to F, $o is the Line D B to B C; and as the Force F is to F and E, $o is the Line D A to A B: Therefore the Forces A and B have to the Forces E and F a proportion compounded of the$e three, namely, of B A to A C, of D B to B C, and of D A A B. But of the two proportions D A to A B, and A B to A C, is compounded the proportion of D A to A C: Therefore the Forces A and B have to the Forces E and F, the proportion com- pounded of this D A to A C, and of the other D B to D C. But the Rectangle A D B hath to the Rectangle A C B, a pro- portion compounded of the $ame D A to A C, and of D B to B C: Therefore the Forces A and B are to the Forces E and F, as the Rectangle A D B is to the Rectangle A C B; which is as much as to $ay, the Re$i$tance again$t Fraction in C, hath the $ame proportion to the Re$i$tance again$t Fraction in D, that the Rectangle A D B hath to the Rectangle A C B: Which was to be demon$trated.</P> <P>In con$equence of this Theorem we may re$olve a Problem of great Curio$ity; and it is this:</P> <foot>Q PROP.</foot> <p n=>114</p> <head>PROP. XIII. PROBL. IV.</head> <P><I>There being given the greate$t Weight that can be $up- ported at the middle of a Cylinder or Pri$me, where the Re$i$tance is leafl; and there being given a Weight greater than that, to find in the $aid Cylin- der, the point at which the given greater Weight may be $upporited as the greate$t Weight.</I></P> <P>Let the given weight greater than the greate$t weight that can be $upported at the middle of the Cylinder A B, have unto the $aid greate$t weight, the proportion of the line E to F: it is required to find the point in the Cylinder at which the $aid given weight commeth to be $upported as the bigge$t. Be- tween E and F let G be a Mean-Proportional; and as E is to G, $o let A D be to S, S $hall be le$$er than A D. Let A D be the Diameter of the Semicircle A H D: in which $uppo$e A H equal to S; and joyn together H and D, and take D R equal to it. I $ay that R is the point $ought, at which the given weight, greater than the greate$t that can be $upported at the middle of the Cylinder D, would become as the greate$t weight. On the length <I>B</I>A de$cribe the Semicircle A N B, and rai$e the Perpendicular RN, and conjoyn N and D: And becau$e the two <fig> Squares N R and R D are equal to the Square N D; that is, to the Square A D; that is, to the two A H and and H D; and H D is equal to the Square D R: There- fore the Square N R, that is, the Rectangle A R B $hall be equal to the Square A H; that is, to the Square S: But the Square S is to the Square A D, as F to E; that is, as the greate$t $upportable Weight at D to the given greater Weight: Therefore this greater $hall be $upported at R, as the greate$t that can be there $u$tained. Which is that that we $ought.</P> <P>SAGR. I under$tand you very well, and am con$idering that the Pri$me A B having alwayes more $trength and re$i$tance a- gain$t Pre$$ion in the parts that more and more recede from the middle, whether in very great and heavy Beams one may take <foot>away</foot> <p n=>115</p> away a pretty big part towards the end with a notable alleviation of the weight; which in Beams of great Rooms would be commo- dious, and of no $mall pro$it. And it would be pretty, to find what Figure that Solid ought to have, that it might have equal Re$i- $tance in all its parts; $o as that it were not with more ea$e to be broken by a weight that $hould pre$$e it in the mid$t, than in any other place.</P> <P>SALV. I was ju$t about to tell you a thing very notable and plea$ant to this purpo$e. I will a$$ume a brief Scheme for the bet- ter explanation of my meaning. This Figure D B is a Pri$m, who$e Re$i$tance again$t Fraction in the term A D by a Force pre$$ing at the term B, is le$$e than the Re$i$tance that would be found in the place C I, by how much the length C B is le$$er than B A; as hath already been demon- $trated. Now $uppo$e the <fig> $aid Pri$me to be $awed Diagonally according to the Line FB, $o that the oppo- $ite Surfaces may be two Triangles, one of which to- wards us is F A B. This So- lid obtains a quality contrary to the Pri$me, to wit, that it le$$e re- $i$teth Fraction by the Force placed in B at the term C than at A, by as much the Length C <I>B</I> is le$$e than <I>B</I> A; Which we will ea $ily prove: For imagining the Section C N O parallel to the other A F D, the Line <I>F</I> A $hall be to C N in the Triangle F A <I>B</I> in the $ame proportion, as the Line A <I>B</I> is to <I>B</I> C: and therefore if we $uppo$e the Fulciment of the two Leavers to be in the Points A and C, who$e Di$tances are <I>B</I> A, A F, <I>B</I> C, and C N, the$e, I $ay, $hall be like: and therefore that Moment which the <I>F</I>orce placed at <I>B</I> hath at the Di$tance <I>B</I> A above the Re$i$tance placed at the Di$tance A <I>F</I>, the $aid <I>F</I>orce at <I>B</I> $hall have at the Di$tance <I>B</I>C above the $ame Re$i$tance, were it placed at the Di$tance C N: <I>B</I>ut the Re$i$tance to be overcome at the <I>F</I>ulciment C, being pla- ced at the Di$tance C N, from the <I>F</I>orce in <I>B</I> is le$$er than the Re$i$tance in A $o much as the Rectangle C O is le$$e than the Rectangle A D; that is, $o much as the Line C N is le$s than A <I>F</I>; that is, C <I>B</I> than B A: Therefore the Re$i$tance of the part O C B again$t <I>F</I>raction in C is $o much le$s than the Re$i$tance of the whole D A O again$t <I>F</I>racture in O, as the Length C B is le$s than A B. We have therefore from the Beam or Pri$me D B, taken away a part, that is half, cutting it Diagonally, and left the Wedge or triangular Pri$m <I>F</I> B A; and they are two Solids of contrary Qualities, namely, that more re$i$ts the more it is $hortned, and this in $hortning lo$eth its toughne$s as fa$t. Now this being granted, <foot>Q 2 it</foot> <p n=>116</p> it $eemeth very rea$onable, nay, nece$$ary, that one may give it a cut, by which taking away that which is $uperfluous, there remai- neth a Solid of $uch a <I>F</I>igure, as in all its parts hath equal Re$i- $tance.</P> <P>SIMP. It mu$t needs be $o; for where there is a tran$ition from the greater to the le$$er, one meeteth al$o with the equal.</P> <P>SAGR. But the bu$ine$$e is to find how we are to guide the Saw for making of this Section.</P> <P>SIMP. This $eemeth to me as if it were a very ea$ie bu$ine$$e; for if in $awing the Pri$m diagonally, taking away half of it, the Figure that remains retaineth a contrary quality to that of the whole Pri$m, $o as that in all places wherein this acquireth $trength, that as fa$t lo$eth it, me thinks, that keeping the middle way, that is, taking only the half of that half, which is the fourth part of the whole, the remaining Figure will not gain or lo$e $trength in any of all tho$e places wherein the lo$$e and the gain of the other two Figures were alwaies equal.</P> <P>SALV. You have not hit the mark, <I>Simplicius</I>; and as I $hall $hew you, it will appear in reality, that that which may be cut off from the Pri$m, and taken away without weakening it is not its fourth part, but the third. Now it remaineth (which is that that was hinted by <I>Sagredus</I>)</P> <head>PROP. XIV. PROBL. V.</head> <P><I>To find according to what Line the Section is to be made; Which I will prove to be a Parabolical Line.</I></P> <P>But fir$t it is nece$$ary to demon$trate a certain Lemma, which is this:</P> <head>LEMMA I.</head> <P><I>If there $hall be two Ballances or Leavers divided by their Fulci- ments in $uch $ort that the two Distances where at the Forces are to be placed, have to each other double the proportion of the Di$tances at which the Re$i$tances $ball be, which Re$i- $tances are to each other as their Di$tances, the $u$taining Powers $hall be equal.</I></P> <P>Let A B and C D be two Leavers divided upon their Fulciments E and F, in $uch $ort that the Di$tance E B hath to F D a pro- portion double to that which the Di$tance E A hath to F C. I $ay, <foot>the</foot> <p n=>117</p> the Powers that in BD $hall $u$tain the Re$i$tances A and C $hall be equal to each other. Let E G be $uppo$ed a Mean-Proporti- onal between E B and F D; therefore as B E is to E G, $o $hall G E be to F D, and A E to C <I>F</I>; and $o is $uppo$ed the Re$i$tance of A to the Re$i$tance of C. And becau$e that as E G is to <I>F</I> D, $o is A E to C <I>F</I>; by Permutation as G E is to E A, $o $hall D <I>F</I> be to <I>F</I> C: And therefore (in regard that the two Leavers <fig> D C and G A are divided pro- portionally in the Points <I>F</I> and E) in ca$e the Power that being placed at D compen$ates the Re$i$tance of C were at G, it would countervail the $ame Re$i$tance of C placed in A: But by what hath been granted, the Re$i$tance of A hath the $ame propor- tion to the Re$i$tance of C, that AE hath to C <I>F</I>; that is, B E hath to E G: Therefore the Power G, or if you will D, placed at B will $u$tain the Re$i$tance placed at A: Which was to be de- mon$trated.</P> <P>This being under$tood: in the Surface <I>F</I> B of the Pri$me D B, let the Parabolical Line <I>F</I> N B be drawn, who$e Vertex is B, ac- cording to which let the $aid Pri$me be $uppo$ed to be $awed, the Solid compri$ed between the Ba$e A D, the Rectangular Plane A G, the Bight Line B G, and the Superficies D G B <I>F</I> being le$t incurvated according to the Curvity of the Parabolical Line <I>F</I> N B. I $ay, that that Solid is through- <fig> out of equal Re$i- $tance. Let it be cut by the Plane C O pa- rallel to A D; and imagine two Leavers divided and $uppor- ted upon the Fulciments A and C; and let the Di$tances of one be B A and A F, and of the other B C, and C N. And becau$e in the Parabola <I>F B</I> A, A <I>B</I> is to <I>B</I> C, as the Square of <I>F</I> A to the Square of C N, it is manife$t, that the Di$tance <I>B</I> A of one Leaver, hath to the Di$tance <I>B</I> C of the other a proportion double to that which the other Di$tance A <I>F</I> hath to the other C N, And be- cau$e the Re$i$tance that is to be equal by help of the Leaver <I>B</I> A hath the $ame proportion to the Re$i$tance that is to be equal by help of the Leaver <I>B</I> C, that the Rectangle D A hath to the Rectangle O C; which is the $ame that the Line A <I>F</I> hath to N C, which are the other two Di$tances of the Leavers; it is ma- nife$t by the fore going Lemma, that the $ame Force that being <foot>applyed</foot> <p n=>118</p> applyed to the Line <I>B</I> G will equal the Re$i$tance D A, will like- wi$e equal the Re$i$tance C O. And the $ame may be demon$tra- ted, if one cut the Solid in any other place: therefore that Parabo- lical Solid is throughout of equal Re$i$tance. In the next place, that cutting the Pri$me according to the Parabolical Line F N B, the third part of it is taken away, appeareth, For that the Semi- Parabola F N <I>B</I> A and the Rectangle F <I>B</I> are Ba$es of two Solids contained between two parallel Planes, that is, between the Rect- angles F B and D G, whereby they retain the $ame Proportion, as tho$e their Ba$es: But the Rectangle F <I>B</I> is Se$quialter to the Se- miparabola F N <I>B</I> A: Therefore cutting the Pri$ine according to the Parabolick Line, we take away the third part of it. Hence we $ee, that <I>B</I>eams may be made with the diminution of their Weight more than thirty three in the hundred, without dimini$hing their Strength in the lea$t; which in great Ships, in particular, for bea- ring the Decks may be of no $mall benefit; for that in $uch kind of Fabricks Lightne$$e is of infinite importance.</P> <P>SAGR. The Commodities are $o many, that it would be tedi- ous, if not impo$$ible, to mention them all. <I>B</I>ut I, laying a$ide the$e, would more gladly under$tand that the alleviation is made according to the a$$igned proportions. That the Section, according to the Diagonal Line, cuts away half of the weight I very well know: but that the other Section according to the Parabolical Line takes away the third part of the Pri$me I can believe upon the word of <I>Salviatus,</I> who evermore $peaks the truth, but in this Ca$e Science would better plea$e me than Faith.</P> <P>SALV. I $ee then that you would have the Demon$tration, whether or no it be true, that the exce$$e of the Pri$me over and above this, which for this time we will call a Parabolical Solid, is the third part of the whole Pri$me. I am certain that I have for- merly demo$trated it; I will try now whether I can put the Demon$tration together again: to which purpo$e I do remember that I made u$e of a Certain Lemma of <I>Archimedes,</I> in$erted by him in his <I>B</I>ook <I>de Spiralibus,</I> and it is this:</P> <head>LEMMA II.</head> <P><I>If any number of Lines at plea$ure $hall exceed one another equal- ly, and the exce$$es be equal to the lea$t of them, and there be as many more, each of them equal to the greate$t; the Squares of all the$e $hall be le$$e than the triple of the Squares of tho$e that exceed one another: but they $hall be more than triple to tho$e others that remain, the Square of the greate$t being $ub- $tracted.</I></P> <foot>This</foot> <p n=>119</p> <P>This being granted: Let the Parabolick Line A <I>B</I> be in$cribed in this Rectangle A C <I>B</I> P: we are to prove the Mixt Triangle <I>B</I> A P, who$e $ides are <I>B</I> P and P A, and <I>B</I>a$e the Parabolical Line <I>B</I> A, to be the third part of the whole Rectangle C P. For if it be not $o, it will be either more than the third part, or le$$e. Let it be $uppo$ed that it may be le$$e, and to that which is <fig> wanting $uppo$e the Space X to be equal. Then di- viding the Rectangle con- tinually into equal parts with Lines parallel to the Sides <I>B</I> P and C A, we $hall in the end arrive at $uch parts, as that one of them $hall be le$$e than the Space X. Now let one of them be the Rectangle O <I>B,</I> and by the Points where the other Parallels inter$ect the Parabolick Line, let the Pa- rallels to A P pa$$e: and here I will $uppo$e a Figure to be cir- cum$cribed about our Mixt-Triangle, compo$ed of Rectangles, which are <I>B</I> O, I N, H M, F L, E K, G A; which Figure $hall al$o yet be le$s than the third part of the Rectangle C P, in regard that the exce$$e of that Figure over and above the Mixed Triangle is much le$$e than the Rectangle <I>B</I> O, which yet again is le$$e than the Space X.</P> <P>SAGR. Softly, I pray you, for I do not $ee how the exce$$e of this circum$cribed Figure above the Mixt Triangle is con$iderably le$$er than the Rectangle <I>B</I> O.</P> <P>SALV. Is not the Rectangle <I>B</I> O equal to all the$e $mall Rect- angles by which our Parabolical Line pa$$eth; I mean the$e, <I>B</I> I, I H, H F, F E, E G, and G A, of which one part only lyeth with- out the Mixt Triangle? And the Rectangle <I>B</I> O, is it not al$o $up- po$ed to be le$$e than the Space X? Therefore if the Triangle to- gether with X did, as the Adver$ary $uppo$eth, equalize the third part of the Rectangle C P the circum$cribed Figure that adjoyns to the Triangle $o much le$$e than the Space X, will remain even yet le$$e than the third part of the $aid Rectangle C P. <I>B</I>ut this cannot be, for it is more than a third part, therefore it is not true that our Mixt Triangle is le$$e than one third of the Rectangle.</P> <P>SAGR. I under$tand the Solution of my Doubt. <I>B</I>ut it is requi$ite now to prove unto us, that the Circum$cribed Figure is more than a third part of the Rectangle C P; which, I believe, will be harder to do.</P> <P>SALV. Not at all. For in the Parabola the Square of the Line <marg><I>The Quadrature of the Parabola $hewn by one $ingle De- mon$tration.</I></marg> D E hath the $ame proportion to the Square of Z G, that the Line <foot>DA</foot> <p n=>120</p> D A hath to A Z; which is the $ame that the Rectangle K E hath to the Rectangle A G, their heights A K and K L being equal. There- fore the proportion that the Square E D hath to the Square Z G; that is, the Square L A hath to the Square A K, the Rectangle K E hath likewi$e to the Rectangle K Z. And in the $elf-$ame manner we might prove that the other Rectangles L F, M H, N I, O B are to one another as the Squares of the Lines M A, N A, O A, P A. Con$ider we in the next place, how the Circum$cribed Figure is compounded of certain Spaces that are to one another as the Squares of the Lines that exceed with Exce$$es equal to the lea$t, and how the Rectangle C P is compounded of $o many other Spa- ces each of them equal to the Greate$t, which are all the Rectan- gles equal to O B. Therefore, by the Lemma of <I>Archimedes,</I> the Circum$cribed Figure is more than the third part of the Rectangle C P: But it was al$o le$$e, which is impo$$ible: Therefore the Mixt-Triangle is not le$$e than one third of the Rectangle C P. I $ay likewi$e, that it is not more: For if it be more than one third of the Rectangle C P, $uppo$e the Space X equal to the ex- ce$$e of the Triangle above the third part of the $aid Rectangle C P, and the divi$ion and $ubdivi$ion of the Rectangle into Rect- angolets, but alwaies equal, being made, we $hall meet with $uch as that one of them is le$$er than the Space X; which let be done: and let the Rectangle <I>B</I> O be le$$er than X; and, having de$cribed the Figure as before, we $hall have in$cribed in the Mixt-Triangle a Figure compounded of the Rectangles V O, T N, S M, N L, Q K, which yet $hall not be le$s <fig> than the third part of the great Rectangle C P, for the Mixt Triangle doth much le$$e exceed the In- $cribed Figure than it doth exceed the third part of the Rectangle C P; Be- cau$e the exce$$e of the Triangle above the third part of the Rectangle C P is equal to the Space X which is greater than the Rectangle <I>B</I> O, and this al- $o is con$iderably greater than the exce$$e of the Triangle above the In$cribed Figure: For to the Rectangle <I>B</I> O, all the Rectan- golets A G, G E, E <I>F,</I> F H, H I, I <I>B</I> are equal, of which the Ex- ce$$es of the Triangle above the In$cribed <I>F</I>igure are le$$e than half: And therefore the Triangle exceeding the third part of the Rectangle C P, by much more (exceeding it by the Space X) than it exceedeth its in$cribed <I>F</I>igure, that $ame <I>F</I>igure $hall al$o be greater than the third part of the Rectangle C P: <I>B</I>ut it is le$$er, by the Lemma pre$uppo$ed: <I>F</I>or that the Rectangle C P, as being <foot>the</foot> <p n=>127</p> the Aggregate of all the bigge$t Rectangles, hath the $ame pro- portion to the Rectangles compounding the In$cribed <I>F</I>igure, that the Aggregate of of all the Squares of the Lines equal to the big- ge$t, hath to the Squares of the Lines that exceed equally, $ub$tra- cting the Square of the bigge$t: And therefore (as it hapneth in Squares) the whole Aggregate of the bigge$t (that is the Rectan- gle C P) is more than triple the Aggregate of the exceeding ones, the bigge$t deducted, that compound the In$cribed <I>F</I>i- gure. Therefore the Mixt-Triangle is neither greater nor le$$er than the third part of the Rectangle C P: It is therefore equal.</P> <P>SAGR. A pretty and ingenuous Demon$tration: and $o much the more, in that it giveth us the Quadrature of the Parabola, $hew- ing it to be <I>Se$quitertial</I> of the Triangle in$cribed in the $ame; proving that which <I>Archimedes</I> demon$trateth by two very diffe- rent, but both very admirable, methods of a great number of Pro- po$itions. As hath likewi$e been demon$trated lately by <I>Lucas Valerius,</I> another $econd <I>Archimedes</I> of our Age, which Demon- $tration is $et down in the Book that he writ of the Center of the Gravity of Solids.</P> <P><I>S</I>ALV. A Treati$e which verily is not to come behind any one that hath been written by the mo$t <I>F</I>amous Geometricians of the pre$ent and all pa$t Ages: which when it was read by our <I>Acade- mick,</I> it made him de$i$t from pro$ecuting his Di$coveries that he was then proceeding to write on the $ame Subject: in regard he $aw the whole bu$ine$s $o happily found and demon$trated by the $aid <I>Valerius.</I></P> <P>SAGR. I was informed of all the$e things by our <I>Academick</I>; and have be$ought him withall that he would one day let me $ee his Demon$trations that he had $ound at the time when he met with the <I>B</I>ook of <I>Valerius:</I> but I never was $o happy as to $ee them.</P> <P>SALV. I have a Copy of them, and will impart them to you, for you will be much plea$ed to $ee the variety of Methods, which the$e two Authors take to inve$tigate the $ame Conclu$ions, and their Demon$trations: wherein al$o $ome of the Conclu$ions have different Explanations, howbeit in effect equally true.</P> <P>SAGR. I $hall be very glad to $ee them, therefore when you re- turn to our wonted Conferences you may do me the favour to bring them with you. <I>B</I>ut in the mean time, this $ame of the Re- fi$tance of the Solid taken from the Pri$me by a Parabolick Secti- on, being an Operation no le$$e ingenuous than beneficial in many Mechanical Works, it would be good that Artificers had $ome ea- $ie and expedite Rule how they may draw the $aid Parabolick Line upon the Plane of the Pri$me.</P> <P>SALV. There are $everal waies to draw tho$e Lines, but two <marg><I>Several waies to de$cribe a Para- bola.</I></marg> that are more expedite than all the re$t, I will de$cribe unto you. <foot>R One</foot> <p n=>122</p> One of which is truly admirable, $ince that thereby, in le$$e time than another can with Compa$$es $lightly draw upon a paper four or $ix Circles of different $izes, I can de$ign thirty or forty Parabolick Lines no le$$e exact, $mall, and $mooth than the Cir- cumferences of tho$e Circles. I have a <I>B</I>all of <I>B</I>ra$$e exqui$itely round, no bigger than a Nut, this thrown upon a Steel Mirrour held, not erect to the Horizon, but $omewhat inclined, $o that the <I>B</I>all in its motion may run along pre$$ing lightly upon it, leaveth a Parabolical Line finely and $moothly de$cribed, and wider or narrower according as the Projection $hall be more or le$s elevated. Whereby al$o we have a clear and $en$ible Experiment that the Motion of Projects is made by Parabolick Lines: an Effect ob$er- ved by none before our <I>Academick,</I> who al$o layeth down the Demon$tration of it in his <I>B</I>ook of Motion, which we will joynt- ly peru$e at our next meeting. Now the <I>B</I>all, that it may de$cribe by its motion tho$e Parabola's, mu$t be rouled a little in the hands that it may be warmed, and $omewhat moy$tned, for by this means it will leave its track more apparent upon the Mirrour. The other way to draw the Line that we de$ire upon the Pri$me is after this manner. Let two Nailes be fa$tned on high in a Wall, at an equal di$tance from the Horizon, and remote from one another twice the breadth of the Rectangle upon which we would trace the Semiparabola, and to the$e two Nails tye a $mall thread of $uch a length that its doubling may reach as far as the length of the Pri$me; this $tring will hang in a Parabolick <I>F</I>igure: $o that tra- cing out upon the Wall the way that the $aid String maketh on it, we $hall have a whole Parabola de$cribed: which a Perpendicular that hangeth in the mid$t between the$e two Nailes will divide into two equal parts. And for the transferring or $etting off of that Line afterwards upon the oppo$ite Surfaces of the Pri$me it is not difficult at all, $o that every indifferent Arti$t will know how to do it. The $ame Line might be drawn upon the $aid Sur- face of the Pri$me by help of the Geometrical Lines delineated up- on the <I>Compa$$e</I> of our <I>Friend,</I> without any more ado.</P> <P>We have hitherto demon$trated $o many Conclu$ions touching the Contemplation of the$e Re$i$tances of Solids again$t Fraction by having fir$t opened the way unto the Science with $uppo$ing the direct Re$i$tance for known, that we may in pur$uance of them proceed forwards to the finding of other, and other Conclu$ions, with their Demon$trations of tho$e which in Nature are infinite. Only at pre$ent, for a final conclu$ion of this daies Conferences, I will add the Speculation of the Re$i$tances of the Hollow Solids which Art, and chiefly Nature, u$eth in an hundred Operations, when without encrea$ing the weight $he greatly augmenteth the $trength: as is $een in the Bones of Birds, and in many Canes that <foot>are</foot> <p n=>123</p> are light and of great Re$i$tance again$t bending and breaking. For if a Wheat Straw that $upports an Ear that is heavier than the whole Stalk, were made of the $ame quantity of matter but were ma$$ie or $olid, it would be much le$$e repugnant to Fraction or Flection. And with the $ame Rea$on Art hath ob$erved, and Ex- perience confirmed, that an hollow Cane, or a Trunk of Wood or Metal, is much more firm and tough than if being of the $ame weight and length it were $olid, which con$equently would be more flender, and therefore Art hath contrived to make Lances hol- low within when they are de$ired to be $trong and light. We will $hew therefore, that</P> <head>PROPOSITION XV.</head> <P><I>The Re$i$tances of two Cylinders, equall, and equally long, one of which is Hollow, and the other Ma$sie, have to each other the $ame proportion, as their Dia- meters.</I></P> <P>Let the Cane or Hollow Cylinder be A E, [<I>as in</I> Fig. 17.] and the Cylinder I N Ma$$ie, and equall in weight and length. I $ay, the Re$i$tance of the Cane A E hath the $ame propor- tion to the Re$i$tance of the $olid Cylinder, as the Diameter A B hath to the Diameter I L. Which is very manife$t; For the Cane and the Cylinder I N being equal, and of equal lengths, the Circle I L that is Ba$e of the Cylinder $hall be equal to the Ring A B that is Ba$e of the Cane A E, (I call the Superficies that re- maineth when a le$$er Circle is taken out of a greater that is Con- centrick with it a Ring:) and therefore their Ab$olute Re$i$tan- ces $hall be equal: but becau$e in breaking cro$$e-waies we make u$e in the Cylinder I N of the length L N for a Leaver, and of the point L for a Fulciment, and of the Semidiameter or Diameter L I for a Counter-Leaver; and in the Cane the part of the Leaver, that is the Line B E is equal to L N; but the Counter-Leaver at the Fulciment B is the Diameter or Semidiameter A B: It is mani- fe$t therefore that the Re$i$tance of the Cane exceedeth that of the Solid Cylinder as much as the Diameter A B exceeds the Dia- meter I L; Which is that that we $ought. Toughne$s therefore is ac- quired in the hollow Cane above the Toughne$s of the $olid Cylin- der according to the proportion of the Diameters: provided al- waies that they be both of the $ame matter, weight, and length.</P> <P>It would be well, that in con$equence of this we try to inve$tigate that which hapneth in other Ca$es indifferently between all Canes and $olid Cylinders of equal length, although unequal in quantity of weight, and more or le$s evacuated. And fir$t we will demon- $trate, that</P> <foot>R 2 PROP</foot> <pb> <fig> <cap><I>Place this at the end of the $econd Dialogue pag:</I> 124,</cap> <p n=>124</p> <head>PROP. XVI. PROBL. VI.</head> <P><I>A Trunk or Hollow Cane being given, a Solid Cylinder may be found equal to it.</I></P> <P>This Operation is very ea$ie. For let the Line A B, be the Dia- meter of the Cane, and C D the Diameter of the Hollow or Cavity. Let the Line A E be $et off upon the greater Circle equal to the Diameter C D, and conjoyn E B. And becau$e in <fig> the Semicircle A E B the Angle E is Right- Angle, the Circle who$e Diameter is A B $hall be equall to the two Circles of the Di- ameters A E and E B: But A E is the Dia- meter of the Hollow of the Cane: Therefore the Circle who$e Diameter is E B, $hall be equal to the Ring A C B D: And therefore the $olid Cylinder, the Circle of who$e Ba$e hath the Diameter E B $hall be equal to the Cane, they being of the $ame length. This demon$trated, we may pre$ently be able</P> <head>PROP. XVII. PROBL. VII.</head> <P><I>To find what proportion is betwixt the Re$i$tances of any what$oever Cane and Cylinder, their lengths be- ing equal.</I></P> <P>LET the Cane A B E, and the Cylinder R S M, be of equal length: it is required to find what proportion the Re$i$tances have to each other. By the precedent let the Cylinder I L N be found equal to the Cane, and of the $ame length; and to the Lines I L and R S (Diameters of the Ba$es of the Cylinders I N and <fig> R M) let the Line V be a fourth Proportional. I $ay, the Re$i$tance of the Cane A E is to the Re$i- $tance of the Cylinder R M, as the Line A B is to V. For the Cane A E being equal to, and of the $ame length with the Cylinder I N, the Re$i$tance of the Cane $hall be to the Re$i$tance of the Cylinder, as the Line A B is to I L: But the Re$i$tance of the Cylinder I N is to the Re$i$tance of the Cylinder R M, as the Cube I L is to the Cube R S; that is, as the Line I L to V: Therefore, <I>ex æquali,</I> the Re$i$tance of the Cane A E hath the $ame proportion to the Re$i$tance of the Cylinder R M, that the Line A B hath to V: Which is that that was $ought.</P> <head><I>The End of the Second Dialogue.</I></head> <p n=>125</p> <head>GALILEUS, HIS DIALOGUES OF MOTION.</head> <head>The Third Dialogue.</head> <head><I>INTERLOCUTORS,</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <head>OF LOCAL MOTION.</head> <P><I>We promote a very new Science, but of a very old Subject. There is nothing in Nature more antient than</I> MOTION, <I>of which many and great Volumns have been written by Philo$ophers: But yet there are $undry Symptomes and Properties in it worthy of our Notice, which I find not to have been hi- therto ob$erved, much le$$e demon$trated by any. Some $light particulars have been no- ted: as that the Natural Motion of Grave Bodies continually accelle-</I> <foot><I>rateth,</I></foot> <p n=>126</p> <I>rateth, as they de$cend towards their Center: but it hath not been as yet declared in what proportion that Acceleration is made. For no man, that I know, hath ever demon$trated, That there is the $ame proportion between the Spaces, thorow which a thing moveth in equal Times, as there is between the Odde Numbers which follow in order after a Vnite. It hath been ob$erved that Projects [or things thrown or darted with vi- olence] make a Line that is $omewhat curved; but that this line is a Pa- rabola, none have hinted: Yet the$e, and $undry other things, no le$$e worthy of our knowledg, will I here demon$trate: And which is more, I will open a way to a mo$t ample and excellent Science, of which the$e our Labours $hall be the Elements: into which more $ubtil and piercing Wits than mine will be better able to dive.</I></P> <P><I>We divide this Treati$e into three parts. In the fir$t part we con$ider $uch things as re$pect Equable or Vniforme Motion. In the $econd we write of Motion naturally accelerate. In the third we treat of Violent Motion, or</I> De Projectis.</P> <head>OF EQVABLE MOTION.</head> <P><I>Concerning Equable or Vniform Motion we have need of onely one Definition, which I thus deliver.</I></P> <head>DEFINITION.</head> <P>By an Equable or Uniform Motion, I under$tand that by which a Moveable in all equal Times pa$$eth thorow equal Spaces.</P> <head>ADVERTISEMENT.</head> <P><I>I thought good to add to the old Definition (which $imply termeth that an Equable Motion, whereby equal Spaces are pa$t in equal Times) this Particle</I> All, <I>that is, any what$oever Times that are equal: for it may happen, that a Moveable may pa$$e thorow equal Spaces in cer- tain equal Times, though the Spaces be not equal which it hath gone in le$$er, though equal parts of the $ame Time. From this our Definition follow the$e four Axiomes:</I> $cilicet,</P> <head>AXIOMEL</head> <P>In the $ame Equable Motion that Space is greater which is pa$$ed in a longer Time, and that le$$er which is pa$t in a $horter.</P> <foot>AXIOME</foot> <p n=>127</p> <head>AXIOME II.</head> <P>In the $ame Equable Motion, the greater the Space is that hath been gone thorow, the longer was the Time in which the Move- able was going it.</P> <head>AXIOME III.</head> <P>The Space which a greater Velocity pa$$eth in any Time, is great- er than the Space which a le$$er Velocity pa$$eth in the $ame Time.</P> <head>AXIOME IV.</head> <P>The Velocity which pa$$eth a greater Space, is greater than the Velocity which pa$$eth a le$$er Space in the $ame Time.</P> <head>THEOR. I. PROP. I.</head> <P>If a Moveable moving with an Equable Motion, and with the $ame Velocity pa$$e two $everal Spaces, the Times of the Motion $hall be to one another as the $aid Spaces.</P> <P><I>Let the Moveable by an Equable Motion with the $ame Velocity paß the two Spaces A B and B C: and let D E be the Time of the Moti- on thorow A B; and let the Time of the Motion thorow B C be E F I $ay that the Time D E to the Time E F, is as the Space A B to the Space B C. Protract the Spaces and Times on both $ides, towards G H and I K, and in A G take any number of Spaces equal to A B,</I> <fig> <I>and in D I the like number of Times equal to D E. Again, in C H take any number of Spaces equal to B C, and in F K take the $ame number of Times equal to the Time E F. This done, the Space B G will con- tain ju$t as many Spaces equal to B A, as the Time E I containeth Times equal to E D, equimultiplied according to what ever Rate; And likewi$e the Space B H will contain as many Spaces equal to B C, as</I> <foot><I>the</I></foot> <p n=>128</p> <I>the Time K E containeth Times equal to F E, at what ever rate equi- multiplied. And fora$much as D E is the Time of the Motion thorow A B, the whole Time E I, $hall be the Time of the whole Space of the Motion thorow B G, by rea$on that the Motion is Equable, and that the number of the Times in E I equal to D E, is the $ame with the number of Spaces in B G, equal to B A: For the $ame rea$on E K is the Time of the Motion thorow H B. Now in regard the Motion is Equable, if the Space G B were equal to H B, the Time I E would be equal to E K: and if G B be greater than B H, I E $hall likewi$e be greater than E K: and if le$$er, le$$er. They are therefore four Magnitudes; A B the fir$t, B C the $econd, D E the third, and E F the Fourth; and the fir$t and third, to wit, the Space A B, and Time D E, there were taken the Time I E, and the Space G B equimultiple, according to any multi- plication; and it hath been demon$trated that the$e do at once either equal, or fall $hort of, or el$e exceed the Time E K, and Space B H, which are equimultiple of the $econd and fourth: Therefore the fir$t bath to the $econd, to wit the Space A B to the Space B C, the $ame proportion that the third hath to the fourth, to wit, the Time D E to the Time E F. Which was to be demon$trated.</I></P> <head>THEOR. II. PROP. II.</head> <P>If a Moveable in equal Times pa$$e thorow two Spaces, the $aid Spaces will be to each other, as the Velocities. And if the Spaces are to each other as the Velocities, the Times will be equal.</P> <P><I>Let us $uppo$e A B and B C in the former Figure, to be two Spaces pa$t, by the Moveable in equal times; the Space A B with the Velocity D E, and the Space B C with the Velocity E F. I $ay, that the Space A B is to the Space B C, as the Velocity D E is to the Velocity E F: and thus I prove it. Take as before, of the Spaces and Velocities equi-multiples, accordieg to any what ever Rate, $ci- licet G B and I E, of A B and D E, and likewi$e H B and K E, of B C and E F: It may be concluded as above, that G B and I E are both at once either equal to, or fall $hort of, or el$e exceed the equi-mul- tiples of D H and E K. Therefore the Propo$ition is proved.</I></P> <foot>THEOR.</foot> <p n=>129</p> <head>THEOR. III. PROP. III.</head> <P>The Times in which the $ame Space is pa$t tho- row by unequal Velocities, have the $ame pro- portion to each other as their Velocities contra- rily taken.</P> <P><I>Let the two unequal Velocities be A the greater, and B the le$$e: and according to both the$e let a Motion be made thorow the $ame Space C D. I $ay the Time in which the Velocity A pa$$eth the Space C D, $hall be to the Time in which the Velocity B pa$$eth the $aid Space, as the Velocity B to the Velocity A. As A is to B, $o let C D be to C E: Then, by the former Propo$ition, the Time in</I> <fig> <I>which the Velocity A pa$$eth C D, $hall be the $ame with the Time in which B pa$$eth C E. But the Time in which the Velocity B pa$$eth C E, is to the Time in which it pa$$eth C D, as C E is to C D: Therefore the Time in which the Velocity A pa$$eth C D, is to the Time in which the Velocity B pa$$eth the $ame C D, as C E is to C D; that is, the Ve- locity B is to the Velocity A: Which was to be proved.</I></P> <head>THEOR. IV. PROP. IV.</head> <P>If two Moveables move with an Equable Mo- tion, but with unequal Velocities, the Spaces which they pa$$e in unequal Times, are to each other in a proportion compounded of the pro- portion of the Velocities, and of the propor- tion of the Times.</P> <P><I>Let the two Moveables moving with an Equable Motion, be E and F: And let the proportion of the Velocity of the Moveable E be to the Velocity of the Moveable F, as A is to B: And let the Time in which E is moved, be unto the Time in which F is moved, as C is to D. I $ay the Space pa$$ed by E, with the Velocity A in the Time C, is to the Space pa$$ed by F, with the Velocity B in the Time D, in a proportion compounded of the proportion of the Velocity A to the Velocity B, and of</I> <foot>S <I>the</I></foot> <p n=>130</p> <I>the proportion of the Time C to the Time D. Let the Space pa$$ed by the Moveable E, with the Velocity A in the Time C, be G: And as the Velocity A is to the Velocity B, <fig> $o let G be to I: And as the Time C is to the Time D, $o let I be to L: It is manife$t, that I is the Space pa$$ed by F in the $ame Time in which E pa$$eth thorow G; $eeing that the Spaces G and I are as the Velocities A and B; and $eeing that as the Time C is to the Time D, $o is I unto L; and $ince that I is the Space pa$$ed by the Moveable F in the Time C: Therefore L $hall be the Space that F pa$$eth in the Time D, with the Velocity B: But the proportion of G to L, is compounded of the proportions of G to I, and of I to L; that is, of the proportions of the Velocity A to the Velocity B, and of the Time C to the Time D: Therefore the Propo$ition is demon$trated.</I></P> <head>THEOR. V. PROP. V.</head> <P>If two Moveables move with an Equable Motion, but with unequal Velocities, and if the Spaces pa$$ed be al$o unequal, the Times $hall be to each other in a proportion compounded of the proportion of the Spaces, and of the proporti- on of the Velocities contrarily taken.</P> <P><I>Let A and B be the two Moveables, and let the Velocity of A be to the Velocity of B, as V to T, and let the Spaces pa$$ed, be as S to R. I $ay the proportion of the Time in which A is moved to the Time in which B is moved, $hall be compounded of the proportions of the Velocity T to the Velocity V, and of the Space S to the Space R. Let C be the Time of the Motion A;</I> <fig> <I>and as the Velocity T is to the Velocity V, $o let the Time C be to the Time E: And for a$much as C is the Time in which A with the Velocity V pa$$eth the Space S; and that the Time C is to the Time E, as the Velocity T of the Moveable B is to the Velocity V, E $hall be the Time in which the Moveable B would pa$$e</I> <foot><I>the</I></foot> <p n=>131</p> <I>the $ame Space S. Again as the Space S is to the Space R, $o let the Time E be to the Time G: Therefore G is the Time in which B would pa$$e the Space R. And becau$e the proportion of C to G is compounded of the proportions of C to E, and of E to G; And $ince the proportion of C to E is the $ame with that of the Velocities of the Moveables A and B contrarily taken; that is, with that of T and V; And the proportion of E to G is the $ame with the proportion of the Spaces S and R: There- fore the Propo$ition is demon$trated.</I></P> <head>THEOR. VI. PROP. VI.</head> <P>If two Moveables move with an Equable Motion, the proportion of their Velocities $hall be com- pounded of the proportion of the Spaces pa$- $ed, and of the proportion of the Times con- trarily taken.</P> <P><I>Let A and B be the two Moveables moving with an Equable Motion; and let the Spaces by them pa$$ed, be as V to T; and let the Times be as S to R. I $ay that the proportion of the Ve- locity of the Moveable A, to that of the Velocity of B, $hall be compounded of the proportions of the Space V to the Space T, and of the Time R to the Time S. Let C be the Velocity with which the Moveable A pa$$eth the Space V in the Time S: And let the Velocity C be to the Velo-</I> <fig> <I>city E, as the Space V is to the Space T; And E $hall be the Veloci- ty with which the Moveable B pa$$eth the Space T in the Time S: Again, let the Velocity E be to the other Velocity G, as the Time R is to the Time S; And G $hall be the Velocity with which the Moveable B pa$$eth the Space T in the Time R. We have therefore the Velocity C, wherewith the Moveable A pa$$eth the Space V in the Time S; and the Velocity G, wherewith the Move- able B pa$$eth the Space T in the Time R: And the proportion of C to G is compounded of the proportions of C to E and of E to G: But the proportion of C to E, is $uppo$ed the $ame with that of the Space V to the Space T; and the proportion of E to G, is the $ame with that of R to S: Therefore the Propo$ition is manifest.</I></P> <foot>S 2 SALV.</foot> <p n=>132</p> <marg>* That is the A- cademick, <I>i. e. Galileus.</I></marg> <P>SALV. This that we have read, is what our ^{*} <I>Author</I> hath written of the Equable Motion. We will pa$s therefore to a more $ubtil and new Contemplation touching the Motion Naturally Accelerate: and behold here the Title and Introduction.</P> <head>OF MOTION NATVRALLY ACCELERATE.</head> <P><I>In the former Book we have con$idered the Accidents which ac- company Equable Motion; we are now to treat of another kind of Motion which we call Accelerate. And fir$t it will be expedient to find out and explain a Definition be$t agreeing to that which Nature makes u$e of. For though it be not nconvenient to feign a Motion at plea- $ure, and then to con$ider the Accidents that attend it (as tho$e have done, who having framed in their imagination Helixes and Conchoi- des, which are Lines ari$ing from certain Motions, although not u$ed by Nature, and upon that Suppo$ition have laudably demon$trated the Symptomes thereof) yet in regard that Nature maketh u$e of a certain kind of Acceleration in the de$cent of Grave Bodies, we are re$olved to $earch out and contemplate the pa$$ions thereof, and $ee whether the Definition that we are about to produce of this our Accelerate Motion, doth aptly and congruou$ly $ute with the E$$ence of Motion Naturally Accelerate. After many long and laborious Studies we have found out a Definition which $eemeth to expre$$e the true nature of this Accelerate Motion, in regard that all the Natural Experiments that fall under the Ob$ervation of our Sen$es, do agree with tho$e its properties that we intend anon to demon$trate. In this Di$qui$ition we have been a$$i- $ted, and as it were led by the hand by that ob$ervation of the u$ual Method and common procedure of Nature her $elf in her other Operati- ons, wherein $he con$tantly makes u$e of the Fir$t, Simple$t, and Ea- $ie$t Means that are: for I believe that no man can think that Swim- ming or flying can be performed in a more $imple or ea$ie way, than that which Fi$hes and Birds do u$e out of a Natural In$tinct. Why there- fore $hall not I be per$waded, that, when I $ee a Stone to acquire conti- nually new additions of Velocity in its de$cending from its Re$t out of $ome high place, this encrea$e made in the $imple$t ea$ie$t and mo$t obvious manner that we can imagine? Now if we $eriou$ly examine all the ways that can be devi$ed, we $hall find no encrea$es, no acqui$itions le$$e intricate or more intelligible than that which ever encrea$eth or makes its additions after the $ame manner. This appeareth by the great</I> Affinity <I>that is between Time and Motion. For as the Equability or Vniformity of Motion is defined and expre$$ed by the Equability of the</I> <foot><I>Times</I></foot> <p n=>133</p> <I>Times and Spaces, (for we call that Motion or Lation Equable, by which equal Spaces are pa$t in equal Times) $o by the $ame Equability of the parts of Time, we may perceive, that the encrea$e of Celerity in the Natu- ral Motion of Grave Bodies, is made after a Simple and plain manner; conceiving in our Mind that their Motion is continually accelerated uni- formly and at the $ame Rate, whil$t equal additions of Celerity are conferred upon them in all equal Times. So that taking any equal par- ticles of Time beginning from the fir$t In$tant in which the Moveable departeth from Re$t, and entereth upon its De$cent, the Degree of Velocity acquired in the fir$t and $econd Particles of Time, is double the degree of Velocity that the Moveable acquired in the fir$t Particle: and the degree of Velocity that it acquireth in three Particles, is triple, and that in four quadruple to the $ame Degree of the fir$t Time: As, for our better under$tanding, if a Moveable $hould continue its Motion according to the degree or moment of Velocity acquired in the fir$t Parti- cle of Time, and $hould extend its cour$e equably with that $ame De- gree; this Motion would be twice as $low as that which it would obtain according to the degree of Velocity acquired in two Particles of Time: So that it will not be improper if we under$tand the Intention of the Ve- locity, to proceed according to the Exten$ion of the Time. From whence we may frame this Definition of the Motion of which we are about to <*>eat.</I></P> <head>DEFINITION.</head> <P>Motion Accelerate in an Equable or Vniform Proportion, I call that which departing from Re$t, $uperaddeth equal moments of Velocity in equal Times.</P> <P>SAGR. Though it were Irrational for me to oppo$e this or any other Definition a$$igned by any what$oever Author, they being all Arbitrary, yet I may very well, without any offence, que$tion whe- ther this Definition, which is under$tood and admitted in Ab$tract, doth $ute, agree, and hold true in that $ort of Accelerate Motion, which Grave Bodies de$cending naturally do exerci$e. And becau$e the Authour $eemeth to promi$e us, that the Natural Motion of Grave Bodies is $uch as he hath defined it, I could wi$h that $ome Scruples were removed that trouble my mind; that $o I might apply my $elf afterwards with greater attention to the Proportions and Demon$trations which are expected.</P> <P>SALV. I like well, that you and <I>Simplicius</I> do propound Doubts as they come in the way: which I do imagine will be the <foot>$ame</foot> <p n=>134</p> $ame that I my $elf did meet with when I fir$t read this Treati$e, and that, either were re$olved by conferring with the Author, or removed by my own con$idering of them.</P> <P>SAGR. Whil$t I am fancying to my $elf a Grave De$cending Moveable to depart from Re$t, that is from the privation of all Velocity, and to enter into Motion, and in that to go encrea- $ing, according to the proportion after which the Time encrea$eth from the fir$t in$tant of the Motion; and to have <I>v. gr.</I> in eight Pul$ations, acquired eight degrees of Velocity, of which in the fourth Pul$ation it had gained four, in the $econd two, in the fir$t one, Time being $ubdivi$ible <I>in infinitum,</I> it followeth, that the Antecedent Velocity alwayes dimini$hing at that Rate, there will bt no degree of Velocity $o $mall, or, if you will, of Tardity $o great, in which the $aid Moveable is not found to be con$ti- tuted, after its departure from infinite Tardity, that is, from Re$t. So that if that degree of Velocity which it had at four Pul- $ations of Time, was $uch, that maintaining it Equable, it would have run two Miles in an hour, and with the degree of Velocity that it had in the $econd Pul$ation, it would have gone one mile an hour, it mu$t be granted, that in the In$tants of Time neeter and neerer to its fir$t In$tant of moving from Re$t, it is $o $low, as that (continuing to move with that Tardity) it would not have pa$$ed a Mile in an hour, nor in a day, nor in a year, nor in a thou$and; nay, nor have gone one $ole foot in a greater time: An accident to which me thinks the Imagination but very unea- $ily accords, $eeing that Sen$e $heweth us, that a Grave Falling Body commeth down $uddenly, and with great Velocity.</P> <P>SALV. This is one of tho$e Doubts that al$o fell in my way upon my fir$t thinking on this affair, but not long after I remo- ved it: and that removal was the effect of the $elf $ame Expe- riment which at pre$ent $tarts it to you. You $ay, that in your opinion, Experience $heweth that the Moveable hath no $ooner departed from Re$t, but it entereth into a very notable Velocity: and I $ay, that this very Experiment proves it to us, that the fir$t Impetus's of the Cadent Body, although it be very heavy, are mo$t $lack and $low. Lay a Grave Body upon $ome yielding mat- ter, and let it continue upon it till it hath pre$$ed into it as far as it can with its $imple Gravity; it is manife$t, that rai$ing it a yard or two, and then letting it fall upon the $ame matter, it $hall with its percu$$ion make a new pre$$ure, and greater than that made at fir$t by its meer weight: and the effect $hall be cau$ed by the falling Moveable conjoyned with the Velocity acquired in the Fall: which impre$$ion $hall be greater and greater, accord- ing as the Percu$$ion $hall come from a greater height; that is, according as the Velocity of the Percutient $hall be greater. We <foot>may</foot> <p n=>135</p> may therefore without mi$take conjecture the quantity of the Ve- locity of a falling heavy Body; by the quality and quantity of the Percu$$ion. But tell me Sirs, that Beetle which being let fall upon a Stake from an height of four yards, driveth it into the ground, <I>v. gr.</I> four inches, comming from an height of two yards, $hall drive it much le$$e, and le$$e from an height of one, and le$$e from a foot; and la$tly lifting it up an inch, what will it do more than if without any blow it were laid upon it? Certainly but very little, and the operation would be wholly impercep- tible, if it were rai$ed the thickne$$e of a leaf. And becau$e the effect of the Percu$$ion is regulated by the Velocity of the Percu- tient, who will que$tion but that the Motion is very $low, and the Velocity extreme $mall, where its operation is impercep- tible? See now of what power Truth is, $ince the $ame Experi- ment that $eemed at the fir$t blu$h to hold forth one thing, be- ing better con$idered, a$certains us of the contrary. But without having recour$e to that Experiment (which without doubt is mo$t per$wa$ive) me-thinks that it is not hard to penetrate $uch a Truth as this by meer Di$cour$e. We have an heavy $tone $u- $tained in the Air at Re$t: let it be di$engaged from its uphol- der, and $et at liberty; and, as being more grave than the Air, it goeth de$cending downwards, and that not with a Motion Equa- ble, but $low in the beginning, and continually afterwards ac- celerate: and $eeing that the Velocity is Augmentable and Di- mini$hable <I>in infinitum,</I> what Rea$on $hall per$wade me, that that Moveable departing from an infinite Tardity (for $uch is Re$t) entereth immediately into ten degrees of Velocity, rather than in one of four, or in this more than in one of two, of one, of half one, or of the hundredth part of one; and to be $hort, in all the infinite le$$er? Pray you hear me. I do not think that you would $cruple to grant me, that the acqui$t of the Degrees of Ve- locity of the falling Stone may be made with the $ame Order as is the Diminution and lo$$e of the $ame degrees, when with an impellent Virtue it is driven upwards to the $ame height: But if that be $o, I do not $ee how it can be $uppo$ed that in the diminu- tion of the Velocity of the a$cendent Stone, $pending it all, it can come to the $tate of Re$t before it hath pa$$ed thorow all the degrees of Tardity.</P> <P>SIMP. But if the greater and greater degrees of Tardity are infinite, it $hall never $pend them all; $o that the a$cendent Grave will never attain to Re$t, but will move <I>ad infinitum,</I> $till retarding: a thing which we $ee not to happen.</P> <P>SALV. This would happen, <I>Simplicius,</I> in ca$e the Moveable $hould $tay for $ome time in each degree: but it pa$$eth thorow them, without $taying longer than an in$tant in any of them. <foot>And</foot> <p n=>136</p> And becau$e in every quantitative Time, though never $o $mall, there are infinite In$tants, therefore they are $ufficient to an$wer to the infinite degrees of Velocity dimini$hed. And that the a$cendent Grave Body per$i$ts not for any quantitative Time in one and the $ame degree of Velocity, may thus be made out: Becau$e, a certain quantitative Time being a$$igned it in the fir$t in$tant of that Time, and likewi$e in the la$t, the Moveable $hould be found to have one and the $ame degree of Velocity, it might by this $econd degree be likewi$e driven upwards $uch an- other Space, like as from the fir$t it was tran$ported to the $e- cond; and by the $ame rea$on it would pa$$e from the $econd to the third, and, in $hort, would continue its Motion Uniform <I>ad infinitum.</I></P> <P>SAGR. From this Di$cour$e, as I conceive, one might derive a very appo$ite Rea$on of the Que$tion controverted among$t Philo. $ophers, Touching what $hould be the Cau$e of the acceleration of the Natural Motion of Grave Moveables. For when I confider in the Grave Body driven upwards, its continual Diminution of that Virtue impre$$ed upon it by the Projicient, which $o long as it was $uperiour to that other contrary one of Gravity, forced it upwards, this and that being come to an <I>Equilibrium,</I> the Move- able cea$eth to ri$e any higher, and pa$$eth thorow the $tate of Re$t, in which the <I>Impetus</I> impre$$ed is not annihilated, but one- ly that exce$$e is $pent, which it before had above the Gravity of the Moveable, whereby prevailing over the $ame, it did drive it upwards. And the Diminution of this forrein <I>Impetus</I> continu- ing, and con$equently the advantage beginning to be on the part of the Gravity, the De$cent al$o beginneth but $low, in regard of the oppo$ition of the Virtue impre$$ed, a con$iderable part of which $till remaineth in the Moveable: but becau$e it doth go continually dimini$hing, and is $till with a greater and greater proportion overcome by the Gravity, hence ari$eth the continual Acceleration of the Motion.</P> <P>SIMP. The conceit is witty, but more $ubtil than $olid: for in ca$e it were concludent, it $alveth onely tho$e Natural Motions to which a Violent Motion preceded, in which part of the extern Virtue $till remains in force: but where there is no $uch remaining impul$e, as where the Moveable departeth from a long Quie$- cence, the $trength of your whole Di$cour$e vani$heth.</P> <P>SAGR. I believe that you are in an Errour, and that this Di- $tinction of Ca$es which you make, is needle$$e, or, to $ay bet- ter, <I>Null.</I> Therefore tell me, whether may there be impre$$ed on the Project by the Projicient $ometimes much, and $ometimes little Vertue; $o as that it may be $tricken upwards an hundred yards, and al$o twenty, or four, or one?</P> <foot>SIMP.</foot> <p n=>137</p> <P>SIMP. No doubt but there may.</P> <P>SAGR. And no le$$e po$$ible is it, that the $aid Virtue impre$$ed $hall $o little $eperate the Re$i$tance of the Gravity, as not to rai$e the Project above an inch: and finally the Virtue of the Projicient may be onely $o much, as ju$t to equalize and com- pen$ate the Re$i$tance of the Gravity, $o as that the Moveable is not driven upwards, but onely $u$tained. So that when you hold a Stone in your hand, what el$e do you, but impre$$e on it $o much Virtue impelling upwards, as is the faculty of its Gra- vity drawing downwards? And this your Virtue, do you not continue to keep it impre$$ed on the Stone all the time that you hold it in your hand? What $ay you, is it dimini$hed by your long holding it? And this $u$tention which impedeth the Stones de$cent, what doth it import, whether it be made by your hand, or by a Table, or by a Rope, that $u$pends it? Doubtle$$e no thing at all. Conclude with your $elf therefore, <I>Simplicius,</I> that the precedence of a long, a $hort, or a Momentary Re$t to the Fall of the Stone, makes no alteration at all, $o that the Stone $hould not alwaies depart affected with $o much Virtue contrary to Gravity, as did exactly $uffice to have kept it in Re$t.</P> <P>SALV. I do not think it a $ea$onable time at pre$ent to enter upon the Di$qui$ition of the Cau$e of the Acceleration of Natu- ral Motion: touching which $undry Philo$ophers have produced $undry opinions: $ome reducing it to the approximation unto the Center others to the le$$e parts of the <I>Medium</I> $ucce$$ively re- maining to be perforated; others to a certain Extru$ion of the Ambient <I>Medium,</I> which in reuniting upon the back of the Moveable, goeth driving and continually thru$ting it; which Fancies, and others of the like nature, it would be nece$$ary to examine, and with $mall benefit to an$wer. It $erveth our Au- thours turn at the pre$ent, that we under$tand that he will de- clare and demon$trate to us $ome Pa$$ions of an Accelerate Mo- tion (be the Cau$e of its Acceleration what it will) $o as that the Moments of its Velocity do go encrea$ing, after its departure from Re$t with that mo$t $imple proportion wherewith the Continua- tion of the Time doth encrea$e: which is as much as to $ay, that in equal Times there are made equal additaments of Velocity. And if it $hall be found, that the Accidents that $hall hereafter be demon$trated, do hold true in the Motion of Naturally De- $cendent and Accelerate Grave Moveables, we may account, that the a$$umed Definition taketh in that Motion of Grave Bo- dies, and that it is true, that their Acceleration doth encrea$e ac- cording as the Time and Duration of the Motion encrea$eth.</P> <P>SAGR. By what as yet is $et before my Intellectuals, it appears to me that one might with (haply) more plainne$$e define, and yet <foot>T never</foot> <p n=>138</p> never alter the Conceit; $aying that, A Motion uniformly accele- rate is that in which the Velocity goeth encrea$ing according as the Space encrea$eth that is pa$$ed thorow: So that, for example, the degree of Velocity acquired by the Moveable in a de$cent of four yards $hould be double to that that it would have after it had de$cended a Space of two, and this double to that acquired in the Space of the fir$t Yard. For I do not think that it can be doubted, but that that Grave Moveable which falleth from an height of $ix yards hath, and percu$$eth with an <I>Impetus</I> double to that which it had when it had de$cended three yards, and triple to that which it had at two, and $extuple to that had in the Space of one.</P> <P>SALV. I comfort my $elf in that I have had $uch a Companion in my Errour: and I will tell you farther, that your Di$cour$e hath $o much of likelihood and probability in it, that our Author him$elf did not deny unto me, when I propo$ed it to him, that he likewi$e had been for $ome time in the $ame mi$take. <I>B</I>ut that which I af- terwards extreamly wondred at, was to $ee in four plain words, di$covered, not only the falfity, but impo$$ibility of two Propo$i- tions that carry with them $o much of $eeming truth, that having propounded them to many, I never met with any one but did freely admit them to be $o.</P> <P>SIMP. Certainly I $hould be of the number, and that the De- $cendent Grave Moveable <I>vires acquir at eundo,</I> encrea$ing its Ve- locity at the rate of the Space, and that the Moment of the $ame Percutient is double, coming from a double height, $eem to me Pro- po$itions to be granted without any hæ$itation or controver$ie.</P> <P>SALV. And yet they are as fal$e and impo$$ible, as that Moti- on is made in an in$tant. And hear a clear proof of the $ame. In ca$e the Velocities have the $ame proportion as the Spaces pa$$ed, or to be pa$$ed, tho$e Spaces $hall be pa$$ed in equal Times: if therefore the Velocities with which the falling Moveable pa$$eth the Space of four yards, were double to the Velocities with which it pa$$eth the two fir$t yards (like as the Space is double to the Space) then the Times of tho$e Tran$itions are equal: but the $ame Move- able's pa$$ing the four yards, and the two in one and the $ame Time, hath place only in In$tantaneous Motion. <I>B</I>ut we $ee, that the falling grave <I>B</I>ody maketh its Motion in Time, and pa$$eth the two yards in a le$$er than it doth the four. Therefore it is fal$e that its Velocity encrea$eth as its Space. The other Propo$ition is demon- $trated to be fal$e with the $ame per$picuity. For that which per- cu$$eth being the $ame, the difference and Moment of the Percu$$ton cannot be determined but by the difference of Velocity; If there- fore the percutient, coming from a double height, make a Percu$$i- on with a double Moment, it is nece$$ary that it $trike with a dou- ble Velocity: <I>B</I>ut the double Velocity pa$$eth the double Space in <foot>the</foot> <p n=>139</p> the $ame Time; and we $ee the Time of the De$cent from the grea- ter altitude to be longer.</P> <P>SAGR. This is too great an Evidence, too great a Facility wherewith you manife$t ab$truce Conclu$ions: this extream ea$i- ne$s rendreth them of le$$e value than they were whil$t they lay hid under contrary appearances. I believe that the Generality of men little pre$$e tho$e Notions which are ea$ily obtained, in compari- $on of tho$e about which men make $o long and inexplicable alter- cations.</P> <P>SALV. To tho$e which with great brevity and clarity $hew the fallacies of Propo$itions that have been commonly received for true by the generality of people, it would be a very tolerable in- jury to return them only $lighting in$tead of thanks: but there is much di$plea$ure and mole$tation in another certain affection $ometimes found in $ome men, that pretending in the $ame Studies at lea$t Parity with any whom$oever, do $ee that they have let pa$s $uch and $uch for true Conclu$ions, which afterwards by another, with a $hort and ea$ie di$qui$ition, have been detected and convicted for fal$e. I will not call that affection Envy, that is ac- cu$tomed to convert in time to hatred and de$pite again$t the di$- coverers of $uch Fallacies, but I will call it an itch, and a de$ire to be able rather to maintain their inveterate Errours, than to per- mit the reception of new-di$covered Truths. Which humour $ome- times induceth them to write in contradiction of tho$e truths which are but too perfectly known unto them$elves only to keep the Reputation of others low in the opinion of the numerous and ill-informed Vulgar. Of $uch fal$e Conclu$ions received for true, and very ea$ie to be confuted, I have heard no $mall number from our <I>Academick,</I> of $ome of which I have kept account.</P> <P>SAGR. And you mu$t not deprive us of them; but in due time impart them to us, when a particular Meeting $hall be appointed for them. For the pre$ent, continuing the di$cour$e we are about, I think that by this time we have e$tabli$hed the Definition of Mo- tion uniformly Accelerate, treated of in the en$uing di$cour$es, and it is this;</P> <P><I>A Motion Equable, or Vniformly Accelerate, we call that which departing from Re$t $uperadds equal Moments of Velocity in equal Times.</I></P> <P>SALV. That Definition being confirmed, the Author asketh and $uppo$eth but one only Principle to be true, namely:</P> <foot>T 2 SUP-</foot> <p n=>140</p> <head>SVPPOSITION.</head> <P><I>I $uppo$e that the degrees of Velocity acquired by the $ame Moveable upon Planes of different inclinations are equal then, when the Elevations of the $aid Planes are equal.</I></P> <P>By the Elevation of an inclined Plane he meaneth the Per- pendicular, which from the higher term of the $aid Plane falleth upon the Horizontal Line produced along by the lower term of the $aid Plane inclined: as for better under$tanding; the Line A B being parallel to the Horizon, upon which let the two <fig> Planes C A, and C D be inclined: the Perpendicular C B falling up- on the Horizontal Line B A the Author calleth the Elevation of the Planes C A and C D; and $uppo$eth that the degrees of Velocity of the $ame Moveable de$cending along the inclined Planes C A and C D, acqui- red in the Terms A and D are equal, for that their Elevation is the $ame C B. And $o great al$o ought the degree of Velocity be under$tood to be which the $ame Moveable falling from the Point C would acquire in the term B.</P> <P>SAGR. The truth is, this Suppo$ition hath in it $o much of pro- bability, that it de$erveth to be granted without di$pute, alwaies pre$uppo$ing that all accidental and extern Impediments are re- moved, and that the Planes be very Solid and Ter$e, and the Move- able in Figure mo$t perfectly Rotund, $o that neither the Plane, nor the Moveable have any unevenne$s. All Contra$ts and Im- pediments, I $ay, being removed, the light of Nature dictates to me without any difficulty, that a Ball heavy and perfectly round de$cending by the Lines C A, C D, and C B would come to the terms A D, and B with equal <I>Impetus's.</I></P> <P>SALV. You argue very probably; but over and above the pro- bability, I will by an Experiment $o increa$e the likelihood, as that it wants but little of being equal to a very nece$$ary Demon$trati- on. Imagine this leafe of Paper to be a Wall erect at Right-angles to the Horizon, and at a Nail, fa$tned in the $ame, hang a Ball or Plummet of Lead, weighing an ounce or two, $u$pended by the $mall thread A B, two or three yards long, perpendicular to the Horizon: and on the Wall draw an Horizontal Line D C, cutting <foot>the</foot> <p n=>141</p> the Perpendicular A B at Right angles, which A B mu$t hang two Inches, or thereabouts, from the Wall: Then transferring the $tring A B with the Ball into C, let go the $aid Ball; which you will <fig> $ee fir$t to de$cend de$cribing C B D, and to pa$s $o far beyond the Term B, that run- ning along the Arch B D it will ri$e almo$t as high as the de$igned Parallel C D, wanting but a very $mall mat- ter of reaching to it, the preci$e arrival thi- ther being denied it by the Impediment of the Air, and of the Thread. From which we may truly conclude, that the <I>Impetus</I> acquired in the point B by the Ball in its de$cent along the Arch C B, was $o much as $ufficed to carry it upwards along $uch another Arch B D unto the $ame height: having made, and often reiterated this Experiment, let us drive into the Wall, along which the Perpendicular A B pa$$eth, another Nail, as in E or in F, which is to $tand out five or $ix In- ches; and this to the end that the thread A B, returning as before to carry back the Ball C along the Arch C B, when it is come to B, the Thread $topping at the Nail E may be con$trained to move along the Circumference B G, de$cribed about the Center E: by which we $hall $ee what that $ame <I>Impetus</I> is able to do, which be- fore, being conceived in the $ame term B, carried the $ame Move- able along the Arch B D unto the height of the Horizontal Line C D. Now, Sirs, you $hall with delight $ee the Ball carried unto the Horizontal Line in the Point G; and the $ame will happen if the $top be placed lower, as in F, where the Ball would de$cribe the Arch B I, evermore terminating its a$cent exactly in the Line C D: and in ca$e the Check were $o low that the overplus of the thread beneath it cannot reach to the height of C D, (which would happen if it were nearer to the point B than to the inter$ection of A B with the Horizontal Line C D) then the thread would whirle and twine about the Nail. This experiment leaveth no place for our doubting of the truth of the Suppo$ition: for the two Arches C B and D B being equall, and $cituate alike, the acqui$t of Moment made along the De$cent in the Arch C B, is the $ame with that made along the De$cent in the Arch D B. But the Moment acquired in <I>B,</I> along the Arch C <I>B,</I> is able to carry the $ame Moveable upwards along the Arch <I>B</I> D: Therefore the Mo- ment acquired in the De$cent D <I>B</I> is equall to that which driveth <foot>the</foot> <p n=>142</p> the $ame Moveable along the $ame Arch from <I>B</I> to D: So that ge- nerally every Moment acquired along the De$cent of an Arch is equall to that which hath power to make the $ame Moveable re- a$cend along the $ame Arch: <I>B</I>ut all the Moments that make the Moveable a$cend along all the Arches <I>B</I> D, <I>B</I> G, <I>B</I> I are equal, $ince they are made by one and the $ame Moment acquired along the De$cent C <I>B,</I> as Experience $hews: Therefore all the Moments that are acquired by the De$cents along the Arches D <I>B,</I> G <I>B,</I> and I <I>B</I> are equal.</P> <P>SAGR. Your Di$cour$e is in my Judgment very Rational, and the Experiment $o appo$ite and pertinent to verifie the <I>Po$tulatum,</I> that it very well de$erveth to be admitted as if it were Demon- $trated.</P> <P>SALV. I will not con$ent, <I>Sagredus,</I> that we take more to our $elves than we ought; and the rather for that we are chiefly to make u$e of this A$$umption in Motions made upon $treight and not curved Superficies; in which the Acceleration proceedeth with degrees very different from tho$e wherewith we $uppo$e it to pro- ceed in $treight Planes. In$omuch, that although the Experiment alledged $hews us, that the de$cent along the Arch C <I>B</I> conferreth on the Moveable $uch a Moment, as that it is able to re-carry it to the $ame height along any other Arch <I>B</I> C, <I>B</I> G, and <I>B</I> I, yet we cannot with the like evidence $hew, that the $ame would hap- pen in ca$e a mo$t exact <I>B</I>all were to de$cend by $treight Planes in- clined according to the inclinations of the Chords of the$e $ame Arches: yea, it is credible, that Angles being formed by the $aid Right Planes in the term <I>B,</I> the <I>B</I>all de$cended along the Declivi- ty according to the Chord C <I>B,</I> finding a $top in the Planes a$cend- ing according to the Chords <I>B</I> D, <I>B</I> G, and <I>B</I> I, in ju$tling again$t them, would lo$e of its <I>Impetus,</I> and could not be able in ri$ing to attain the height of the Line C D. <I>B</I>ut the Ob$tacle being remo- ved, which prejudiceth the Experiment, I do believe, that the un- der$tanding may conceive, that the <I>Impetus</I> (which in effect de- riveth vigour from the quantity of the De$cent) would be able to remount the Moveable to the $ame height. Let us therefore take this at pre$ent for a <I>Po$tulatum</I> or Petition, the ab$olute truth of which will come to be e$tabli$hed hereafter by $eeing other Con- clu$ions rai$ed upon this Hypothe$is to an$wer, and exactly jump with the Experiment. The Author having $uppo$ed this only Prin- ciple, he pa$$eth to the Propo$itions, demon$tratively proving them; of which the fir$t is this;</P> <foot>THEOR.</foot> <p n=>143</p> <head>THEOR. I. PROP. I.</head> <P>The time in which a Space is pa$$ed by a Movea- ble with a Motion Vniformly Accelerate, out of Re$t, is equal to the Time in which the $ame Space would be pa$t by the $ame Moveable with an Equable Motion, the degree of who$e Velocity is $ubduple to the greate$t and ulti mate degree of the Velocity of the former Vni- formly Accelerate Motion.</P> <P><I>Let us by the exten$ion A B repre$ent the Time, in which the Space</I> C D <I>is pa$$ed by a Moveable with a Motion Vniformly Accelerate, out of Re$t in C: and let the greate$t and la$t de-</I> <fig> <I>gree of Velocity acquired in the In$tants of the Time</I> A B <I>be repre$ented by</I> E B; <I>and con$titute at plea- $ure upon</I> A B <I>any number of parts, and thorow the points of divi$ion draw as many Lines, continued out unto the Line</I> A E, <I>and equidi$tant to</I> B E, <I>which will repre$ent the encrea$e of the degrees of Velocity after the fir$t In$tant A. Then divide</I> B E <I>into two equall parts in</I> F, <I>and draw</I> F G <I>and</I> A G <I>parallel to B A and</I> B F<I>: The Parallelogram</I> A G F B <I>$hall be equall to the Triangle</I> A E B, <I>its Side</I> G F <I>dividing</I> A E <I>into two equall parts in I: For if the Parallels of the Triangle</I> A E <I>B be continued out unto</I> I G F, <I>we $hall have the Aggregate of all the Parallels contained in the Quadrilatural Figure equal to the Aggregate of all the Parallels compre- hended in the Triangle</I> A E <I>B; For tho$e in the Triangle</I> I E F <I>are equal to tho$e contained in the Triangle</I> G I A, <I>and tho$e that are in the</I> Tra- pezium <I>are in common. Now $ince all and $ingular the In$tants of Time do an$wer to all and $ingular the Points of the Line A B; and $ince the Parallels contained in the Triangle</I> A E <I>B do repre$ent the degrees of Ac- celeration or encrea$ing Velocity, and the Parallels contained in the Pa- rallelogram do likewi$e repre$ent as many degrees of Equable Motion or unencrea$ing Velocity: It appeareth, that as many Moments of Velocity pa$$ed in the Accelerate Motion according to the encrea$ing Parallels of the Triangle A E B, as in the Equable Motion according to the Parallels of the Parallelogram G B: Becau$e what is wanting in the fir$t half of the</I> <foot><I>Accelerate</I></foot> <p n=>144</p> <I>Accelerate Motion of the Velocity of the Equable Motion (which defi- cient Moments are repre$ented by the Parallels of the Triangle A</I> G I) <I>is made up by the moments repre$ented by the Parallels of the Triangle</I> I E F. <I>It is manife$t, therefore, that tho$e Spaces are equal which are in the $ame Time by two Moveables, one whereof is moved with a Mo- tion uniformly Accelerated from Re$t, the other with a Motion Equable according to the Moment $ubduple of that of the greate$t Velocity of the Accelerated Motion: Which was to be demon$trated.</I></P> <head>THEOR. II. PROP. II.</head> <P>If a <I>M</I>oveable de$cend out of Re$t with a <I>M</I>oti- on uniformly Accelerate, the Spaces which it pa$$eth in any what$oever Times are to each other in a proportion Duplicate of the $ame Times; that is, they are as the Squares of them.</P> <P><I>Let</I> A B <I>repre$ent a length of Time beginning at the fir$t In$tant A; and let</I> A D <I>and</I> A E <I>repre$ent any two parts of the $aid Time; and let</I> H I <I>be a Line in which the Moveable out of H, (as the fir$t beginning of the Motion) de$cendeth uniformly accelerating; and let the</I> <fig> <I>Space</I> H L <I>be pa$$ed in the fir$t Time</I> A D; <I>and let</I> H M <I>be the Space that it $hall de$cend in the Time</I> A E. <I>I $ay, the Space</I> M H <I>is to the Space</I> H L <I>in duplicate propor- tion of that which the Time</I> E A <I>hath to the Time</I> A D<I>: Or, if you will, that the Spaces</I> M H <I>and</I> H L <I>are to one another in the $ame proportion as the Squares</I> E A <I>and</I> A D. <I>Draw the Line</I> A C <I>at any Angle with</I> A B, <I>and from the points D and E draw the Parallels</I> D O <I>and</I> P E<I>: of which</I> D O <I>will repre$ent the greate$t degree of Velocity acquired in the In$tant D of the Time</I> A D; <I>and</I> P <I>the greate$t degree of Velocity acquired in the In- $tant E of the Time</I> A E. <I>And becau$e we have de- mon$trated in the la$t Propo$ition concerning Spaces, that tho$e are equal to one another, of which two Moveables have pa$t in the $ame Time, the one by a Moveable out of Re$t with a Motion uniformly Accelerate, and the other by the $ame Moveable with an Equable Motion, who$e Velocity is $ubduple to the greate$t acquired by the Accelerate Motion: Therefore</I> M H <I>and</I> H L <I>are the Spaces that two Lquable Motions, who$e Velocities $hould be as the half of</I> P E, <I>and</I> <foot><I>half</I></foot> <p n=>145</p> <I>half of</I> O D, <I>would pa$$e in the Times</I> E A <I>and</I> D A. <I>If it be proved therefore that the$e Spaces</I> M H <I>and</I> L H <I>are in duplicate proportion to the Times</I> E A <I>and</I> D A; <I>We $hall have done that which was intended. But in the fourth Propo$ition of the Fir$t Book we have demon$trated: That the Spaces pa$t by two Moveables with an Equable Motion are to each other in a proportion compounded of the proportion of the Velo- cities and of the proportion of the Times: But in this ca$e the propor- tion of the Velocities and the proportion of the Times is the $ame</I> (<I>for as the half of</I> P E <I>is to the half of</I> O D, <I>or the whole</I> P E <I>to the whole</I> O D, <I>$o is</I> A E <I>to</I> A D<I>: Therefore the proportion of the Spaces pa$- $ed is double to the proportion of the Times. Which was to be demon- $trated.</I></P> <P><I>Hence likewi$e it is manife$t, that the proportion of the $ame Spaces is double to the proportions of the greate$t degrees of Velocity: that is, of the Lines</I> P E <I>and</I> O D<I>: becau$e</I> P E <I>is to</I> O D, <I>as</I> E A <I>to</I> D A.</P> <head>COROLARY I.</head> <P><I>Hence it is manife$t, that if there were many equal Times taken in or- der from the fir$t In$tant or beginniug of the Motion, as $uppo$e</I> A D, D E, E F, F G, <I>in which the Spaces</I> H L, L M, M N, N I <I>are pa$$ed, tho$e Spaces $hall be to one another as the odd numbers from an Vnite:</I> $cilicet, <I>as 1, 3, 5, 7. For this is the Rate or pro- portion of the exce$$es of the Squares of Lines that equally exceed one another, and the exce$$e of which is equal to the least of them, or, if you will, of Squares that follow one another, beginning</I> ab Unitate. <I>Whil$t therefore the degree of Velocity is encrea$ed ac- cording to the $imple Series of Numbers in equal Times, the Spaces pa$t in tho$e Times make their encrea$e according to the Series of odd Numbers from an Vnite.</I></P> <P>SAGR. Be plea$ed to $tay your Reading, whil$t I do paraphra$e touching a certain Conjecture that came into my mind but even now; for the explanation of which, unto your under- $tanding and my own, I will de$cribe a $hort Scheme: in which I fan$ie by the Line A I the continuation of the Time after the fir$t In$tant, applying the Right Line A F unto A according to any Angle: and joyning together the Terms I F, I divide the Time A I in half at C, and then draw C B parallel to I F. <I>A</I>nd then con$ide- ring B C, as the greate$t degree of Velocity which beginning from Re$t in the fir$t In$tant of the Time <I>A</I> goeth augmenting accord- ing to the encrea$e of the Parallels to B C, drawn in the Triangle <I>A</I> B C, (which is all one as to encrea$e according to the encrea$e of the Time) I admit without di$pute, upon what hath been $aid already, That the Space pa$t by the falling Moveable with the <foot>V Velocity</foot> <p n=>146</p> Velocity encrea$ed in the manner afore$aid would be equal to the Space that the $aid Moveable would pa$$e, in ca$e it were in the $ame Time <I>A</I> C, moved with an Uniform Motion, who$e degree of Velocity $hould be equal to E C, the half of B C. I now proceed farther, and imagine the Moveable; having de$cended with an <I>A</I>ccelerate Motion, to have in the In$tant C the degree of Velocity B C: It is ma- <fig> nife$t, that if it did continue to move with the $ame degree of Velocity B C, without farther <I>A</I>cceleration, it would pa$$e in the following Time C I, a Space double to that which it pa$$ed in the equal Time <I>A</I> C, with the degree of Uniform Velocity E C, the half of the Degree B C. But becau$e the Moveable de$cendeth with a Velocity encrea$ed alwaies Uni- formly in all equal Times; it will add to the degree C B in the following Time C I, tho$e Tame Moments of Velocity that encrea$e according to the Parallels of the Triangle B F G, equal to the Triangle <I>A</I> B C. So that adding to the degree of Velocity G I, the half of the degree F G, the greate$t of tho$e ac- quired in the <I>A</I>ccelerate Motion, and regulated by the Parallels of the Triangle B F G, we $hall have the degree of Velocity I N, with which, with an Uniform Motion, it would have moved in the Time C I: Which degree I N, being triple the degree E C, pro- veth that the Space pa$$ed in the $econd Time C I ought to be tri- ple to that of the fir$t Time C <I>A. A</I>nd if we $hould $uppo$e to be added to <I>A</I> I another equal part of Time I O, and the Triangle to be enlarged unto <I>A</I> P O; it is manife$t, that if the Motion $hould continue for all the Time I O with the degree of Velocity I F, acquired in the <I>A</I>ccelerate Motion in the Time <I>A</I> I, that degree I F being Quadruple to E C, the Space pa$$ed would be Quadruple to that pa$$ed in the equal fir$t Time <I>A</I> C: But continuing the encrea$e of the Uniform <I>A</I>cceleration in the Triangle F P Q like to that of the Triangle <I>A</I> B C, which being reduced to equable Motion addeth the degree equal to E C, Q R being added, equal to E C, we $hall have the whole Equable Velocity exerci$ed in the Time I O, quintuple to the Equable Velocity of the fir$t Time <I>A</I> C, and therefore the Space pa$$ed quintuple to that pa$t in the fir$t Time <I>A</I> C. We $ee therefore, even by this familiar computation, That the Spaces pa$$ed in equal Times by a Moveable which departing from Re$t goeth acquiring Velocity, according to the encrea$e of the Time, are to one another as the odd Numbers <I>ab</I> <foot><I>unitate</I></foot> <p n=>147</p> <I>unitate 1, 3, 5: A</I>nd that the Spaces pa$$ed being conjunctly taken, that pa$$ed in the double Time is quadruple to that pa$$ed in the $ubduple, that pa$$ed in the triple Time is nonuple; and, in a word, that the Spaces pa$$ed are in duplicate proportion to their Times; that is, as the Squares of the $aid Times.</P> <P>SIMP. I mu$t confe$$e that I have taken more plea$ure in this plain and clear di$cour$e of <I>Sagredus,</I> than in the to-me-more ob$cure Demon$tration of the <I>A</I>uthor: $o that I am very well $atisfied, that the bu$ine$$e is to $ucceed as hath been $aid, the Definition of Uniformly <I>A</I>ccelerate Motion being $uppo$ed, and granted. But whether this be the <I>A</I>cceleration of which Nature maketh u$e in the Motion of its de$cending Grave Bodies, I yet make a que$tion: and therefore for information of me, and of others like unto me, me thinks it would be $ea$onable in this place to produce $ome Experiment among$t tho$e which were $aid to be many, which in $undry Ca$es agree with the Conclu$ions demon- $trated.</P> <P>SALV. You, like a true <I>A</I>rti$t, make a very rea$onable demand, and $o it is u$ual and convenient to do in Sciences that apply Mathematical Demon$trations to Phy$ical Conclu$ions, as we $ee in the Profe$$ors of Per$pection, <I>A</I>$tronomy, Mechanicks, Mu$ick, and others, who with Sen$ible Experiments confirm tho$e their Principles that are as the foundations of all the following Structure: and therefore I de$ire that it may not be thought $uperfluous, that we di$cour$e with $ome prolixity upon this fir$t and grand funda- mental on which we lay the weight of the Immen$e Machine of infinite Conclu$ions, of which we have but a very $mall part $et down in this Book by our <I>A</I>uthor, who hath done enough to open the way and door that hath been hitherto $hut unto all Specula- tive Wits. Touching Experiments, therefore, the <I>A</I>uthor hath not omitted to make $everal; and to a$$ure us, that the <I>A</I>ccelerati- on of natural-de$cending Graves hapneth in the afore$aid propor- tion, I have many times in his company $et my $elf to make a triall thereof in the following Method.</P> <P>In a pri$me or Piece of Wood, about twelve yards long, and half a yard broad one way, and three Inches the other, we made, upon the narrow Side or edge a Groove of little more than an Inch wide; we $hot it with the Grooving Plane very $traight, and to make it very $mooth and $leek, we glued upon it a piece of Vellum, poli$hed and $moothed as exactly as can be po$$ible: and in it we have let a brazen Ball, very hard, round, and $mooth, de$cend. Having placed the $aid Pri$me Pendent, rai$ing one of its ends above the Horizontal Plane a yard or two at plea$ure, we have let the Ball (as I $aid) de$cend along the Grove, ob$erving, in the manner that I $hall tell you pre$ently, the Time which it $pent in <foot>V 2 running</foot> <p n=>148</p> runing it all; repeating the $ame ob$ervation again and again to a$$ure our $elves of the Time, in which we never found any diffe- rence, no not $o much as the tenth part of one beat of the Pul$e. Having done, and preci$ely ordered this bu$ine$$e, we made the $ame Ball to de$cend only the fourth part of the length of that Grove: and having mea$ured the time of its de$cent, we alwaies found it to be punctually half the other. And then making trial of other parts, examining one while the Time of the whole Length with the Time of half the Length, or with that of 2/3, or of 3/4, or, in brief, with any whatever other Divi$ion, by Experiments repeated near an hundred Times, we alwaies found the Spaces to be to one another as the Squares of the Times. And this in all Inclinations of the Plane, that is, of the Grove in which the Ball was made to de$cend. In which we ob$erved moreover, that the Times of the De$cents along $undry Inclinations did retain the $ame proportion to one another, exactly, which anon you will $ee a$$igned to them, and demon$trated by the Author. And as to the mea$uring of the Time; we had a good big Bucket full of Water hanged on high, which by a very $mall hole, pierced in the bottom, $pirted, or, as we $ay, $pin'd forth a $mall thread of Water, which we received with a $mall cup all the while that the Ball was de$cending in the Grove, and in its parts; and then weighing from time to time the $mall parcels of Water, in that manner gathered, in an exact pair of $cales, the differences and proportions of their Weights gave ju$tly the differences and proportions of the Times; and this with $uch exactne$$e, that, as I $aid before, the trials being many and many times repeated, they never differed any con$iderable matter.</P> <P>SIMP. I $hould have received great $atisfaction by being pre$ent at tho$e Experiments: but being confident of your diligence in making them, and veracity in relating them, I content my $elf, and admit them for true and certain.</P> <P>SALV. We may, then, rea$$ume our Reading, and go on.</P> <head>COROLLARY II.</head> <P>It is collected in the $econd place, that if any two Spaces are ta- ken from the beginning of the Motion, pa$$ed in any Times, tho$e Times $hall be unto each other as one of them is to a Space that is the Mean proportional between them.</P> <P><I>For taking two Spaces S T, and S V from the beginning of the Mo- tion S, to which S X is a Mean-proportional, the Time of the de$cent along S T, $hall be to the Time of the de$cent along S V, as S T to S X; or, if you will, the Time along S V $hall be to the Time along S T,</I> <foot><I>as</I></foot> <p n=>149</p> <fig> <I>as VS is to SX. For it is demon$trated, that Spaces pa$$ed are in duplicate proportion to the Times, or, (which is the $ame) are as the Squares of the Times: But the pro- portion of the Space VS to the Space ST is double to the proportion of V S to SX, or is the $ame that V S, and S X $quared have to one another: Therefore, the proportion of the Times of the Motion by V S, and ST, is as the Spaces or Lines V S to S X.</I></P> <head>SCHOLIUM.</head> <P><I>That which is demon$trated in Motions that are made Perpendicu- larly, may be under$tood al$o to hold true in the Motions made along Planes of any whatever Inclination; for it is $uppo$ed, that in them the degree of Acceleration encrea$eth in the $ame proportion; that is, according to the encrea$e of the Time; or, if you will, according to the $imple and primary Series of Numbers.</I></P> <P>SALV. Here I de$ire <I>Sagredus,</I> that I al$o may be allowed, al- beit perhaps with too much tediou$ne$$e in the opinion of <I>Simplici- us,</I> to defer for a little time the pre$ent Reading, untill I may have explained what from that which hath been already $aid and de- mon$trated, and al$o from the knowledge of certain Mechanical Conclu$ions heretofore learnt of our <I>Academick,</I> I now remember to adjoyn for the greater confirmation of the truth of the Princi- ple, which hath been examined by us even now with probable Rea$ons and Experiments: and, which is of more importance, for the Geometrical proof of it, let me fir$t demon$trate one $ole Ele- mental <I>Lemma</I> in the Contemplation of <I>Impetus's.</I></P> <P>SAGR. If our advantage $hall be $uch as you promi$eus, there is no time that I would not mo$t willingly $pend in di$cour$ing about the confirmation and thorow e$tabli$hing the$e Sciences of Motion: and as to my own particular, I not only grant you liber- ty to $atisfie your $elf in this particular, but moreover entreat you to gratifie, as $oon as you can, the Curio$ity which you have begot in me touching the $ame: and I believe that <I>Simplicius</I> al$o is of the $ame mind.</P> <P>SIMP. I cannot deny what you $ay.</P> <P><I>S</I>ALV. Seeing then that I have your permi$$ion, I will in the fir$t place con$ider, as an Effect well known, That</P> <foot>PROP.</foot> <p n=>150</p> <head>LEMMA.</head> <P><I>That the Moments or Velocities of the $ame Moveable are different upon different Inclinations of Planes, and the greate$t is by the Line elevated perpendicularly above the Horizon, and by the others inclined, the $aid Velocity dimini$heth according as they more and more depart from Perpendicularity, that is, as they in- cline more obliquely: $o that the Impetus, Talent, Energy, or, we may $ay, Moment of de$cending is dimini$hed in the Moveable by the $ubjected Plane, upon which the $aid Moveable lyeth and de$cendeth.</I></P> <P>And the better to expre$s my $elf, let the Line A B be perpen- dicularly erected upon the Horizon A C: then $uppo$e the $ame to be declined in $undry Inclinations towards the Horizon, as in A D, A E, A F, <I>&c.</I> I $ay, that the greate$t and total <I>Impetus</I> of the Grave Body in de$cending is along the Perpendicular B A, and le$s than that along D A, <fig> and yet le$s along E A; and $ucce$$ively dimini$hing along the more inclined F <I>A,</I> and fi- nally is wholly extinct in the Horizontal C <I>A,</I> where the Moveable is indifferent either to Motion or Re$t, and hath not of it $elf any Inclination to move one way or other, nor yet any Re$i$tance to its being mo- ved: for as it is impo$$i- ble that a Grave Body, or a Compound thereof $hould move naturally upwards, receding from the Common Center, towards which all Grave Matters con$pire to go, $o it is impo$$ible that it do $pontaneou$ly move, unle$s with that Motion its particular Center of Gravity do acquire Proxi- mity to the $aid Common Center: $o that upon the Horizontal which here is under$tood to be a Superficies equidi$tant from the $aid Center, and therefore altogether void of Inclination, the <I>Im- petus</I> or Moment of that $ame Moveable $hall be nothing at all. Having under$tood this mutation of <I>Impetus,</I> I am to explain that which, in an old Treati$e of the Mechanicks, written heretofore in <I>Padona</I> by our <I>Academick,</I> only for the u$e of his Scholars, was diffu$ely and demon$tratively proved, upon the occa$ion of con- $idering the Original and Nature of the admirable In$trument cal- led the Screw, and it is, With what proportion that mutation of <foot><I>Impetus,</I></foot> <p n=>151</p> <I>Impetus</I> is made along $everal Inclinations or Declivities of Planes.</P> <P>As, for example, in the inclined Plane A F, drawing its Eleva- tion above the Horizontal, that is, the Line F C, along the which the <I>Impetus</I> of a Grave Body, and the Moment of De$cent is the greate$t; it is $ought what proportion this Moment hath to the Moment of the $ame Moveable along the Declivity F A: Which Proportion, I $ay, is Reciprocal to the $aid Lengths. And this is the <I>Lemma</I> that was to go before the Theorem, which I hope to be able anon to Demon$trate. Hence it is manife$t, That the <I>Impetus</I> of De$cent of a Grave Body is as much as the Re$i$tance or lea$t force that $ufficeth to arre$t and $tay it. For this Force or Re$i- $tance, and its mea$ure, I will make u$e of the Gravity of another Moveable. Let us now upon the Plane F A put the Moveable G tyed to a thread which $liding over F hath fa$tned at its other end the Weight H: and let us con$ider that the Space of the De$cent or A$cent of the Weight H along the Perpendicular, is alwaies equal to the whole A$cent or De$cent of the other Moveable G along the ^{*} Declivity A F, but yet not to the A$cent or De$cent <marg>* Or inclined Plane.</marg> along the Perpendicular, in which only the $aid Moveable G (like as every other Moveable) exerci$eth its Re$i$tance. Which is manife$t: for con$idering in the Triangle AFC the Motion of the Moveable G, as for example, upwards from A to F, to be com- po$ed of the tran$ver$e Horizontal Line A C, and of the Perpendi- cular C F: <I>A</I>nd in regard, that as to the Horizontal Plane along which the Moveable, as hath been $aid, hath no Re$i$tance to mo- ving (it not making by that Motion any lo$s, nor yet acqui$t in regard of its particular di$tance from the Common Center of Grave Matters, which in the Horizon continueth $till the $ame) it remai- neth that the Re$i$tance be only in re$pect of the <I>A</I>$cent that it is to make along the Perpendicular C F. Whil$t therefore the Grave Moveable G, moving from <I>A</I> to F, hath only the Perpendicular Space C F to re$i$t in its <I>A</I>$cent, and whil$t the other Grave Move- able H de$cendeth along the Perpendicular of nece$$ity as far as the whole Space F <I>A,</I> and that the $aid proportion of <I>A</I>$cent and De$cent maintains it $elf alwaies the $ame, be the Motion of the $aid Moveables little or much (by rea$on they are tyed toge- ther) we may confidently affirm, that in ca$e there were an <I>Equi- librium,</I> that is Re$t, to en$ue betwixt the $aid Moveables, the Mo- ments, the Velocities, or their Propen$ions to Motion, that is the Spaces which they would pa$s in the $ame Time $hould an$wer re- ciprocally to their Gravities, according to that which is demon$tra- ted in all ca$es of Mechanick Motions: $o that it $hall $uffice to impede the de$cent of G, if H be but $o much le$s grave than it, as in proportion the Space C F is le$$er than the Space F <I>A.</I> Therefore <foot>$uppo$e</foot> <p n=>152</p> $uppo$e that the Moveable G is to the Moveable H, as F <I>A</I> is to F C; and then the <I>Equilibrium</I> $hall follow, that is, the Moveables H and G $hall have equal Moments, and the Motion of the $aid Moveables $hall cea$e. <I>A</I>nd becau$e we $ee that the <I>Impetus,</I> Energy, Moment, or Propen$ion of a Moveable to Motion is the $ame as is the Force or $malle$t Re$i$tance that $ufficeth to $top it; and becau$e it hath been concluded, that the Grave Body H is $uf. ficient to arre$t the Motion of <fig> the Grave Body G: Therefore the le$$er Weight H, which in the Perpendicular F C imploy- eth its total Moment, $hall be the preci$e mea$ure of the par- tial Moment that the greater Weight G exerci$eth along the inclined Plane F <I>A</I>: But the mea$ure of the total Moment of the $aid Grave Body G, is the $elf $ame, ($ince that to impede the Perpendicular De$cent of a Grave Body there is required the oppo$ition of $uch another Grave Body, which likewi$e is at liberty to move Perpendicularly:) Therefore the partial <I>Impetus</I> or Moment of G along the inclined Plane F A $hall be to the grand and total <I>Impetus</I> of the $ame G along the Perpendicular F C, as the Weight H to the Weight G: that is, by Con$truction, as the $aid Perpendicular F C, the Eleva- tion of the inclined Plane, is to the $ame inclined Plane F A: Which is that that by the <I>Lemma</I> was propo$ed to be demon- $trated, and which by our Author, as we $hall $ee, is $uppo$ed as known in the $econd part of the Sixth Propo$ition of the pre$ent Treati$e.</P> <P>SAGR. From this that you have already concluded I conceive one may ea$ily deduce, arguing <I>ex æquali</I> by perturbed Proportion, that the Moments of the $ame Moveable, along Planes variou$ly inclined (as F A and F I) that have the $ame Elevation, are to each other in Reciprocal proportion to the $ame Planes.</P> <P>SALV. <I>A</I> mo$t certain Conclu$ion. This being agreed on, we will pa$s in the next place to demon$trate the <I>Theoreme,</I> namely, that</P> <foot>THEOR.</foot> <p n=>153</p> <head>THEOREM.</head> <P><I>The degrees of Velocity of a Moveable de$cending with a Natural Motion from the $ame height along Planes in any manner inclined at the arrival to the Horizon are alwaies equal, Impediments be- ing removed.</I></P> <P>Here we are in the fir$t place to adverti$e you, that it having been proved, that in any Inclination of the Plane the Move- able from its rece$$ion from Quie$$ence goeth encrea$ing its Ve- locity, or quantity of its <I>Impetus,</I> with the proportion of the Time (according to the Definition which the Author giveth of Motion naturally Accelerate) whereupon, as he hath by the pre- cedent Propo$ition demon$trated, the Spaces pa$$ed are in dupli- cate proportion to the Times, and, con$equently, to the degrees of Velocity: look what the <I>Impetus's</I> were in that which was fir$t moved, $uch proportionally $hall be the degrees of Velocity gai- ned in the $ame Time; $eeing that both the$e and tho$e encrea$e with the $ame proportion in the $ame Time.</P> <P>Now let the inclined Plane be A B, its elevation above the Ho rizon the Perpendicular A C, and the Horizontal Plane C B: and becau$e, as was even now concluded, the <I>Impetus</I> of a Moveable along the Perpendicular A C is to the <I>Impetus</I> of the $ame along the inclined Plane A B, as A B is to A C, let there be taken in the inclined Plane A B, A D a third proportional to A B and A C: The <I>Impetus,</I> therefore, along A C is to the <I>Impetus</I> along A B, that is along A D, as A C is to <fig> A D: And therefore the Move- able in the $ame Time that it would pa$s the Perpendicular Space AC, $hall likewi$e pa$s the Space A D, in the inclined Plane A B, (the Moments being as the Spaces:) And the degree of Velocity in C $hall have the $ame proportion to the degree of Velocity in D, as A C hath to A D: But the degree of Velocity in B is to the $ame degree in D, as the Time along A B is to the Time along AD, by the definition of Accelerate Motion; And the Time along AB is to the Time along A D, as the $ame A C, the Mean Proportional between B A and A D, is to A D, by the la$t Corollary of the $econd Propo$ition: Therefore the degrees of Velocity in B and in C have to the de- gree in D, the $ame Proportion as A C hath to A D; and therefore are equal: Which is the <I>Theorem</I> intended to be demon$trated.</P> <P>By this we may more concludingly prove the en$uing third <foot>X Propo$i-</foot> <p n=>154</p> Propo$ition of the Author, in which he makes u$e of this Princi- ple; and it is, That the Time along the inclined Plane, hath to the Time along the Perpendicular, the $ame proportion as the $aid In- clined Plane and Perpendicular. For if we put the ca$e that BA be the Time along A B, the Time along A D $hall be the Mean between them, that is A C, by the $econd Corollary of the $econd Propo$ition: But if C A be the Time along A D, it $hall likewi$e be the Time along <I>A</I> C, by rea$on that <I>A</I> D and <I>A</I> C are pa$t in equal Times: And therefore in ca$e B <I>A</I> be the Time along A B, <I>A</I> C $hall be the Time along <I>A</I> C: Therefore, as <I>A</I> B is to A C, $o is the Time along <I>A</I> B to the Time along <I>A</I> C.</P> <P>By the $ame di$cour$e one $hall prove, that the Time along <I>A</I> C is to the Time along the inclined Plane <I>A</I> E, as <I>A</I> C is to <I>A</I> E: Therefore, <I>ex æquali,</I> the Time along the inclined Plane <I>A B</I> is, Directly, to the Time along the inclined Plane <I>A</I> E as <I>A B</I> to <I>A E, &c.</I></P> <P>One might al$o by the $ame application of the <I>Theorem,</I> as <I>Sa- gredus</I> $hall very evidently $ee anon, immediately demon$trate the $ixth Propo$ition of the <I>A</I>uthor: <I>B</I>ut let this Digre$$ion $uffice for the pre$ent, which he perhaps thinketh too tedious, though in- deed it is of $ome importance in the$e matters of Motion.</P> <P>SAGR. You may $ay extreamly delightful, and mo$t nece$$ary to the perfect under$tanding of that Principle.</P> <P>SALV. I will go on, then, in my Reading of the Text.</P> <head>THEOR. III. PROP. III.</head> <P>If a Moveable departing from Re$t do move along an Inclined Plane, and al$o along the Perpendi- cular who$e heights are the $ame, the Times of their Motions $hall be to one another as the Lengths of the $aid Plane and Perpendicular.</P> <P><I>Let the inclined Plane be A C, and the Perpendicular A B, who$e heights are the $ame above the Horizon C B, to wit, the $elf $ame Line B A. I $ay, that the Time of the De$cent of the $ame Moveable upon the Plane A C, hath the $ame Proporti- on to the Time of the De$cent along the Perpendicular A B, as the Length of the Plane A C hath to the Length of the $aid Perpendi- cular. For let any number of Lines D G, E I, F L, be drawn, Paral- lel to the Horizon C B: It is manife$t from the A$$umption fore- going, that the degrees of Velocity of the Moveable, departing from A the beginning of Motion, acquired in the Points G and D are</I> <foot><I>equal,</I></foot> <p n=>155</p> <I>equal, their exce$$e or elevation above the Horizon being equal; and $o the degrees in the Points I and E; as al$o the degrees in L and F. And if not only the$e Parallels, but many more were $up- po$ed to be drawn from all the points imagined to be in the Line A B, untill they meet the Line A C, the Mo-</I> <fig> <I>ments, or degrees of the Velocities along the extreams [or ends] of every one of tho$e Parallels, $hall be alwaies equal to one ano- ther: Therefore the two Spaces A C and A B are pa$t with the $ame degree of Velocity: But it hath been demon$trated, that if two Spaces be pa$$ed by a Moveable with one and the $ame degree of Velocity, the Times of the Motions have the $ame proportion as tho$e Spaces: Therefore the Time of the Motion along A C is to the Time along A B, as the Length of the Plane A C to the length of the Perdendicular A B. Which was to be demon$trated.</I></P> <P>SAGR. It $eemeth to me, that the $ame might very clearly and conci$ely be concluded, it having fir$t been proved that the $um of the Accelerate Motion of the Tran$itions along A C and A B, is as much as the Equable Motion, who$e degree of Velocity is $ub- duple to the greate$t degree C B: Therefore the two Spaces AC and A B being pa$$ed with the $ame Equable Motion, it hath been $hewn, by the Fir$t Propo$ition of the fir$t, that the Times of the Tran$itions $hall be as the $aid Spaces.</P> <head>COROLLARY.</head> <P>Hence is collected, that the Times of the De$cents along Planes of different Inclination, but of the $ame Elevation, are to one another according to their Lengths.</P> <P><I>For if we $uppo$e another Plane A M, coming from A, and ter- minated by the $ame Horizontal C B; it $hall in like manner be demon$trated, that the Time of the De$cent along A M, is to the Time along A B, as the Line A M to A B: But as the Time A B is to the Time along A C, $o is the Line A B to A C: Therefore,</I> ex æquali, <I>as A M is to A C, $o is the Time along A M to the Time along A C.</I></P> <foot>X 2 PROP.</foot> <p n=>156</p> <head>THEOR. IV. PROP. IV.</head> <P>The Times of the Motions along equal Planes, but unequally inclined, are to each other in $ubduple proportion of the Elevations of tho$e Planes Reciprocally taken.</P> <P><I>Let there proceed from the term B two equal Planes, but une- qually inclined, B A and B C, and let A E and C D be Hori- zontal Lines, drawn as far as the Perpendicular B D: Let the Elevation of the Plane B A be B E; and let the Elevation of the Plane B C be B D: And let B I be a Mean Proportional between the Elevations D B and B E: It is manife$t</I> <fig> <I>that the proportion of D B to B I, is $ub- duple the proportion of D B to B E. Now I $ay, that the proportion of the Times of the De$cents or Motions along the Planes B A and B C, are the $ame with the proportion of D B to B I Reciprocal- ly taken: So that to the Time B A the Elevation of the other Plane B C, that is B D be Homologal; and to the Time along B C, B I be Homologal: Therefore it is to be demon$trated, That the Time along B A is to the Time along B C, as D B is to B I. Let I S be drawn equidi$tant from D C. And becau$e it hath been demon$trated that the Time of the De$cent along B A, is to the Time of the De$cent along the Perpendicular B E, as the $aid B A is to B E; and the Time along B E is to the Time along B D, as B E is to B I; and the Time along B D is to the Time along B C, as B D to B C, or as B I to B S: Therefore,</I> ex æqua- li, <I>the Time along B A $hall be to the Time along B C as B A to B S, or as C B to BS: But C B is to B S, as D B to B I: Therefore the Propo$ition is manife$t:</I></P> <foot>THEOR.</foot> <p n=>157</p> <head>THEOR. V. PROP. V.</head> <P>The proportion of the Times of the De$cents along Planes that have different Inclinations and Lengths, and the Elivations unequal, is compounded of the proportion of the Lengths of tho$e Planes, and of the $ubduple proporti- on of their Elevations Reciprocally taken.</P> <P><I>Let A B and A C be Planes inclined after different manners, who$e Lengths are unequal, as al$o their Elevations. I $ay, the proportion of the Time of the De$cent along A C to the Time along A B, is compounded of the proportion of the $aid A C to A B, and of the $ubduple proportion of their Elevation Recipro- cally taken. For let the Perpendicular A D be drawn, with which let the Horizontal Lines B G and C D inter$ect, and let A L be a Mean-proportional between C A and A E; and from the point L let a Parallel be drawn to the Horizon inter$ecting</I> <fig> <I>the Plane A C in F; and A F $hall be a Mean proportional between C A and A E. And becau$e the Time along A C is to the Time along A E, as the Line F A to A E; and the Time along A E is to the Time along A B, as the $aid A E to the $aid A B: It is manife$t that the Time along A C is to the Time along A B, as A F to A B. It remaineth, therefore, to be demon$trated, that the proportion of A F to A B is compounded of the proportion of C A to A B, and of the proportion of G A to A L; which is the $ubduple proportion of the Elevati- ons D A and A G Reciprocally taken. But that is manife$t, C A being put between F A and A B: For the proportion of F A to A C is the $ame as that of L A to A D, or of G A to A L; which is $ub- duple of the proportion of the Elevations G A and A D; and the proportion of C A to A B is the proportion of the Lengths: Therefore the Propo$ition is manife$t.</I></P> <foot>THEOR.</foot> <p n=>158</p> <head>THEOR. VI. PROP. VI.</head> <P>If from the highe$t or lowe$t part of a Circle, erect upon the Horizon, certain Planes be drawn inclined towards the Circumference, the Times of the De$cents along the $ame $hall be equal.</P> <P><I>Let the Circle be erect upon the Horizon G H, who$e Diameter recited upon the lowe$t point, that is upon the contact with the Horizon, let be F A, and from the highe$t point A let certain Planes A B and A C incline towards the Circumference: I $ay that the Times of the De$cents along the $ame are equal. Let B D and C E be two Perpendiculars let fall unto the Diameter; and let A I be a Mean- Proportional between the Altitudes</I> <fig> <I>of the Planes E A and A D. And becau$e the Rectangles F A E and F A D are equal to the Squares of A C and A B; And al$o becau$e that as the Rectangle F A E, is to the Rectangle F A D, $o is E A to A D. Therefore as the Square of C A is to the Square of B A, $o is the Line E A to the Line A D. But as the Line E A is to D A, $o is the Square of I A to the Square of A D: Therefore the Squares of the Lines C A and A B are to each other as the Squares of the Lines I A and A D: And therefore as the Line C A is to A B, $o is I A to A D: But in the precedent Propo$ition it hath been demon- $trated that the proportion of the Time of the De$cent along A C to the Time of the De$cent by A B, is compounded of the proportions of C A to A B, and of D A to A I, which is the $ame with the proportion of B A to A C: Therefore the proportion of the Time of the De$cent along A C, to the Time of the De$cent along A B, is compounded of the pro- portions of C A to A B, and of B A to A C: Therefore the proporti- on of tho$e Times is a proportion of equality: Therefore the Propo$ition is evident.</I></P> <P><I>The $ame is another way demon$trated from the Mechanicks: Name- ly that in the en$uing Figure the Moveable pa$$eth in equal Times along C A and D A. For let B A be equal to the $aid D A, ond let fall the Perpendiculars B E and D F: It is manife$t by the Elements of the</I> <foot><I>Mechanicks:</I></foot> <p n=>159</p> <I>Mechanicks: That the Moment of the Weight elevated upon the Plane according to the Line A B C, is to its total Moment, as B E to B A;</I> <fig> <I>And that the Moment of the $ame Weight upon the Elevation A D, is to its total Moment, as D F to D A or B A: Therefore the Mo- ment of the $aid Weight upon the Plane inclined according to D A, is to the Moment upon the Plane inclined according to A B C, as the Line D F to the Line B E: Therefore the Spaces which the $aid Weight $hall pa$$e in equal Times along the Inclined Planes C A and D A, $hall be to each other as the Line B E to D F; by the $econd Propo$ition of the Fir$t Book: But as B E is to D F, $o A C is demon$trated to be to D A: Therefore the $ame Moveable will in equal Times pa$$e the Lines C A and D A.</I></P> <P><I>And that C A is to D A as B E is to D F, is thus demon$trated.</I></P> <P><I>Draw a Line from C to D; and by D and B draw the Lines D G L, (cutting C A in the point I) and B H, Parallels to A F: And the Angle A D I $hall be equal to the Angle D C A, for that the parts L A and A D of the Circumference $ubtending them, are equal, and the Angle D A C common to them both: Therefore of the equiangled Triangles C A D and D A I, the $ides about the equal Angles $hall be proportional: And as C A is to A D, $o is D A to A I, that is B A to A I, or H A to A G; that is, B E to D F: Which was to be proved.</I></P> <P><I>Or el$e the $ame $hall be demon$trated more $peedily thus.</I></P> <P><I>Vnto the Horizon A B, let a Circle be erect, who$e Diameter is perpendicular to the Horizon: and from the highe$t Term D let a Plane</I> <fig> <I>at plea$ure D F, be inclined to the Circumference. I $ay that the De- $cent along the Plane D F, and the Fall along the Diameter B C, will be pa$$ed by the $ame Moveable in equal Times. For let F G be drawn parallel to the Horizon A B, which $hall be perpendicular to the Diameter D C, and let a Line conjoyn F and C: and becau$e the Time of the Fall along D C, is to the Time of the Fall along D G, as the Mean Proportional between C D and D G, is to the $aid D G; and the</I> <foot><I>Mean</I></foot> <p n=>160</p> <I>Mean between C D and D G being D F, (for that the Angle D F C in the Semicircle, is a Right Angle, and F G perpendicular to D C:) Therefore the Time of the Fall along D C is to the Time of the Fall along D G, as the Line F D to D G: But it hath been demon$trated that the Time of the De$cent along D F, is to the Time of the Fall along D G, as the $ame Line D F is to D G: The Times, therefore, of the De$cent along D F and Fall along D C, are to the Time of the Fall along the $aid D G in the $ame proportion: Therefore they are equal. It will likewi$e be demon$trated, if from the lowe$t Term C, one $hould rai$e the Chord C E, and draw E H parallel to the Hori- zon, and conjoyn E and D, that the Time of the De$cent along E C equals the Time of the Fall along the Diameter D C.</I></P> <head>COROLLARY I.</head> <P>Hence is collected that the Times of the De$cents along all the Chords drawn from the Terms C or D are equal to one another.</P> <head>COROLLARY II.</head> <P>It is al$o collected that if the Perpendicular and inclined Plane de$cend from the $ame point along which the De$cents are made in equal Times, they are in a Semicircle who$e Dia- meter is the $aid Perpendicular.</P> <head>COROLLARY III.</head> <P>Hence it is collected that the Times of the Motions along inclined Planes, are then equal, where the Elevations of equal parts of tho$e Planes $hall be to one another as their Longitudes.</P> <P><I>For it hath been $hewn that the Times C A and D A in the la$t Fi- gure $ave one are equal, the Elevation of the part A B being equal to A D, that is, that B E $hall be to the Elevation D F, as C A to D A.</I></P> <P>SAGR. Pray you Sir be plea$ed to $tay your Reading of what followeth until that I have $atisfied my $elf in a Contemplation that ju$t now cometh into my mind, which if it be not a delu$i- on, is not far from being a plea$ing diverti$ement: as are all $uch that proceed from Nature or nece$$ity.</P> <P>It is manife$t, that if from a point a$$igned in an Horizontal Plane, one $hall produce along the $ame Plane infinite right Lines every way, upon each of which a point is under$tood to move with an Equable Motion, all beginning to move in the $ame in$tant <foot>of</foot> <p n=>161</p> of Time from the a$$igned point, and the Velocities of them all being equal, there $hall con$equently be de$cribed by tho$e move- able points Circumferences of Circles alwayes bigger and bigger, all concentrick about the fir$t point a$$igned: ju$t in the $ame manner as we $ee it done in the Undulations of $tanding Water, when a $tone is dropt into it; the percu$$ion of which $erveth to give the beginning to the Motion on every $ide, and remaineth as the Center of all the Circles that happen to be de$igned $ucce$- $ively bigger and bigger by the $aid Undulations. But if we ima- gine a Plane erect unto the Horizon, and a point be noted in the $ame on high, from which infinite Lines are drawn inclined, ac- cording to all inclinations, along which we fancy grave Movea- bles to de$cend, each with a Motion naturally Accelerate with tho$e Velocities that agree with the $everal Inclinations; $uppo$ing that tho$e de$cending Moveables were continually vi$i- ble, in what kind of Lines $hould we $ee them continually di$po$ed? Hence my wonder ari$eth, $ince that the precedent Demon$trati- ons a$$ure me, that they $hall all be alwayes $een in one and the $ame Circumference of Circles $ucce$$ively encrea$ing, according as the Moveables in de$cending go more and more $ucce$$ively re- ceding from the highe$t point in which their Fall began: And the better to declare my $elf, let the chiefe$t point A be marked, from which Lines de$cend according to any Inclinations A F, A H, and the Perpendicular A B, in which taking the points C and D, de- $cribe Circles about them that pa$s by <fig> the point A, inter$ecting the inclined Lines in the points F, H, B, and E, G, I. It is manife$t, by the fore-going Demon$trations, that Moveables de- $cendent along tho$e Lines departing at the $ame Time from the term A, one $hall be in E, the other $hall be in G, and the other in I; and $o con- tinuing to de$cend they $hall arrive in the $ame moment of Time at F, H, and B: and the$e and infinite others continuing to move along the infinite differing Inclinations, they $hall alwayes $ucce$$ively arrive at the $elf-$ame Circumferences made bigger & bigger <I>in infinitum.</I> From the two Species, therefore, of Motion of which Nature makes u$e, ari$eth, with admirable harmonious variety, the generation of in- $inite Circles. She placeth the one as in her Seat, and original be- ginning, in the Center of infinite concentrick Circles; the other is con$tituted in the $ublime or highe$t Contact of infinite Circum- ferences of Circles, all excentrick to one another: Tho$e proceed from Motions all equal and Equable; The$e from Motions all al- <foot>Y wayes</foot> <p n=>162</p> wayes Inequable to them$elves, and all unequal to one another, that de$cend along the infinite different Inclinations. But we fur- ther adde, that if from the two points a$$igned for the Emanations, we $hall $uppo$e Lines to proceed, not onely along two Superfi- cies Horizontal and Upright [or erect] but along all every ways like as from tho$e, beginning at one $ole point, we pa$$ed to the production of Circles from the lea$t to the greate$t, $o beginning from one $ole point we $hall $ucce$$ively produce in$inite Spheres, or we may $ay one Sphere, that $hall <I>gradatim</I> increa$e to infinite bigne$$es: And this in two fa$hions; that is, either with placing the original in the Center, or el$e in the Circumference of tho$e Spheres.</P> <P>SALV. The Contemplation is really ingenuous, and adequate to the Wit of <I>Sagredus.</I></P> <P>SIMP. Though I am at lea$t capable of the Speculation, ac- cording to the two manners of the production of Circles and Spheres, with the two different Natural Motions, howbeit I do not perfectly under$tand the production depending on the Acce- lerate Motion and its Demon$tration, yet notwith$tanding that licence of a$$igning for the place of that Emanation as well the lowe$t Center, as the highe$t Spherical Superficies, maketh me to think that its po$$ible that $ome great Mi$tery may be contained in the$e true and admirable Conclu$ions: $ome Mi$tery I $ay touching the Creation of the Univer$e, which is held to be of Spherical form, and concerning the Re$idence of the Fir$t Cau$e.</P> <P>SALV. I am not unwilling to think the $ame: but $uch pro- found Speculations are to be expected from Sharper Wits than ours. And it $hould $uffice us, that if we be but tho$e le$$e noble Workmen that di$cover and draw forth of the Quarry the Marbles, in which the Indu$trious Statuaries afterwards make wonderful Images appear, that lay hid under rude and mi$haped Cru$ts. Now, if you plea$e, we will go on.</P> <head>THEOR. VII. PROP. VII.</head> <P>If the Elevations of two Planes $hall have a pro- portion double to that of their Lengths, the Motions in them from Re$t $hall be fini$hed in equal Times.</P> <P><I>Let A E and A B be two unequal Planes, and unequally inclined, and let their Elevations be F A and D A, and let F A have the $ame proportion to D A, as A E hath to A B. I $ay that the Times of the Motions along the Planes A E and A B, out of Re$t in A are</I> <foot><I>equal</I></foot> <p n=>163</p> <I>equal. Draw Horizontal Parallels to the Line of Elevation E F and B D, which cutteth A E in G. And be-</I> <fig> <I>cau$e the proportion of F A to A D, is double the proportion of E A to A B; and as F A to A D, $o is E A to A G: There- fore the proportion of E A to A G, is dou- ble the proportion of E A to A B: There- fore A B is a Mean-Proportional between E A and A G: And becau$e the Time of the De$cent along A B, is to the Time of the De- $cent along A G, as A B to A G; and the Time of the De$cent along AG, is to the Time of the De$cent along A E, as A G is to the Mean-proportional between A G and A E, which is A B: Therefore</I> ex equali, <I>the Time along A B is to the Time along A E, as A B unto it $elf: Therefore the Times are equal: Which was to be demon$trated.</I></P> <head>THEOR. VIII. PROP. VIII.</head> <P>In Planes cut by the $ame Circle, erect to the Horizon, in tho$e which meet with the end of the erect Diameter, whether upper or lower, the Times of the Motions are equal to the Time of the Fall along the Diameter: and in tho$e which fall $hort of the Diameter, the Times are $horter; and in tho$e which inter- $ect the Diameter, they are longer.</P> <P><I>Let A B be the Perpendicular Diameter of the Circle erect to the Horizon. That the Times of the Motions along the Planes pro- duced out of the Terms A and B unto the Circumference are equal, hath already been demon$trated: That the Time of the De$cent along the Plane D F, not reaching to the</I> <fig> <I>Diameter is $borter, is demon$trated by drawing the Plane D B, which $hall be both longer and le$$e decli- ning than D F. Therefore the Time along D F is $horter than the Time along D B, that is, along A B. And that the Time of the De$cent along the Plane that inter$ecteth the Dia- meter, as C O is longer, doth in the $ame manner appear, for that it is longer and le$$e declining than C B: Therefore the Propo$ition is de- mon$trated.</I></P> <foot>Y 2 THEOR.</foot> <p n=>164</p> <head>THEOR. IX. PROP. IX.</head> <P>If two Planes be inclined at plea$ure from a point in a Line parallel to the Horizon, and be inter- $ected by a Line which may make Angles Al- ternately equal to the Angles contained be- tween the $aid Planes and Horizontal Parallel, the Motion along the parts cut off by the $aid Line, $hall be performed in equal Times.</P> <P><I>From off the point C of the Horizontal Line X, let any two Planes be inclined at plea$ure C D and C E, and in any point of the Line C D make the Angle C D F equal to the Angle X C E: and let the Line D F cut the Plane C E in F, in $uch a manner that the Angles C D F and C F D may be equal to the Angles X C E, L C D Alternately taken. I $ay, that</I> <fig> <I>the Times of the De$cents along C D and C F are equal. And that (the Angle C D F being $uppo$ed equal to the Angle X C E) the Angle C F D is equal to the Angle D C L, is manife$t. For the Common An- gle D C F being taken from the three Angles of the Triangle C D F equal to two Right An- gles, to which are equal all the Angles made with to the Line L X at the point C, there remains in the Triangle two Angles C D F and C F D, equal to the two Angles X C E and L C D: But it was $up- po$ed that C D F is equal to the Angle X C E: Therefore the remaining Angle C F D is equal to the remaining angle D C L. Let the Plane C E be $uppo$ed equal to the Plane C D, and from the points D and E rai$e the Perpendiculars D A and E B, unto the Horizontal Paral- lel X L; and from C unto D F let fall the Perpendicular C G. And becau$e the Angle C D G is equal to the Angle E C B; and becau$e D G C and C B E are Right Angles; The Triangles C D G and C B E $hall be equiangled: And as D C is to C G, $o let C E be to E B: But D C is equal to C E: Therefore C G $hall be equal to E B. And inregard that of the Triangles D A C and C G F, the An- gles C and A are equal to the Angles F and G: Therefore as C D is to D A, $o $hall F C be to C G; and Alternately, as D C is to C F, $o</I> <foot><I>is</I></foot> <p n=>165</p> <I>is D A to C G, or B E. The proportion therefore of the Elevations of the Planes equal to C D and C E, is the $ame with the proportion of the Longitudes D C and C E: Therefore, by the fir$t Corollary of the precedent Sixth Propo$ition, the Times of the Dc$cent along the $ame $hall be equal: Which mas to be proved.</I></P> <P><I>Take the $ame another way: Draw F S perpendicular to the Horizontal Parallel A S. Becau$e the Triangle C S F is like to the Triangle D G C, it $hall be, that as S F is to F C, $o is G C to C D. And becau$e the Triangle C F G is like to the Triangle D C A, it $hall be, that as F C is to C G, $o is C D to D A: Therefore,</I> ex æquali, <I>as S F is to C G, $o is C G to D A: Thorefore C G is a</I> <fig> <I>Mean-proportional between S F and D A: And as DA is to S F, $o is the Square D A unto the Square C G Again, the Triangle A C D being like to the Triangle C G F, it $hall be, that as D A is to D C, $o is G C to C F: and, Alternately, as D A is to G C, $o is D C to C F; and as the Square of D A is to the Square of C G, $o is the Square of D C to the Square of C F. But it hath been proved that the Square D A is to the Square C G as the Line D A is to the Line F S: Therefore, as the Square D C is to the Square C F, $o is the Line D E to F S: There- fore, by the $eventh fore-going, in regard that the Elevations D A and F S, of the Planes C D, and C F are in double proportion to their Planes; the Times of the Motions along the $ame $hall be equal.</I></P> <head>THEOR. X. PROP. X.</head> <P>The Times of the Motions along $everal Inclina- tions of Planes who$e Elevations are equal, are unto one another as the Lengths of tho$e Planes, whether the Motions be made from Re$t, or there hath proceeded a Motion from the $ame height.</P> <P><I>Let the Motions be made along A B C, and along A B D, until they come to the Horizon D C, in $uch $ort as that the Motion along A B precedeth the Motions along B D and B C. I $ay, that the Time of the Motion along B D, is to the Time along B C, as</I> <foot><I>the</I></foot> <p n=>166</p> <I>the Length B D is to B C. Let A F be drawn parallel to the Ho- rizon, to which continue out D B, meeting it in F; and let F E be a Mean-proportional between D F and F B; and draw E O parallel to D C, and A O $hall be a Mean-proportional between C A and A B: But if we $uppo$e the Time along A B, to be as A B, the Time a-</I> <fig> <I>long F B $hall be as F B. And the Time along all A C, $hall be as the Mean-proportional A O; and along all F D $hall be F E: Wherefore the Time along the remainder B C $hall be B O; and along the remainder B D $hall be B E. But as B E is to B O, $o is B D to B C: Therefore the Times along B D and B C, after the De$cent along A B and F B, or which is the $ame, along the Common part A B, $hall be to one another as the Lengths B D and B C: But that the Time along B D, is to the Time along B C, out of Re$t in B, as the Length B D to B C, hath already been demon$trated. Therefore the Times of the Motions along different Planes who$e Elevations are equal, are to one another as the Lengths of the $aid Planes, whether the Motion be made along the $ame out of Re$t, or whether another Motion of the $ame Altitude do precede tho$e Motions: Which was to be de- mon$trated.</I></P> <head>THEOR. XI. PROP. XI.</head> <P>If a Plane, along which a Motion is made out of Re$t, be divided at plea$ure, the Time of the Motion along the fir$t part, is to the Time of the Motion along the $econd, as the $aid fir$t part is to the exce$$e whereby the $ame part $hall be exceeded by the Mean-Propor- tional between the whole Plane and the $ame fir$t part.</P> <P><I>Let the Motion be along the whole Plane A B, ex quiete in A, which let be divided at plea$ure in C; and let A F be a Mean proportional between the whole B A and the fir$t part A C; C F $hall be the exce$$e of the Mean proportional F A above the part A C. I $ay the Time of the Motion along A C is to the Time of the following Motion along C B, as A C to C F. Which is manife$t;</I> <foot><I>For</I></foot> <p n=>167</p> <I>For the Time along A C is to the Time along all A B, as A C to the Mean-proportional A F: There-</I> <fig> <I>fore, by Divi$ion, the Time along A C, $hall be to the Time along the remainder C B as A C to C F: If therefore the Time along A C be $uppo$ed to be the $aid A C, the Time along C B $hall be C F: Which was the Propo$ition.</I></P> <P><I>But if the Motion be not made along the continu- ate Plane A C B, but by the inflected Plane A C D until it come to the Horizon B D, to which from F a Parallel is drawn F E. It $hall in like manner be</I> <fig> <I>demon$trated, that the Time along A C is to the Time along the reflected Plane C D, as A C is to C E. For the Time along A C is to the Time a- long C B, as A C is to C F: But the Time along C B, after A C hath been demon$trated to be to the Time along C D, after the $aid De$oent along A C, as C B is to C D; that is, as C F to C E: Therefore,</I> ex æquali, <I>the Time along A C $hall be to the Time along C D, as the Line A C to C E.</I></P> <head>THEOR. XII. PROP. XII.</head> <P>If the Perpendicular and Plane Inclined at plea- $ure, be cut between the $ame Horizontal Lines, and Mean-Proportionals between them and the parts of them contained betwixt the common Section and upper Horizontal Line be given; the Time of the Motion a- long the Perpendicular $hall have the $ame proportion to the Time of the Motion along the upper part of the Perpendicular, and af- terwards along the lower part of the inter$e- cted Plane, as the Length of the whole Per- pendicular hath to the Line compounded of the Mean-Proportional given upon the Per- pendicular, and of the exce$$e by which the whole Plane exceeds its Mean-Proporttonal.</P> <foot><I>Let</I></foot> <p n=>168</p> <P><I>Let the Horizontal Lines be A F the upper, and C D the low- er; between which let the Perpendicular A C, and inclined Plane D F, be cut in B; and let A R be a Mean-Proportional between the whole Perpendicular C A, and the upper part A B; and let F S be a Mean-proportional between the whole Inclined Plane D F, and the upper part B F. I $ay, that the Time of the Fall along the whole Perpendicular A C hath the $ame proportion to the Time along its upper part A B, with the lower of the Plane, that is, with B D, as A C hath to the Mean-proporti- onal of the Perpendicular, that is</I> <fig> <I>A R, with S D, which is the ex- ce$$e of the whole Plane D F above its Mean-proportional F S. Let a Line be drawn from R to S, which $hall be parallel to the two Horizon- tal Lines. And becau$e the Time of the Fall along all A C, is to the Time along the part A B, as C A is to the Mean proportional A R, if we $uppo$e A C to be the Time of the Fall along A C, A R $hall be the Time of the Fall along A B, and R C that along the remainder B C. For if the Time along A C be $uppo$ed, as was done, to be A C it $elf the Time along F D $hall be F D; and in like manner D S may be concluded to be the Time a- long B D, after F B, or after A B. The Time therefore along the whole A C, is A R, with R C; And the Time along the inflected Plane A B D, $hall be A R, with S D: Which was to be proved.</I></P> <P><I>The $ame happeneth, if in$tead of the Perpendicular, another Plane were taken, as $uppo$e N O; and the Demonstration is the $ame.</I></P> <head>PROBL I. PROP. XIII.</head> <P>A Perpendicular being given, to Inflect a Plane unto it, along which, when it hath the $ame Elevation with the $aid Perpendicular, it may make a Motion after its Fall along the Per- pendicular in the $ame Time, as along the $ame Perpendicular <I>ex quiete.</I></P> <P><I>Let the Perpendicular given be A B, to which extended to C, let the part B C be equal; and draw the Horizontal Lines C E and A G. It is required from B to inflect a Plane reach- ing to the Horizon C E, along which a Motion, after the Fall out</I> <foot><I>of</I></foot> <p n=>169</p> <I>of A, $hall be made in the $ame Time, as along A B from Re$t in A. Let C D be equal to C B, and drawing B D, let B E be applied equal to both B D and D C. I $ay B E is the Plane required. Continue out E B to meet the Horizontal Line A G in G;</I> <fig> <I>and let G F be a Mean-Proportional be- tween the $aid E G and G B. E F $hall be to F B, as E G is to G F; and the Square E F $hall be to the Square F B, as the Square E G is to the Square G F; that is as the Line E G to G B: But E G is double to G B: Therefore the Square of E F is double to the Square of F B: But al$o the Square of D B is double to the Square of B C: Therefore, as the Line E F is to F B, $o is D B to B C: And by Compo$ition and Permutation, as E B is to the two D B and B C, $o is B F to B C: But B E is equal to the two D B and B C: Therefore B F is equal to the $aid B C, or B A. If there- fore A B be under$tood to be the Time of the Fall along A B, G B $hall be the Time along G B, and G F the Time along the whole G E: There- fore B F $hall be the Time along the remainder B E, after the Fall from G, or from A, which was the Propo$ition.</I></P> <head>PROBL. II. PROP. XIV.</head> <P>A <I>P</I>erpendicular and a <I>P</I>lane inclined to it being given, to find a part in the upper <I>P</I>erpendicu- lar which $hall be pa$t <I>ex quiete</I> in a Time equal to that in which the inclined <I>P</I>lane is pa$t after the Fall along the part found in the Perpendicular.</P> <P><I>Let the Perpendicular be D B, and the Plane inclined to it A C. It is required in the Perpendicular A D to find a part which $hall be pa$t</I> ex quiete <I>in a Time equal to that in which the Plane A C is pa$t after the Fall along the $aid part. Draw the Horizontal Line C B; and as B A more twice A C is to A C, $o let E A be to A R; And from R let fall the Perpendicular R X unto D B. I $ay X is the point requi- red. And becau$e as B A more twice A C is to A C, $o is C A to A E, by Divi$ion it $hall be that as B A more A C is to A C, $o is C E to E A: And becau$e as B A is to A C, $o is E A to A R, by Compo$ition it $hall be that as B A more A C is to A C, $o is E R to R A: But as B A more A C is to A C, $o is C E to E A: Therefore, as C E is to E A, $o is E R, to R A, and both the Antecedents to both the Con$equents, that is, C R</I> <foot>Z <I>to</I></foot> <p n=>170</p> <I>to R E: Therefore C R, R E, and R A are Proportionals. Farther- more, becau$e as B A is to A C, $o E A is $uppo$ed to be to A R, and,</I> <fig> <I>in regard of the likene$$e of the Triangles, as B A is to A C, $o is X A to A R: There- fore, as E A is to A R, $o is X A to A R: Therefore E A and X A are equal. Now if we under$tand the Time along R A to be as R A, the Time along R C $hall be R E, the Mean-Proportional between C R and R A: And A E $hall be the Time along A C after R A or after X A: But the Time along X A is X A, $o long as R A is the Time along R A: But it hath been proved that X A and A E are equal: Therefore the Propo$ition is proved.</I></P> <head>PROBL. III. <I>P</I>RO<I>P.</I> XV.</head> <P>A <I>P</I>erpendicular and a <I>P</I>lane inflected to it being given, to find a part in the <I>P</I>erpendicular ex- tended downwards which $hall be pa$$ed in the $ame. Time as the inflected <I>P</I>lane after the Fall along the given Perpendicular.</P> <P><I>Let the Perpendicular be A B, and the Plane In$lected to it B C. It is required in the Perpendicular extended downwards to find a part which from the Fall out of A $hall be pa$t in the $ame Time as B C is pa$$ed from the $ame Fall out of A. Draw the Horizontal Line A D, with which let C B meet extended to D; and let D E be a Mean- proportional between C D and D B;</I> <fig> <I>and let B F be equal to B E; and let A G be a third Proportional to B A and A F. I $ay, B G is the Space that after the Fall A B $hall be pa$t in the $ame Time, as the Plane B C $hall be pa$t af- ter the $ame Fall. For if we $uppo$e the Time along A B to be as A B, the Time along D B $hall be as D B: And becau$e D E is the Mean-proportional between B D and D C, the $ame D E $hall be the Time along the whole D C, and B E the Time along the Part or Remainder B C</I> ex quiete, <I>in D, or</I> ^{*} ex ca$u <I>A B: And it may in</I> <marg>* From or after the Fall A B.</marg> <I>like manner be proved, that B F is the Time along B G, after the $ame Fall: But B F is equal to B E: Which was the Propo$ition to be proved.</I></P> <foot>THEOR.</foot> <p n=>171</p> <head>THEOR. XIII. <I>P</I>RO<I>P.</I> XVI.</head> <P>If the parts of an inclined <I>P</I>lane and <I>P</I>erpendicu- lar, the Times of who$e Motions <I>ex quiete</I> are equal, be joyned together at the $ame point, a Moveable coming out of any $ublimer Height $hall $ooner pa$$e the $aid part of the inclined <I>P</I>lane, than that part of the Perpendicular.</P> <P><I>Let the Perpendicular be E B, and the Inclined Plane C E, joyned at the $ame Point E, the Times of who$e Motions from off Re$t in E are equal, and in the Perpendicular continued out, let a $ublime point A be taken at plea$ure, out of which the Moveables may be let fall. I $ay, that the Inclined Plane E C $hall be pa$$ed in a le$$e Time than the Perpendicular E B, after the Fall A E. Draw a Line from C to B, and having drawn the Horizontal Line A D continue out C E till it meet the $ame in D; and let D F be a Mean-Proportional between C D and D E; and let A G be a</I> <fig> <I>Mean-Proportional between B A and A E; and draw F G and D G. And becau$e the Time of the Motion along E C and E B out of Re$t in E are equal, the Angle C $hall be a Right Angle, by the $econd Corollary of the Sixth Propo$ition; and A is a Right Angle, and the Vertical Angles at E are equal: Therefore the Tri- angles A E D and C E B are equian- gled, and the Sides about equal An- gles are Proportionals: Therefore as B E is to E C, $o is D E to E A. Therefore the Rectangle B E A is equal to the Rectangle C E D: And becau$e the Rectangle C D E ex- ceedeth the Rectangle C E D, by the Square E D, and the Rectangle B A E doth exceed the Rectangle B E A, by the Square E A: The exce$$e of the Rectangle C D E above the Rectangle B A E, that is of the Square F D above the Square A G $hall be the $ame as the exce$$e of the Square D E above the Square A E; which exce$s is the Square D A: Therefore the Square F D is equal to the two Squares G A and A D, to which the Square G D is al$o equal: Therefore the</I> <foot>Z 2 <I>Line</I></foot> <p n=>172</p> <I>Line D F is equal to D G, and the Angle D G F is equal to the An- gle D F G, and the Angle E G F is le$$c than the Angle E F G, and the oppo$ite Side E F le$$e than the Side E G. Now if we $uppo$e the Time of the Fall along A E to be as A E, the Time by D E $hall be as D E; and A G being a Mean-Proportional between B A and A E, A G $hall be the Time along the whole A B, and the part E G $hall be the Time along the Part E B</I> ex quiete <I>in A. And it may in like man- ner be proved that E F is the Time along E C after the De$cent D E, or after the Fall A E: But E F is proved to be le$$er than E G: Therefore the Propo$ition is proved.</I></P> <head>COROLLARY.</head> <P>By this and the precedent it appears, that the Space that is pa$- $ed along the Perpendicular after the Fall from above in the $ame Time in which the Inclined Plane is pa$t, is le$$e than that which is pa$t in the $ame Time as in the Inclined, no fall from above preceding, yet greater than the $aid Inclined Plane.</P> <P><I>For it having been proved, but now, that of the Moveables coming from the $ublime Term A the Time of the Conver$ion along E C is $horter than the Time of the Progre$$ion along E B; It is manife$t that the Space that is pa$t along E B in a Time equal to the Time along E C is le$s than the whole Space E B. And that the $ame Space along the Perpendicular is greater than E C is mani- fe$ted by rea$$uming the Figure of the pre-</I> <fig> <I>cedent Propo$ition, in which the part of the Perpendicular B G hath been demon$trated to be pa$$ed in the $ame Time as B C after the Fall A B: But that B G is greater than B C is thus collected. Becau$e B E and F B are equal, and B A le$$er than B D, F B, hath greater proportion to B A, than E B hath to B D: And, by Compo$ition, F A hath greater proportion to A B, than E D to D B: But as F A is to A B, $o is G F to F B, (for A F is the Mean-Proportional between B A and A G:) And in like man- ner, as E D is to B D, $o is C E to E B: Therefore G B hath greater proportion to B F, than C B hath to B E: Therefore G B is greater than B C.</I></P> <foot>PROEL.</foot> <p n=>173</p> <head>PROBL. IV. <I>P</I>RO<I>P.</I> XVII.</head> <P>A <I>P</I>erpendicular and <I>P</I>lane Inflected to it being given, to a$$ign a part in the given <I>P</I>lane, in which after the Fall along the <I>P</I>erpendicular the Motion may be made in a Time equal to that in which the Moveable <I>ex quiete</I> pa$$eth the <I>P</I>erpendicular given.</P> <P><I>Let the Perpendicular be A B, and a Plane Inflected to it B E: It is required in B E to a$$ign a Space along which the Moveable af- ter the Fall along A B may move in a Time equal to that in which the $aid Perpendicular A B is pa$$ed</I> ex quiete. <I>Let the Line A D be parallel to the Horizon, with which let the Plane prolonged meet in D; and $uppo$e F B equal to B A; and as B D is to D F, $o let F D be to D E. I $ay, that</I> <fig> <I>the Time along B E after the Fall along A B equalleth the Time along A B, out of Re$t in A. For if we $uppo$e A B to be the Time along A B, D B $hall be the Time along D B. And becau$e, as B D is to D F, $o is F D to D E, D F $hall be the Time along the whole Plane D E, and B F along the part B E out of D: But the Time along B E after D B, is the $ame as after A B: Therefore the Time along B E after A B $hall be B F, that is, equal to the Time</I> ex quiete <I>in A: Which was the Propo$ition.</I></P> <head><I>P</I>ROBL. V. PROP. XVIII.</head> <P>Any Space in the Perpendicular being given from the a$$igned beginning of Motion that is pa$$ed in a Time given, and any other le$$er Time being al$o given, to find another Space in the $aid Perpendicular that may be pa$$ed in the given le$$er Time.</P> <foot><I>Let</I></foot> <p n=>174</p> <P><I>Let the Perpendicular be A D, in which let the Space a$$igned be A B, who$e Time from the beginning A let be A B: and let the Horizon be C B E, and let a Time be given le$s than A B, to which let B C be noted equal in the Horizon: It is required in the $aid Perpendicular to find a Space equal to the $ame A B that $hall be pa$$ed in the Time B C. Draw a Line from A to</I> <fig> <I>C. And becau$e B C is le$$e than B A, the Angle B A C $hall be le$$e than the Angle B C A. Let C A E be made equal to it, and the Line A E meet with the Horizon in the Point E, to which $up- po$e E D a Perpendicular, cutting the Perpendi- cular in D, and let D F be cut equal to B A. I $ay, that the $aid F D is a part of the Perpendi- cular along which the Lation from the beginning of Motion in A, the Time B C given will be $pent. For if in the Right-angled Triangle A E D, a Perpendicular to the oppo$ite Side A D, be drawn E B, A E $hall be a Mean-Proportional betwixt D A and A B, and B E a Mean-Proportional betwixt D B and B A, or betwixt F A and A B (for F A is equal to D B.) And in regard A B hath been $uppo$ed to be the Time along A B, A E, or E C $hall be the Time along the whole A D, and E B the Time along A F: There- fore the part B C $hall be the Time along the part F D: Which was intended.</I></P> <head>PROBL. VI. PROP. XIX.</head> <P>Any Space in the Perpendicular pa$$ed from the beginning of the Motion being given, and the Time of the Fall being a$$igned, to find the Time in which another Space. equal to the gi- ven one, and taken in any part of the $aid Per- pendicular, $hall be afterwards pa$t by the $ame Moveable.</P> <P><I>In the Perpendicular A B let A C be any Space taken from the be- ginning of the Motion in A, to which let D B be another equal Space taken any where at plea$ure, and let the Time of the Motion along A C be given, and let it be A C. It is required to $ind the Time of the</I> <foot><I>Motion</I></foot> <p n=>175</p> <I>Motion along D B after the Fall from A. About the whole A B de- $cribe a Semicircle A E B, and from</I> <fig> <I>C let fall C E <*> Perpendicular to A B, and draw a Line from A to E; which $hall be greater than E C. Let E F be out equall to E C: I $ay, that the remainder F A is the Time of the Motion along D B. For be- cau$e A E is a Mean-proportional be- twixt B A and and A C, and A C is the Time of the Fall along A C; A E $hall be the Time along the Whole A B. And becau$e C E is a Mean-proportional betwixt D A and A C, (for D A is equal to B C) C E, that is E F $hall be the Time along A D: Therefore the Remainder A F $hall be the Time along the Remainder B B: Which is the Propo$ition.</I></P> <head>COROLLARY.</head> <P>Hence is gathered, that if the Time of any Space <I>ex quiete</I> be as the $aid Spaec, the Time thereof after another Space is ad- ded $hall be the exce$$e of the Mean-proportional betwixt the Addition and Space taken together, and the $aid Space above the Mean-proportional betwixt the fir$t Space and the Addition.</P> <P><I>As for example, it being $uppo$ed that the Time along</I> <fig> <I>A B, out of Re$t in A, be A B; A S being another Space added, The Time along A B after S A $hall be the exce$$e of the Mean-proportional betwixt S B and B A above the Mean-proportional betwixt B A and A S.</I></P> <head>PROBL VII. PROP. XX.</head> <P>Any Space and a part therein after the begining of the Motion being given, to find another part towards the end that $hall be pa$t in the $ame Time as the fir$t part given.</P> <P><I>Let the Space be C B, and let the part in it given after the begin- ing of the Motion in C be C D. It is required to find another part towards the end B, which $hall be pa$t in the $ame Time as</I> <foot><I>the</I></foot> <p n=>176</p> <I>the given part C D. Take a Mean-proportional betwixt B C and C D, to which $uppo$e B A equal; and let C E be a third proportional be-</I> <fig> <I>tween B C and C A. I $ay, that E B is the Space that after the Fall out of C $hall be past in the $ame Time as the $aid C D is pa$$ed. For if we $uppo$e the Time along C B to be as C B; B A (that is the Mean-proportional betwixt B C and C D) $hall be the Time along C D. And becau$e C A is the Mean proportional betwixt B C and C E, C A $hall be the Time along C E: But the whole B C is the Time along the Whole C B: Therefore the part B A $hall be the Time along the part E B, after the Fall out of C: But the $aid B A was the Time along C D: Therefore C D and E B $hall be pa$t in equal Times out of Re$t in C: Which was to be done.</I></P> <head>THEOR. XIV. PROP. XXI.</head> <P>If along the Perpendicular a Fall be made <I>ex quie- te,</I> in which from the begining of the Motion a part is taken at plea$ure, pa$$ed in any Time, after which an Inflex Motion followeth along any Plane however Inclined, the Space which along that Plane is pa$$ed in a Time equal to the Time of the Fall already made along the Perpendicular $hall be to the Space then pa$- $ed along the Perpendicular more than double, and le$$e than triple.</P> <P><I>From the Horizon A E let fall a Perpendicular A B, along which from the begining A let a Fall be made, of which let a part A C be taken at plea$ure; then out of C let any Plane G be inclined at plea$ure: along which after the Fall along A C let the Motion be con- tinued. I $ay, the Space pa$$ed by that Motion along C G in a Time equall to the Time of the Fall along A C, is more than double, and le$s than triple that $ame Space A C. For $uppo$e C F equal to A C, and extending out the Plane G C as far as the Horizon in E, and as C E is to E F, $o let F E be to E G. If therefore we $uppo$e the Time of</I> <foot><I>the</I></foot> <p n=>177</p> <I>the Fall along A C to be as the Line A C; C E $hall be the Time along E C, and C F or C A the Time of the Motion along C G. Therefore it is to be proved that the</I> <fig> <I>Space C G is more than double, and le$$e than triple the $aid C A. For in regard that as C E is to E F, $o is F E to E G; therefore al$o $o is C F to F G. But E C is le$$e than E F: Therefore C F $hall be le$$e than F G, and G C more than double to F C or A C. And moreover, in regard that F E is le$$e than double to E C, (for E C is greater than C A or C F) G F $hall al$o be le$$e than double to F C, and G C le$$e than triple to C F or C A: Which was to be demon$trated.</I></P> <P><I>And the $ame may be more generally propounded: for that which hapneth in the Perpendicular and Inclined Plane, holdeth true al$o if after the Motion a Plane $omewhat inclined it be inflected along a more inclining Plane, as is $een in the other Figure: And the Demon$tration is the $ame.</I></P> <head><I>P</I>ROBL. VIII. <I>P</I>RO<I>P.</I> XXII.</head> <P>Two unequall Times being given, and a Space that is pa$t <I>ex quiete</I> along the Perpendicular in the $horte$t of tho$e given Times, to inflect a Plane from the highe$t point of the Perpen- dicular unto the Horizon, along which the Moveable may de$cend in a Time equal to the longe$t of tho$e Times given.</P> <P><I>Let the unsqual Times be A the greater, and B the le$$er; and let the Space that is pa$t</I> ex quiete <I>along the Perpendicular in the Time B, be C D. It is required from the Term C to inflect</I> [or <fig> bend] <I>a Plane untill it reach the Horizon that may be pa$$ed in the</I> <foot>Aa <I>Time</I></foot> <p n=>178</p> <I>Time A. As B is to A, $o let C D be to another Line, to which let C X be equal that de$cendeth from C unto the Horizon: It is manife$t that the Plane C X is that along which the Moveable de$cendeth in the Gi- ven Time A. For it hath been demon$trated, that the Time along the inclined Plane hath the $ame proportion to the Time along its ^{*} Eleva-</I> <marg>* Or Perpendi- cular.</marg> <I>tion, as the Length of the Plane hath to the Length of its Elevation,. The Time, therefore, along C X is to the Time along C D, as C X is to C D, that is, as the Time A is to the Time B: But the Time B is that in which the Perpendicular is pa$t</I> ex quiete: <I>Therefore the Time A is that in which the Plane C X is pa$$ed.</I></P> <head><I>P</I>ROBL. IX. PROP. XXIII.</head> <P>A Space pa$t <I>ex quiete</I> along the Perpendicular in any Time being given, to inflect a Plane from the lowe$t term of that Space, along which, after the Fall along the Perpendicular, a Space equal to any Space given may be pa$$ed in the $ame Time: which neverthele$$e is more than double, and le$$e than triple the Space pa$$ed along the Perpendicular.</P> <P><I>Along the Perpendicular A S, in the Time A C, let the Space A C be pa$t</I> ex quiete <I>in A; to which let I R be more than double, and le$$e than triple. It is required from the Terme C to inflect a Plane, along which a Moveable after the Fall along A C may in the $ame Time A C pa$$e a Space equal to the $aid I R. Let R N, and N M be equal to A C: And look what proportion the part I M hath to M N, the $ame $hall the Line A C have to another, equal to which draw C E from C to</I> <fig> <I>the Horizon A E, which con- tinue out towards O, and take C F, F G, and G O, equal to the $aid R N, N M, and M I. I $ay, that the Time along the inflected Plane C O, after the Fall A G, is equal to the Time A C out of Re$t in A. For in regard that as O G is to G F, $o is F C to C E by Compo$ition it $hall be that as O F is to F G or F C, $o is F E to E C; and as one of the Antecedents is to one of the Con- $equents, $o are all to all; that is, the whole O E is to E F as F E to E C: Therefore O E, E F, and E C are Continual Proportionals:</I> <foot><I>And</I></foot> <p n=>179</p> <I>And $ince it was $uppo$ed that the Time along A C is as A C, C E $hall be the Time along E C; and E F the Time along the whole E O; and the part C F that along the part C O: But C F is equal to the $aid C A: Therefore that is done which was required: For the Time C A is the Time of the Fall along A C</I> ex quiete <I>in A; and C F (which is equal to C A) is the Time along C O, after the De$cent along E C, or after the Fall along A C: Which was the Propo$ition.</I></P> <P><I>And here it is to be noted, that the $ame may happen if the preceding Motion be not made along the Perpendicular, but along an Inclined Plane: As in the following Figure, in which let the preceding Lation be made along the inclined Plane A S beneath the Horizon A E: And the Demon- $tration is the very $ame.</I></P> <head>SCHOLIUM.</head> <P>If one ob$erve well, it $hall be manife$t, that the le$$e the given Line I R wanteth of being triple to the $aid A C, the nearer $hall the Inflected Plane, along which the $econd Motion is to be made, which $uppo$e to be C O, come to the Perpen- dicular, along which in a Time equal to A C a Space $hall be pa$$ed triple to A C.</P> <P><I>For in ca$e I R were very near the triple of A C, I M $hould be well- near equal to M N: And if, as I M is to M N by Con$truction, $o A C is to C E, then it is evident that the $aid C E will be found but little bigger than C A, and, which followeth of con$equence, the point E $hall be found very near the point A, and C O to containe a very acute</I> <fig> <I>Angle with C S, and almo$t to concur both in one Line. And on the contrary, if the $aid I R were but a very little more than double the $aid A C, I M $hould be a very $hort Line. Hence it may happen al$o that A C may come to be very $hort in re$pect of C E which $hall be very long, and $hall ap- proach very near the Horizontal Parallel drawn from C. And from hence we may collect, that if in the pre$ent Figure after the De$cent along the inclined Plane A C, a Reflexion be made along the Horizontal Line, as</I> v. gr. <I>C T, the Space along which the Moveable afterwards moved in a Time equal to the Time of the De$cent along A C would be exactly double to the Space A C. And it appears that the like Di$cour$e may be here applied: For it is apparent by what hath been $aid, that $ince O E</I> <foot><I>Aa</I> 2 <I>is</I></foot> <p n=>180</p> <I>is to E F, as F E is to E C, that F C determineth the Time along C O: And if a part of the Horizontal Line T C double to C A be divided in two equal parts in V, the exten$ion towards X $hall be prolonged</I> in in- finitum, <I>whil$t it $eeks to meet with the prolonged Line A E: And the proportion of the Infinite Line T X to the Infinite Line V X, $hall be no other than the proportion of the Infinite Line V X to the Infinite Line X C.</I></P> <P><I>We may conclude the $elf-$ame thing another way by rea$$uming the $ame Rea$oning that we u$ed in the Demon$tration of the fir$t Propo$i- tion. For re$uming the Triangle A B C, repre$enting to us by its Pa- rallels to the Ba$e B C the Degrees of Velocity continually encrea$ed ac- cording to the encrea$es of the Time; from which, $ince they are infi- nite, like as the Points are infinite in the Line A C, and the In$tants in any Time, $hall re$ult the Superficies of that $ame Triangle, if we under$tand the Motion to continue for $uch another Time, but no far- ther with an Accelerate, but with an Equable Motion, according to the greate$t degree of Velocity acquired, which degree is repre$ented by the Line B C. Of $uch degrees $hall be made up an Aggregate like to a Parallelogram A D B C, which is the double of</I> <fig> <I>the Triangle A B C. Wherefore the Space which with degrees like to tho$e $hall be pa$$ed in the $ame Time, $hall be double to the Space pa$t with the de- grees of Velocity repre$ented by the Triangle A B C: But along the Horizontal Plane the Motion is Equa- ble, for that there is no cau$e of Acceleration, or Re- tardation: Therefore it may be concluded that the Space C D, pa$$ed in a Time equall to the Time A C is double to the Space A C: For this Motion is made</I> ex quiete <I>Accelerate according to the Parallels of the Triangle; and that according to the Parallels of the Parallelogram, which, becau$e they are infinite, are donble to the infinite Parallels of the Triangle.</I></P> <P><I>Moreover it may farther be ob$erved, that what ever degree of $wiftne$s is to be found in the Moveable, is indelibly impre$$ed upon it of its own nature, all external cau$es of Acceleration or Retardation being removed; which hapneth only in Horizontal Planes: for in de- clining Planes there is cau$e of greater Acceleration, and in the ri$ing Planes of greater Retardation. From whence in like manner it fol- loweth that the Motion along the Horizontal Plane is al$o Perpetual: for if it be Equable, it can neither be weakned nor retarded, nor much le$$e de$troyed. Farthermore, the degree of Celerity acquired by the Moveable in a Natural De$cent, being of its own Nature Indelible and Penpetual, it is worthy con$ideration, that if after the De$cent along a declining Plane a Reflexion be made along another Plane that is ri$ing, in this latter there is cau$e of Retardation, for in the$e kind of Planes</I> <foot><I>the</I></foot> <p n=>181</p> <I>the $aid Moveable doth naturally de$cend; whereupon there re$ults a mixture of certain contrary Affections, to wit, that degree of Celerity acquired in the precedent De$cent, which would of it $elf carry the Move- able uniformly</I> in infinitum, <I>and of Natural Propen$ion to the Motion of De$cent according to that $ame proportion of Acceleration wherewith it alwaies moveth. So that it will be but rea$onable, if, enquiring what accidents happen when the Moveable after the De$cent along any incli- ned Plane is Reflected along $ome ri$ing Plane, we take that greate$t de- gree acquired in the De$cent to keep it $elf perpetually the $ame in the A$cending Plane; But that there is $uperadded to it in the A$cent the Natural Inclination downwards, that is the Motion from Re$t Accelerate according to the received proportion: And le$t this $hould, perchance, be $omewhat intricate to be under$tood, it $hall be more clearly explained by a Scheme.</I></P> <P><I>Let the De$cent therefore be $uppo$ed to be made along the Declining Plane A B, from which let the Reflex Motion be continued along another Ri$ing Plane B C: And in the fir$t place let the Planes be equal, and elevated at equal Angles to the Horizon G H. Now it is manife$t, that the Moveable</I> ex quiete <I>in A de$cending along A B acquireth degrees of Velocity according to the increa$e of its Time, and that the degree in B is the greate$t of tho$e acquired and by Nature immutably impre$$ed, I mean the Cau$es of new Acceleration or Retardation being removed: of Acceleration, I $ay, if it $hould pa$$e any farther along the extended Plane; and of Retardation, whil$t the Reflection is making along the Acclivity B C: But along the Horizontal Plane G H the Equable Mo- tion according to the de-</I> <fig> <I>gree of Velocity acquired from A unto B would ex- tend</I> in infinitum. <I>And $uch a Velocity would that be which in a Time equal to the Time of the De$cent along A B would pa$$e a Space in double the Horizon to the $aid A B. Now let us $uppo$e the $ame Moveable to be Equably moved with the $ame degree of Swiftne$$e along the Plane B C, in $uch $ort that al$o in this Time equal to the Time of the De$cent along A B a Space may be pa$$ed a long B C extended double to the $aid A B. And let us under- $tand that as $oon as it beginneth to a$cend there naturally befalleth the $ame that hapneth to it from A along the Plane A B, to wit, a certain De$cent</I> ex quiete <I>according to tho$e degrees of Acceleration, by vertue of which, as it befalleth in A B, it may de$cend as much in the $ame Time along the Reflected Plane as it doth along A B: It is manife$t, that by this $ame Mixture of the Equable Motion of A$cent, and the Acce- lerate of De$cent the Moveable may be carried up to the Term C along the Plane B C according to tho$e degrees of Velocity, which $hall be</I> <foot><I>equal.</I></foot> <p n=>182</p> <I>equal. And that two points at plea$ure D and E being taken, equally remote from the Angle B, the Tran$ition along D B is made in a Time equal to the Time of the Reflection along B E, we may collect from hence: Draw D F, which $hall be Parallel to B C; for it is manife$t that the De$cent along A D is reflected along D F: And if after D the Move- able pa$$e along the Horizontal Plane D E, the</I> Impetus <I>in E $hall be the $ame as the</I> Impetus <I>in D: Therefore it will a$cend from E to C: And therefore the degree of Velocity in D is equal to the degree in E. From the$e things, therefore, we may rationally affirm, that, if a de- $cent be made along any inclined Plane, after which a Reflection may follow along an elevated Plane, the Moveable may by the conceived</I> Impetus <I>a$cend untill it attain the $ame beight, or Elevation from the Horizon. As if a De$cent be made along A B, the Moveable would pa$$e along the Reflected Plane B C, untill it arrive at the Horizon A C D; and that not only when the Inclinations of the Planes are equal, but al$o when they are unequal, as is the Plane B D: For it was first $uppo$ed, that the degrees of Velocity are equal, which are acqui- red upon Planes unequally inclined, $o long as the Elevation of tho$e Planes above the Horizon was the $ame: But, if there being the $ame Inclination of the Planes E B and B D, the De$cent along E B $ufficeth to drive the Moveable along the Plane BD as far as D, $eeing this Impul$e</I> <fig> <I>is made by the</I> Impe- tus <I>of Velocity in the point B; and if the</I> Impetus <I>be the $ame in B, whether the Moveable de$cend a- long A B, or along E B: It is manife$t, that the Moveable $hall be in the $ame manner driven along B D, after the De$cent along A B, and after that along E B: But it will happen that the Time of the A$cent along B D $hall be longer than along B C, like as the De$cent along E B is made in a longer time than along A B: But the Proportion of tho$e Times was before demon$trated to be the $ame as the Lengths of tho$e Planes. Now it follows, that we $eek the proportion of the Spaces pa$t in equal Times along Planes, who$e Inclinations are different, but their Elevations the $ame; that is, which are comprehended between the $ame Horizontal Parallels. And this hapneth according to the fol- lowing Propo$ition.</I></P> <foot>THEOR.</foot> <p n=>183</p> <head>THEOR. XV. PROP. XXIV.</head> <P>There being given between the $ame Horizontal Parallels a Perpendicular and a <I>P</I>lane eleva- ted from its lowe$t term, the Space that a Moveable after the Fall along the <I>P</I>erpendi- cular pa$$eth along the Elevated <I>P</I>lane in a Time equal to the Time of the Fall, is greater than that <I>P</I>erpendicular, but le$$e than double the $ame.</P> <P><I>Between the $ame Horizontal Parallels B C and H G let there be the Perpendicular A E; and let the Elevated Plane be E B, along which after the Fall along the Perpendicular A E out of the Term E let a Reflexion be made towards B. I $ay, that the Space, along which the Moveable a$cendeth in a Time equal to the Time of the De$cent A E, is greater than A E, but le$$e than double the $ame A E. Let E D be equal to A E, and as E B is to B D, $o let D B be to B F. It $hall be proved, fir$t that the point F is the Term at which the Moveable with a Reflex Motion along E B arriveth in a Time equal to the Time A E: And then, that E F is greater than E A, but le$$e than double the $ame. If we $uppo$e the Time of the De$cent along A E to be as A E, the Time of the De$cent along B E, or A$cent along E B $hall be as the $ame Line B E: And D B being a Mean-Proportional betwixt E B and B F, and B E being the Time of De$cent along the whole B E, B D $hall be the Time of the De$cent along B F, and the Remaining part D E the Time of the</I> <fig> <I>De$cent along the Re- maining part F E: But the Time along F E</I> ex quiete <I>in B, and the Time of the A$cent a- long E F is the $ame, $ince that the Degree of Velocity in E was acqui- red along the De$cent B E, or A E: Therefore the $ame Time D E $hall be that in which the Moveable after the Fall out of A along A E, with a Reflex Motion along E B $hall reach to the Mark F: But it hath been $uppo$ed that E D is equal to the $aid A E: Which was fir$t to be proved. And becau$e that as the whole E B is to the whole B D, $o is the part taken away D B to the part taken away B F, therefore, as the whole E B is to the whole B D, $o $hall the Remainder E D be to D F: But E B is greater than B D: Therefore E D is greater than D F, and E F le$$e than double to D E or A E: Which was to be proved.</I></P> <foot><I>And</I></foot> <p n=>184</p> <P><I>And the $ame al$o hapneth if the precedent Motion be not made along the Perpendicular, but along an Inclined Plane; and the Demon- $tration is the $ame, provided that the Reflex Plane be le$$e ri$ing, that is, longer than the declining Plane.</I></P> <head>THEOR. XVI. <I>P</I>RO<I>P.</I> XXV.</head> <P>If after the De$cent along any Inclined Plane a Motion follow along the Plane of the Hori- zon, the Time of the De$cent along the Incli- ned Plane $hall be to the Time of the Motion along any Horizontal Line; as the double Length of the Inclined Plane is to the Line ta- ken in the Horizon.</P> <P><I>Let the Horizontal Line be C B, the inclined Plane A B, and after the De$cent along A B let a Motion follow along the Horizon, in which take any Space B D. I $ay, that the Time of the De$cent along A B to the Time of the Motion along B D is as the double of A B to B D. For B C being $uppo$ed the double of A B, it is manife$t by</I> <fig> <I>what hath already been demon$tra- ted that the Time of the De$cent along A B is equal to the Time of the Motion along B C: But the Time of the Motion along B C is to the Time of the Motion along B D, as the Line C B is to the Line B D: Therefore the Time of the Motion along A B is the Time along B D, as the Double of A B is to B D: Which was to be proved.</I></P> <head>PROBL X. PROP. XXVI.</head> <P>A Perpendicular between two Horizontal <I>P</I>aral- lel Lines, as al$o a Space greater than the $aid <I>P</I>erpendicular, but le$$e than double the $ame, being given, to rai$e a <I>P</I>lane between the $aid <I>P</I>arallels from the lowe$t Term of the <I>P</I>er- pendicular, along which the Moveable may with a Reflex Motion after the Fall along the <I>P</I>erpendicular pa$$e a Space equal to the Space given, and in a Time equal to the Time of the Fall along the <I>P</I>erpendicular.</P> <foot><I>Let</I></foot> <p n=>185</p> <P><I>Let A B be a Perpendicular between the Horizontal Parallels A O and B C; and let F E be greater than B A, but le$$e than double the $ame. It is required between the $aid Parallels from the point B to rai$e a Plane, along which the Moveable after the Fall from A to B may with a Reflex Motion in a Time equal to the Time of the Fall along A B pa$$e a Space a$cending equal to the $aid E F. Suppo$e E D equall to A B, the Remaining Part D F $hall be le$$e, for that the whole E F is le$$e than double to A B: Let D I be equal to D F, and as E I is to I D, $o let D F be to another Space F X, and out of B let the Right-</I> <fig> <I>Line B O be reflected, equal to E X. I $ay, that the Plane along B O is that along which after the Fall A B a Moveable in a Time equal to the Time of the Fall along A B pa$$eth a$cending a Space equal to the given Space E F. Suppo$e B R and R S equal to the $aid E D and D F. And becau$e that as E I is to I D, $o is D F to F X; therefore, by Compo$ition, as E D is to D I, $o $hall D X be to X F; that is, as E D is to D F, $o $hall D X be to X F, and E X to X D; that is, as B O is to O R, $o $hall R O be to O S: And if we $uppo$e the Time along A B to be A B, the Time along O B $hall be the $ame O B, and R O the Time along O S, and the Remaining Part B R the Time along the Remaining Part S B, de$cending from O to B: But the Time of the De$cent along S B from Rest in O, is equal to the Time of the A$cent from B to S after the Fall A B: Therefore B O is the Plane ele- vated from B, along which after the Fall along A B the Space B S equal to the given Space E F is pa$$ed in the Time B R or B A: Which was required to be done.</I></P> <foot>Bb THEOR-</foot> <p n=>186</p> <head>THEOR. XVII. PROP. XXVII.</head> <P>If a Moveable de$cend along unequal <I>P</I>lanes, who$e Elevation is the $ame, the Space that $hall be pa$t along the lower part of the longe$t in a Time equal to that in which the whole $horter <I>P</I>lane is pa$$ed, is equal to the Space that is compounded of the $aid $horter <I>P</I>lane and of the part to which that $horter <I>P</I>lane hath the $ame <I>P</I>roportion that the longer <I>P</I>lane hath to the Exce$$e by which the longe$t exceedeth the $horte$t.</P> <P><I>Let A C be the longer Plane, and A B the $horter, who$e Elevation A D is the $ame; and in the lower part of A C take the Space C E, equal to the $aid A B; and as C A is to A E, (that is to the exce$$e of the Plane C A above A B) $o let C E be to E F. I $ay, that the Space F C is that which is pa$t after the De$cent out of A in a Time equal to the Time of</I> <fig> <I>the De$cent along A B. For the whole C A, being to the whole A E, as the part taken away C E is to the part taken away E F, therefore the re- maining part E A $hall be to the remaining part A F, as the whole C A is to the whole A E: Therefore the three Spaces C A, A E, and A F are three Continual proportionals. And if the Time along A B be $uppo$ed to be as A B, the Time along A C $hall be as A C, and the Time along A F $hall be as A E, and along the remain- ing part F C $hall be as E C: But E C is equal to the $aid A B: There- fore the Propo$ition is manife$t.</I></P> <head>THEOR. XVIII. PROP. XXVIII.</head> <P><I>Let the Horizontal Line A G be Tangent to a Circle, and from the point of Contact let A B be the Diameter, and A E B two Chords at plea$ure: We are to a$$ign the proportion of the Time of the Fall along A B to the Time of the De$cent along both the Chords A E B. Let B E be continued out till it meet the Tangent in G, and</I> <foot><I>let</I></foot> <p n=>187</p> <I>let the Angle B A E be cut in two equal parts, and draw A F. I $ay, that the Time along A B is to the Time along A E B, as A E is to A E F. For in regard the Angle F A B is equal to the Angle F A E, and the An- gle E A G to the Angle A B F, the whole Angle G A F $hall be equal to the two Angles F A B, and A B F; to which al$o the Angle G F A</I> <fig> <I>is equal: Therefore the Line G F is equal to G A. And becau$e the Rectangle B G E is equal to the Square of G A, it $hall likewi$e be equal to the Square of G F, and the three Lines B G, G F, and G E $hall be proportionals. And if we $uppo$e A E to be the Time along A E, G E $hall be the Time along G E, and G F the Time along the whole G B, and E F the Time along E B, after the De$cent out of G, or out of A, along A E: The Time, therefore, along A E, or along A B $hall be to the Time along A E B, as A E is to A E F: Which was to be determined.</I></P> <P><I>More briefly thus. Let G F be cut equal to G A: It is manife$t that G F is the Mean-proportional between B G, and G E. The re$t as before.</I></P> <head>PROBL. <I>XI. P</I>RO<I>P. XXIX.</I></head> <P>Any Horizontal Space being given upon the end of which a Perpendicular is erected, in which a part is taken equal to half of the Space given in the Horizontal a Moveable fal- ling from that height, and turned along the Horizon, $hall pa$$e the Horizontal Space to- gether with the Perpendicular in a $horter Time than any other Space of the <I>P</I>erpendi- cular with the $ame Horizontal Space.</P> <P><I>Let there be an Horizontal Space in which let any Space be given B C, and on B let there be a Perpendicular erected, in which let B A be the half of the fore$aid B C. I $ay, that the Time in which a Moveable let fall out of A pa$$eth both the Spaces A B and B C is the $horte$t of all Times in which the $aid Space B C with a part of the Perpendicular, whether greater or le$$er than the part A B, $hall be pa$- $ed. Let a greater be taken, as in the $ir$t Figure, or le$$er, as in the</I> <foot><I>Bb 2 $econd</I></foot> <p n=>188</p> <I>$econd, which let be E B. It is to be proved that the Time in which the Spaces E B and B C are pa$$ed is longer than the Time in which A B and B C are pa$$ed. Let the Time along A B be as A B; the $ame $hall be the Time of the Motion along the Horizontal Space B G; becau$e B C is double to A B, and the Time along both the Spaces A B C $hall be double of O B A. Let B O</I> <fig> <I>be a Mean-proportional between E B and B A. B O $hall be the Time of the Fall along E B. Again, let the Horizontal Space B D be double to the $aid B E: It is manife$t that the Time of it after the Fall E B is the $ame B O. As D B is to B C, or as E B is to B A, $o let O B be to B N: and in regard the Motion along the Horizontal Plane is Equable, and O B being the Time along B D after the Fall out of E, therefore N B $hall be the Time along B C after the Fall from the $ame Altitude E. Hence it is manife$t, that O B, together with B N is the Time along E B C; and becau$e the double of B A is the Time along A B C; it remains to be proved, that O B, to- gether with B N is more than double B A. Now becau$e O B is a Mean between E B and B A, the proportion of E B to B A is double the pro- portion of O B to B A: and, in regard that E B is to B A, as O B is to B N, the proportion of O B to B N $hall al$o be double the proportion of O B to B A: But that proportion of O B to B N is compounded of the proportions of O B to B A, and of A B to B N: therefore the proportion of A B to B N is the $ame with that of O B to B A. Therefore B O, B A, and B N are three continual Proportionals, and O B, together with B N, are greater than double B A: Whereupon the Propo$ition is ma- nife$t.</I></P> <foot>THEOR.</foot> <p n=>189</p> <head>THEOR. <I>XIX.</I> PROP. <I>XXX.</I></head> <P>If a Perpendicular be let fall from any point of the Horizontal Line, and out of another point in the $ame Horizontal Line a Plane be drawn forth untill it meet the Perpendicular, along which a Moveable de$cendeth in the $horte$t time unto the $aid Perpendicular, this Plane $hall be that which cutteth off a part equall to the di$tance of the a$$igned point from the end of the Perpendicular.</P> <P><I>Let the Perpendicular B D be let fall from the point B of the Ho- rizontal Line A C, in which let there be any point C, and in the Perpendicular let the Di$tance B E be $uppo$ed equal to the Di- $tance B C, and draw C E. I $ay, that of all Planes inclined out of the point C till they meet the Perpendicular C E is that, along which in the $horte$t of all Times the De$cent</I> <fig> <I>is made unto the Perpendicular. For let the Planes C F and C G be inclined above and below, and draw I K a Tan- gent unto the Semidiameter B C of the de$cribed Circle in C, which $hall be equidi$tant from the Perpendicular; and unto the $aid C F let E K be Paral- lel cutting the Circumference of the Cir- cle in L: It is manife$t that the Time of the De$cent along L E is equal to the Time of the De$cent along C E: But the Time along K E is longer than along L E: Therefore the Time along K E is longer than that along C E: But the Time along K E is equal to the Time a- long C F, they being equal, and drawn according to the $ame Inclination: Likewi$e $ince C G, and I E are equal, and inclined according to the $ame Inclination, the Times of the Motions along them $hall be equal: But H E being $horter than I E, the Time along it is al$o $horter than I E: Therefore the Time al$o along C E, (which is equal to the Time along H E) $hall be $horter than the Time along I E: The Propo$ition, therefore, is manife$t.</I></P> <foot>THEOR.</foot> <p n=>190</p> <head>THEOR. <I>XX.</I> PROP. <I>XXXI.</I></head> <P>If a Right-Line $hall be in any manner inclined upon the Horizontal Line, the Plane produced from a given point in the Horizon untill it meet with the Inclined Plane, along which the De$cent is made in the $horte$t of all Times, is that which $hall divide the Angle contained between the two <I>P</I>erpendiculars drawn from the given <I>P</I>oint, the one unto the Horizontal Line, the other to the Inclined Line, into two equal parts.</P> <P><I>Let C D be a Line inclined in any manner upon the Hori- zontal Line A B, and let any point A be given in the Hori- zon, and from it let A C be drawn Perpendicular to A B, and A E Perpendicular to C D, and let the Line F A divide the Angle C A E into two equal parts. I $ay, that of all Planes incli- ned out of any point of the Line C D to the point A that $ame pro- duced along F A is it along</I> <fig> <I>which the De$cent is made in the $horte$t of all Times. Let F G be drawn Parallel to AE; the alternate Angles G F A and F A E $hall be equal: But E A F is equal to that other F A G: Therefore of the Tri- angle the Sides F G and G A $hall be equal. If therefore about the Center G, at the di- $tance G A, a Circle be de$cri- bed it $hall pa$$e by F, and $hall touch the Horizontal, and the Inclined Lines in the points A and F: For the Angle G F C is a Right Angle, and likewi$e G F is equidi$tant to A E: Whence it is manife$t that all Lines produced from the point A unto the inclined Plane do extend beyond the Circumference, and, which followeth of con$equence, that the Motions along the $ame do take up more Time than along F A. Which was to be demon$trated.</I></P> <foot>LEMMA</foot> <p n=>191</p> <head>LEMMA.</head> <P>If two Circles touch one another within, the innermo$t of which toucheth $ome Right Line, and the exteriour one cutteth it, three Lines produced from the Contact of the Circles unto three points of the Tangent Right-Line, that is, to the Con- tact of the interiour Circle, and to the Sections of the exte- riour $hall contain equall Angles in the Contact of the Circles.</P> <P><I>Let two Circles touch one another in the point A, of which let the Centers be B, that of the le$$er, and C that of the greater; and let the interiour Circle touch any Line F G in the point H, and let the grea- ter cut it in the points F and G, and connect the three Lines A F, A H, and A G. I $ay, that the Angles by</I> <fig> <I>them contained F A H and G A H are equal. Produce A H untill it meeteth the Circumference in I, and from the Centers draw B H and C I, and thorow the $aid Centers let B C be drawn, which continued forth $hall meet with the Contact A, and with the Circum- ferences of the Circles in O and N. And becau$e the Angles I C N and H O B are equal, for as much as either of them is double to the Angle I A N, the Lines B H and C I $hall be Parallels: And becau$e B H drawn from the Center to the Contact is Perpendicular to F G; C I $hall al$o be Perpendicular to the $ame, and the Arch F I equal to the Arch I G, and, which followeth of con$equence, the Angle F A I to the Angle I A G: Which was to be demon$trated.</I></P> <foot>THEOR.</foot> <p n=>192</p> <head>THEOR. <I>XXI.</I> PROP. <I>XXXII.</I></head> <P>If two points be taken in the Horizon, and any Line $hould be inclined from one of them to- wards the other, out of which a Right-Line is drawn unto the Inclined Line, cutting off a part thereof equal to that which is included between the points of the Horizon, the De- $cent along this la$t drawn $hall be $ooner per- formed, than along any other Right Lines pro- duced from the $ame point unto the $aid Incli- ned Line. And along other Lines which are on each hand of this by equal Angles a De- $cent $hall be made in equal Times.</P> <P><I>In the Horizon let there be two points A and B, and from B incline the Right Line B C, in which from the Term B take B D equal to the $aid B A, and draw a Line from A to D. I $ay, that the De- $cent along A D is more $wiftly made, than along any other what$oever drawn from the point A unto the inclined Line B C. For out of the points A and D unto B A and</I> <fig> <I>B D draw the Perpendiculars A E and D E, inter$ecting one another in E: and fora$much as in the equicrural Triangle A B D the Angles B A D and B D A are equal, the remainders to the Right-Angles D A E and E D A $hall be equal. Therefore a Circle de$cribed about the Center E at the di$tance A E $hall al$o pa$$e by D; and the Lines B A and B D will touch it in the points A and D. And $ince A is the end of the Perpendicular A E, the De$cent along A D $hall be $ooner performed, than along any other produced from the $ame Term A unto the Line B C beyond the Circumference of the Circle: Which was fir$t to be proved.</I></P> <P><I>But if in the Perpendicular A E being prolonged any Center be taken as F, and at the di$tance F A the Circle A G C be de$cribed cutting the Tangent Line in the points G and C; drawing A G and A C they $hall make equal Angles with the middle Line A D by what hath been afore</I> <foot><I>demon-</I></foot> <p n=>193</p> <I>demon$trated, and the Motions thorow them $hall be performed in equal Times $eeing that they terminate in A unto the Circumference of the Circle A G O from the highe$t point of it A.</I></P> <head>PROBL. XII. PROP. <I>XXXIII.</I></head> <P>A Perpendicular and Plane inclined to it being given, who$e height is one and the $ame, as al- $o the highe$t term, to find a point in the Per- pendicular above the common term, out of which if a Moveable be demitted that $hall afterwards turn along the inclined Plane, the $aid Plane may be pa$t in the $ame Time in which the <I>P</I>erpendicular <I>ex quiete</I> would be pa$$ed.</P> <P><I>Let the Perpendicular and inclined Plane, who$e Altitude is the $ame, be A B and A C. It is required in the Perpendicular B A, continued out from the point A to find a Point out of which a Moveable de$cending may pa$$e the Space A C in the $ame Time in which it will pa$$e the $aid Perpendicular A B out of Re$t in A. Draw D C E at Right-Angles to A C, and let C D be cut equal to A B, and draw a Line from A to D: The Angle A D C $hall be greater than the Angles C A D: (for C A is greater than A B or C D:) Let the Angle D A E be equal to the Angle A D E; and to A E let E F an in- clined Plane be Perpen-</I> <fig> <I>dicular, and let both be- ing prolonged meet in F, and unto both A I and A G $uppo$e C F to be equal, and by G draw G H equidi$tant to the Horizon. I $ay, that H is the point which is $ought. For $uppo$ing the Time of the Fall along the Perpendicular A B to be A B, the Time along A C ex quiete in A $hall be the $ame A C. And becau$e in the Right- angled Triangle A E F, from the Right Angle E unto the Ba$e A F, E C is a Perpendicular, A E $hall be a Mean-Proportional betwixt F A and A C, and C E a Mean betwixt A C and C F, that is, betwixt C A and A I: and fora$much as the Time of A C out of A is A C, A E</I> <foot>Cc <I>$hall</I></foot> <p n=>194</p> <I>$hall be the Time of the whole A F, and E C the Time of A I: And be- cau$e in the Equicrural Triangle A E D the Side A E is equal to the Side E D, E D $hall be the Time along A F, and E C is the Time along A I: Therefore C D, that is A B $hall be the Time along A F</I> ex qui- ete <I>in A; which is the $ame as if we $aid, that A B is the Time along A G out of G, or out of H: Which was to be done.</I></P> <head>PROBL. <I>XIII. P</I>RO<I>P. XXXIV.</I></head> <P>An inclined <I>P</I>lane and Perpendicular who$e $ub- lime term is the $ame being given, to find a more $ublime point in the Perpendicular pro- longed out of which a Moveable falling, and being turned along the inclined <I>P</I>lane, may pa$$e them both in the $ame Time, as it doth the $ole inclined <I>P</I>lane <I>ex quiete</I> in its $uperi- our Term.</P> <P><I>Let the inclined Plane and Perpendicular be A B and A C, who$e Term A is the $ame. It is required in the Perpendicular prolonged from A to find a $ublime point, out of which the Moveable de$cen- ding, and being turned along the Plane A B, may pa$$e the a$$igned part of the Perpendicular and the Plane A B in the $ame Time, as it would the $ole Plane A B out of Re$t in A.</I></P> <fig> <P><I>Let the Ho- rizontal Line be B C, and let A N be cut equal to A C; and as A B is to B N, $o let A L be to L C: and unto A L let A I be equal, and unto A C and B I let C E be a third proportional, marked in the Perpendicular A C produced. I $ay, that C E is the Space acquired; $o that the Perpendicular being extended above A, and the part A X equal to C E being taken, a Moveable out of X will pa$$e both the</I> <foot><I>Spaces</I></foot> <p n=>195</p> <I>Spaces X A B in the $ame Time as it would the $ole Space A B out of A. Draw the Horizontal Line X R Parallel to B C, with which let B A being prolonged meet in R, and then A B being continued out unto D draw E D Parallel to C B, and upon A D de$cribe a Semicircle, and from B, and Perpendicular to D A, erect B F till it meet with the Cir- cumference. It is manife$t that F B is a Mean-proportional betwixt A B and B D, and that the Line drawn from F to A is a Mean-propor- tional betwixt D A and A B. Suppo$e B S equal to B I, and F H equal to F B: And becau$e, as A B is to B D, $o is A C to C E, and becau$e B F is a Mean-proportional betwixt A B and B D, and becau$e B I is a Mean-proportional betwixt A C and C E; therefore as B A is to A C, $o is F B to B S. And becau$e as B A is to A C, or A N, $o is F B to B S, therefore, by Conver$ion of the proportion, B F is to F S, as A B is to B N, that is, A L to L C; therefore the Rectangle under F B and C L, is equal to the Rectangle under A L, and S F: But this Rectangle A L, and S F, is the exce$$e of the Rectangle under A L and F B, or A I and B F, over and above the Triangle A I and B S, or A I B; and the Rectangle F B and L C is the exce$$e of the Rectangle A C and B F over and above the Rectangle A L and B F: But the Rectangle A C and B F is equal to the Rectangle A B I; (for as B A is to A C, $o is F B to B I:) The exce$$e, therefore, of the Rectangle A B I above the Rectan- gle A I and B F, or A I and F H, is equal to the exce$$e of the Rectangle A I and F H above the Rectangle A I B: Therefore twice the Rectan- gle A I and F H is equal to the two Rectangles A B I and A I B; that is twice A I B with the Square of B I. Let the Square A I be common to both, and twice the Rectangle A I B with the two Squares A I, and I B, (that is, the Square A B) $hall be equal to twice the Rectangle A I and F H, with the Square A I: Again, taking in commonly the Square B F; the two Squares A B and B F, that is the $ole Square A F $hall be equal to twice the Rectangle A I and F H, with the two Squares A I and F B, that is A I and F H: But the $ame Square A F is equal to twice the Rectangle A H F, with the two Squares A H and H F: Therefore twice the Rectangle A I and F H, with the Squares A I and F H, are equal to twice the Rectangle A H F, with the Squares A H and H F: And, the Common Square H F being taken away, twice the Rectangle A I and F H, with the Square A I, $hall be equal to twice the Rectangle A H F, with the Square A H. And becau$e that in all the Rectangles F H is the Common Side, the Line A H $hall be equal to A I: For if it $hould be greater or le$$er, then the Rectangles F H A and the Square H A would al$o be greater or le$$er than the Rectangles F H and I A, and the Square I A: Contrary to what hath been demon$trated.</I></P> <P><I>Now if we $uppo$e the Time of the De$cent along A B to be as A B, the Time along A C $hall be as A C, and I B the Mean-proportional be- twixt A C and C E $hall be the Time along C E, or along X A from Re$t in X: And becau$e betwixt D A and A B, or R B and B A the</I> <foot>Cc 2 <I>Mean-</I></foot> <p n=>196</p> <I>Mean-proportional is A F, and between A B and B D, that is, R A and A B the Mean is B F, to which F H is equal; Therefore,</I> exprædemon- $tratis, <I>the exce$$e A H $hall be the Time along A B</I> ex quiete <I>in R, or after the Fall out of X; $ince the Time along the $aid A B</I> ex quiete <I>in A, $hall be A B. Therefore the Time along X A is I B; and along A B after R A, or after X A, is A I: Therefore the Time along X A B $hall be as A B, namely the $elf-$ame with the Time along the $ole A B</I> ex qui- ete <I>in A. Which was the Propo$ition.</I></P> <head>PROBL. XIV. PROP. XXXV.</head> <P>An Inflected Line unto a given <I>P</I>erpendicular be- ing a$$igned, to take part in the Inflected Line, along which alone <I>ex quiete</I> a Motion may be made in the $ame Time, as it would be along the $ame together with the Perpendicular.</P> <P><I>Let the Perpendicular be A B, and a Line inflected to it B C. It is required in B C to take a part, along which alone out of Re$t a Motion may be made in the $ame Time as it would along the $ame together with the Perpendicular A B. Draw the Horizon A D, with which let the Inclined Line C B prolonged meet in E; and $uppo$e B F equal to B A, and on the Center E at the di$tance E F de$cribe the Circle F I G; and continue out F E unto the Circumference in G; and as G B is to B F, $o let B H be to H F; and let H I touch the Circle in I. Then out of B erect B K</I> <fig> <I>Perpendicular to F C, with which let the Line E I L meet in L; and la$t of all let fall L M Perpendicular to E L, meeting B C in M. I $ay, that along the Line B M from Rest in B a Motion may be made in the $ame Time, as it would be</I> ex quiete <I>in A along both A B and B M. Let E N be made equal to E L. And becau$e as G B is to B F, $o is B H to H F; there- fore, by Permutation as G B is to B H, $o will B F be to F H; and, by Divi$ion, G H $hall be to H B, as B H is to H F: Wherefore the Rect- angle G H F $hall be equal to the Square H B: But the $aid Rectangle is al$o equal to the Square H I: Therefore B H is equal to the $ame H I.</I> <foot><I>And</I></foot> <p n=>197</p> <I>And becau$e in the Quadrilateral Figure I L B H the Sides H B and H I are equal, and the Angles B and I Right Angles, the Side B L $hall likewi$e be equal to the Side L I: But E I is equal to E F: Therefore the whole Line L E, or N E is equal to the two Lines L B and E F: Let the Common Line E F be taken away, and the remainder F N $hall be equal to L B: And F B was $uppo$ed equal to B A: Therefore L B $hall be equal to the two Lines A B and B N. Again, if we $uppo$e the Time along A B to be the $aid A B, the Time along E B $hall be equal to E B; and the Time along the whole E M $hall be E N, namely, the Mean-proportional betwixt M E and E B: I berefore the Time of the De$cent of the remaining part B M after E B, or after A B, $hall be the $aid B N: But it hath been $uppo$ed, that the Time along A B is A B: Therefore the Time of the Fall along both A B and B M is A B N: And becau$e the Time along E B</I> ex quiete <I>in E is E B, the Time along B M</I> ex quiete <I>in B $hall be the Mean-proportional between B E and B M; and this is B L: The Time, therefore, along both A B M</I> ex quiete <I>in A is A B N: And the Time along B M only</I> ex quiete <I>in B is B L: But it was proved that B L is equal to the two A B and B N: Therefore the Propo$ition is manife$t.</I></P> <P><I>Otherwi$e with more expedition.</I></P> <P><I>Let B C be the Inclined Plane, and B A the Perpendicular. Continue out C B to E, and unto E C erect a Perpendicular at B, which being prolonged $uppo$e B H equal to the exce$$e of B E above B A; and to the Angle B H E let the Angle H E L be equal; and let E L continued out meet with B K in L; and from L erect the Perpendicular L M unto E L meeting B C in M. I $ay, that</I> <fig> <I>B M is the Space acquired in the Plane B C. For becau$e the Angle M L E is a Right- Angle, therefore B L $hall be a Mean-proportional betwixt M B and B E; and L E a Mean proportional betwixt M E and E B; to which E L let E N be cut equal: And the three Lines N E, E L, and L H $hall be equal; and H B $hall be the exce$$e of N E above B L: But the $aid H B is al$o the exce$$e of N E above N B and B A: Therefore the two Lines N B and B A are equal to B L. And if we $uppo$e E B to be the Time along E B, B L $hall be the Time along B M</I> ex quiete <I>in B; and B N $hall be the Time of the $ame B M after E B or after A B; and A B $hall be the Time along A B: Therefore the Times along A B M, namely, A B N, are equal to the Times along the $ole Line B M</I> ex quiete <I>in B: Which was intended.</I></P> <foot>LEMMA</foot> <p n=>198</p> <head>LEMMAI.</head> <P><I>Let D C be Perpendicular to the Diameter B A; and from the Term B continue forth B E D at plea$ure, and draw a Line from F to B. I $ay, that F B is a Mean-proportional be-</I> <fig> <I>twixt D B and B E. Draw a Line from E to F, and by B draw the Tangent B G; which $hall be Parallel to the former C D: Wherefore the Angle D B G $hall be equal to the Angle F D B, like as the $ame G B D is equal al$o to the Angle E F B in the al- tern Portion or Segment: Therefore the Triangles F B D and F E B are alike: And, as B D is to B F, $o is F B to B E.</I></P> <head>LEMMA II.</head> <P><I>Let the Line A C be greater than D F; and let A B have greater proportion to B C, than D E hath to E F. I $ay, that A B is greater than D E. For becau$e A B hath to B C</I> <fig> <I>greater proportion than D E hath to D F, therefore look what proportion A B hath to B C, the $ame $hall D E have to a Line le$- $er than E F; let it have it to E G: And becau$e A B to B C, is as D E, to E G, there- fore, by Compo$ition, and by converting the Proportion, as C A is to A B, $o is G D to D E: But C A is greater than G D: Therefore B A $hall be greater than D E.</I></P> <head>LEMMA III.</head> <fig> <P><I>Let A C I B be the Quadrant of a Circle: and to A C let B E be drawn from B Pa- rallel: And out of any Center taken in the $ame de$cribe the Circle B O E S, touching A B in B, and cutting the Circumference of the Quadrant in I; and draw a Line from C to B, and another from C to I continued out to S. I $ay, that the Line C I is alwaies le$$e than C O. Draw a Line from A to I; which toucheth the Circle B O E. And if D I be drawn it $hall be equal to D B: And becau$é D B toucheth the Quadrant, the $aid D I $hall likewi$e touch it; and $hall be Per-</I> <foot><I>pendicular</I></foot> <p n=>199</p> <I>pendicular to the Diameter A I: Wherefore al$o A I toucheth the Cir- cle B O E in I. And, becau$e the Angle A I C is greater than the An- gle A B C, as in$i$ting on a larger Periphery: Therefore the Angle S I N $hall be al$o greater than the $ame A B C: Therefore the Portion I E S is greater than the Portion B O; and the Line C S, nearer to the Center, greater than C B: Therefore al$o C O is greater than C I; for that S C is to C B, as O C is to C I.</I></P> <P><I>And the $ame al$o would happen to be greater, if (as in the other Figure) the Quadrant B I C were</I> <fig> <I>le$$er: For the Perpendicular D B will cut the Circle C I B: Wherefore D I al$o is equal to the $aid D B; and the Angle D I A $hall be Obtu$e, and therefore A I N will al$o cut B I N: And becau$e the Angle A B C is le$$e than the Angle A I C, which is equal to S I N; and this now is le$$e than that which would be made at the Contact in I by the Line S I: Therefore the Porti- on S E I is much greater than the Por- tion B O: Wherefore,</I> &c. <I>Which was to be demon$trated.</I></P> <head>THEOR. <I>XXII.</I> PROP. <I>XXXVI.</I></head> <P>If from the lowe$t point of a Circle erect unto the Horizon a Plane $hould be elevated $ub- tending a Circumference not greater than a Quadrant, from who$e Terms two other Planes are Inflected to any point of the Cir- cumference, the De$cent along both the Infle- cted Planes would be performed in a $horter Time than along the former elevated Plane alone, or than along but one of the other two, namely, along the lower.</P> <P><I>Let C B D be the Circumference not greater than a Quadrant of a Circle erect unto the Horizon on the lower point C, in which let C D be an elevated Plane; and let two Planes be inflected from the Terms D and C to any point in the Circumference taken at plea$ure, as B. I $ay, that the Time of the De$cent along both tho$e Planes D B C is $horter than the Time of the De$cent along the $ole Plane D C, or along the other only B C</I> ex quiete <I>in B. Let the Horizontal Line M D A</I> <foot><I>be</I></foot> <p n=>200</p> <I>be drawn by D, with which let C B prolonged meet in A; and let fall the Perpendiculars D N and M C to M D, and B N to B D; and about the Right-angled Triangle D B N de$cribe the Semicircle D F B N, cutting D C in F; and let D O be a Mean-proportional betwixt C D and D F; and A V a Mean-proportional betwixt C A and A B: And let P S be the time in which the whole D C, or B C, $hall be pa$$ed; (for it is manife$t that they $hall be both pa$t in the $ame Time;) And look what proportion C D hath to D O, the $ame $hall the Time S P have to the Time P R: the Time P R $hall be that in which a Movea- ble out of D will pa$$e D F; and R S that in which it $hall pa$$e the re- mainder F C. And becau$e P S is al$o the Time in which the Movea- ble out of B $hall pa$$e B C; if it be $uppo$ed that as B C is to C D, $o is S P to P T, P T $hall be the Time of the De$cent out of A to C: by rea$on D C is a Mean-proportional betwixt A C and C B, by what was before demon$trated: La$t of all, as C A is to A V, $o let T P be to</I> <fig> <I>P G: P G $hall be the Time, in which thé Moveable out of A de$cendeth to B. And becau$e of the Circle D F N the Diameter erect to the Horizon is D N, the Lines D F and D B $hall be pa$- $ed in equal Times. So that if it $hould be demon$tra- ted that the Moveable would $ooner pa$$e B C after the De$cent D B, than F C after the Lation D F; we $hould have our in- tent. But the Moveable will with the $ame Celerity of Time pa$$e B C coming out of D along D B, as if it came out of A along A B: for that in both the De$cents D B and A B it acquireth equal Moments of Velo- city: Therefore it $hall re$t to be demon$trated that the Time is $horter in which B C is pa$$ed after A B, than that in which F C is pa$t after D F. But it hath been demon$trated, that the Time in which B C is pa$$ed after A B is G T; and the Time of F C after D F is R S. It is to be proved therefore, that R S is greater than G T: Which is thus done. Becau$e as S P is to P R, $o is C D to D O, therefore, by Conver- $ion of proportion, and by Inver$ion, as R S is to S P, $o is O C to C D: and as S P is to P T, $o is D C to C A: And, becau$e as T P is to PG, $o is C A to A V: Therefore al$o, by Conver$ion of the proportion, as P T is to T G, $o is A C to C V: therefore, ex equali, as R S is to G T, $o is O C to C V. But O C is greater than C V, as $hall anon be de- mon$trated: Therefore the Time R S is greater than the Time G T: Which it was required to demon$trate. And becau$e C F is greater than C B, and F D le$$e than B A, therefore C D $hall have greater propor- tion to D F than C A to A B: And as C D is to D F, $o is the Square</I> <foot><I>C O to</I></foot> <p n=>201</p> <I>C O to the Square O F; fora$much as C D, D O, and O F are Propor- tionals: And as C A is to A B, $o is the Square C V to the Square V B: Therefore C O hath greater proportion to O F, than C V to V B: Therefore, by the foregoing Lemma, C O is greater than C V. It is manife$t moreover, that the Time along D C is to the Time along D B C, as D O C is to D O together with C V.</I></P> <head>SCHOLIUM.</head> <P>From the$e things that have been demon$trated may evidently be gathered, that the $wifte$t of all Motions betwixt Term and Term is not made along the $horte$t Line, that is by the Right, but along a portion of a Circle.</P> <P><I>For in the Quadrat B A E C, who$e Side B C is erect to the Hori- zon, let the Arch A C be divided into any number of equal parts, A D, D E, E F, F G, G C; and let Right-lines be drawn from C to the Points A, D, E, F, G, H; and al$o by Lines joyn A D, D E, E F, F G. and G C. It is manifest, that the Motion along the two Lines A D C is $ooner performed than along the</I> <fig> <I>$ole Line A C, or D C out of Re$t in D: But out of Re$t in A, D C is $ooner pa$t than the two A D C: But along the two D E C out of Re$t in A the De$cent is likewi$e $ooner made than along the $ole C D: Therefore the De$cent along the three Lines A D E C $hall be performed $ooner than along the two A D C. And in like manner the De$cent along A D E preceding, the Motion is more $peedily con- $ummated along the two EFC than along the $ole FC: Therfore along the four A D E F C the Motion is quicklier accompli$hed than along the three A D E C: And $o, in the la$t place, along the two F G C after the precedent De$cent along A D E F the Motion will be $ooner con$umma- ted than along the $ole F C: Therefore along the five A D E F G C the De$cent $hall be effected in a yet $horter Time than along the four A D E F C: Whereupon the nearer by in$cribed Poligons we approach the Circumference, the $ooner will the Motion be performed between the two a$$igned points A C.</I></P> <P><I>And that which is explained in a Quadrant, holdeth true likewi$e in a Circumference le$$e than the Quadrant: and the Ratiocination is the $ame.</I></P> <foot>Dd PROBL.</foot> <p n=>202</p> <head>PROBL.XV. PROP. XXXVII.</head> <P>A Perpendicular and Inclined Plane of the $ame Elevation being given, to find a part in the In- clined Plane that is equal to the Perpendicu- lar, and pa$$ed in the $ame Time as the $aid Perpendicular.</P> <P><I>LET A B be the Perpendicular, and A C the Inclined Plane. It is required in the Inclined to find a part equal to the Perpendicular A B, that after Re$t in A may be pa$$ed in a Time equal to the Time in which the Perpendicular is pa$$ed. Let A D be equal to A B, and cut the Remainder B C in two equal parts in I; and as A C is to</I> <fig> <I>C I, $o let C I be to another Line A E; to which let D G be equal: It is manife$t that E G is equal to A D and to A B. I $ay moreover, that this $ame E G is the $ame that is pa$$ed by the Moveable coming out of Re$t in A in a Time equal to the Time in which the Moveable fall eth along A B. For becau$e that as A C is to C I, $o is C I to A E, or I D to D G; Therefore by Conver$ion of the proportion, as C A is to A I, $o is D I to I G. And becau$e as the whole C A is to the whole A I, $o is the part taken away C I to the part I G; therefore the Remaining part I A $hall be to the Remainder A G, as the whole C A is to the whole A I: Therefore A I is a Mean-propor- tional betwixt C A and A G; and C I a Mean-proportional betwixt C A and A E: If therefore we $uppo$e the Time along A B to be as A B; A C $hall be the Time along A C, and C I or I D the Time along A E: And becau$e A I is a Mean-proportional betwixt C A and A G; and C A is the Time along the whole A C: Therefore A I $hall be the Time along. A G; and the Remainder I C that along the Remainder G C: But D I was the Time along A E: Therefore D I and I C are the Times along both the Spaces A E and C G: Therefore the Remainder D A $hall be the Time along E G, to wit, equal to the Time along A B. Which was to be done.</I></P> <head>COROLLARIE.</head> <P>Hence it is manife$t, that the Space required is an intermedial be- tween the upper and lower parts that are pa$t in equal Times.</P> <foot>PROBL.</foot> <p n=>203</p> <head><I>P</I>ROBL. XVI. <I>P</I>RO<I>P.</I> XXXVIII.</head> <P>Two Horizontal Planes cut by the Perpendicular being given, to find a $ublime point in the <I>P</I>er- pendicular, out of which Moveables falling and being reflected along the Horizontal <I>P</I>lanes may in Times equal to the Times of the De$cents along the $aid Horizontal <I>P</I>lanes, namely, along the upper and along the lower, pa$$e Spaces that have to each other any given proportion of the le$$er to the greater.</P> <P><I>LET the Planes C D and B E be inter$ected by the Perpendicular A C B, and let the given proportion of the le$$e to the greater be N to F G. It is required in the Perpendicular A B to find a point on high, out of which a Moveable falling, and reflected along C D may in a Time equal to the Time of its Fall, pa$$e a Space, that $hall have unto the Space pa$$ed by the other Moveable coming out of the $ame $ub- lime point in a Time equal to the Time of its Fall with a Reflex Motion along the Plane B E the $ame proportion as the given Line N batb to</I> <fig> <I>F G. Let G H be made equal to the $aid N; and as F H is to H G, $o let B C be to C L. I $ay, L is the $ublime point required. For taking C M double to C L, draw L M meeting the Plane B E in O; B O $hall be double to B L: And becau$e, as F H is to H G, $o is B C to C L; therefore, by Compo$ition and In- ver$ion, as H G, that is, N is to G F, $o is C L to L B, that is, C M to B O: But becau$e C M is double to L C; let the Space C M be that which by the Moveable coming from L after the Fall L C is pa$$ed along the Plane C D; and by the $ame rea$on B O is that which is pa$$ed after the Fall L B in a Time equal to the Time of the Fall along L B; fora$- much as B O is double to B L: Therefore the Propo$ition is manife$t.</I></P> <foot>Dd 2 SAGR.</foot> <p n=>204</p> <P>SAGR. Really me thinks that we may ju$tly grant our <I>Acade- mian</I> what he without arrogance a$$umed to him$elf in the begining of this his Treati$e of $hewing us a <I>New Science</I> about <I>a very old Subject.</I> And to $ee with what Facility and Per$picuity he deduceth from one $ole Principle the Demon$trations of $o many Propo$iti- ons, maketh me not a little to wonder how this bu$ine$s e$caped unhandled by <I>Archimedes, Apollonius, Euclid,</I> and $o many other <I>I</I>llu$trious Mathematicians and Phylo$ophers: e$pecially $ince there are found many great Volumns of <I>Motion.</I></P> <P>SALV. There is extant a $mall Fragment of <I>Euclid</I> touching <I>Motion,</I> but there are no marks to be $een therein of any $teps that he took towards the di$covery of the Proportion of <I>Acceleration,</I> and of its Varieties along different <I>I</I>nclinations. So that indeed one may $ay, that never till now was the door opened to a new Con- templation fraught with infinite and admirable Conclu$ions, which in times to come may bu$ie other Wits.</P> <P>SAGR. <I>I</I> verily believe, that as tho$e few Pa$$ions (<I>I</I> will $ay for example) of the Circle demon$trated by <I>Euclid</I> in the third of his <I>Elements</I> are an introduction to innumerable others more ab- $truce, $o tho$e produced and demon$trated in this $hort Tractate, when they $hall come to the hands of other Speculative Wits, $hall be a manuduction unto infinite others mote admirable: and it is to be believed that thus it will happen by rea$on of the Nobility of the Argument above all others Phy$ical.</P> <P>This daies Conference hath been very long and laborious; in which <I>I</I> have ta$ted more of the $imple Propo$itions than of their Demon$trations; many of which, <I>I</I> believe, will co$t me more than an hour a piece well to comprehend them: a task that <I>I</I> re$erve to my $elf to perform at lea$ure, you leaving the Book in my hands $o $oon as we $hall have heard this part that remains about the Moti- on of Projects: which $hall, if you $o plea$e, be to morrow.</P> <P>SALV. <I>I</I> $hall not fail to be with you.</P> <head><I>The End of the Third Dialogue.</I></head> <p n=>205</p> <head>GALILEUS, HIS DIALOGUES OF MOTION.</head> <head>The Fourth Dialogue.</head> <head><I>INTERLOCUTORS,</I></head> <head>SALVIATUS, SAGREDUS, and SIMPLICIUS.</head> <P>SALVIATUS.</P> <P><I>Simplicius</I> likewi$e cometh in the nick of time, therefore without interpo$ing any <I>Re$t</I> let us proceed to <I>Motion</I>; and $ee here the <I>Text</I> of our <I>Author.</I></P> <head>OF THE MOTION OF PROJECTS.</head> <P><I>What accidents belong to</I> Equable Motion, <I>as al$o to the</I> Na- turally Accelerate <I>along all whatever Inclinations of Planes, we have con$idered above. In this Contemplation which we are now entering upon, I will attempt to declare, and with $olid Demon$trations</I> <foot><I>to</I></foot> <p n=>206</p> <I>to e$tabli$h $ome of the principal Symptomes, and tho$e worthy of know- ledge, which befall a Moveable whil$t it is moved with a Motion com- pounded of a twofold Lation, to wit, of the Equable and Naturally- Accelerate: and this is that Motion, which we call the Motion of Pro- jects: who$e Generation I constitute to be in this manner.</I></P> <P><I>I fancy in my mind a certain Moveable projected or thrown along an Horizontal Plane, all impediment $ecluded: Now it is manife$t by what we have el$ewhere $poken at large, that that Motion will be Equa- ble and Perpetual along the $aid Plane, if the Plane be extended</I> in in- finitum<I>: but if we $uppo$e it terminate, and placed on high, the Move- able, which I conceive to be endued with Gravity, being come to the end of the Plane, proceeding forward, it addeth to the Equable and Indeli- ble fir$t Lation that propen$ion downwards which it receiveth from its Gravity, and from thence a certain Motion doth re$ult compounded of the Equable Horizontal, and of the De$cending naturally. Accellerate Lations: which I call</I> Projection. <I>Some of who$e Accidents we will de- mon$trate; the fir$t of which $hall be this.</I></P> <head>THEOR.I. PROP.I.</head> <P><I>A Project, when it is moved with a Motion compounded of the Horizontal Equable, and of the Naturally- Accelerate downwards, $hall de$cribe a Semipara- bolical Line in its Lation.</I></P> <P>SAGR. It is requi$ite, <I>Salviatus,</I> in favour of my $elf, and, as I believe, al$o of <I>Simplicius,</I> here to make a pau$e; for I am not $o far gone in Geometry as to have $tudied <I>Apol- lonius,</I> $ave only $o far as to know that he treateth of the$e Para- bola's, and of the other Conick Sections, without the knowledge of which, and of their Pa$$ions, I do not think that one can under- $tand the Demon$trations of other Propo$itions depending on them. And becau$e already in the very fir$t Propo$ition it is pro- po$ed by the Author to prove the Line de$cribed by the Project to be Parabolical, I imagine to my $elf, that being to treat of none but $uch Lines, it is ab$olutely nece$$ary to have a perfect know- ledge, if not of all the Pa$$ions of tho$e Figures that are demon- $trated by <I>Apollonius,</I> at lea$t of tho$e that are nece$$ary for the Sci- ence in hand.</P> <P>SALV. You undervalue your $elf very much, to make $trange of tho$e Notions, which but even now you admitted as very well under$tood: I told you heretofore, that in the Treati$e of Re$i- $tances we had need of the knowledge of certain Propo$itions of <foot><I>Apollonius,</I></foot> <p n=>207</p> <I>Apollonius,</I> at which you made no $eruple.</P> <P>SAGR. It may be either that I knew them by chance, or that I might for once gue$$e at, and take for granted $o much as $erved my turn in that Tractate: but here where I imagine that we are to hear all the Demon$trations that concern tho$e Lines, it is not con- venient, as we $ay, to $wallow things whole, lo$ing our time and pains.</P> <P>SIMP. But as to what concerns me, although <I>Sagredus</I> were, as I believe he is, well provided for his occa$ions, the very fir$t Terms already are new to me: for though our Philo$ophers have handled this Argument of the Motion of Projects, I do not remem- ber that they have confined them$elves to de$ine what the Lines are which they de$cribe, $ave only in general that they are alwaies Curved Lines, except it be in Projections Perpendicularly upwards. Therefore in ca$e that little Geometry that I have learnt from <I>Eu- clid</I> $ince the Time that we have had other Conferences, be not $uf- ficient to render me capable of the Notions requi$ite for the under- $tanding of the following Demon$trations, I mu$t content my $elf with bare Propo$itions believed, but not under$tood.</P> <P>SALV. But I will have you to know them by help of the Au- thor of this Book him$elf, who when he heretofore granted me a $ight of this his Work, becau$e I al$o at that time was not perfect in the Books of <I>Apollonius,</I> took the pains to demon$trate to me two mo$t principal Pa$$ions of the Parabola without any other Pre- cognition, of which two, and no more, we $hall $tand in need in the pre$ent Treati$e; which are both likewi$e proved by <I>Apollonius,</I> but after many others, which it would take up a long time to look over, and I am de$irous that we may much $horten the Journey, ta- king the fir$t immediately from the pure and $imple generation of the $aid Parabola, and from this al$o immediately $hall be deduced the Demon$tration of the $econd. Coming therefore to the fir$t;</P> <P>De$cribe the Right Cone, who$e Ba$e let be the Circle I B K C, and Vertex the point L, in which, cut by a Plane parallel to the <fig> Side L K, ari$eth the Section B A C called a Parabola; and let its Ba$e B C cut the Diameter I K of the Circle I B K C at Right-Angles; and let the Axis of the Parabola A D be Parallel to the $ide L K; and taking any point F in the Line B F A, draw the Right-Line F E parallel to B D. I $ay, that the Square of B D hath to the Square of F E the $ame proportion that the Axis D A hath to the part A E. Let a Plane parallel to the Circle I B K C <foot>be</foot> <p n=>208</p> be $uppo$ed to pa$$e by the Point E, which $hall make in the Cone a Circular Section, who$e Diameter is G E H. And becau$e upon the Diameter I K of the Circle I B K, B D is a Perpendicular, the Square of B D $hall be equal to the Rectangle made by the parts I D and D K: And likewi$e in the upper Circle which is under$tood to pa$$e by the points G F H, the Square of the Line F E is equal to the Rectangle of the parts G E H: Therefore the Square of B D hath the $ame proportion to the Square of F E, that the Rectangle I D K hath to the Rectangle G E H. And becau$e the Line E D is Parallel to H K, E H $hall be equal to D K, which al$o are Parallels: And therefore the Rectangle I D K $hall have the $ame proportion to the Rectangle G E H, as I D hath to G E; that is, that D A hath to A E: Therefore the Rectangle I D K to the Rectangle G E H, that is, the Square B D to the Square F E, hath the $ame proportion that the Axis D A hath to the part A E: Which was to be de- mon$trated.</P> <P>The other Propo$ition, likewi$e nece$$ary to the pre$ent Tract, we will thus make out. Let us de$cribe the Parabola, of which let the Axis C A be prolonged out unto D; and taking any point B, let the Line B C be $uppo$ed to be continued out by the $ame Parallel un- <fig> to the Ba$e of the $aid Parabola; and let D A be $uppo$ed equal to the part of the Axis C A. I $ay, that the Right-Line drawn by the points D and B, falleth not within the Parabola, but without, $o as that it only toucheth the $ame in the $aid point B: For, if it be po$$ible for it to fall within, it cutteth it above, or being pro- longed, it cutteth it below. And in that Line let any point G be taken, by which pa$$eth the Right Line F G E. And becau$e the Square F E is greater than the Square G E, the $aid Square F E $hall have greater proportion to the Square B C, than the $aid Square G E hath to the $aid B C. And becau$e, by the precedent, the Square F E is to the Square B C as E A is to A C; therefore E A hath greater proportion to A C, than the Square G E hath to the Square B C; that is, than the Square E D hath to the Square D C: (becau$e in the Triangle D G E as G E is to the Parallel B C, $o is E <I>D</I> to <I>D</I> C:) But the Line E A to A C, that is, to A <I>D</I> hath the $ame proportion that four Rectangles E A <I>D</I> hath to four Squares of A <I>D,</I> that is, to the Square C <I>D,</I> <foot>(which</foot> <p n=>209</p> (which is equal to four Squares of A D:) Therefore four Rectan- gles E A D $hall have greater proportion to the Square C D, than the Square E D hath to the Square D C: Therefore four Rectan- gles E A D $hall be greater than the Square E D: which is fal$e, for they are le$$e; becau$e the parts E A and A D of the Line E D are not equal: Therefore the Line D B toucheth the Parabola in B, and doth not cut it: Which was to be demon$trated.</P> <P>SIMP. You proceed in your Demon$trations too $ublimely, and $till, as far as I can perceive, $uppo$e that the Propo$itions of <I>Euclid</I> are as familiar and ready with me, as the fir$t Axioms them- $elves, which is not $o. And the impo$ing upon me, ju$t now, that four Rectangles E A <I>D</I> are le$s than the Square <I>D</I> E becau$e the parts E A and A <I>D</I> of the Line E <I>D</I> are not equal, doth not $atis$ie me, but leaveth me in doubt.</P> <P>SALV. The truth is, all the Mathematicians that are not vulgar $uppo$e that the Reader hath ready by heart the Elements of <I>Euclid</I>: And here to $upply your want, it $hall $u$fice to remember you of a Propo$ition in the $econd Book, in which it is demon$trated that when a Line is cut into equal parts, and into unequal, the Rectangle of the unequal parts is le$s than the Rectangle of the equal, (that is, than the Square of the half) by $o much as is the Square of the Line comprized between the Sections. Whence it is manife$t, that the Square of the whole, which continueth four Squares of the Half, is greater than four Rectangles of the unequal parts. Now it is nece$$ary that we bear in mind the$e two Propo$i- tions which have been demon$trated, taken from the Conick Ele- ments, for the better under$tanding the things that follow in the pre$ent Treati$e: for of the$e two, and no more, the Author makes u$e. Now we may rea$$ume the Text to $ee in what manner he doth demon$trate his fir$t Propo$ition, in which he intendeth to prove unto us, That the Line de$cribed by the Grave Moveable, when it de$cends with a Motion compounded of the Equable Horizontal, and of the Natural <I>D</I>e$cending is a Semiparabola.</P> <P><I>Suppo$e the Horizontal Line or Plane A B placed on high; upon</I> [or along] <I>which let the Moveable pa$$e with an Equable Motion out of A unto B: and the $upport of the Plane failing in B let there be derived upon the Moveable from its own Gravity a Motion naturally downwards according to the Perpendicular B N. Let the Line B E be $uppo$ed applyed unto the Plane A B right out, as if it were the Efflux or mea$ure of the Time, on which at plea$ure note any equal parts of Time, B C, C D, D E: And out of the points B C D E $uppo$e Per- pendicular Lines to be let fall equidi$tant or parallel to B N: In the fir$t of which take any part C I, who$e quadruple take in the following one D F, nonuple E H, and $o in the re$t that follow according to the propor-</I> <foot>Ee <I>tion</I></foot> <p n=>210</p> <I>tion of the Squares of C B, D B, E B, or, if you will, in the doubled proportion of the Lines. And if unto the Moveable moved beyond B towards C with the Equable Lation we $uppo$e the Perpendicular De$cent to be $uperadded according to the quantity C I, in the Time B C it $hall be found con$tituted in the Term I. And proceeding farther,</I> <fig> <I>in the Time D B, namely, in the double of B C, the Space of the De$cent down- wards $hall be quadruple to the fir$t Space C I: For it hath beendemon$trated in the fir$t Trastate, that the Spaces pa$$ed by GraveBo- dies with a Motion Natu- rally Accelerate are in du- plicate proportion of their Times. And it likewi$e followeth, that the Space E H pa$$ed in the Time B E, $hall be as G. So that it is manife$tly proved, that the Spaces E H, D F, C I, are to one another as the Squares of the Lines E B, D B, C B. Now from the points I, F, and H draw the Right Lines I O, F G, H L, Parallel to the $aid E B; and each of the Lines H L, F G, and I O $hall be equal to each of the other Lines E B, D B, and C B; as al$o each of tho$e B O, B G, and B L, $hall be equal to each of tho$e C I, D F, and E H: And the Square H L $hall be to the Square F G, as the Line L B to B G: And the Square F G $hall be to the Square I O, as G B to B O: Therefore the Points I, F, and H are in one and the $ame Parabolical Line. And in like manner it $hall be demon$trated, any equalparticles of Time of what$oever Mag- nitude being taken, that the place of the Moveable who$e Motion is compounded of the like Lations, is in the $ame Times to be found in the $ame Parabolick Line: Therefore the Propo$ition is manife$t.</I></P> <P>SALV. This Conclu$ion is gathered from the Conver$ion of the fir$t of tho$e two Propo$itions that went before, for the Parabola being, for example, de$cribed by the points B H, if either of the two F or I were not in the de$cribed Parabolick Line, it would be within, or without; and by con$equence the Line F G would be either greater or le$$er than that which $hould determine in the Pa- rabolick Line; Wherefore the Square of HL would have, not to the Square of F G, but to another greater or le$$er, the $ame pro- portion that the Line L B hath to BG, but it hath the $ame propor- tion to the Square of F G: Therefore the point F is in the Parabo- lick Line: And $o all the re$t, <I>&c.</I></P> <P>SAGR. It cannot be denied but that the Di$cour$e is new, in- genious and concludent, arguing <I>ex $uppo$itione,</I> that is, $uppo$ing that the Tran$ver$e Motion doth continue alwaies Equable, and <foot>that</foot> <p n=>211</p> that the Natural <I>Dcor$um</I> do likewi$e keep its tenour of continu- ally Accelerating according to a proportion double to the Times; and that tho$e Motions and their Velocities in mingling be not al- tered, di$turbed, and impeded, $o that finally the Line of the Pro- ject do not in the continuation of the Motion degenerate into an- other kind; a thing which $eemeth to me to be impo$$ible. For, in regard that the Axis of our Parabola, according to which we $up- po$e the Natural Motion of Graves to be made, being Perpendicu- lar to the Horizon, doth terminate in the Center of the Earth; and in regard that the Parabolical Line doth $ucce$$ively enlarge from its Axis, no Project would ever come to terminate in the Center, or if it $hould come thitherwards, as it $eemeth nece$$ary that it mu$t, the Line of the Project $hould de$cribe another mo$t different from that of the Parabola.</P> <P>SIMP. I add to the$e difficulties $everal others; one of which is that we $uppo$e, that the Horizontal Plane which hath neither accli- vity or declivity is a Right Line; as if that $uch a Line were in all its parts equidi$tant from the Center, which is not true: for depart- ing from its middle it goeth towards the extreams, alwaies more and more receding from the Center, and therefore alwaies a$cending: which of con$equence rendereth it Impo$$ible that its Motion $hould be perpetual, or that it $hould for any time continue Equa- ble, and nece$$itates it to grow continually more and more weak. Moreover, it is, in my Opinion, impo$$ible to avoid the Impedi- ment of the <I>Medium,</I> but that it will take away the Equability of the Tran$ver$e Motion, and the Rule of the Acceleration in falling Grave Bodies. By all which difficulties it is rendred very improba- ble that the things demon$trated with $uch incon$tant Suppo$i- tions $hould afterwards hold true in the practical Experiments.</P> <P>SALV. All the Objections and Difficulties alledged are $o well grounded, that I e$teem it impo$$ible to remove them; and for my own part I admit them all, as al$o I believe the Author him$elf would do. And I grant that the Conclu$ions thus demon- $trated in Ab$tract, do alter and prove fal$e, and that $o egregiou$- ly, in Concrete, that neither is the Tran$ver$e Motion Equable, nor is the Acceleration of the Natural in the proportion $uppo$e, nor is the Line of the Project Parabolical, <I>&c. B</I>ut yet on the contrary, I de$ire that you would not $cruple to grant to this our Author that which other famous Men have $uppo$ed, although fal$e. And the $ingle Authority of <I>Archimedes</I> may $atisfie every one: who in his Mechanicks, and in the fir$t Quadrature of the Parabola, taketh it as a true Principle, that the <I>B</I>eam of the <I>B</I>allance or Stilliard is a Right Line in all its points equidi$tant from the Common Center of Grave <I>B</I>odies, and that the Scale-ropes, to which the Weights are hanged, are parallel to one another. Which <foot>Ee 2 Liberty</foot> <p n=>212</p> Liberty of his hath been excu$ed by $ome, for that in our practices the In$truments we u$e, and the Di$tances which we take are $o $mall in compari$on of our great remotene$s from the Center of the Terre$trial Globe, that we may very well take a Minute of a degree of the great Circle as if it were a Right Line, and two Per- pendiculars that $hould hang at its extreams as if they were Paral- lels. For if we were in practical Operations to keep account of $uch like Minutes, we $hould begin to reprove the Architects, who with the Plumb Line $uppo$e that they rai$e very high Towers between Lines equidi$tant. And I here add, that we may $ay that <I>Archimedes,</I> and others $uppo$e in their Contemplations that they were con$tituted remote at an infinite di$tance from the Center; in which ca$e their A$$umptions were not fal$e: And that therefore they did conclude by Ab$olute Demon$tration. Again, if we will practice the demon$trated Conclu$ions in terminate Di$tances, by $uppo$ing an immen$e Di$tance, we ought to defalk from the truth demon$trated that which our Di$tance from the Center doth import, not being really infinite, but yet $uch as that it may be termed Immen$e in compari$on of the Artifices that we make u$e of, the greate$t of which will be the Ranges of Projects, and among$t the$e that only of Canon $hot; which though it be great, yet $hall it not exceed four of tho$e Miles of which we are remote from the Center well-nigh $o many thou$ands: and the$e coming to deter- mine in the Surface of the Terre$trial Globe may very well only in- $en$ibly alter that Parabolick Figure, which we grant would be extreamly transformed in going to determine in the Center. In the next place as to the perturbation proceeding from the Impedi- ment of the <I>Medium,</I> this is more con$iderable, and, by rea$on of its $o great multiplicity of Varieties, incapable of being brought under any certain Rules, and reduced to a Science: for if we $hould propo$e to con$ideration no more but the Impediment which the Air procureth to the Motions con$idered by us, this alone $hall be found to di$turb all, and that infinite waies, according as we infinite waies vary the Figures, Gravities, and Velocities of the Moveables. For as to the Velocity, according as this $hall be grea- ter, the greater $hall the oppo$ition be that the Air makes again$t them, which $hall yet more impede the $aid Moveable according as they are le$s Grave: $o that although the de$cending Grave Body ought to go Accelerating in a duplicate proportion to the Duration of its Motion, yet neverthele$s, albeit the Moveable were very Grave, in coming from very great heights, the Impediment of the Air $hall be $o great, as that it will take from it all power of far- ther encrea$ing its Velocity, and will reduce it to an Uniform and Equable Motion: And this Adequation $hall be $o much the $ooner obtained, and in $o much le$$er heights, by how much the Moveable <foot>$hall</foot> <p n=>213</p> $hall be le$s Grave. That Motion al$o which along the Horizontal Plane, all other Ob$tacles being removed, ought to be Equable and perpetual, $hall come to be altered, and in the end arre$ted by the Impediment of the Air: and here likewi$e $o much the $ooner, by how much the Moveable $hall be Lighter. Of which Accidents of Gravity, of Velocity, and al$o of Figure, as being varied $eve- ral waies, there can no fixed Science be given. And therefore that we may be able Scientifically to treat of this Matter it is requi$ite that we ab$tract from them; and, having found and demon$trated the Conclu$ions ab$tracted from the Impediments, that we make u$e of them in practice with tho$e Limitations that Experience $hall from time to time $hew us. And yet neverthele$s the benefit $hall not be $mall, becau$e $uch Matters, and their Figures $hall be made choice of as are le$s $ubject to the Impediments of the <I>Medium</I>; $uch are the very Grave, the Rotund: and the Spaces, and the Velocities for the mo$t part will not be $o great, but that their ex- orbitances may with ea$ie ^{*} Allowance be reduced to a certainty. <marg>* Tarra.</marg> Yea more, in Projects practicable by us, that are of Grave Matters, and of Round Figure, and al$o that are of Matters le$$e Grave, and of Cylindrical Figure, as Arrows, $hot from Slings or Bows, the variation of their Motion from the exact Parabolical Figure $hall be altogether in$en$ible. Nay, (and I will a$$ume to my $elf a little more freedom) that in ^{*} In$truments that are practicable by <marg>* Artifizii.</marg> us, their $malne$s rendreth the extern and accidental Impediments, of which that of the <I>Medium</I> is mo$t con$iderable, to be but of very $mall note, I am able by two experiments to make manife$t. I will con$ider the Motions made thorow the Air, for $uch are tho$e chiefly of which we $peak: again$t which the $aid Air in two man- ners exerci$eth its power. The one is by more impeding the Movea- bles le$s Grave, than tho$e very Grave. The other is in more oppo- $ing the greater than the le$s Velocity of the $ame Moveable. As to the fir$t; Experience $hewing us that two Balls of equal bigne$s, but in weight one ten or twelve times more Grave than the other, as, for example, one of Lead and another of Oak would be, de$cending from an height of 150, or 200 Yards, arrive to the Earth with Velocity very little different, it a$$ureth us that the Im- pediment or Retardment of the Air in both is very $mall: for if the <I>B</I>all of Lead departing from on high in the $ame Moment with that of Wood, were but little retarded, and this much, the Lead at its coming to the ground $hould leave the Wood a very con$idera- ble Space behind, $ince it is ten times more Grave; which never- thele$s doth not happen: nay, its Anticipation $hall not be $o much as the hundredth part of the whole height. And between a <I>B</I>all of Lead, and another of Stone which weighs a third part, or half $o much as it, the difference of the Times of their coming to <foot>the</foot> <p n=>214</p> the ground would be hardly ob$ervable. Now becau$e the <I>Impe- tus</I> that a <I>B</I>all of Lead acquireth in falling from an height of 200 Yards (which is $o much that continuing it in an Equable Moti- on it would in a like Time run 400 Yards) is very con$iderable in compari$on of the Velocity that we confer with <I>B</I>ows or other Ma- chines, upon our Projects (excepting the <I>Impetus's</I> that depend on the Fire) we may without any notable Errour conclude and account the Propo$itions to be ab$olutely true that are demon$tra- ted without any regard had to the alteration of the <I>Medium.</I> In the next place as touching the other part, that is to $hew, that the Impediment that the $aid Moveable receiveth from the Air whil$t it moveth with great Velocity is not much greater than that which oppo$eth it in moving $lowly, the en$uing Experiment giveth us full a$$urance of it. Su$pend by two threads both of the $ame length, <I>v. gr.</I> four or five Yards, two equal <I>B</I>alls of Lead: and having fa$tned the $aid threads on high, let both the <I>B</I>alls be re- moved from the $tate of Perpendicularity; but let the one be re- moved 80. or more degrees, and the other not above 4 or 5: $o that one of them being left at liberty de$cendeth, and pa$$ing be- yond the Perpendicular, de$cribeth very great Arches of 160, 150, 140, <I>&c.</I> degrees, dimini$hing them by little and little: but the other $winging freely pa$$eth little Arches of 10, 8, 6, <I>&c.</I> this al$o dimini$hing them in like manner by little and little. Here I $ay, in the fir$t place, that the fir$t <I>B</I>all $hall pa$s its 180, 160, <I>&c.</I> degrees in as much Time as the other doth its 10, 8, <I>&c.</I> From whence it is manife$t, that the Velocity of the fir$t <I>B</I>all $hall be 16 and 18 times greater than the Velocity of the $econd: $o that in ca$e the greater Velocity were to be more impeded by the Air than <marg>Or $ewer.</marg> the le$$er, the Vibrations $hould be more ^{*} rare in the greate$t Arches of 180, or 160 degrees, <I>&c.</I> than in the lea$t of 10, 8, 4, and al$o of 2, and of 1; but this is contradicted by Experience: for if two A$$i$tants $hall $et them$elves to count the Vibrations, one the greate$t, the other the lea$t, they will find that they $hall number not only tens, but hundreds al$o, without di$agreeing one $ingle Vibration, yea, or one $ole point. And this ob$ervati- on joyntly a$$ureth us of the two Propo$itions, namely, that the greate$t and lea$t Vibrations are all made one after another under equal Times, and that the Impediment and Retardment of the Air operates no more in the $wifte$t Motion, than in the $lowe$t: contrary to that which before it $eemed that we our $elves al$o would have judged for company.</P> <P>SAGR. Rather, becau$e it cannot be denied but that the Air impedeth both tho$e and the$e, $ince they both continually grow more languid, and at la$t cea$e, it is requi$ite to $ay that tho$e Re- tardations are made with the $ame proportion in the one and in the <foot>other</foot> <p n=>215</p> other Operation. And then, the being to make greater Re$i$tance at one time than at another, from what other doth it proceed, but only from its being a$$ailed at one time with a greater <I>Impetus</I> and Velocity, and at another time with le$$er? And if this be $o then the $ame quantity of the Velocity of the Moveable is at once the Cau$e and the Mealure of the quantity of the Re$i$tance. Therefore all Motions, whether they be $low or $wift, are retarded and impe- ded in the $ame proportion: a Notion in my judgment not con- temptible.</P> <P>SALV. We may al$o in this $econd ca$e conclude, That the Fallacies in the Conclu$ions, which are demon$trated, ab$tracting from the extern Accidents, are in our In$truments of very $mall con$ideration, in re$pect of the Motions of great Velocities of which for the mo$t part we $peak, and of the Di$tances which are but very $mall in relation to the Semidiameter and great Circles of the Terre$trial Globe.</P> <P>SIMP. I would gladly hear the rea$on why you $eque$trate the Projects from the <I>Impetus</I> of the Fire, that is, as I conceive from the force of the Powder, from the other Projects made by Slings, Bows, or Cro$s-bows, touching their not being in the $ame manner $ubject to the Acceleration and Impediment of the Air.</P> <P>SALV. I am induced thereto by the exce$$ive, and, as I may $ay, Supernatural Fury or Impetuou$ne$s with which tho$e Projects are driven out: For indeed I think that the Velocity with which a <I>B</I>ul- let is $hot out of a Musket or Piece of Ordinance may without any Hyperbole be called Supernatural. For one of tho$e <I>B</I>ullets de- $cending naturally thorow the Air from $ome immen$e height, its Velocity, by rea$on of the Re$i$tance of the Air will not go in- crea$ing perpetually: but that which in Cadent <I>B</I>odies of $mall Gravity is $een to happen in no very great ^{*} Space, I mean their <marg>* Or Way.</marg> being reduced in the end to an Equable Motion, $hall al$o happen after a De$cent of thou$ands of yards, in a <I>B</I>all of Iron or Lead: and this determinate and ultimate Velocity may be $aid to be the greate$t that $uch a <I>B</I>ody can obtain or acquire thorow the Air: which Velocity I account to be much le$$er than that which cometh to be impre$$ed on the $ame <I>B</I>all by the fired Powder. And of this a very appo$ite Experiment may adverti$e us. At an height of an hundred or more yards let off a Musket charged with a Leaden <I>B</I>ullet perpendicularly downwards upon a Pavement of Stone; and with the $ame Musket $hoot again$t $uch another Stone at the Di- $tance of a yard or two, and then $ee which of the two <I>B</I>ullets is more flatted: for if that coming from on high be le$s ^{*} dented than <marg>* Or battered.</marg> the other, it $hall be a $ign that the Air hath impeded it, and dimi- ni$hed the Velocity conferred upon it by the Fire in the beginning of the Motion: and that, con$equently, $o great a Velocity the Air <foot>would</foot> <p n=>216</p> would not $uffer it to gain coming from never $o great an height: for in ca$e the Velocity impre$$ed upon it by the Fire $hould not exceed that which it might acquire of its $elf de$cending naturally, the battery downwards ought rather to be more valid than le$s. I have not made $uch an Experiment, but incline to think that a Musket or Cannon Bullet falling from never $o great an height, will not make that percu$$ion which it maketh in a Wall at a Di- $tance of a few yards, that is of $o few that the $hort perforation, or, if you will, Sci$$ure to be made in the Air $ufficeth not to ob- viate the exce$s of the $upernatural impetuo$ity impre$$ed on it by the Fire. This exce$$ive <I>Impetus</I> of $uch like forced $hots may cau$e $ome deformity in the Line of the Projection; making the beginning of the Parabola le$s inclined or curved than the end. <I>B</I>ut this can be but of little or no prejudice to our Author in practical Operations: among$t the which the principal is the com- po$ition of a Table for the Ranges, or Flights, which containeth the di$tances of the Falls of <I>B</I>alls $hot according to all Elevations. And becau$e the$e kinds of Projections are made with Mortar- Pieces, and with no great charge; in the$e the <I>Impetus</I> not being $upernatural, the Ranges de$cribe their Lines very exactly.</P> <P><I>B</I>ut for the pre$ent let us proceed forwards in the Treati$e, where the Author de$ireth to lead us to the Contemplation and Inve$tigation of the <I>Impetus</I> of the Moveable whil$t it moveth with a Motion compounded of two. And fir$t of that compoun- ded of two Equable Motions; the one Horizontal, and the other Perpendicular.</P> <head>THEOR. II. PROP. II.</head> <P>If any Moveable be moved with a twofold Equa- ble Motion, that is, Horizontal and Perpen- dicular, the <I>Impetus</I> or Moment of the Lation compounded of both the Motions $hall be <I>po- tentia</I> equal to both the Moments of the fir$t Motions.</P> <P><I>For let any Moveable be moved Equably with a double Lation, and let the Mutations of the Perpendicular an$wer to the Space A B, and let B C an$wer to the Horizontal Lation pa$$ed in the $ame Time. Fora$much therefore as the Spa-</I> <fig> <I>ces A B, and B C are pa$$ed by the Equable Mo- tion in the $ame Time, their Moments $hall be to cach other as the $aid A B and B C. But the Moveable which is moved according to the$e two Mutations $hall de-</I> <foot><I>$cribe</I></foot> <p n=>217</p> <I>$cribe the Diagonal A C, and its Moment $hall be as A C. But A C is</I> potentia <I>equal to the $aid A B and B C: therefore the Moment com- pounded of both the Moments A B and B C, is</I> potentia <I>equal to them both taken together: Which was to be demon$trated.</I></P> <P>SIMP. It is nece$$ary that you ea$e me of one Scruple that cometh into my mind, it $eemeth to me that this which is now con- cluded oppugneth another Propo$ition of the former Tractate: in which it is affirmed, That the <I>Impetus</I> of the Moveable coming from A into B is equal to that coming from A into C; and now it is concluded, that the <I>Impetus</I> in C is greater than that in B.</P> <P>SALV. The Propo$itions, <I>Simplicius,</I> are both true, but very different from one another. Here the Author $peaks of one $ole Moveable moved with one $ole Motion, but compounded of two, both Equable; and there he $peaks of two Moveables moved with Motions Naturally Accelerated, one along the Perpendicular A B, and the other along the Inclined Plane A C: and moreover, the Times there are not $uppo$ed equal, but the Time along the Inclined Plane A C is greater than the Time along the Perpen- dicular A B: but in the Motion $poken of at pre$ent, the Motions along A B, B C and A C are under$tood to be Equable, and made in the $ame Time.</P> <P>SIMP. Excu$e me, and go on, for I am $atisfied.</P> <P>SALV. The Author proceeds to $hew us that which hapneth concerning the <I>Impetus</I> of a Moveable moved in like manner with one Motion compounded of two, that is to $ay, the one Horizon- tal and Equable, and the other Perpendicular but Naturally-Acce- lerate, of which in fine the Motion of the Project is compounded, and by which the Parabolick Line is de$cribed; in each point of which the Author endeavours to determine what the <I>Impetus</I> of the Project is; for under$tanding of which he $heweth us the manner, or, if you will, Method of regulating and mea$uring that $ame <I>Im- petus</I> upon the $aid Line, along which the Motion of the Grave Moveable de$cending with a Natural-Accelerate Motion departing from Re$t is made, $aying:</P> <head>THEOR. III. PROP. III.</head> <P><I>Let a Motion be made along the Line A B out of Re$t in A, and take in $ome point C; and $uppo$e the $aid A C to be the Time or Mea$ure of the Time of the $aid Fall along the Space A C, as al$o the Mea$ure of the</I> Impetus <I>or Moment in the Point C acquired by the De$cent along A C. Now let there be taken in the $aid Line A B any other Point, as $uppo$e B, in which we are to determine of the</I> Impetus <I>acquired by the Moveable along the Fall A B, in proportion to</I> <foot>Ff <I>the</I></foot> <p n=>218</p> <I>the</I> Impetus, <I>which it obtaineth in C, who$e Mea$ure is $uppo$ed to be A C, Let A S be a Mean-proportional betwixt B A and A C. We will demon$trate that the</I> Impetus <I>in B is to the</I> Impetus <I>in C, as S A is to A C. Let the Horizontal Line C D be double to the $aid A C; and B E double to B A. It appeareth by what hath been demon$trated, That the Cadent along A C being turned along the Horizon C D, and according to the</I> Impetus <I>acquired in C, with an Equable Motion, $hall pa$s the Space C D in a Time equal to that in which the $aid A C is pa$$ed</I> <fig> <I>with an Accelerate Motion; and likewi$e that B E is pa$$ed in the $ame time as A B: But the Time of the De$cent along A B is A S: There- fore the Horizontal Line B E is pa$$ed in A S. As the Time S A is to the Time A C, $o let E B be to B L. And becau$e the Motion by B E is Equable, the Space B L $hall be pa$$ed in the Time A C ac- cording to the Moment of Celerity in B: But in the $ame Time A C the Space C D is pa$$ed, according to the Moment of Velocity in C: the Moments of Velocity therefore are to one another as the Spaces which according to the $ame Moments are pa$$ed in the $ame Time: Therefore the Moment of Velocity in C is to the Moment of Celerity in B, as D C is to B L. And becau$e as D C is to B E, $o are their halfs, to wit, C A to A B: but as E B is to B L, $o is B A to A S: Therefore,</I> exæquali, <I>as D C is to B L, $o is C A to A S: that is, as the Moment of Velocity in C is to the Moment of Velocity in B, $o is C A to A S; that is, the Time along C A to the Time along A B. I he manner of Mea$u- ring the</I> Impetus, <I>or the Moment of Velocity upon a Line along which it makes a Motion of De$cent is therefore manife$t; which</I> Impetus <I>is indeed $uppo$ed to encrea$e according to the Proportion of the Time.</I></P> <P><I>But this, before we proceed any farther, is to be premoni$hed, that in regard we are to $peak for the future of the Motion compounded of the Equable Horizontal, and of the Naturally Accelerate downwards, (for from this Mixtion re$ults, and by it is de$igned the Line of the Project, that is a Parabola;) it is nece$$ary that we define $ome common mea$ure according to which we may mea$ure the Velocity,</I> Impetus, <I>or Moment of both the Motions. And $eeing that of the Equable Motion the de- grees of Velocity are innumerable, of which you may not take any promi$cuou$ly, but one certain one which may be be compared and con- joyned with the Degree of Velocity naturally Accelerate. I can think of no more ea$ie way for the electing and determining of that, than by a$- $uming another of the $ame kind. And that I may the better expre$s my meaning; Let A C be Perpendicular to the Horizon C B; and A C</I> <foot><I>to</I></foot> <p n=>219</p> <I>to be the Altitude, and C B the Amplitude of the Semiparabola A B; which is de$cribed by the Compo$ition of two Lations; of which one is that of the Moveable de$cending along A C with a Motion Naturally Acceler ate</I> ex quiete <I>in A; the other is the Equable Tran$ver$al Moti- on according to the Horizontal Line A D. The</I> Impetus <I>acquired in C along the De$cent A C is determined by the quantity of the $aid height A C; for the</I> Impetus <I>of a Moveable</I> <fig> <I>falling from the $ame height is alwaies one and the $ame: but in the Horizontal Line one may a$$ign not one, but innume- rable Degrees of Velocities of Equable Motions: out of which multitude that I may $ingle out, and as it were point with the finger to that which I make choice of, I extend or prolong the Altitude C A</I> in $ublimi, <I>in which, as was done before, I will pitch upon A E; from which if I conceive in my mind a Moveable to fall</I> ex quiete <I>in E, it appeareth that its</I> Im- petus <I>acquired in the Time A, is one with which I conceive the $ame Moveable being turned along A D to be moved; and its degree of Vclocity to be that, which in the Time of the De$cent along E A pa$$eth a Space in the Horizon double to the $aid E A. This Præmonition I judged nece$$ary.</I></P> <P><I>It is moreover to be advertized that the Amplitude of the Semi- parabola A B $hall be called by me the Horizontal Line</I> [or Plane] <I>C B.</I></P> <P><I>The Altitude, to with A C, the Axis of the $aid Parabola.</I></P> <P><I>And the Line E A, by who$e De$cent the Horizontal</I> Impetus <I>is de- termined, I call the Sublimity, or height.</I></P> <P><I>The$e things being declared and defined, I proceed to Demon$tra- tion.</I></P> <P>SAGR. Stay, I pray you, for here me thinks it is convenient to adorn this Opinion of our Author with the conformity of it to the Conceit of <I>Plato</I> about the determining the different Veloci- ties of the Equable Motions of the Revolutions of the Cœle$tial Bodies; who, having perhaps had a conjecture that no Moveable could pa$$e from Re$t into any determinate degree of Velocity in which it ought afterwards to be perpetuated, unle$s by pa$$ing thorow all the other le$$er degrees of Velocity, or, if you will, greater degrees of Tardity, which interpo$e between the a$$igned degree, and the highe$t degree of Tardity, that is of Re$t, $aid that God after he had created the Moveable Cœle$tial <I>B</I>odies that he might a$$ign them tho$e Velocities wherewith they were afterwards <foot>Ff 2 to</foot> <p n=>220</p> to be perpetually moved with an Equable Circular Motion, made them, they departing from Re$t, to move along determinate Spaces with that Natural Motion in a Right Line, according to which we $en$ibly $ee our Moveables to move from the $tate of Re$t $ucce$- $ively Accelerating. And he addeth, that having made them to acquire that degree in which it plea$ed him that they $hould after- wards be perpetually con$erved, he converted their Right or direct Motion into Circular; which only is apt to con$erve it $elf Equa- ble, alwaies revolving without receding from, or approaching to any prefixed term by them de$ired. The Conceit is truly worthy of <I>Plato</I>; and is the more to be e$teemed in that the grounds there- of pa$$ed over in $ilence by him, and di$covered by our Author by taking off the Mask or Poetick Repre$entation, do $hew it to be in its native a$pect a true Hi$tory. And I think it very credible that we having by the Doctrine of A$tronomy $ufficiently competent Knowledge of the Magnitudes of the Orbes of the Planets, and of their Di$tances from the Center about which they move, as al$o of their Velocities, our Author (to whom <I>Plato's</I> Conjecture was not unknown) may $ometime for his curio$ity have had $ome thought of attempting to inve$tigate whether one might a$$ign a determinate Sublimity from which the <I>B</I>odies of the Planets depar- ting, as from a $tate of Re$t, and moved for certain Spaces with a Right and Naturally Accelerate Motion, afterwards converting the Acquired Velocity into Equable Motions, they might be found to corre$pond with the greatne$s of their Orbes, and with the Times of their Revolutions.</P> <P><I>S</I>ALV. I think I do remember that he hath heretofore told me, that he had once made the Computation, and al$o that he found it exactly to an$wer the Ob$ervations; but that he had no mind to $peak of them, doubting le$t the two many Novelties by him di$- covered, which had provoked the di$plea$ure of many again$t him, might blow up new $parks. <I>B</I>ut if any one $hall have the like de- $ire he may of him$elf by the Doctrine of the pre$ent Tract give him$elf content. <I>B</I>ut let us pur$ue our bu$ine$s, which is to $hew;</P> <head>PROBL. I. PROP. IV.</head> <P>How in a Parabola given, de$cribed by the Pro- ject, the <I>Impetus</I> of each $everal point may be determined.</P> <P><I>Let the Semiparabola be B E C, who$e Amplitude is C D and Al- titude D B, with which continued out on high the Tangent of the Parabola C A meeteth in A; and along the</I> Vertex <I>B let B I be</I> <foot><I>an</I></foot> <p n=>221</p> <I>an Horizontal Line, and parallel to C D. And if the Amplitude C D be equal to the whole Altitude D A, B I $hall be equal to B A and B D. And if the Time of the Fall along A B, and the Moment of Velocity acquired in B along the De$cent A B</I> ex quiete <I>in A be $uppo$ed to be mea$ured by the $aid A B, then D C (that is twice B I) $hall be the Space which $hall be pa$$ed by the</I> Impetus A <I>B turned along the Hori- zontal Line in the $ame Time: But in the $ame Time falling along B D out of Re$t in B, it $hall pa$s the Altitude B D: Therefore the Movea- ble falling out of Re$t in A along A B, being converted with the</I> Impetus <I>A B</I> <fig> <I>along the Horizontal Parallel $hall pa$s a Space equal to D C. And the Fall along B D $upervening, it pa$$eth the Altitude B D, and de$cribes the Parabola B C; who$e</I> Impetus <I>in the Term C is compounded of the Equable Tran$ver$al who$e Moment is as A B, and of another Moment acquired in the Fall B D in the Term D or C; which Moments are Equal. If therefore we $uppo$e A B to be the Mea$ure of one of them, as $uppo$e of the Equa- ble Tran$ver$al; and B I, which is equal to B D, to be the Mea$ure of the</I> Impetus <I>acquired in D or C; then the Subten$e I A $hall be the quantity of the Moment compound of them both: Therefore it $hall be the quantity or Mea$ure of the whole Moment which the Project de$cend- ing along the Parabola B C $hall acquire of</I> Impetus <I>in C. This pre- mi$ed, take in the Parabola any point E, in which we are to determine of the</I> Impetus <I>of the Project. Draw the Horizontal Parallel E F, and let B G be a Mean-proportional between B D and B F. And fora$- much as A B or B D is $uppo$ed to be the Mea$ure of the Time, and of the Moment of the Velocity in the Fall B D</I> ex quiete <I>in B: B G $hall be the Time, or the Mea$ure of the Time, and of the</I> Impetus <I>in F, coming out of B. If therefore B O be $uppo$ed equal to B G, the Diagonal drawn from A to O $hall be the quantity of the</I> Impetus <I>in E; for A B hath been $uppo$ed the determinator of the Time, and of the</I> Impe- tus <I>in B, which turned along the Horizontal Parallel doth alwaies continue the $ame: And B O determineth the</I> Impetus <I>in F or in E along the De$cent</I> ex quiete <I>in B in the Altitude B F: But the$e two A B and B O are</I> potentia <I>equal to the Power A O. Therefore that is manife$t which was $ought.</I></P> <P>SAGR. The Contemplation of the Compo$ition of the$e diffe- rent <I>Impetus's,</I> and of the quantity of that <I>Impetus</I> which re$ults from this mixture, is $o new to me, that it leaveth my mind in no $mall confu$ion. I do not $peak of the mixtion of two Motions <foot>Equable,</foot> <p n=>222</p> Equable, though unequal to one another, made the one along the Horizontal Line, and the other along the Perpendicular, for I very well comprehend that there is made a Motion of the$e two <I>poten- tia</I> equal to both the Compounding Motions, but my confu$ion ari$eth upon the mixing of the Equable-Horizontal and Perpendi- cular-Naturally-Accelerate Motion. Therefore I could wi$h we might toge ther a little better con$ider this bu$ine$s.</P> <P>SIMP. And I $tand the more in need thereof in that I am not yet $o well $atisfied in Mind as I $hould be, in the Propo$itions that are the fir$t foundations of the others that follow upon them. I will add, that al$o in the Mixtion of the two Motions Equable Horizontal, and Perpendicular, I would better under$tand that <I>Potentia</I> of their Compound. Now, <I>Salviatus,</I> you $ee what we want and de$ire.</P> <P>SALV. Your de$ire is very rea$onable: and I will e$$ay whe- ther my having had a longer time to think thereon may facilitate your $atisfaction. But you mu$t bear with and excu$e me if in di$- cour$ing I $hall repeat a great part of the things hitherto delivered by our Author.</P> <P>It is not po$$ible for us to $peak po$itively touching Motions and their Velocities or <I>Impetus's,</I> be they Equable, or be they Naturally Accelerate, unle$s we fir$t agree upon the Mea$ure that we are to u$e in the commen$uration of tho$e Velocities, as al$o of the Time. As to the Mea$ure of the Time, we have already that which is commonly received by all of Hours, Prime-Minutes, and Se- conds, <I>&c.</I> and as for the mea$uring of Time we have that com- mon Mea$ure received by all, $o it is requi$ite to a$$ign another Mea$ure for the Velocities that is commonly under$tood and re- ceived by every one; that is, which every where is the $ame. The Author, as hath been declared, adjudged the Velocity of Naturally de$cending Grave-Bodies to be fit for this purpo$e; the encrea$ing Velocities of which are the $ame in all parts of the World. So that that $ame degree of Velocity which (for example) a Ball of Lead of a pound acquireth in having, departing from Re$t, de$cended Per- pendicularly as much as the height of a Pike, is alwaies, and in all places the $ame, and therefore mo$t commodious for explicating the quantity of the <I>Impetus</I> that is derived from the Natural De- $cent. Now it remains to find a way to determine likewi$e the Quantity of the <I>Impetus</I> in an Equable Motion in $uch a manner, that all tho$e which di$cour$e about it may form the $ame conceit of its greatne$s and Velocity; $o that one may not imagine it more $wift, and another le$s; whereupon afterwards in conjoyning and mingling this Equable Motion imagined by them with the e$tabli- $hed Accelerate Motion $everal men may form $everal Conceits of $everal greatne$$es of <I>Impetus's.</I> To determine and repre$ent this <foot><I>Impetus,</I></foot> <p n=>223</p> <I>Impetus,</I> and particular Velocity our Author hath not found any way more commodious, than the making u$e of the <I>Impetus</I> which the Moveable from time to time acquires in the Naturally-Accele- rate Motion, any acquired Moment of which being reduced into an Equable Motion retaineth its Velocity preci$ely limited, and $uch, that in $uch another Time as that wherein it did De$cend, it pa$$eth double the Space of the Height from whence it fell. But becau$e this is the principal point in the bu$ine$s that we are upon, it is good to make it to be perfectly under$tood by $ome particular Example. Rea$$uming therefore the Velocity and <I>Impetus</I> acqui- red by the Cadent Moveable, as we $aid before, from the height of a Pike, of which Velocity we will make u$e for a Mea$ure of other Velocities and <I>Impetu$$es</I> upon other occa$ions, and $uppo- $ing, for example, that the Time of that Fall be four $econd Mi- nutes of an hour, to find by this $ame Mea$ure how great the <I>Im- petus</I> of the Moveable would be falling from any other height greater, or le$$er, we ought not from the proportion that this other height hath to the height of a Pike to argue and conclude the quan- tity of the <I>Impetus</I> acquired in this $econd height, thinking, for example, that the Moveable falling from quadruple the height hath acquired quadruple Velocity, for that it is fal$e: for that the Velocity of the Naturally-Accelerate Motion doth not increa$e or decrea$e according to the proportion of the Spaces, but according to that of the Times, than which that of the Spaces is greater in a duplicate proportion, as was heretofore demon$trated. Therefore when in a Right Line we have a$$igned a part for the Mea$ure of the Velocity, and al$o of the Time, and of the Space in that Time pa$$ed (for that for brevity $ake all the$e three Magnitudes are often repre$ented by one $ole Line,) to find the quantity of the Time, and the degree of Velocity that the $ame Moveable would have acquired in another Di$tance we $hall obtain the $ame, not immediataly by this $econd Di$tance, but by the Line which $hall be a Mean-proportional betwixt the two Di$tances. But I will better declare my $elf by an Example. In the Line A C Perpendi- cular to the Horizon let the part A B be under$tood to be a Space pa$$ed by a Moveable naturally de$cending <fig> with an Accelerate Motion: the Time of which pa$- $age, in regard I may repre$ent it by any Line, I will, for brevity, imagine it to be as much as the $ame Line A B and likewi$e for a Mea$ure of the <I>Impetus</I> and Velocity acquired by that Motion, I again take the $ame Line A B; $o that of all the Spaces that are in the progre$s of the Di$cour$e to be con$idered the part A B may be the Mea$ure. Having all our plea$ure e$tabli$hed under one $ole Magnitude A B the$e three Mea$ures of different kinds of <foot>Quantities,</foot> <p n=>224</p> Quantities, that is to $ay, of Spaces, of Times, and of <I>Impetus's,</I> let it be required to determine in the a$$igned Space, and at the height A C, how much the Time of the Fall of the Moveable from A to C is to be, and what the <I>Impetus</I> is that $hall be found to have been acquired in the $aid Term C, in relation to the Time and to the <I>Impetus</I> mea$ured by A B. Both the$e que$tions $hall be re$olved taking A D the Mean-proportional betwixt the two Lines A C and A B; affirming the Time of the Fall along the whole Space A C to be as the Time A D is in relation to A B, a$$igned in the beginning for the Quantity of the Time in the Fall A B. And like- wi$e we will $ay that the <I>Impetus,</I> or degree of Velocity that the Cadent Moveable $hall obtain in the Term C, in relation to the <I>Impetus</I> that it had in B, is as the $ame Line A D is in relation to A B, being that the Velocity encrea$eth with the $ame proportion as the Time doth: Which Conclu$ion although it was a$$umed as a <I>Po$tulatum,</I> yet the Author was plea$ed to explain the Applicati- on thereof above in the third Propo$ition.</P> <P>This point being well under$tood and proved, we come to the Con$ideration of the <I>Impetus</I> derived from two compound Moti- ons: whereof let one be compounded of the Horizontal and alwaies Equable, and of the Perpendicular unto the Horizon, and it al$o Equable: but let the other be compounded of the Horizontal like- wi$e alwaies Equable, and of the Perpendicular Naturally-Accele- rate. If both $hall be Equable, it hath been $een already that the <I>Impetus</I> emerging from the compo$ition of both is <I>potentia</I> equal to both, as for more plainne$s we will thus Exemplifie. Let the Move- able de$cending along the Perpendicular A B be $uppo$ed to have, for example, three degrees of Equable <I>Impetus,</I> but being tran$- ported along A B towards C, let the $aid Velocity and <I>Impetus</I> be $uppo$ed four degrees, $o that in the $ame Time that falling it would pa$s along the Perpendicular, <I>v. gr.</I> three yards, <fig> it would in the Horizontal pa$s four, but in that compounded of both the Velocities it cometh in the $ame Timefrom the point A un- to the Term C, de$cending all the way along the Diagonal Line A C, which is not $even yards long, as that $hould be which is com- pounded of the two Lines A B, 3, and B C, 4, but is 5; which 5 is <I>potentia</I> equal to the two others, 3 and 4: For having found the Squares of 3 and 4, which are 9 and 16, and joyning the$e together, they make 25 for the Square of A C, which is equal to the two Squares of A B and B C: whereupon A C $hall be as much as is the Side, or, if you will, Root of the Square 25, which is 5. For a con$tant and certain Rule therefore, when it is required to a$$ign the Quantity of the <I>Impetus</I> re$ulting from two <I>Impetus's</I> given, the one Horizontal, and the other Perpendicular, and both Equable, <foot>they</foot> <p n=>225</p> they are each of them to be $quared, and their Squares being put together the Root of the Aggregate is to be extracted, which $hall give us the quantity of the <I>Impetus</I> compounded of them both. And thus in the foregoing example, that Moveable that by vertue of the Perpendicular Motion would have percu$$ed upon the Hori- zon with three degrees of Force, and with only the Horizontal Mo- tion would have percu$$ed in C with four degrees, percu$$ing with both the <I>Impetus's</I> conjoyned, the blow $hall be like to that of the Percutient moved with five degrees of Velocity and Force. And this $ame Percu$$ion would be of the $ame Impetuo$ity in all the points of the Diagonal A C, for that the compounded <I>Impetus's</I> are alwaies the $ame, never encrea$ing or dimini$hing.</P> <P>Let us now $ee what befalls in compounding the Equable Hori- zontal Motion with another Perpendicular to the Horizon which beginning from Re$t goeth Naturally Accelerating. It is already manife$t, that the Diagonal, which is the Line of the Motion com- pounded of the$e two, is not a Right Line, but Semiparabolical, as hath been demon$trated; ^{*} in which the <I>Impetus</I> doth go con- <marg>* Or along which.</marg> tinually encrea$ing by means of the continual encrea$e of the Ve- locity of the Perpendicular Motion: Wherefore, to determine what the <I>Impetus</I> is in an a$$igned point of that Parabolical Diagonal, it is requi$ite fir$t to a$$ign the Quantity of the Uniform Horizontal <I>Impetus,</I> and then to find what is the <I>Impetus</I> of the falling Movea- ble in the point a$$igned: the which cannot be determined without the con$ideration of the Time $pent from the beginning of the Compo$ition of the two Motions: which Con$ideration of the Time is not required in the Compo$ition of Equable Motions, the Velocities and <I>Impetus's</I> of which are alwaies the $ame: but here where there is in$erted into the mixture a Motion which beginning from extream Tardity goeth encrea$ing in Velocity according to the continuation of the Time, it is nece$$ary that the quantity of the Time do $hew us the quantity of the degree of Velocity in the a$$igned point: for, as to the re$t, the <I>Impetus</I> compounded of the$e two (as in Uniform Motions) is <I>potentia</I> equal to both the others compounding. But here again I will better explain my meaning by an example. In A C the Perpendicular to the Horizon let any part be taken A B; the which I will $uppo$e to $tand for the Mea$ure of the Space of the Natural Motion made along the $aid Perpen- dicular, and likewi$e let it be the Mea$ure of the Time, and al$o of the degree of Velocity, or, if you will, of the <I>Impetus's.</I> It is ma- nife$t in the fir$t place, that if the <I>Impetus</I> of the Moveable in B <I>ex quiete</I> in A $hall be turned along B D parallel to the Horizon in an Equable Motion, the quantity of its Velocity $hall be $uch that in the Time A B it $hall pa$s a Space double to the Space A B, which let be the Line B D. Then let B C be $uppo$ed equal to B A, and <foot>Gg let</foot> <p n=>226</p> let C E be drawn parallel and equal to B D, and thus by the Points B and E we $hall de$cribe the Parabolick Line B E I. And becau$e that in the Time A B with the <I>Impetus</I> A B the Horizontal Line B D or C E is pa$$ed, double to A B, and in $uch another Time the Per- pendicular B C is pa$$ed with an acqui$t of <I>Impetus</I> in C equal to the $aid Horizontal Line; therefore the Moveable in $uch another Time as A B $hall be found to have pa$$ed from B to E along the Parabola B E with an <I>Impetus</I> compounded of two, each equal to the <I>Impetus</I> A B. And becau$e one of them is Horizontal, and the other Perpendicular, the <I>Impetus</I> compound of them $hall be equal in Power to them both, that is <fig> double to one of them. So that $uppo$ing B F equal to B A, and drawing the Diagonal A F, the <I>Impetus</I> or the Percu$$ion in E $hall be greater than the Percu$- $ion in B of the Moveable fal- ling from the Height A, or than the Percu$$ion of the Horizon- tal <I>Impetus</I> along B D, according to the proportion of A F to A B. But in ca$e, $till retaining B A for the Mea$ure of the Space of the Fall from Re$t in A unto B, and for the Mea$ure of the Time and of the <I>Impetus</I> of the falling Moveable acquired in B, the Altitude B O $hould not be equal to, but greater than A B, taking B G to be a Mean-propor- tional betwixt the $aid A B and B O, the $aid B G would be the Mea$ure of the Time and of the <I>Impetus</I> in O, acquired in O by the Fall from the height B O; and the Space along the Horizontal Line, which being pa$$ed with the <I>Impetus</I> A B in the Time A B would be double to A B, $hall, in the whole duration of the Time B G, be $o much the greater, by how much in proportion B G is greater than B A. Suppo$ing therefore L B equal to B G, and draw- ing the Diagonal A L, it $hall give us the quantity compounded of the two <I>Impetus's</I> Horizontal and Perpendicular, by which the Parabola is de$cribed; and of which the Horizontal and Equable is that acquired in B by the fall of A B, and the other is that acquired in O, or, if you will, in I by the De$cent B O, who$e Time, as al$o the quantity of its Moment was B G. And in this Method we $hall inve$tigate the <I>Impetus</I> in the extream term of the Parabola, in ca$e its Altitude were le$$er than the Sublimity A B, taking the Mean- proportional betwixt them both: which being $et off upon the Ho- rizontal Line in the place of B F, and the Diagonal drawn, as A F, we $hall hereby have the quantity of the <I>Impetus</I> in the extream term of the Parabola.</P> <foot>And</foot> <p n=>227</p> <P>And to what hath hitherto been propo$ed touching <I>Impetus's,</I> Blows, or if you plea$e, Percu$$ions of $uch like Projects, it is ne- ce$$ary to add another very nece$$ary Con$ideration; and this it is: That it doth not $uffice to have regard to the Velocity only of the Project for the determining rightly of the Force and Violence of the Percu$$ion, but it is requi$ite likewi$e to examine apart the State and Condition of that which receiveth the Percu$$ion, in the effica- cy of which it hath for many re$pects a great $hare and intere$t. And fir$t there is no man but knows that the thing $mitten doth $o much $uffer violence from the Velocity of the Percutient by how much it oppo$eth it, and either totally or partially checketh its Motion: For if the Blow $hall light upon $uch an one as yieldeth to the Velocity of the Percutient without any Re$i$tance, that Blow $hall be nullified: And he that runneth to hit his Enemy with his Launce, if at the overtaking of him it $hall fall out that he moveth, giving back with the like Velocity, he $hall make no thru$t, and the Action $hall be a meer touch without doing any harm.</P> <P>But if the Percu$$ion $hall happen to be received upon an Object which doth not wholly yield to the Percutient, but only partially, the Percu$$ion $hall do hurt, though not with its whole <I>Impetus,</I> but only with the exce$s of the Velocity of the $aid Percutient above the Velocity of the recoile and rece$$ion of the Object percu$$ed: $o that, if <I>v. g.</I> the Percutient $hall come with 10 degrees of Velo- city upon the Percu$$ed Body, which giving back in part retireth with 4 degrees, the <I>Impetus</I> and Percu$$ion $hall be as if it were of 6 degrees. And la$tly, the Percu$$ion $hall be entire and perfect on the part of the Percutient when the thing percu$$ed yieldeth not, but wholly oppo$eth and $toppeth the whole Motion of the Percu- tient; if haply there can be $uch a ca$e. And I $ay on the part of the Percutient, for when the Body percu$$ed moveth with a contra- ry Motion towards the Percutient, the Blow and Shock $hall be $o much the more Impetuous by how much the two Velocities uni- ted are greater than the $ole Velocity of the Percutient. More- over, you are likewi$e to take notice, that the more or le$s yielding may proceed not only from the quality of the Matter more or le$s hard, as if it be of Iron, of Lead, or of Wooll, <I>&c.</I> but al$o from the Po$ition of the Body that receiveth the Percu$$ion. Which Po- $ition if it $hall be $uch as that the Motion of the Percutient hap- neth to hit it at Right-Angles, the <I>Impetus</I> of the Percu$$ion $hall be the greate$t: but if the Motion $hall proceed obliquely, and, as we $ay, a$lant, the Percu$$ion $hall be weaker; and that more, and more according to its greater and greater Obliquity: for an Ob- ject in that manner $cituate, albeit of very $olid matter, doth not damp or arre$t the whole <I>Impetus</I> and Motion of the Percutient, which $lanting pa$$eth farther, continuing at lea$t in $ome part to <foot>Gg 2 move</foot> <p n=>228</p> move along the Surface of the oppo$ed Body Re$i$ting. When therefore we have even now determined of the greatne$s of the <I>Impetus</I> of the Project in the end of the Parabolicall Line, it ought to be under$tood to be meant of the Percu$$ion received upon a Line at Right Angles with the $ame Parabolick Line, or with the Line that is Tangent to the Parabola in the fore$aid point: for although that $ame Motion be compounded of an Horizontal and a Perpendicular Motion, the <I>Impetus</I> is not at the greate$t either upon the Horizontal Plane, or upon that erect to the Horizon, be- ing received upon them both obliquely.</P> <P>SAGR. Your $peaking of the$e Blows, and the$e Percu$$ions hath brought into my mind a Problem, or, if you will, Que$tion in the Mechanicks, the $olution whereof I could never find in any Author, nor any thing that doth dimini$h my admiration, or $o much as in the lea$t afford my judgment $atisfaction. And my doubt and wonder lyeth in my not being able to comprehend whence that Immen$e Force and Violence $hould proceed, and on what Principle it $hould depend, which we $ee to con$i$t in Per- cu$$ion, in that with the $imple $troke of an Hammer, that doth not weigh above eight or ten pounds, we $ee $uch Re$i$tances to be overcome as would not yield to the weight of a Grave Body that without Percu$$ion hath an <I>Impetus</I> only by pre$$ing and bearing upon it, albeit the weight of this be many hundreds of pounds more. I would likewi$e find out a way to mea$ure the Force of this Percu$$ion, which I do not think to be infinite, but rather hold that it hath its Term in which it may be compared, and in the end Regulated with other Forces of pre$$ing Gravities, either of Lea- vers, or of Screws, or of other Mechanick In$truments, of who$e multiplication of Force I am thorowly $atisfied.</P> <P>SALV. You are not alone in the admirablene$s of the effect, and the ob$curity of the cau$e of $o $tupendious an Accident. I ruminated a long time upon it in vain, my $tupifaction $till encrea- $ing; till in the end meeting with our <I>Academian,</I> I received from him a double $atisfaction: fir$t in hearing that he al$o had been a long time at the $ame lo$s; and next in under$tanding that after he had at times $pent many thou$ands of hours in $tudying and con- templating thereon, he had light upon certain Notions far from our fir$t conceptions, and therefore new, and for their Novelty to be admired. And becau$e that I already $ee that your Curio$ity would gladly hear tho$e Conceits which are Remote from common Conjecture, I $hall not $tay for your entreaty, but I give you my word that $o $oon as we $hall have fini$hed the Reading of this Treati$e of Projects, I will $et before you all tho$e Fancies, or, I might $ay, Extravagancies that are yet left in my memory of the Di$cour$es of the Academick. In the mean time let us pro$ecute the Propo$itions of our Author.</P> <foot>PROBL.</foot> <p n=>229</p> <head>PROBL. II. PROP. V.</head> <P>In the Axis of a given Parabola prolonged to find a $ublime point out of which the Moveable falling $hall de$cribe the $aid Parabola.</P> <P><I>Let the Parabola be A B, its Amplitude H B, and its prolonged Axis H E; in which a Sublimity is to be found, out of which the Moveable falling, and converting the</I> Impetus <I>conceived in A along the Horizontal Line, de$cribeth the Parabola A B. Draw the Horizontal Line A G, which $hall be Parallel to B H, and $uppo$ing A F equal to A H draw the Right Line F B, which toucheth the Parabola in B, and cutteth the Horizontal Line A G in G; and unto F A and A G let A E be a third Proportional. I $ay, that E is the $ublime Point re- quired, out of which the Moveable falling</I> ex quiete <I>in E, and the</I> Im- petus <I>conceived in A being converted along the Horizontal Line over- taking the</I> Impetus <I>of the De$cent</I> <fig> <I>in H</I> ex quiete <I>in A, de$cribeth the Parabola A B. For if we $uppo$e E A to be the Mea$ure of the Time of the Fall from E to A, and of the</I> Impetus <I>acquired in A, A G (that is a Mean-proportional be- tween E A and A F) $hall be the Time and the</I> Impetus <I>coming from F to A, or from A to H. And becau$e the Moveable coming out of E in the Time E A with the</I> Impetus <I>acquired in A pa$$eth in the Ho- rizontal Lation with an Equable Motion the double of E A; There- fore likewi$e moving with the $ame</I> Impetus <I>it $hall in the Time A G pa$s the double of G A, to wit, the Mean-proportional B H (for the Spaces pa$$ed with the $ame Equable Motion are to one another as the Times of the $aid Motions:) And along the Perpendicular A H $hall be pa$$ed with a Motion</I> ex quiete <I>in the $ame Time G A: Therefore the Amplitude H B, and Altitude A H are pa$$ed by the Moveable in the $ame Time: Therefore the Parabola A B $hall be de$cribed by the De$cent of the Project coming from the Sublimity E: Which was re- quired.</I></P> <head>COROLLARY.</head> <P>Hence it appeareth that the half of the <I>B</I>a$e or Amplitude of the Semiparabola (which is the fourth part of the Amplitude of the whole Parabola) is a Mean-proportional betwixt its Al- titude and the Sublimity out of which the Moveable falling de$cribeth it.</P> <foot>PRO<I>B</I>L.</foot> <p n=>230</p> <head>PROBL. III. <I>P</I>RO<I>P.</I> VI.</head> <P>The Sublimity and Altitude of a Semiparabola being given to find its Amplitude.</P> <P><I>Let A C be perpendicular to the</I> <fig> <I>Horizontal Line D C, in which let the Altitude C B and the Sublimity B A be given: It is required in the Horizontal Line D C to find the Amplitude of the Semiparabola that is de$cribed out of the Sublimity B A with the Alti- tude B C. Take a Mean proportional between C B and B A, to which let C D be double, I $ay, that C D is the Amplitude required. The which is manife$t by the precedent Propo$ition.</I></P> <head>THEOR. IV. PROP. VII.</head> <P>In Projects which de$cribe Semiparabola's of the $ame Amplitude, there is le$s <I>Impetus</I> required in that which de$cribeth that who$e Ampli- tude is double to its Altitude, than in any other.</P> <P><I>For let the Semiparabola be B D, who$e Amplitude C D is dou- ble to its Altitude C B; and in its Axis extended on high let B A be $uppo$ed equal to the Altitude B C; and draw a Line from A to D which toucheth the Semiparabola in D, and $hall cut the Hori- zontal Line B E in E; and B E $hall be equal to B C or to B A: It is manife$t that it is de$cribed by the Project who$e Equable Horizontal</I> Impetus <I>is $uch as is that gained in B of a thing falling from Re$t in A, and the</I> Impetus <I>of the Natural Motion downwards, $uch as is that of a thing coming to C</I> ex quiete <I>in B. Whence it is manife$t, that the</I> Impetus <I>compounded of them, and that $triketh in the Term D is as the Diagonal A E, that is</I> potentia <I>equal to them both. Now let there be another Semiparabola G D, who$e Amplitude is the $ame C D, and the Altitude C G le$s, or greater than the Altitude B C, and let H D touch the $ame, cutting the Horizontal Line drawn by G in the point K; and as H G is to G K, $o let K G be to G L: by what hath been demon$trated G L $hall be the Altitude from which the Project falling de$cribeth the</I> <foot><I>Parabola</I></foot> <p n=>231</p> <I>Parabola G D. Let G M be a Mean-proportional betwixt A B and G L; G M $hall be the Time, and the Moment or</I> Impetus <I>in G of the Project falling from L, (for it hath been $uppo$ed that A B is the Mea- $ure of the Time and</I> Impetus.) <I>Again, let G N be a Mean-propor- tional betwixt B C and C G: this G N $hall be the Mea$ure of the Time and the</I> Impetus <I>of the</I> <fig> <I>Project falling from G to C. If therefore a Line be drawn from M to N it $hall be the the Mea$ure of the</I> Impetus <I>of the Project a- long the Para- bola B D, $cri- king in the term D. Which</I> Impetus, <I>I $ay, is greater than the</I> Impetus <I>of the Project along the Parabola B D, who$e quantity was A E. For becau$e G N is $uppo$ed the Mean-pro- portional betwixt B C and C G, and B C is equal to B E, that is to H G; (for they are each of them $ubduple to D C:) Therefore as C G is to G N, $o $hall N G be to G K: and, as C G or H G is to G K, $o $hall the Square N G be to the Square of G K: But as H G is to G K, $o was K G $uppo$ed to be to G L: Therefore as N G is to the Square G K, $o is K G to G L: But as K G is to G L, $o is the Square K G unto the Square G M, (for G M is the Mean between K G and G L:) Therefore the three Squares N G, K G, and G M are continual proportionals: And the two extream ones N G and G M taken together, that is the Square M N is greater than double the Square K G, to which the Square A E is double: Therefore the Square M N is greater than the Square A E: and the Line M N greater than the Line A E: Which was to be de- mon$trated.</I></P> <head>CORROLLARY I.</head> <P>Hence it appeareth, that on the contrary, in the Project out of D along the Semiparabola D B, le$s <I>Impetus</I> is required than along any other according to the greater or le$$er Elevation of the Semiparabola B D, which is according to the Tan- gent A D, containing half a Right-Angle upon the Hori- zon.</P> <foot>COROLLARY</foot> <p n=>232</p> <head>COROLLARRY II.</head> <P>And that being $o, it followeth, that if Projections be made with the $ame <I>Impetus</I> out of the Term D, according to $everal Elevations, that $hall be the greate$t Projection or Amplitude of the Semiparabola or whole Parabola which followeth at the Elevation of a ^{*} Semi-Right-Angle; and the re$t, made <marg>* Or, at the Ele- vation of 45 de- grees.</marg> according to greater or le$$er Angles, $hall be greater or le$$er.</P> <P>SAGR. The $trength of Nece$$ary Demon$trations are full of plea$ure and wonder; and $uch are only the Mathematical. I un- der$tood before upon tru$t from the Relations of $undry Gunners, that of all the Ranges of a Cannon, or of a Mortar-piece, the grea- te$t, <I>$cilicet</I> that which carryeth the Ball farthe$t was that made at the Elevation of a Semi-Right-Angle, which they call, of the Sixth point of the Square: but the knowledge of the Cau$e whence it hapneth infinitely $urpa$$eth the bare Notion that I received upon their atte$tation, and al$o from many repeated Experiments.</P> <P>SALV. You $ay very right: and the knowledge of one $ingle Effect acquired by its Cau$es openeth the Intellect to under$tand and a$certain our $elves of other effects, without need of repairing unto Experiments, ju$t as it hapneth in the pre$ent Ca$e; in which having found by demon$trative Di$cour$e the certainty of this, That the greate$t of all Ranges is that of the Elevation of a Semi- Right-Angle, the Author demon$trates unto us that which po$$ibly hath not been ob$erved by Experience: and that is, that of the other Ranges tho$e are equal to one another who$e Elevations ex- ceed or fall $hort by equal Angles of the Semi-right: $o that the Balls $hot from the Horizon, one according to the Elevation of $e- ven Points, and the other of 5, $hall light upon the Horizon at equal Di$tances: and $o the Ranges of 8 and of 4 points, of 9 and of 3, <I>&c.</I> $hall be equal. Now hear the Demon$tration of it.</P> <head>THEOR. V. PROP. VIII.</head> <P>The Amplitudes of Parabola's de$cribed by Pro- jects expul$ed with the $ame <I>Impetus</I> according to the Elevations by Angles equidi$tant above, <marg>* Or Angle of 45.</marg> and beneath from the ^{*} Semi-right, are equal to each other.</P> <p n=>233</p> <P><I>Of the Triangle M C B, about the Right-Angle C, let the Ho- rizontal Line B C and the Perpendicular C M be equal; for $o the Angle M B C $hall be Semi-right; and prolonging C M to D, let there be con$tituted in B two equal Angles above and below the Diagonal M B,</I> viz. <I>M B E, and M B D. It is to be demon$trated that the Amplitudes of the Parabola's de$cribed by the Projects be- ing emitted</I> [or $hot off] <I>with the $ame</I> Impetus <I>out of the Term B, according to the Elevations of the Angles E B C and D B C, are equal. For in regard that the extern Angle B M C, is equal to the two intern M D B and M B D, the Angle M B C $hall al$o be equal to them. And if we $uppo$e M B E in$tead of the Angle M B D, the $aid Angle M B C $hall be equal to the two</I> <fig> <I>Angles M B E and B D C: And taking away the common Angle M B E, the remaining An- gle B D C $hall be equal to the remaining An- gle E B C: Therefore the Triangles D C B and B C E are alike. Let the Right Lines D C and E C be divided in the mid$t in H and F; and draw H I and F G parallel to the Ho- rizontal Line C B; and as D H is to H I, $o let I H be to H L: the Triangle I H L $hall be like to the Triangle I H D, like to which al$o is E G F. And $eeing that I H and G F are equal (to wit, halves of the $ame B C:) There- fore F E, that is F C, $hall be equal to H L: And, adding the common Line F H, C H $hall be equal to F L. If therefore we under$tand the Se- miparabola to be de$cribed along by H and B, who$e Altitude $hall be H C, and Sublimity H L, its Amplitude $hall be C B, which is double to HI, that is, the Mean betwixt D H, or C H, and HL: And D B $hall be a Tangent to it, the Lines C H and H D being equal. And if, again, we conceive the Parabola to be de$cribed along by F and B from the Sublimity FL, with the Altitude F C, betwixt which the Mean- proportional is F G, who$e double is the Horizontal Line C B: C B, as before, $hall be its Amplitude; and E B a Tangent to it, $ince E F and F C are equal: But the Angles D B C and E B C</I> ($cilicet, <I>their Eleva- tions) $hall be equidi$tant from the Semi-Right Angle: Therefore the Propo$ition is demon$trated.</I></P> <head>THEOR. VI. <I>P</I>RO<I>P.</I> IX.</head> <P>The Amplitudes of Parabola's, who$e Altitudes and Sublimities an$wer to each other <I>è contra- rio,</I> are equall.</P> <foot>Hh <I>Let</I></foot> <p n=>234</p> <P><I>Let the Altitude G F of the Parabola F H have the $ame proporti- on to the Altitude C B of the Parabola B D, as the Sublimity B A hath to the Sublimity F E. I $ay, that the Amplitude H G is equal to the Amplitude D C. For $ince the fir$t G F hath the $ame propor- tion to the $econd C B, as the third B A hath to the fourth F E; There- fore, the Rectangle</I> <fig> <I>G F E of the fir$t and fourth, $hall be equal to the Rectangle C B A of the $econd and third: Therefore the Squares that are equal to the$e Rectangles $hall be equal to one another: But the Square of half of G H is equal to the Rectangle G F E; and the Square of half of C D is equal to the Rectangle C B A: There- fore the$e Squares, and their Sides, and the doubles of their Sides $hall be equal: But the$e are the Amplitudes G H and C D: Therefore the Propo$ition is manife$t.</I></P> <head>LEMMA <I>pro $equenti.</I></head> <P>If a Right Line be cut according to any proportion, the Squares of the Mean-proportionals between the whole and the two parts are equal to the Square of the whole.</P> <P><I>Let A B be cut according to any proportion in C. I $ay, that the Squares of the Mean-proportional Lines between the whole A B and the parts A C and C B, being taken together are equal to the Square of the whole A B. And this appeareth, a Semi-</I> <fig> <I>circle being de$cribed upon the whole Line B A, and from C a Perpendicular being ere- cted C D, and Lines being drawn from D to A, and from D to B. For D A is the Mean- proportional betwixt A B and A C; and D B is the Mean-proporti- onal between A B and B C: And the Squares of the Lines D A and D B taken together are equal to the Square of the whole Line A B, the Angle A D B in the Semicircle being a Right-Angle: Therefore the Propo$ition is manifest.</I></P> <foot>THEOR.</foot> <p n=>235</p> <head>THEOR. VII. PROP. X.</head> <P>The <I>Impetus</I> or Moment of any Semiparabola is equal to the Moment of any Moveable falling naturally along the Perpendicular to the Ho- rizon that is equal to the Line compounded of the Sublimity and of the Altitude of the Se- miparabola.</P> <P><I>Let the Semiparabola be A B, its Sublimity D A, and Altitude A C, of which the Perpendicular D C is compounded. I $ay, that the</I> Impetus <I>of the Semiparabola in B is equal to the Moment of the Moveable Naturally falling from D to C. Suppo$e D C it $elf to be the Mea$ure of the Time and of the</I> Impetus; <I>and take a Mean-pro- portional betwixt C D and D A, to which let</I> <fig> <I>C F be equal; and withal let C E be a Mean- proportional between D C and C A: Now C F $hall be the Mea$ure of the Time and of the Mo- ment of the Moveable $alling along D A out of Re$t in D; and C E $hall be the Time and Mo- ment of the Moveable falling along A C, out of Re$t in A, and the Moment of the Diagonal E F $hall be that compounded of both the others,</I> $cil. <I>that of the Semiparabola in B. And becau$e D C is cut according to any proportion in A, and becau$e C F and C E are Mean-Proportionals between C D and the parts D A and A C; the Squares of them taken together $hall be equal to the Square of the whole; by the Lemma aforegoing: But the Squares of them are al$o equal to the Square of E F: Therefore D F is equal al$o to the Line D C: Whence it is manife$t that the Moments along D C, and along the Se- miparabola A B, are equal in C and B: Which was required.</I></P> <head>COROLLARY.</head> <P>Hence it is manife$t, that of all Parabola's who$e Altitudes and Sublimities being joyned together are equal, the <I>Impetus's</I> are al$o equal.</P> <foot>Hh 2 PROBL.</foot> <p n=>236</p> <head>PROBL. IV. PROP. XI.</head> <P>The <I>Impetus</I> and Amplitude of a Semiparabola be- ing given, to find its Altitude, and con$equently its Sublimity.</P> <P><I>Let the</I> Impetus <I>given be defined by the Perpendicular to the Ho- rizon A B; and let the Amplitude along the Horizontal Line be B C. It is required to find the Altitude and Sublimity of the Parabola who$e</I> Impetus <I>is A B, and Amplitude B C. It is manife$t, from what hath been already demon$trated, that half the Amplitude B C will be a Mean-proportional betwixt the Altitude and the Sublimity of the $aid Semiparabola, who$e</I> Impetus, <I>by the precedent Propo$ition, is the $ame with the</I> Impetus <I>of the Moveable falling from Re$t in A along the whole Perpendicular A B: Wherefore B A is $o to be cut that the Rectangle contained by its parts may be equal to the Square of half of B C, which let be B D. Hence it appeareth to be nece$$ary that D B do not exceed the</I> <fig> <I>half of B A; for of Rectangles contained by the parts the greate$t is when the whole Line is cut into two equal parts. Therefore let B A be divided into two equal parts in E. And if B D be equal to B E the work is done; and the Altitude of the Semipara- bola $hall be B E, and its Sublimity E A: (and $ee here by the way that the Amplitude of the Parabola of a Semi-right Elevation, as was demon$trated above, is the greate$t of all tho$e de$cribed with the $ame</I> Impetus.) <I>But let B D be le$s than the half of B A, which is $o to be cut that the Rectangle under the parts may be equal to the Square B D. Upon E A de$cribe a Semicircle, upon which out of A $et off A F equal to B D, and draw a Line from F to E, to which cut a part equal E G. Now the Rectangle B G A, together with the Square E G, $hall be equal to the Square E A; to which the two Squares A F and F E are al$o equal: Therefore the equal Squares G E and F E be- ing $ub$tracted, there remaineth the Rectangle B G A equal to the Square A F,</I> $cilicet, <I>to B D; and the Line B D is a Mean-proportional betwixt B G and G A. Whence it appeareth, that of the Semipa- rabola who$e Amplitude is B C, and</I> Impetus <I>A B, the Altitude is B G, and the Sublimity G A. And if we $et off B I below equal to G A, this $hall be the Altitude, and I A the Sublimity of the Semiparabola I C. From what hath been already demon$trated we are able,</I></P> <foot>PROBL.</foot> <p n=>237</p> <head>PROBL. V. PROP. XII.</head> <P>To collect by Calculation of the Amplitudes of all Semiparabola's that are de$cribed by Projects expul$ed with the $ame <I>Impetus,</I> and to make Tables thereof.</P> <P><I>It is obvious, from the things demon$trated, that Parabola's are de- $cribed by Projects of the $ame</I> Impetus <I>then, when their Subli- mities together with their Altitudes do make up equal Perpendicu- lars upon the Horizon. The$e Perpendiculars therefore are to be com- prehended between the $ame Horizontal Parallels. Therefore let the Horizontal Line C B be $uppo$ed equal to the Perpendicular B A, and draw the Diagonal from A to C. The Angle A C B $hall be Semi- right, or 45 Degrees. And the Perpendicular B A being divided into two equal parts in D, the Semiparabola D C $hall be that which is de- $cribed from the Sublimity A D together with the Altitude D B: and its</I> Impetus <I>in C $hall be as great as that of the Moveable coming out of Re$t in A along the Perpendicular A B is in B. And if A G be drawn parallel to B C, the united Altitudes and Sublimities of all other re- maining Semiparabola's who$e future</I> Impetus's <I>are the $ame with tho$e now mentioned mu$t be bounded by the Space between the Parallels</I> <fig> <I>A G and B C. Farthermore, it having been but now demon$trated, that the Am- plitudes of the Semiparabola's who$e Tangents are equidi$tant either above or below from the Semi right Elevation are equal, the Calculations that we frame for the greater Elevations will likewi$e $erve for the le$$er. We choo$e moreover a number of ten thou$and parts for the greate$t Amplitude of the Projection of the Semiparabola made at the Elevation of 45 degrees: $o much therefore the Line B A, and the Amplitude of the Semipa- rabola B C, are to be $uppo$ed. And we make choice of the number 10000, becau$e we in our Calculation u$e the Table of Tangents, in which this number agreeth with the Tangent of 45 degrees. Now, to come to the bu$ine$s, let C E be drawn, contain- ing the Angle E C B greater (Acute neverthele$s,) than the Angle A C B; and let the Semiparabola be de$cribed which is touched by the Line E C, and who$e Sublimity united with its Altitude is equal to B A. In the Table of Tangents take the $aid B E for the Tangent at the</I> <foot><I>given</I></foot> <p n=>238</p> <I>given Angle B C E, which divide into two equal parts at F. Then find a third Proportional to B F and B C, (or to the half of B C,) which $hall of nece$$ity be greater than F A; therefore let it be F O: Of the Semiparabola, therefore, in$cribed in the Triangle E C B, ac- cording to the Tangent C E, who$e Amplitude is C B, the Altitude B F, and the Sublimity F O is found: But the whole Line B O ri$eth above the Parallels A G and C B, whereas our work was to bound it between them: For $o both it and the Semiparabola D C $hall be de$cribed by the Projects out of C expelled with the $ame</I> Impetus. <I>Therefore we are to $eek another like to this, (for innumerable greater and $maller, like to one another, may be de$cribed within the Angle B C E) to who$e united Sublimity and Altitude B A $hall be equal. Therefore as O B is to B A, $o let the Amplitude B C be to C R: and C R $hall be found,</I> $cilicet <I>the Amplitude of the Semiparabola according to the Elevation of the Angle B C E, who$e conjoyned Sublimity and Altitude is equal to the Space contained between the Parallels G A and C B: Which was required. The work, therefore, $hall be after this manner.</I></P> <P><I>Take the Tangent of the given Angle B C E, to the half of which add the third Proportional of it, and half of B C, which let be F O: Then as O B is to B A, $o let B C be to another, which let be C R, to wit, the Amplitude $ought. Let us give an Example.</I></P> <P><I>Let the Angle E C B be 50 degrees, its Tangent $hall be 11918, who$e half, to wit, B F, is 5959, and the half of B C is 5000, the third proportional of the$e halves is 4195, which added to the $aid B F maketh 10154: for the $aid B O. Again, as O B is to B A, that is as 10154 is to 10000, $o is B E, that is 10000 (for each of them is the Tangent of 45 degrees) to another: and that $hall give us the required Altitude R C 9848, of $uch as B C (the greate$t Amplitude) is 10000. To the$e the Amplitudes of the whole Parabola's are double,</I> $cilicet <I>19696 and 20000. And $o much likewi$e is the Amplitude of the Parabola according to the Elevation of 40 degrees, $ince it is equal- ly di$tant from 45 degrees.</I></P> <P>SAGR. For the perfect under$tanding of this Demon$tration I mu$t be informed how true it is, that the Third Proportional to B F and B I, is (as the Author $aith) nece$$arily greater than F A.</P> <P>SALV. That inference, as I conceive, may be deduced thus. The Square of the Mean of three proportional Lines is equal to the Rectangle of the other two: whence the Square of B I, or of B D equal to it, ought to be equal to the Rectangle of the fir$t F B multiplied into the third to be found: which third is of nece$$ity to be greater than F A, becau$e the Rectangle of B F multiplied into F A is le$s than the Square B D: and the Defect is as much as the Square of D F, as <I>Euclid</I> demon$trates in a Propo$ition of his <foot>Second</foot> <p n=>239</p> Second Book. You mu$t al$o know, that the point F which divi- deth the Tangent E B in the middle, will many other times fall above the point A, and once al$o in the $aid A: In which ca$es it is evident of it $elf, that the third proportional to the half of the Tan- gent, and to B I (which giveth the Sublimity) is all above A. But the Author hath taken a Ca$e in which it was not manife$t that the $aid third Proportional is alwaies greater than F A: and which therefore being $et off above the point F pa$$eth beyond the Paral- lel A G. Now let us proceed.</P> <P><I>It will not be unprofitable if by help of this Table we compo$e ano- ther, $hewing the Altitudes of the $ame Semiparabola's of Projects of the $ame</I> Impetus. <I>And the Con$truction of it is in this manner.</I></P> <head>PROBL. VI. PROP. XIII.</head> <P>From the given Amplitudes of Semiparabola's in the following Table $et down, keeping the common <I>Impeius</I> with which every one of them is de$cribed, to compute the Altitudes of each $everal Semiparabola.</P> <P><I>Let the Amplitude given be B C, and of the</I> Impetus, <I>which is $uppo$ed to be alwaies the $ame, let the Mea$ure be O B, to wit, the Aggregate of the Altitude and Sublimity. The $aid Altitude is required to be found and di$tingui$hed. Which $hall then be done when B O is $o divided as that the Rectangle contained under its parts is equal to the Square of half the Amplitude B C. Let that $ame divi- $ion fall in F; and let both O B and B C be cut in the mid$t at D and I.</I> <fig> <I>The Square I B, therefore, is equal to the Rectangle B F O: And the Square D O is equal to the $ame Rectangle together with the Square F D. If therefore from the Square D O we deduct the Square B I, which is equal to the Rectangle B F O, there $hall remain the Square F D; to who$e Side D F, B D be- ing added it $hall give the de$ired Altitude Altitude B F. And it is thus compounded</I> ex datis. <I>From half of the Square B O known $ub$tract the Square B I al$o known, of the remainder take the Square Root, to which add D B known; and you $hall have the Altitude $ought B F. For example. The Altitude of the Parabola de$cribed at the Elevation of 55 degrees is to be found. The Amplitude, by the follow- ing Table is 9396, its half is 4698, the Square of that is 22071204,</I> <foot><I>this</I></foot> <p n=>240</p> <I>this $ub$tracted from the Square of the half B O, which is alwaies the $ame, to wit, 2500000, the remainder is 2928796, who$e Square Root is 1710 very near, this added to the half of B O, to wit, 5000, gives 67101, and $o much is the Altitude B F. It will not be unprofi- table, to give the Third Table, containing the Altitudes and Sublimi- ties of Semiparabola's, who$e Amplitude $hall be alwaies the $ame.</I></P> <P>SAGR. This I would very gladly $ee $ince by it I may come to know the Difference of the <I>Impetus's,</I> and of the Forces that are required for carrying the Project to the $ame Di$tance with Ranges which are called at Random: which Difference I believe is very great according to the different Elevations [<I>or Mountures:</I>] $o that if, for example, one would at the Elevation of 3 or 4 degrees, or of 87 or 88 make the Ball to fall where it did, being $hot at the Ele- vation of <I>gr.</I> 45. (where, as hath been $hewn, the lea$t <I>Impetus</I> is required) I believe that it would require a very much greater Force.</P> <P>SALV. You are in the right: and you will find that to do the full execution in all the Elevations it is requi$ite to make great Pro- gre$$ions towards an infinite <I>Impetus.</I> Now let us $ee the Con$tru- ction of the Table.</P> <foot>The</foot> <p n=>241</p> <table> <row><col>Degrees of Elevation.</col><col></col><col></col></row> <row><col>The Amplitudes of the Semipara- bola's, de$cribed with the $ame <I>Impetus.</I></col><col></col><col></col></row> <row><col>Gr.</col><col></col><col>Gr.</col></row> <row><col>45</col><col>10000</col><col></col></row> <row><col>46</col><col>9994</col><col>44</col></row> <row><col>47</col><col>9976</col><col>43</col></row> <row><col>48</col><col>9945</col><col>42</col></row> <row><col>49</col><col>9902</col><col>41</col></row> <row><col>50</col><col>9848</col><col>40</col></row> <row><col>51</col><col>9782</col><col>39</col></row> <row><col>52</col><col>9704</col><col>38</col></row> <row><col>53</col><col>9612</col><col>37</col></row> <row><col>54</col><col>9511</col><col>36</col></row> <row><col>55</col><col>9396</col><col>35</col></row> <row><col>56</col><col>9272</col><col>34</col></row> <row><col>57</col><col>9136</col><col>33</col></row> <row><col>58</col><col>8989</col><col>32</col></row> <row><col>59</col><col>8829</col><col>31</col></row> <row><col>60</col><col>8659</col><col>30</col></row> <row><col>61</col><col>8481</col><col>29</col></row> <row><col>62</col><col>8290</col><col>28</col></row> <row><col>63</col><col>8090</col><col>27</col></row> <row><col>64</col><col>7880</col><col>26</col></row> <row><col>65</col><col>7660</col><col>25</col></row> <row><col>66</col><col>7431</col><col>24</col></row> <row><col>67</col><col>7191</col><col>23</col></row> <row><col>68</col><col>6944</col><col>22</col></row> <row><col>69</col><col>6692</col><col>21</col></row> <row><col>70</col><col>6428</col><col>20</col></row> <row><col>71</col><col>6157</col><col>19</col></row> <row><col>72</col><col>5878</col><col>18</col></row> <row><col>73</col><col>5592</col><col>17</col></row> <row><col>74</col><col>5300</col><col>16</col></row> <row><col>75</col><col>5000</col><col>15</col></row> <row><col>76</col><col>4694</col><col>14</col></row> <row><col>77</col><col>4383</col><col>13</col></row> <row><col>78</col><col>4067</col><col>12</col></row> <row><col>79</col><col>3746</col><col>11</col></row> <row><col>80</col><col>3420</col><col>10</col></row> <row><col>81</col><col>3090</col><col>9</col></row> <row><col>82</col><col>2756</col><col>8</col></row> <row><col>83</col><col>2419</col><col>7</col></row> <row><col>84</col><col>2079</col><col>6</col></row> <row><col>85</col><col>1736</col><col>5</col></row> <row><col>86</col><col>1391</col><col>4</col></row> <row><col>87</col><col>1044</col><col>3</col></row> <row><col>88</col><col>698</col><col>2</col></row> <row><col>89</col><col>349</col><col>1</col></row> </table> <table> <row><col>Degrees of Elevation.</col><col></col><col></col><col></col></row> <row><col>The Altitudes of the Se- miparabola's, who$e <I>Impetus</I> is the $ame.</col><col></col><col></col><col></col></row> <row><col>Gr.</col><col></col><col>Gr.</col><col></col></row> <row><col>1</col><col>3</col><col>46</col><col>5173</col></row> <row><col>2</col><col>13</col><col>47</col><col>5346</col></row> <row><col>3</col><col>28</col><col>48</col><col>5523</col></row> <row><col>4</col><col>50</col><col>49</col><col>5698</col></row> <row><col>5</col><col>76</col><col>50</col><col>5868</col></row> <row><col>6</col><col>108</col><col>51</col><col>6038</col></row> <row><col>7</col><col>150</col><col>52</col><col>6207</col></row> <row><col>8</col><col>194</col><col>53</col><col>6379</col></row> <row><col>9</col><col>245</col><col>54</col><col>6546</col></row> <row><col>10</col><col>302</col><col>55</col><col>6710</col></row> <row><col>17</col><col>365</col><col>56</col><col>6873</col></row> <row><col>12</col><col>432</col><col>57</col><col>7033</col></row> <row><col>13</col><col>506</col><col>58</col><col>7190</col></row> <row><col>14</col><col>585</col><col>59</col><col>7348</col></row> <row><col>15</col><col>670</col><col>60</col><col>7502</col></row> <row><col>16</col><col>760</col><col>61</col><col>7649</col></row> <row><col>17</col><col>855</col><col>62</col><col>7796</col></row> <row><col>18</col><col>955</col><col>63</col><col>7939</col></row> <row><col>19</col><col>1060</col><col>64</col><col>8078</col></row> <row><col>20</col><col>1170</col><col>65</col><col>8214</col></row> <row><col>21</col><col>1285</col><col>66</col><col>8346</col></row> <row><col>22</col><col>1402</col><col>67</col><col>8474</col></row> <row><col>23</col><col>1527</col><col>68</col><col>8597</col></row> <row><col>24</col><col>1685</col><col>69</col><col>8715</col></row> <row><col>25</col><col>1786</col><col>70</col><col>8830</col></row> <row><col>26</col><col>1922</col><col>71</col><col>8940</col></row> <row><col>27</col><col>2061</col><col>72</col><col>9045</col></row> <row><col>28</col><col>2204</col><col>73</col><col>9144</col></row> <row><col>29</col><col>2351</col><col>74</col><col>9240</col></row> <row><col>30</col><col>2499</col><col>75</col><col>9330</col></row> <row><col>31</col><col>2653</col><col>76</col><col>9415</col></row> <row><col>32</col><col>2810</col><col>77</col><col>9493</col></row> <row><col>33</col><col>2967</col><col>78</col><col>9567</col></row> <row><col>34</col><col>3128</col><col>79</col><col>9636</col></row> <row><col>35</col><col>3289</col><col>80</col><col>9698</col></row> <row><col>36</col><col>3456</col><col>81</col><col>9755</col></row> <row><col>37</col><col>3621</col><col>82</col><col>9806</col></row> <row><col>38</col><col>3793</col><col>83</col><col>9851</col></row> <row><col>39</col><col>3962</col><col>84</col><col>9890</col></row> <row><col>40</col><col>4132</col><col>85</col><col>9924</col></row> <row><col>41</col><col>4302</col><col>86</col><col>9951</col></row> <row><col>42</col><col>4477</col><col>87</col><col>9972</col></row> <row><col>43</col><col>4654</col><col>88</col><col>9987</col></row> <row><col>44</col><col>4827</col><col>89</col><col>9998</col></row> <row><col>45</col><col>5000</col><col>90</col><col>10000</col></row> </table> <table> <row><col>A Table containing the Altitudes and Subli- mities of the Semiparabola's, who$e Am- plitudes are the $ame, that is to $ay, of 10000 parts, calculated to each Deg. of Elevation.</col><col></col><col></col><col></col><col></col><col></col></row> <row><col>Gr.</col><col>Altit.</col><col>Sublim.</col><col>Gr.</col><col>Altit.</col><col>Sublim.</col></row> <row><col>1</col><col>87</col><col>286533</col><col>46</col><col>5177</col><col>4828</col></row> <row><col>2</col><col>175</col><col>142450</col><col>47</col><col>5363</col><col>4662</col></row> <row><col>3</col><col>262</col><col>95802</col><col>48</col><col>5553</col><col>4502</col></row> <row><col>4</col><col>349</col><col>71531</col><col>49</col><col>5752</col><col>4345</col></row> <row><col>5</col><col>437</col><col>57142</col><col>50</col><col>5959</col><col>4106</col></row> <row><col>6</col><col>525</col><col>47573</col><col>51</col><col>6174</col><col>4048</col></row> <row><col>7</col><col>614</col><col>40716</col><col>52</col><col>6399</col><col>3006</col></row> <row><col>8</col><col>702</col><col>35587</col><col>53</col><col>6635</col><col>3765</col></row> <row><col>9</col><col>792</col><col>31565</col><col>54</col><col>6882</col><col>3632</col></row> <row><col>10</col><col>881</col><col>28367</col><col>55</col><col>7141</col><col>3500</col></row> <row><col>11</col><col>972</col><col>25720</col><col>56</col><col>7413</col><col>3372</col></row> <row><col>12</col><col>1063</col><col>23518</col><col>57</col><col>7699</col><col>3247</col></row> <row><col>13</col><col>1154</col><col>21701</col><col>58</col><col>8002</col><col>3123</col></row> <row><col>14</col><col>1246</col><col>20056</col><col>59</col><col>8332</col><col>3004</col></row> <row><col>11</col><col>1339</col><col>18663</col><col>60</col><col>8600</col><col>2887</col></row> <row><col>16</col><col>1434</col><col>17405</col><col>61</col><col>9020</col><col>2771</col></row> <row><col>17</col><col>1529</col><col>16355</col><col>62</col><col>9403</col><col>2658</col></row> <row><col>18</col><col>1624</col><col>15389</col><col>63</col><col>9813</col><col>2547</col></row> <row><col>19</col><col>1722</col><col>14522</col><col>64</col><col>10251</col><col>2438</col></row> <row><col>20</col><col>1820</col><col>13736</col><col>65</col><col>10722</col><col>2331</col></row> <row><col>21</col><col>1919</col><col>13024</col><col>66</col><col>11220</col><col>2226</col></row> <row><col>22</col><col>2020</col><col>12376</col><col>67</col><col>11779</col><col>2122</col></row> <row><col>23</col><col>2123</col><col>11778</col><col>68</col><col>12375</col><col>2020</col></row> <row><col>24</col><col>2226</col><col>11230</col><col>69</col><col>13025</col><col>1919</col></row> <row><col>25</col><col>2332</col><col>10722</col><col>70</col><col>13237</col><col>1819</col></row> <row><col>26</col><col>2439</col><col>10253</col><col>71</col><col>14521</col><col>1721</col></row> <row><col>27</col><col>2547</col><col>9814</col><col>72</col><col>15388</col><col>1624</col></row> <row><col>28</col><col>2658</col><col>9404</col><col>73</col><col>16354</col><col>1528</col></row> <row><col>29</col><col>2772</col><col>9020</col><col>74</col><col>17437</col><col>1413</col></row> <row><col>30</col><col>2887</col><col>8659</col><col>75</col><col>18660</col><col>1339</col></row> <row><col>31</col><col>3008</col><col>8336</col><col>76</col><col>20054</col><col>1246</col></row> <row><col>32</col><col>3124</col><col>8001</col><col>77</col><col>21657</col><col>1154</col></row> <row><col>33</col><col>3247</col><col>7699</col><col>78</col><col>23523</col><col>1062</col></row> <row><col>34</col><col>3373</col><col>7413</col><col>79</col><col>25723</col><col>972</col></row> <row><col>35</col><col>3501</col><col>7141</col><col>80</col><col>28356</col><col>881</col></row> <row><col>36</col><col>3633</col><col>6882</col><col>81</col><col>31560</col><col>792</col></row> <row><col>37</col><col>3768</col><col>6635</col><col>82</col><col>35577</col><col>702</col></row> <row><col>38</col><col>3906</col><col>6395</col><col>83</col><col>40222</col><col>613</col></row> <row><col>39</col><col>4049</col><col>6174</col><col>84</col><col>47572</col><col>525</col></row> <row><col>40</col><col>4196</col><col>5959</col><col>85</col><col>57150</col><col>437</col></row> <row><col>41</col><col>4246</col><col>5752</col><col>86</col><col>71503</col><col>349</col></row> <row><col>42</col><col>4502</col><col>5553</col><col>87</col><col>95405</col><col>262</col></row> <row><col>43</col><col>4662</col><col>5362</col><col>88</col><col>143181</col><col>174</col></row> <row><col>44</col><col>4828</col><col>5177</col><col>89</col><col>286499</col><col>87</col></row> <row><col>45</col><col>5000</col><col>5000</col><col>90</col><col>Infinite</col><col></col></row> </table> <foot>Ii</foot> <p n=>242</p> <head>PROBL. VII. PROP. XIV.</head> <P>To find the Altitudes and Sublimities of Semipa- rabola's who$e Amplitudes $hall be equal for each degree of Elevation.</P> <P><I>This we $hall ea$ily do. For $uppo$ing the Amplitude of the Semi- par abola to be of 10000 parts, the half of the Tangent of each degree of Elevation $hews the Altitude. As for example, of the Semiparabola who$e Elevation is 30 degrees, and Amplitude, as is $uppo$ed, 10000 parts, the Altitude $hall be 2887, for $o much, very near, is the half of the Tangent. And having found the Altitude the Sublimity is to be known in this manner. For a$much as it hath been demon$trated that the half of the Amplitude of a Semiparabola is the Mean proportional betwixt the Altitude and Sublimity, and the Alti- tude being already found, and the half of the Amplitude being alwaies the $ame, to wit, 5000 parts, if we $hall divide the Square thereof by the Altitude found, the de$ired Sublimity $hall come forth. As in the Example: The Altitude found was 2887; The Square of the 5000 parts is 25000000; which being divided by 2887, giveth 8659, ve- ry near, for the Sublimity $ought.</I></P> <P>SALV. Now here we $ee, in the $ir$t place, that the Conje- cture is very true which was mentioned afore, that in different Elevations the farther one goeth from the middlemo$t, whether it be in the Higher, or in the Lower, $o much greater <I>Impetus</I> and Vio- lence is required to carry the Project to the $ame Di$tance. For the <I>Impetus</I> lying in the mixture of the two Motions, Equable, Hori- zontal, and Perpendicular Naturally-Accelerate, of which <I>Impetus</I> the Aggregate of the Altitude and Sublimity is the Mea$ure, we do $ee in the propounded Table that that $ame Aggregate is lea$t in the Elevation of <I>gr.</I> 45, in which the Altitude and Sublimity are equal, <I>$cilicet</I> each 5000, and their Aggregate 10000. But if we $hould look on any greater Elevation, as, for example, of <I>gr.</I> 50, we $hould $ind the Altitude to be 5959, and the Sublimity 4196, which added together make 10155. And $o much al$o we $hould find the <I>Impetus</I> of <I>gr.</I> 40 to be, this and that Elevation being equally re- mote from the middlemo$t. Where we are to note, in the $econd place, that it is true, That equal <I>Impetus's</I> are $ought by two, and two in the Elevations equidi$tant from the middlemo$t, with this pretty variation over and above that the Altitudes and the Subli- mities of the ^{*} $uperiour Elevations an$wer alternally to the Sub- <marg>* <I>i.e.</I> Tho$e above 45 deg.</marg> limities and Altitudes of the Inferiour: $o that whereas in the <foot>example</foot> <p n=>243</p> example propo$ed, in the Elevation of <I>gr.</I> 50. the Altitude is 5959 and the Sublimity 4196, in the Elevation of <I>gr.</I> 40. it falls out on the contrary that the Altitude is 4196, and the Sublimity 5959: And the $ame happens in all others without any difference; $ave only that for the avoyding of tediou$ne$s in Calculations we have kept no account of $ome fractions, which in $o great $ums are of no value, but may without any prejudice be omitted.</P> <P>SAGR. I am ob$erving that of the two <I>Impetus's</I> Horizontal and Perpendicular in Projections, the more Sublime they are, they need $o much the le$s of the Horizontal, and the more of the Perpendi- cular. Moreover in tho$e of $mall Elevation, great mu$t be the Force of the Horizontal <I>Impetus,</I> which is to carry the Project in a little Altitude. But although I comprehend very well that in the Total Elevation of <I>gr.</I> 90, all the force in the world $ufficeth not to drive the Project one $ingle Inch from the Perpendicular, but that it mu$t of nece$$ity fall in the $ame place whence it was expel- led; yet dare I not with the like certainty affirm that likewi$e in the nullity of Elevation, that is in the Horizontal Line, the Project cannot by any Force le$s than infinite, be driven to any di- $tance: So, as that, for example, a Culverin it $elf $hould not be able to carry a Ball of Iron Horizontally, or, as they $ay, at Point blank, that is at no point, which is when it hath no Elevation. I $ay, in this ca$e I $tand in $ome doubt; and that I do not re$olute- ly deny the thing, the rea$on depends on another Accident which $eems no le$s $trange, and yet I have a very nece$$ary Demon$trati- on for it. And the Accident is this, the Impo$$ibility of di$tending a Rope, $o, as that it may be $tretched right out, and parallel to the Horizon, but that it alwaies $wayes and bendeth, nor is there any Force that can $tretch it otherwi$e.</P> <P>SALV. So then, <I>Sagredus,</I> your wonder cea$eth in this ca$e of the Rope becau$e you have the Demon$tration of it. But if we $hall well con$ider the matter, it may be we $hall find $ome corre- $pondence between the Accident of the Project and this of the Rope. The Curvity of the Line of the Horizontal Projection $eem- eth to be derived from two Forces, of which one, (which is that of the Projicient) driveth it Horizontally, and the other, (which is the Gravity of the Project) draweth it downwards Perpendicu- larly. Now $o in the $tretching of the Rope, there are the Forces of tho$e that pull it Horizontally, and there is al$o the weight of the Rope it $elf, which naturally inclineth it downwards. The$e two effects are very much alike in the generation of them. And if you allow the weight of the Rope $o much $trength and power as to be able to oppo$e and overcome any whatever Immen$e Force, that would di$tend it right out, why will you deny the like to the weight of the Bullet? But be$ides, I $hall tell you, and at once procure your <foot>Ii 2 wonder</foot> <p n=>244</p> wonder, and delight, that the Rope thus tentered, and $tretcht little or much, doth $hape it $elf into Lines that come very near to Para- bolical, and the re$emblance is $o great, that if you draw a Para- bolical Line upon a plain Superficies that is erect unto the Horizon, and holding it rever$ed, that is with the Vertex downwards and with the Ba$e Parallel to the Horizon, you cau$e a Chain to be held pendent, and $u$tained at the extreams of the Ba$e of the De$cribed Parabola, you $hall $ee the $aid Chain, as you $laken it more or le$s, to incurvate and apply it $elf to the $ame Parabola, and this $ame Application $hall be $o much the more exact, when the de$cribed Parabola is le$s curved, that is more di$tended: So that in Parabola's de$cribed with Elevations under <I>gr.</I> 45, the Chain an$wereth the Parabola almo$t to an hair.</P> <P>SAGR. It $eems then that with $uch a Chain wrought into $mall Links one might in an in$tant trace out many Parabolick Lines up- on a plain Superficies.</P> <P>SALV. One might, and that al$o with no $mall commodity, as I $hall tell you anon.</P> <P>SIMP. But before you pa$s any farther, I al$o would gladly be a$certained at lea$t in that Propo$ition of which you $ay there is a very nece$$ary Demon$tration, I mean that of the Impo$$ibility of di$tending a Rope, by any whatever immen$e Force, right out and equidi$tant from the Horizon.</P> <P>SAGR. I will $ee if I remember the Demon$tration, for under- $tanding of which it is nece$$ary, <I>Simplicius,</I> that you $uppo$e for true, that which in all Mechanick In$truments is confirmed, not on- ly by Experience, but al$o by Demon$tration: and this it is, That the Velocity of the Mover, though its Force be very $mall, may overcome the Re$i$tance, though very great, of a Re$i$ter, which mu$t be moved $lowly when ever the Velocity of the Mover hath greater proportion to the Tardity of the Re$i$ter, than the Re$i- $tance of that which is to be moved hath to the Force of the Mo- ver.</P> <P>SIMP. This I know very well, and it is demon$trated by <I>Ari- $totle</I> in his Mechanical Que$tions, and is manife$tly $een in the Lea- ver and in the Stiliard, in which the Roman which weigheth not above 4 pounds, will lift up a weight of 400 in ca$e the di$tance of the $aid Roman from the Center on which the Beam turneth be more than an hundred times greater than the di$tance of that point at which the great weight hangeth from the $ame Center: and this cometh to pa$s becau$e in the de$cent which the Roman maketh pa$$eth a Space above an hundred times greater than the Space which the great weight mounteth in the $ame Time: Which is all one as to $ay, that the little Roman moveth with a Velocity above an hundred times greater than the Velocity of the great Weight.</P> <foot>SAGR.</foot> <p n=>245</p> <P>SAGR. You argue very well, and make no $eruple at all of granting, that be the Force of the Mover never $o $mall it $hall $u- perate any what ever great Re$i$tance at all times when that $hall more exceed in Velocity than this doth in Force and Gravity. Now come we to the ca$e of the Rope. And drawing a $mall Scheme be plea$ed to under$tand for once that this Line A B, re$t- ing upon the two fixed and $tanding points A and B, to have hang- ing at its ends, as you $ee, two immen$e Weights C and D, which drawing it with great Force make it to $tand directly di$tended, it being a $imple Line without any gravity. And here I proceed, and tell you, that if at the mid$t of that which is the point E, you $hould hang any never $o little a Weight, as is this H, the Line A B would yield, and inclining towards the point F, and by con$equence lengthening, will con$train the two great Weights C and D to a$cend upwards: which I demon$trate to you in this manner: About the two points A and B as Centers I de$cribe two Quadrants E F G, and E L M, and in regard that the two Semidiameters AI and B L are equal to the two Semidiameters A E and E B, the exce$- $es F I and F L $hall be the quantity of the prolongations of the parts A F and F B, above A E and E B; and of con$equence $hall <fig> determine the A$cents of the Weights C and D, in ca$e that the Weight H had had a power to de$cend to F: which might then be in ca$e the Line E F, which is the quantity of the De$cent of the $aid Weight H, had greater proportion to the Line F I which de- termineth the A$cent of the two Weights C & D, than the pondero- $ity of both tho$e Weights hath to the pondero$ity of the Weight H. But this will nece$$arily happen, be the pondero$ity of the Weights C and D never $o great, and that of H never $o $mall; for the exce$s of the Weights C and D above the Weight His not $o great, but that the exce$s of the Tangent E F above the part of the Secant F I may bear a greater proportion. Which we will prove thus: Let there be a Circle who$e Diameter is G A I; and look what proportion the pondero$ity of the Weights C and D have to the pondero$ity of H, let the Line B O have the $ame proportion to another, which let be C, than which let D be le$$er: So that B O <foot>$hall</foot> <p n=>246</p> $hall have greater proportion to D, than to C. Unto O B and D take a third proportional B E; and as O E is to E B, $o let the Dia- meter G I (prolonging it) be to I F: and from the Term F draw the Tangent F N. And becau$e it hath been pre$uppo$ed, that as O E is to E B, $o is G I to I F: therefore, by Compo$ition, as O B is to B E, $o is G F to F I: But betwixt O B and B E the Mean- proportional is D; and betwixt G F and F I the Mean-proporti- onal is N F: Therefore N F hath the $ame proportion to F I that O B hath to D: which proportion is greater than that of the Weights C and D to the Weight H. Therefore, the De$cent or Velocity of the Weight H having greater proportion to the A$cent or Velocity of the Weights C and D, than the pondero$ity of the $aid Weights C and D hath to the pondero$ity of the Weight H: It is manife$t, that the Weight H $hall de$cend, that is, that the Line A B $hall depart from Horizontal Rectitude. And that which befalleth the right Line A B deprived of Gravity in ca$e any $mall Weight H cometh to be hanged at the $ame in E, happens al$o to the $aid Rope A B, $uppo$ed to be of ponderous Matter, without the addition of any other Grave Body; for that the Weight of the Matter it $elf compounding the $aid Rope AB is $u$pended thereat.</P> <P>SIMP. You have fully $atisfied me; therefore <I>Salviatus</I> may ac- cording to his promi$e declare unto us, what the Commodity is that may be drawn from $uch like Chains, and after that relate unto us tho$e Speculations which have been made by our <I>Accademian</I> touching the Force of Percu$$ion.</P> <P><I>S</I>ALV. We are for this day $ufficiently employed in the Con- templations already delivered, and the Time, which is pretty late, would not be enough to carry us through the matters you mention; therefore we $hall defer our Conference till $ome more convenient time.</P> <P>SAGR. I concur with you in opinion, for that by $undry di$- cour$es that I have had with the Friends of our <I>Academick</I> I have learnt that this Argument of the Force of Percu$$ion is very ob- $cure, nor hath hitherto any one that hath treated thereof penetra- ted its intricacies, full of darkne$s, and altogether remote from mans fir$t imaginations: and among$t the Conclu$ions that I have heard of, one runs in my mind that is very extravagant and odde, namely, That the Force of Percu$$ion is Interminate, if not Infi- nite. We will therefore attend the lea$ure of <I>Salviatus.</I> But for the pre$ent, tell me what things are tho$e which are written at the end of the Treati$e of Projects?</P> <P>SALV. The$e are certain Propo$itions touching the Center of Gravity of Solids, which our <I>Academick</I> found out in his youth, <marg>* <I>Fredericus Co- mandinus.</I></marg> conceiving that what ^{*} <I>Frederico Comandino</I> had writ touching the <foot>$ame</foot> <p n=>247</p> $ame was not altogether without Imper$ection. He therefore thought that with the$e Propo$itions, which here you $ee written, he might $upply that which is wanting in the Book of <I>Comandine</I>; and he applyed him$elf to the $ame at the In$tance of the mo$t Illu$trious Lord Marque$s <I>Guid' Vbaldo dal Monte,</I> the mo$t ex- cellent Mathematician of his Time, as his $everal Printed Works do $peak him; and gave a Copy thereof to that Noble Lord with thoughts to have pur$ued the $ame Argument in other Solids not mentioned by <I>Comandine:</I> But he chanced after $ome Time to meet with the ^{*} Book of <I>Signore Luca Valerio,</I> a mo$t famous <marg>* <I>De.</I></marg> Geometrician, and $aw that he re$olveth all the$e matters with- out omi$$ion of any thing, he proceeded no farther, although his Agre$$ions were by methods very different from the$e of <I>Signore Valerio.</I></P> <P>SAGR. It would be a favour, therefore, if, for this time, which interpo$eth between this and our next Meeting, you would plea$e to leave the Book in my hands: for I $hall all the while be read- ing and $tudying the Propo$itions that are con$equently therein writ.</P> <P>SALV. I $hall very willingly obey your Command; and hope that you will take plea$ure in the$e Propo$itions.</P> <foot>APPEN-</foot> <p n=>248</p> <head>AN APPENDIX, In which is contained certain THE OREMS and their DEMONSTRATIONS: Formerly written by the $ame Author, touching the <I>CENTER</I> of <I>GRAVITY,</I> of SOLIDS.</head> <head>POSTVLATVM.</head> <P><I>We pre$uppo$e equall Weights to be alike di$po- $ed in $ever all Ballances, if the Center of Gra- vity of $ome of tho$e Compounds $hall divide the Ballance according to $ome proportion, and the Ballance $hall al$o divide their Center of Gravity according to the $ame proportion.</I></P> <head>LEMMA.</head> <P><I>Let the line A B be cut in two equall parts in C, who$e half A C let be divided in E, $o that as B E is to E A, $o may A E be to E C. I $ay that B E is double</I> <fig> <I>to E A. For as B E is to E A, $o is E A to E C: there- fore by Compo$ition and by Permutation of Proportion, as B A is to A C, $o is A E to E C: But as A E is to E C, that is, B A to A C, $o is B E to E A: Wherefore B E is double to E A.</I></P> <P><I>This $uppo$ed, we will Demon$trate, That,</I></P> <foot>PROP.</foot> <p n=>249</p> <head>PROPOSITION.</head> <P>If certain Magnitudes at any Rate equally exceed- ing one another, and who$e exce$s is equal to the lea$t of them, be $o di$po$ed in the Balance, as that they hang at equal di$tances, to divide the Center of Gravity of the whole Balance $o, that the part towards the le$$er Magnitudes be double to the remainder.</P> <P><I>In the ^{*} Ballance A B, therefore, let there be $u$pended at equal di-</I> <marg>* Or Beam.</marg> <I>$tances any number of Magnitudes, as hath been $aid, F, G, H, K, N; of which let the lea$t be N, and let the points of the Su$pen$ions be A, C, D, E, B, and let the Center of Gravity of all the Magnitudes $o di$po$ed be X. It is to be proved that the part of the Ballance B X towards the le$$er Magnitudes is double to the remaining part X A.</I></P> <P><I>Let the Ballance be divided in two equal parts in D, for it mu$t ei- ther fall in $ome point of the Su$pen$ions, or el$e in the middle point be- tween two of the points of the Su$pen$ions: and let the remaining di- $tances of the Su$pen$ions which fall between A and D, be all divided into halves by the Points M and I; and let all the Magnitudes be divi-</I> <fig> <I>ded into parts equal to N: Now the parts of F $hall be $o many in num- ber, as tho$e Magnitudes be which are $u$pended at the Ballance, and the parts of G one fewer, and $o of the re$t. Let the parts of F therefore be N, O, R, S, T, and let tho$e of G be N, O, R, S, tho$e of H al$o N, O, R, then let tho$e of K be N, O: and all the Magnitudes in which are N $hall be equal to F; and all the Magnitudes in which are O $hall be equal to G; and all the Magnitudes in which are R $hall be equal to H; and tho$e in which S $hall be equal to K; and the Magnitude T is equal to N. Becau$e therefore all the Magnitudes in which are N are equal to one another, they $hall equiponderate in the point D, which divideth the Ballance into two equal parts; and for the $ame cau$e all the Magnitudes in which are O do equiponderate in I; and tho$e in which are R in C; and in which are S in M do equi- ponderate; and T is $u$pended in A. Therefore in the Ballance A D at the equal di$tances D, I, C, M, A, there are Magnitudes $u$pended ex- ceeding one another equally, and who$e exce$s is equal to the lea$t: and the greate$t, which is compounded of all the N N hangeth at D, the</I> <foot>Kk <I>lea$t</I></foot> <p n=>250</p> <I>lea$t which is T hangeth at A; and the re$t are ordinately di$po$ed. And again there is another Ballance A B in which other Magnitudes equal in number and Magnitude to the former are di$po$ed in the $ame order. Wherefore the Ballances A B and A D are divided by the Cen- ter of all the Magnitudes according to the $ame proportion: But the Center of Gravity of the afore$aid Magnitudes is X: Wherefore X divideth the Ballances B A and A D according to the $ame proportion; $o that as B X is to X A, $o is X A to X D: Wherefore B X is double to X A, by the Lemma aforegoing: Which was to be proved.</I></P> <head>PROPOSITION.</head> <P>If in a Parabolical Conoid Figure be de$cribed, and another circum$cribed by Cylinders of equal Altitude; and the Axis of the $aid Co- noid be divided in $uch proportion that the part towards the Vertex be double to that to- wards the Ba$e; the Center of Gravity of the in$cribed Figure of the Ba$e portion $hall be neare$t to the $aid point of divi$ion; and the Center of Gravity of the circum$cribed from the Ba$e of the Conoid $hall be more remote: and the di$tance of either of tho$e Centers from that $ame point $hall be equal to the Line that is the $ixth part of the Altitude of one of the Cylinders of which the Figures are com- po$ed.</P> <P><I>Take therefore a Parabolical Conoid, and the Figures that have been mentioned: let one of them be in$cribed, the other circum- $cribed; and let the Axis of the Conoid, which let be A E, be di- vided in N, in $uch proportion as that A N be double to N E. It is to be proved that the Center of Gravity of the in$cribed Figure is in the Line N E, but the Center of the circum$cribed in the Line A N. Let the Plane of the Figures $o di$po$ed be cut through the Axis, and let the Section be that of the Parabola B A C: and let the Section of the cutting Plane, and of the Ba$e of the Conoid be the Line B C; and let the Sections of the Cylinders be the Rectangular Figures; as ap- peareth in the de$cription. Fir$t, therefore, the Cylinder of the in$cri- bed who$e Axis is D E, hath the $ame proportion to the Cylinder who$e Axis is D Y, as the Quadrate I D hath to the Quadrate S Y; that is, as D A hath to A Y: and the Cylinder who$e Axis is D Y is</I> potentia <foot><I>to</I></foot> <p n=>251</p> <I>to the Cylinder Y Z as S Y to R Z, that is, as Y A to A Z: and, by the $ame rea$on, the Cylinder who$e Axis is Z Y is to that who$e Axis is Z V, as Z A is to A V. The $aid Cylinders, therefore, are to one ano- ther as the Lines D A, A Y; Z A, A V: But the$e are equally exceed- ing to one another, and the exce$s is equal to the lea$t, $o that A Z is double to A V; and A Y is triple the</I> <fig> <I>$ame; and D A Quadruple. Tho$e Cylinders, therefore, are certain Mag- nitudes in order equally exceeding one another, who$e exce$s is equal to the lea$t of them, and is the Line X M, in which they are $u$pended at equal di$tances (for that each of the Cy- linders hath its Center of Gravity in the mia$t of the Axis.) Wherefore, by what hath been above demon$tra- ted, the Center of Gravity of the Mag- nitude compounded of them all divi- deth the Line X M $o, that the part towards X is double to the re$t. Divide it, therefore, and, let X</I> <G>a</G> <I>be double</I> <G>a</G> <I>M: therefore is</I> <G>a</G> <I>the Center of Gravity of the in$cribed Fi- gure. Divide A V in two equal parts in</I> <G>e</G>: <G>e</G> <I>X $hall be double to M E: But X</I> <G>a</G> <I>is double to</I> <G>a</G> <I>M: Wherefore</I> <G>e</G> <I>E $hall be triple E</I> <G>a.</G> <I>But</I> <G>a</G> <I>E is triple E N: It is manife$t, therefore, that E N is greater than E X; and for that cau$e</I> <G>a,</G> <I>which is the Center of Gravity of the in- $cribed Figure, cometh nearer to the Ba$e of the Conoid than N. And becau$e that as A E is to E N, $o is the part taken away</I> <G>e</G> <I>E to the part taken away E</I> <G>a</G>: <I>and the remaining part $hall be to the remaming part, that is, A</I> <G>e</G> <I>to N</I> <G>a,</G> <I>as A E to E N. Therefore</I> <G>a</G> <I>N is the third part of A</I> <G>e,</G> <I>and the $ixt part of A V. And in the $ame manner the Cylinders of the circum$cribed Figure may be demon$trated to be equally exceeding one another, and the exce$s to me equal to the least; and that they have their Centers of Gravity at equal di$tances in the Line</I> <G>e</G> <I>M. If therefore</I> <G>e</G> <I>M be divided in</I> <G>p,</G> <I>$o as that</I> <G>e p</G> <I>be double to the remaining part</I> <G>p</G> <I>M;</I> <G>p</G> <I>$hall be the Center of Gravity of the whole circum$cribed Magnitude. And $ince</I> <G>e p</G> <I>is double to</I> <G>p</G> <I>M; and A</I> <G>e</G> <I>le$s than double EM: (for that they are equal:) the whole A E $hall be le$s than triple E</I> <G>p</G><I>: Where- fore E</I> <G>p</G> <I>$hall be greater than E N. And, $ince</I> <G>e</G> <I>M is triple to M</I> <G>p,</G> <I>and M E with twice</I> <G>e</G> <I>A is likewi$e triple to M E: the whole A E with A</I> <G>e</G> <I>$hall be triple to E</I> <G>p</G><I>: But A E is triple to E N: Wherefore the remaining part A</I> <G>e</G> <I>$hall be triple to the remaining part</I> <G>p</G> <I>N. Therefore N</I> <G>p</G> <I>is the $ixth part of A V. And the$e are the things that were to be demon$trated.</I></P> <foot>Kk 2 COROL-</foot> <p n=>252</p> <head>COROLLARY.</head> <P>Hence it is manife$t, that a Conoid may be in$cribed in a Para- bolical Figure, and another circum$cribed, $o, as that the Centers of their Gravities may be di$tant from the point N le$s than any Line given.</P> <P><I>For if we a$$ume a Line $excuple of the propo$ed Line, and make the Axis of the Cylinders, of which the Figures are compounded given le$$er than this a$$umed Line, there $hall fall Lines between the Centers of Gravities of the$e Figures and the mark N that are le$s than the Line propo$ed.</I></P> <P>The former Propo$ition another way.</P> <P><I>Let the Axis of the Conoid (which let be C D) be divided in O, $o, as that C O be double to O D. It is to be proved that the Center of Gravity of the in$cribed Figure is in the Line O D; and the Center of the circum$cribed in C O. Let the Plane of the Fi- gures be cut through the Axis and C, as hath been $aid. Becau$e there- fore the Cylinders S N, T M, V I,</I> <fig> <I>X E are to one another as the Squares of the Lines S D, T N, V M, X I; and the$e are to one another as the Lines N C, C M, C I, C E: but the$e do exceed one another equally; and the exce$s is equal to the lea$t, to wit, C E: And the Cylinder T M is equal to the Cylinder Q N; and the Cylinder V I equal to P N; and X E is equal to L N: Therefore the Cylin- ders S N, Q N, P N, and L N do equally exceed one another, and the exce$s is equal to the lea$t of them, namely, to the Cylinder L N. But the exce$s of the Cylinder S N, above the Cylinder Q N is a Ring who$e height is Q T; that is, N D; and its breadth S Q. And the exce$s of the Cylinder Q N above P N, is a Ring, who$e breadth is Q P. And the exce$s of the Cylinder P N above L N is a Ring, who$e breadth is P L. Wherefore the $aid Rings S Q, Q P, P L, are equal to another, and to the Cylinder L N. Therefore the Ring S T equalleth the Cylinder X E: the Ring Q V, which is double to S T, equalleth the Cylinder V I; which likewi$e is double to the</I> <foot><I>Cylinder</I></foot> <p n=>253</p> <I>Cylinder X E: and for the $ame cau$e the Ring P X is equal to the Cylinder T M; and the Cylinder L E $hall be equal to the Cylinder S N. In the Beam or Ballance, therefore, K F connecting the middle points of the Right-lines E I and D N, and cut into equal parts in the points H and G, are certain Magnitudes $u$pended, to wit the Cylinders S N, T M, V I, X E; and the Center of Gravity of the fir$t Cylinder is K; and of the $econd H; of the third G; of the fourth F. And we have another Ballance M K, which is the half of the $aid F K, and a like number of points di$tributed into equal parts, to wit, M H, H N, N K, and on it other Magnitudes, equal in number and bigne$s to tho$e which are on the Beam F K, and having the Centers of Gravity in the points M, H, N, and K, and di$po$ed in the $ame order. For the Cylinder L E hath its Center of Gravity in M; and is equal to the Cylinder S N that hath its Center in K: And the Ring P X hath the Center H; and is equal to the Cylinder T M, who$e Center is H: And the Ring Q V ha- ving the Center N is equal to the V I who$e Center is G: And la$tly, the Ring S T having the Center K, is equal to the Cylinder X E who$e Center is F. Therefore the Center of Gravity of the $aid Magnitudes divideth the Beam in the $ame proportion: But the Center of them is one, and therefore $ome point common to both the Beams or Ballance, which let be Y. Therefore F Y and Y K $hall be as K Y and Y M. F Y therefore is double to Y K: and C E being divided into two equal parts in Z, Z F, $hall be double to K D: and for that cau$e Z D triple to D Y: But to the Right Line D O C D is triple: Therefore the Right Line D O is greater than D Y: And for the like cau$e Y the Center of the in$cribed Figure approacheth nearer the Ba$e than the point O. And becau$e as C D is to D O, $o is the part taken away Z D to the part ta- ken away D Y; the remaining part C Z $hall be to the remaining part Y O, as C D is to D O; that is Y O $hall be the third part of C Z; that is, the $ixth part of C E. Again we will, by the $ame rea$on, de- mon$trate the Cylinders of the circum$cribed Figure to exceed one ano- ther equally, and that the exce$s is equal to the lea$t, and that their Centers of Gravity are con$tituted in equal di$tances upon the Beam K Z: and likewi$e that the Rings equal to tho$e $ame Cylinders are in like manner di$po$ed on another Beam K G, the half of the $aid K Z, and that therefore the Center of Gravity of the circum$cribed Figure, which let be R, $o divideth the Beam, as that Z R is to R K, as K R is to R G. Therefore Z R $hall be double to R K: But C Z is equal to the Right Line K D, and not double to it. The whole C D $hall be le$$er than triple to D R: Wherefore the Right Line D R is greater than D O; that is to $ay, the Center of the circum$cribed Figure recedeth from the Ba$e more than the point O. And becau$e Z K is triple to K R; and K D with twice Z C is triple to K D; the whole C D with C Z $hall be triple to D R: But C D is triple to D O: Wherefore the remaining part C Z $hall be triple to the remaining part R O; that is, O R</I> <foot><I>is</I></foot> <p n=>254</p> <I>is the $ixth part of E C: Which wa<*> the Propo$ition.</I></P> <P><I>This being pre-demon$trated, we will prove that</I></P> <head>PROPOSITION.</head> <P>The Center of Gravity of the Parabolick Conoid doth $o divide the Axis, as that the part towards the Vertex is double to the re- maining part towards the Ba$e.</P> <P><I>Let there be a Parabolick Conoid who$e Axis let be A B divided in N $o as that A N be double to N B. It is to be proved that the Cen- ter of Gravity of the Conoid is the point N. For if it be not N, it $hall be either above or below it. Fir$t let it be below; and let it be X: And $et off upon $ome place by it $elf the Line L O equal to N X; and let L O be divided at plea$ure in S: and look what proportion B X and O S both together have to O S, and the $ame $hall the Conoid have to the Solid R. And in the Conoid let Figures be de$cribed by Cylinders having equal Altitudes, $o, as that that which lyeth between the Center of Gravity and the point N be le$s than L S: and let the exce$s of the Conoid above it be le$s than the Solid R: and that this may be done is clear. Take therefore the in$cribed, who$e Center of Gravity let be I: now I X $hall be greater than S O: And becau$e that as X B with S O is to S O, $o is the Conoid to the Solid R: (and R is greater than the exce$s by which the Conoid exceeds the in$cribed Figure:) the proporti- on of the Conoid to the $aid exce$s $hall be greater than both B X and O S unto S O: And, by Divi$ion, the in$cribed Figure $hall have grea- ter proportion to the $aid exce$s than B X to S O: But B X hath to X I a proportion yet le$s than to S O: Therefore the in$cribed Figure $hall have much greater proportion to the re$t of the proportions than B X to X I: Therefore what proportion the in$cribed Figure hath to there$t of the portions, the $ame $hall a certain other Line have to X I: which $hall nece$$arily be greater than B X: Let it, therefore, be M X. We have therefore the Center of Gravity of the Conoid X: But the Center of Gravity of the Figure in$cribed in it is I: of the re$t of the portions by which the Conoid exceeds the in$cribed Figure the Center of Gravity $hall be in the Line X M, and in it that point in which it $hall be $o terminated, that look what proportion the in$cribed Figure hath to the exce$s by which the Conoid exceeds it, the $ame it $hall have to X I: But it hath been proved, that this proportion is that which M X hath to X I: Therefore M $hall be the Center of Gravity of tho$e pro- portions by which the Conoid exceeds the in$cribed Figure: Which certainly cannot be. For if along by M a Plane be drawn equidi$tant to the Ba$e of the Conoid, all tho$e proportions $hall be towards one and</I> <foot><I>the</I></foot> <p n=>255</p> <I>the $ame part, and not by it divided. Therefore the Center of Gravity of the $aid Conoid is not below the point N: Neither is it above. For, if it may, let it be H: and again, as before, $et the Line L O by it $elf equalto the $aid H N, and divided at plea$ure in S: and the $ame pro- portion that B N and S O both together have to S L, let the Conoid have to R: and about the Conoid let a Figure be circum$cribed con$i- $ting of Cylinders, as hath been $aid: by which let it be exceeded a le$s quantity than that of the Solid R: and let the Line betwixt the Center of Gravity of the circum$cribed Figure and the point N be le$$er than S O: the remainder V H $hall be greater than S L. And becau$e that as both B N and O S is to SL, $o is the</I> <fig> <I>Conoid to R: (and R is greater than the exce$s by which the circum- $cribed Figure exceeds the Conoid:) Therefore B N and S O hath le$s pro- portion to S L than the Conoid to the $aid exce$s. And B V is le$$er than both B N and S O; and V H is grea- ter than S L: much greater proporti- on, therefore, hath the Conoid to the $aid proportions, than B V hath to V H. Therefore whatever proporti- on the Conoid hath to the $aid pro- portions, the $ame $hall a Line greater than B V have to V H. Let the $ame be M V: And becau$e the Center of Gravity of the circum$cribed Figure is V, and the Center of the Conoid is H. and $ince that as the Conoid to the re$t of the proportions, $ois M V to V H, M $hall be the Center of Gravity of the remaining proportions: which likewi$e is impo$$ible: Therefore the Center of Gravity of the Conoid is not above the point N: But it hath been de- mon$trated that neither is it beneath: It remains, therefore, that it ne- ce$$arily be in the point N it $elf. And the $ame might be demon$trated of Conoidal Plane cut upon an Axis not erect. The $ame in other terms, as appears by what followeth:</I></P> <head>PROPOSITION.</head> <P>The Center of Gravity of the Parabolick Co- noid falleth betwixt the Center of the cir- cum$cribed Figure and the Center of the in- $cribed.</P> <foot><I>Let</I></foot> <p n=>256</p> <P><I>Let there be a Conoid who$e Axis is A B, and the Center of the circum$cribed Figure C, and the Center of the in$cribed O. I $ay the Center of the Conoid is betwixt the points C and O. For if not, it $hall be either above them, or below them, or in one of them. Let it be below, as in R. And becau$e R is the Center of Gravity of the whole Conoid; and the Center of Gravity of the in$cribed Figure is O: Therefore of the remaining proportions by which the Conoid exceeds the in$cribed Figure the Center of Gravity $hall be in the Line O R ex- tended towards R, and in that point in which it is $o determined, that, what proportion the $aid proportions have to the in$cribed Figure, the $ame $hall O R have to the Line falling betwixt R and that falling point. Let this proportion be that of O R to R X. Therefore X falleth either without the Conoid or within, or in its</I> <fig> <I>Ba$e. That it falleth without, or in its Ba$e it is already manife$t to be an ab$ur- dity. Let it fall within: and becau$e X R is to R O, as the in$cribed Figure is to the exce$s by which the Conoid exceeds it; the $ame proportion that B R hath to R O, the $ame let the in$cribed Figure have to the Solid K: Which nece$$arily $hall be le$$er than the $aid exce$s. And let another Figure be in$cribed which may be exceeded by the Conoid a le$s quantity than is K, who$e Center of Gravity falleth betwixt O and C. Let it be V. And, becau$e the fir$t Figure is to K as B R to R O, and the $e- cond Figure, who$e Center V is greater than the fir$t, and exceeded by the Conoid a le$s quantity than is K; what proportion the $econd Figure hath to the exce$s by which the Conoid exceeds it, the $ame $hall a Line greater than B R have to R V. But R is the Center of Gra- vity of the Conoid; and the Center of the $econd in$cribed Figure V: The Center therefore of the remaining proportions $hall be without the Conoid beneath B: Which is impo$$ible. And by the $ame means we might demon$trate the Center of Gravity of the $aid Conoid not to be in the Line C A. And that it is none of the points betwixt C and O is manife$t. For $ay, that there other Figures de$cribed, greater $omething than the in$cribed Figure who$e Center is O, and le$s than that circum$cribed Figure who$e Center is C, the Center of the Conoid would fall without the Center of the$e Figures: Which but now was concluded to be impo$$ible: It re$ts therefore that it be betwixt the Cen- ter of the circum$cribed and in$cribed Figure. And if $o, it $hall ne- ce$$arily be in that point which divideth the Axis, $o as that the part towards the Vertex is double to the remainder; $ince N may circum- $cribe and in$cribe Figures, $o, that tho$e Lines which fall between</I> <foot><I>their</I></foot> <p n=>257</p> <I>their Centers and the $aid points, may be le$$er than any other Lines. To expre$s the $ame in other terms, we have reduced it to an impo$$ibi- lity, that the Center of the Conoid $hould not fall betwixt the Centers of the in$cribed and circum$cribed Figures.</I></P> <head>PROPOSITION.</head> <P>Suppo$ing three proportional Lines, and that what proportion the lea$t hath to the exce$s by which the greate$t exceeds the lea$t, the $ame $hould a Line given have to two thirds of the exce$s by which the greate$t exceeds the middlemo$t: and moreover, that what pro- portion that compounded of the greate$t, and of double the middlemo$t, hath unto that com- pounded of the triple of the greate$t and mid- dlemo$t, the $ame hath another Line given, to the exce$s by which the greate$t exceeds the middle one; both the given Lines taken toge- ther $hall be a third part of the greate$t of the proportional Lines.</P> <P><I>Let A B, B C, and B F, be three proportional Lines; and what proportion B F hath to F A, the $ame let M S have to two thirds of C A. And what proportion that compounded of A B and the double of B C hath to that compounded of the triple of both A B and B C, the $ame let another, to wit S N, have to A C. Becau$e therefore that A B, B C, and C F,</I> <fig> <I>are proportionals, A G and C F $hall, for the $ame rea$on, be likewi$e $o. Therefore, as A B is to B C, $o is A C to C F: and as the triple of A B is to the triple of B C, $o is A C to C F: Therefore, what proportion the triple of A B with the triple of B C hath to the triple of C B, the $ame $hall A C have to a Line le$s than C F. Let it be C O. Wherefore by Compo$ition and by Conver$ion of proportion, O A $hall have to A C, the $ame proportion, as triple A B with Sextuple B C, hath to triple A B with triple B C. But A C hath to S N the $ame proportion, that triple A B with triple B C hath to A B with double B C: Therefore,</I> ex equali, <I>O A to NS $hall have the $ame proportion, as triple A B with Sexcuple B C hath to A B with</I> <foot>Ll <I>double</I></foot> <p n=>258</p> <I>double B C: But triple A B with $excuple B C, are triple to A B with double B C. Therefore A O is triple to S N.</I></P> <P><I>Again, becau$e O C is to C A as triple C B is to triple A B with tri- ple C B: and becau$e as C A is to A F, $o is triple A B to triple B C: Therefore,</I> ex equali, <I>by perturbed proportion, as O C is to C F, $o $hall triple A B be to triple A B with treble B C: And, by Conver$ion of proportion, as O F is to F C, $o is triple B C to triple A B with triple B C: And as C F is to F B, $o is A C to C B, and triple A C to triple C B: Therefore,</I> ex equali, <I>by Perturbation of proportion, as O F is to F B, $o is triple A C to the triple of both A B and A C together. And becau$e F C and C A are in the $ame proportion as C B and B A; it $hall be that as F C is to C A, $o $hall B C be to B A. And, by Com- po$ition, as F A is to A C, $o are both B A and B C to B A: and $o the triple to the triple: Therefore as F A is to A C, $o the compound of tri- ple B A and triple B C is to triple A B. Wherefore, as F A is to two thirds of A C, $o is the compound of triple B A and triple B C to two thirds of triple B A; that is, to double B A: But as F A is to two thirds of A C, $o is F B to M S: Therefore, as F B is to M S, $o is the compound of triple B A and triple B C to double B A: But as O B is to F B, $o was Sexcuple A B to triple of both A B and B C: Therefore,</I> ex equa- li, <I>O B $hall have to M S the $ame proportion as Sexcuple A B hath to double B A. Wherefore M S $hall be the third part of O B: And it hath been demon$trated, that S N is the third part of A O: It is mani- fe$t therefore, that MN is a third part likewi$e of A B: And this is that which was to be demon$trated.</I></P> <head>PROPOSITION.</head> <P>Of any <I>Fru$tum</I> or Segment cut off from a Para- bolick Conoid the Center of Gravity is in the Right Line that is Axis of the <I>Fru$tum</I>; which being divided into three equal parts the Cen- ter of Gravity is in the middlemo$t and $o di- vides it, as that the part towards the le$$er Ba$e hath to the part towards the greater Ba$e, the $ame proportion that the greater Ba$e hath to the le$$er.</P> <P><I>From the Conoid who$e Axis is R B let there be cut off the Solid who$e Axis is B E; and let the cutting Plane be equidi$taut to the Ba$e: and let it be cut in another Plane along the Axis erect upon the Ba$e, and let it be the Section of the Parabola V R C: R B $hall be the Diameter of the proportion, or the equidi$tant Diameter</I> <foot>LM, V C:</foot> <p n=>259</p> <I>L M, V C: they $hall be ordinately applyed. Divide therefore E B in- to three equal parts, of which let the middlemo$t be Q Y: and divide this $o in the point I that Q I may have the $ame proportion to I Y, as the Ba$e who$e Diameter is V C hath to the Ba$e who$e Diameter is L M; that is, that the Square V C hath to Square L M. It is to be de- mon$trated that I is the Center of Gravity of the Fru$trum L M C. Draw the Line N S, by the by, equall to B R: and let S X be equal to E R: and unto N S and S X a$$ume a third proportional S G: and as N G is to G S, $o let B Q be to I O. And it nothing matters whether the point O fall above or below L M. And becau$e in the Section V R C the Lines L M and V C are ordinately</I> <fig> <I>applyed, it $hall be that as the Square V C is to the Square L M, $o is the Line B R to R E: And as the Square V C is to the Square L M, $o is Q I to I Y: and as B R is to R E, $o is N S to S X: There- fore Q I is to I Y, as R S is to S X. Where- fore as G Y is to Y I, $o $hall both N S and S X be to S X: and as E B is to Y I, $o $hall the compound of triple N S and tri- ple S X be to S X: But as E B is to B Y, $o is the compound of triple N S and S X both together to the compound of N S and S X: Therefore, as E B is to B I, $o is the compound of triple N S and triple S X to the compound of N S and double S X. Therefore N S, S X, and S G are three proporti- onal Lines: And as S G is to G N, $o is the a$$umed O I to two thirds of E B; that is, to N X: And as the compound of N S and double S X is to the compound of triple N S and triple S X, $o is another a$$u- med Line I B to B E; that is, to N X. By what therefore hath been above demon$trated, tho$e Lines taken together are a third part of N S; that is, of R B: Therefore R B is triple to B O: Wherefore O $hall be the Center of Gravity of the Conoid v R C. And let it be the Cen- ter of Gravity of the</I> Fru$trum <I>L R M of the Conoid: Therefore the Center of Gravity of V L M C is in the Line O B, and in that point which $o terminates it, that as V L M C of the</I> Fru$trum <I>is to the proportion L R M, $o is the Line A O to that which intervenes betwixt O and the $aid point. And becau$e R O is two thirds of R B; and R A two thirds of R E; the remaining part A O $hall be two thirds of the remaining part E B. And becau$e that as the</I> Fru$tum <I>V L M C is to the proportion L R M, $o is N G to G S: and as N G to G S, $o is two thirds of E B to O I: and two thirds of E B is equal to the Line A O: it $hall be that as the</I> Fru$tum <I>V L M O is to the proportion L R M, $o is A O to O I. It is manife$t therefore that of the</I> Fru$tum <I>V L M C the Center of Gravity is the point I, and $o divideth the Axis, <*> that the part towards the le$$er Ba$e is to the part towards the grea-</I> <foot>Ll 2 <I>ter</I></foot> <p n=>260</p> <I>ter, as the double of the greater Ba$e together with the Le$$er is to the double of the le$$er together with the greater. Which is the Propo$ition more elegantly expre$$ed.</I></P> <head>PROPOSITION.</head> <P>If any number of Magnitudes $o di$po$ed to one another, as that the $econd addeth unto the fir$t the double of the fir$t, the third addeth unto the $econd the triple of the fir$t, the fourth addeth unto the third the quadruple of the fir$t, and $o every one of the following ones addeth unto the next unto it the magnitude of the fir$t multiplyed according to the number which it $hall hold in order<*> if, I $ay, the$e Magnitudes be $u$pended ordinarily on the Ballance at equal di$tances; the Center of the <I>Equilibrium</I> of all the compounding Magni- tudes $hall $o divide the Beam, as that the part towards the le$$er Magnitudes is triple to the remainder.</P> <P><I>Let the Beam be L T, and let $uch Magnitudes as were $poken of hang upon it; and let them be A, F, G, H, K; of which A is in the fir$t place $u$pended at T. I $ay, that the Center of the</I> Equi- librium <I>$o cuts the Beam T L as that the part towards T is triple to the re$t. Let T L be triple to L I; and S L triple to L P: and Q L to L N,</I> <fig> <I>and L P to L O: I P, P N, N O, and O L $hall be equal. And in F let a Magnitude be placed double to A; in G another trebble to the $ame; in H ano- ther Quadruple; and $o of the re$t: and let tho$e Magnitudes be taken in which there is A; and let the $ame be done in the Magni- tudes F, G, H, K. And becau$e in F the remaining Magnitude, to wit B, is equal to A; take it</I> <foot><I>double</I></foot> <p n=>261</p> <I>double in G, triple in H, &c. and let tho$e Magnitudes be taken in which there is B: and in the $ame manner let tho$e be taken in which is C, D, and E: now all tho$e in which there is A $hall be equal to K: and the compound of all the B B $hall equal H; and the compound of C C $hall equal G; and the compound of all the D D $hall equal F; and E $hall equal A. And becau$e T I is double to I L, I $hall be the point of the</I> Equilibrium <I>of the Magnitudes compo$ed of all the A A: and likewi<*>e $ince S P is double to P L, P $hall be the point of the</I> Equilibri- um <I>of the compost of B B: and for the $ame cau$e N $hall be the point of the</I> Equilibrium <I>of the compo$t of C C: and O of the compound of D D: and L that of E. Therefore T L is a Beam on which at equal di$tances certain Magnitudes K, H, G, F, A do hang. And again L I is another Ballance, on which, at di$tances in like manner equal, do hang $uch a number of Magnitudes, and in the $ame order equal to the former. For the compound of all the A A, which hang on I, is equal to K hanging at L; and the compo$t of all B B, which is $u$pended at P, is equal to H hanging at P; and likewi$e the compound of C C, which hangeth at N do equal G; and the compo$t of D, which hang on O, are equal to F; and E, hanging on L, is equal to A. Wherefore the Ballances are divided in the $ame proportion by the Center of the com- pounds of the Magnitudes And the Center of the compound of, the $aid Magnitudes is one. Therefore the common point of the Right Line T L, and of the Right Line L I $hall be the Center, which let be X. Therefore as T X is to X L, $o $hall L X be to X I; and the whole T L to the whole L I. But T L is triple to L I: Wherefore T X $hall al$o be triple to X L.</I></P> <head>PROPOSITION.</head> <P>If any number of Magnitudes be $o taken, that the $econd addeth unto the fir$t the triple of the fir$t, and the third addeth unto the $econd the quintuple of the fir$t, and the fourth addeth unto the third the $eptuple of the fir$t, and $o the re$t, every one encrea$ing above the next to it, and proceedeth $till to a new multiplex of the fir$t Magnitude according to the con$e- quent odd numbers, like as the Squares of Lines equally exceeding one another do pro- ceed, whereof the exce$s is equal to the lea$t, and if they be $u$pended on a Ballance at equal Di$tances, the Center of <I>Equilibrium</I> of all the compound Magnitudes $o divideth the Beam <foot>that</foot> <p n=>262</p> that the part towards the le$$er Magnitudes is more than triple the remaining part; and al$o one may take a di$tance that is to the $ame le$s than triple.</P> <P><I>In the Ballance B E let there be Magnitudes, $uch as were $poken off, from which let there be other Magnitudes taken away that were to one another as they were di$po$ed in the precedent, and let it be of the compound of all the A A: the re$t</I> <fig> <I>in which are C $hall be di$tributed in the $ame order, but the greate$t de- ficient. Let E D be triple to D B; and G F triple to F B. D $hall be the Center of the</I> Equilibrium <I>of the compound con- $i$ting of all the A A; and F that of the compound of all the C C. Wherefore the Center of the com- pound of both A A and C C falleth be- tween D and F. Let it be O. It is there- fore manife$t that E O is more than triple to O B; but G O le$s thantriple to the $ame O B: Which was to be demon$trated.</I></P> <foot>PROP</foot> <p n=>263</p> <head>PROPOSITION.</head> <P>If to any Cone or portion of a Cone a Eigure con- $i$ting of Cylinders of equal heights be in$cri- bed and another circum$cribed; and if its Axis be $o divided as that the part which lyeth be- twixt the point of divi$ion and the Vertex be triple to the re$t; the Center of Gravity of the in$cribed Figure $hall be nearer to the Ba$e of the Cone than that point of divi$ion: and the Center of Gravity of the circum$cribed $hall be nearer to the Vertex than that $ame point.</P> <P><I>Take therefore a Cone, who$e Axis is N M. Let it be divided in S $o, as that N S be triple to the remainder S M. I $ay, that the Center of Gravity of any Figure in$cribed, as was $aid, in a Cone doth con$i$t in the Axis N M, and approacheth nearer to the Ba$e of the Cone than the point S: and that the Center of Gravity of the Circum$cribed is likewi$e in the Axis N M, and nearer to the Vertex than is S. Let a Figure therefore be $uppo$ed to be in$cribed by the Cy- linders who$e Axis M C, C B, B E, E A are equal. Fir$t therefore the Cylinder who$e Axis is M C hath</I> <fig> <I>to the Cylinder who$e Axis is C B the $ame proportion as its Ba$e hath to the Ba$e of the other (for their Alti- tudes are equal.) But this propor- tion is the $ame with that which the Square C N hath to the Square N B. And $o we might prove, that the Cy- linder who$e Axis is C B hath to the Cylinder who$e Axis is B E the $ame proportion, as the Square B N hath to the Square N E: and the Cylinder who$e Axis is B E hath to the Cylin- der who$e Axis is E A the $ame pro- portion that the Square E N hath to the Square N A. But the Lines N C, N B, E N, and N A equally exceed one another, and their exce$s equalleth the lea$t, that is N A. Therefore they are certain Magnitudes, to wit, in- $cribed Cylinders having con$equently to one another the $ame proporti- on as the Squares of Lines that equally exceed one another, and the ex-</I> <foot><I>ce$s</I></foot> <p n=>264</p> <I>ce$s of which is equal to the lea$t: and they are $o di$po$ed on the Beam T I that their $everal Centers of Gravity con$i$t in it, and that at equal di$tances. Therefore by the things above demon$trated it appeareth that the Center of Gravity of all $o compo$ed Magnitudes do $o divide the Balance T I, that the part to wards T is more than triple to the remain- der. Let this Center be O. T O therefore is more than triple to O I. But T N is triple to I M. Therefore the whole M O will be le$s than a fourth part of the whole M N, who$e fourth part was $uppo$ed to be M S. It is manife$t, therefore, that the point O doth nearer approach the Ba$e of the Cone than S. And let the circum$cribed Figure be com- po$ed of the Cylinders who$e Axis M C, C B, B E, E A and A N are equal to each other, and, like as in tho$e in$cribed, let them be to one another as the Squares of the Lines M N, N C, B N, N E, A N, which equally exceed one another, and the exce$s is equal to the lea$t A N. Wherefore, by the premi$es, the Center of Gravity of all the Cy- linders $o di$po$ed, which let be V, doth $o divide the Beam R I, that the part towards R, to wit R V, is more than triple to the remaining part V I: but T V $hall be le$s than triple to the $ame. But N T is triple to all I M: Therefore all V M is more than the fourth part of all M N, who$e fourth part was $uppo$ed to be M S. Therefore the point V is nearer to the Vertex than the Point S. Which was to be demon$tra- ted.</I></P> <head>PROPOSITION.</head> <P>About a given Cone a Figure may be circum$cri- bed and another in$cribed con$i$ting of Cylin- ders of equal height, $o, as that the Line which lyeth betwixt the Center of Gravity of the circum$cribed, and the Center of Gravity of the in$cribed, may be le$$er than any Line given.</P> <P><I>Let a Cone be given, who$e Axis is A B; and let the Right Line given be K. I $ay; Let there be placed by the Cylinder L equal to that in$cribed in the Cone, having for its Altitude half of the Axis A B: and let A B be divided in C, $o as that A C be tri- ple to C B: And as A C is to K, $o let the Cylinder L be to the Solid X. And about the Cone let there be a Figure circum$cribed of Cylin- ders that have equal Altitude, and let another be in$cribed, $o as that the circum$cribed exceed the in$cribed a le$s quantity than the Solid X. And let the Center of Gravity of the circum$cribed be E; which falls above C: and let the Center of the in$cribed be S, falling beneath C.</I> <foot><I>I $ay,</I></foot> <p n=>265</p> <I>I $ay now, that the Line E S is le$$er than K. For if not, then let C A be $uppo$ed equal to E O. Becau$e therefore O E hath to K the $ame proportion that L hath to X; and the in$cribed Figure is not le$s than the Cylinder L; and the exce$s with which the $aid Figure is exceeded by the circum$cribed is le$s than the Solid X: therefore the in$cribed Figure $hall have to the $aid exce$s</I> <fig> <I>greater proportion than O E hath to K: But the proportion of O E to K is not le$s than that which O E hath to E S with E S. Let it not be le$s than K. Therefore the in$cribed Figure hath to the exce$s of the circum$cri- bed Figure above it greater propor- tion than O E hath to E S. Therefore as the in$cribed is to the $aid exce$s, $o $hall it be to the Line E S. Let E R be a Line greater than E O; and the Center of Gravity of the in$cribed Figure is S; and the Center of the cir- cum$cribed is E. It is manife$t there- fore, that the Center of Gravity of the remaining proportions by which the circum$cribed exceedeth the in $cribed is in the Line R E, and in that point by which it is $o termina- ted, that as the in$cribed Figure is to the $aid proportions, $o is the Line included betwixt E and that point to the Line E S. And this propor- tion hath R E to E S. Therefore the Center of Gravity of the remain- ing proportions with which the circum$cribed Figure exceeds the in- $cribed $hall be R, which is impo$$ible. For the Plane drawn thorow R equidi$tant to the Ba$e of the Cone doth not cut tho$e proportions. It is therefore fal$e that the Line E S is not le$$er than K. It $hall therefore be le$s. The $ame al$o may be done in a manner not unlike this in Pyra- mides, as ne could demon$trate.</I></P> <head>COROLLARY.</head> <P>Hence it is manife$t, that a given Cone may circum$cribe one Figure and in$cribe another con$i$ting of Cylinders of equal Altitudes $o, as that the Lines which are intercepted betwixt their Centers of Gravity and the point which $o divides the Axis of the Cone, as that the part towards the Vertex is tri- ple to the le$t, are le$s than any given Line.</P> <P><I>For, $ince it hath been demon$trated, that the $aid point dividing the Axis, as was $aid, is alwaies found betwixt the Centers of Gravity</I> <foot>Mm <I>of</I></foot> <p n=>266</p> <I>of the Circum$cribed and in$cribed Figures: and that it's po$$ible, that there be a Line in the middle betwixt tho$e Centers that is le$s than any Line a$$igned; it followeth that the $ame given Line be much le$s that lyeth betwixt one of the $aid Centers and the $aid point that divides the Axis.</I></P> <head>PROPOSITION.</head> <P>The Center of Gravity divideth the Axis of any Cone or Pyramid $o, that the part next the Vertex is triple to the remainder.</P> <P><I>Let there be a Cone who$e Axis is A B. And in C let it be divided, $o that A C be triple to the remaining part C B. It is to be proved, that C is the Center of Gravity of the Cone. For if it be not, the Cone's Center $hall be either above or below the point C. Let it be fir$t beneath, and let it be E. And draw the Line L P, by it $elf, equal to C E; which divided at plea$ure in N. And as both B E and P N to- gether are to P N, $o let the Cone be to the Solid X: and in$cribe in the Cone a Solid Figure of Cylinders that have equal Ba$es, who$e Center of Gravity is le$s di$tant from the point C than is the Line L N, and the exce$s of the Cone above it le$s than the Solid X. And that this may be done is manife$t from what hath been already demon$trated. Now let the in$cribed Figure be $uch as</I> <fig> <I>was required, who$e Center of Gravity let be I. The Line I E therefore $hall be greater than N P together with L P. Let C E and I C le$s L N be equal: And be- cau$e both together B E and N P is to N P as the Cone to X: and the exce$s by which the Cone exceeds the in$cribed Figure is le$s than the Solid X: Therefore the Cone $hall have greater proportion to the $aid X S than both B E and N P to N P: and, by Divi$ion, the in$cribed Figure $hall have greater proportion to the exce$s by which the Cone exceeds it, than B E to N P: But B E hath le$s proportion to E I than to N P with I E. Let N P be greater. Then the in$cribed Fi- gure hath to the exce$s of the Cone above it much greater proportion than B E to E I. Therefore as the in$cribed Figure is to the $aid exce$s, $o $hall a Line bigger than B E be to E I. Let that Line be M E. Becau$e, therefore, M E is to E I as the in$cribed Figure is to the exce$s of the Cone above the $aid Figure, and D is the Center of Gravity of the Cone, and I the Center of Gravity of the in$cribed Figure: Therefore</I> <foot>M <I>$hall</I></foot> <p n=>267</p> <I>M $hall be the Center of Gravity of the remaining proportions by which the Cone exceeds the in$cribed Figure. Which is impo$$ible. Therefore the Center of Gravity of the Cone is not below the point C. Nor is it above it. For if it may be, let it be R. And again a$$ume L P cut at plea$ure in N: And as both B C and N P together are to N L, $o let the Cone be to X. And let a Figure be, in like manner, circum$cribed about the Cone, which exceeds the $aid Cone a le$s quantity than the Solid X. And let the Line which intercepts bet wixt its Center of Gravity and C, be le$$er than N P. Now take the circum$cribed Figure, who$e Center let be O; the remainder O R $hall be greater than the $aid N L. And becau$e, as both together B C and P N is to N L, $o is the Cone to X: And the exce$s by which the circum$cribed exceeds the Cone is le$$er than X: And B O is le$$er than B C and P N together: And O R grea- ter than L N: The Cone therefore $hall have much greater proportion to the remaining proportions by which it was exceeded by the circum$cribed Figure, than B O to O R. Let it be as M O is to O R. M O $hall be greater than B C; and M $hall be the Center of Gravity of the pro- portions by which the Cone is exceeded by the circum$cribed Figure. Which is inconvenient. Therefore the Center of Gravity of the Cone is not above the point C. But neither is it below it; as hath been proved. Therefore it $hall be C it $elf. And $o in like manner may it be demon- $trated in any Pyramid.</I></P> <head>PROPOSITION.</head> <P>If there were four Lines continual proportionals; and as the lea$t of them were to the exce$s by which the greate$t exceeds the lea$t, $o a Line taken at plea$ure $hould be to 3/4 the exce$s by which the greate$t exceeds the $econd; and as the Line equal to the$e (<I>viz.</I> to the greate$t, double of the $econd, and triple of the third) is to the Line equal to the quadruple of the fourth, the quadruple of the $econd, and the quadruple of the third, $o $hould another Line taken be to the exce$s of the greate$t above the $econd: the$e two Lines taken together $hall be a fourth part of the greate$t of the propor- tionals.</P> <foot>Mm 2 <I>FOR</I></foot> <p n=>268</p> <P><I>For let A B, B C, B D, and B E be four proportional Lines. And as B E is to E A, $o let F G be to 3/4 of A C. And as the Line equal to A B and to double B C and to triple B D is to the Line equal to the quadruples of A B, B C, and B D, $o let H G be to A C. It is to be proved, that H F is a fourth part of A B. Fora$much therefore as A B, B C, B D, and B E</I> <fig> <I>are proportionals, A C, C D, and D E $hall be in the $ame proportion: And as the quadruple of the $aid A B, B C, and B D is to A B with the double of B C and triple of B D, $o is the quadruple of A C, C D, and D E; that is, the quadruple of A E; to A C with the double of C D, and triple of D E. And $o is A C to H G. Therefore as the triple of A E is to A C, with the double of C D and triple of D E, $o is 3/4 of A C to H G. And as the triple of A E is to the triple of E B, $o is 3/4 A C to G F: Therefore, by the Conver$e of the twenty fourth of the fifth, As triple A E is to A C with double C D and tri- ple D B, $o is 3/4 of A C to H F: And as the quadruple of A E is to A C with the double of C D and triple of D B; that is, to A B with C B and B D, $o is A C to H F. And, by Permutation, as the quadruple of A E is to A C, $o is A B with C B and B D to H F. And as A C is to A E, $o is A B to A B with C B and B D. Therefore,</I> ex æquali, <I>by Perturbed proportion, as quadruple A E is to A E, $o is A B to H F. Wherefore it is manife$t that H F is the fourth part of A B.</I></P> <head>PROPOSITION.</head> <P>The Center of Gravity of the <I>Fru$tum</I> of any Py- ramid or Cone, cut equidi$tant to the Plane of the Ba$e, is in the Axis, and doth $o divide the $ame, that the part towards the le$$er Ba$e is to the remainder, as the triple of the greater Ba$e, with the double of the mean Space be- twixt the greater and le$$er Ba$e, together with the le$$er Ba$e is to the triple of the le$$er Ba$e, together with the $ame double of the mean Space, as al$o of the greater Ba$e.</P> <foot><I>From</I></foot> <p n=>269</p> <P><I>From a Cone or Pyramid who$e Axis is A D, and equidi$tant to the Plane of the Ba$e, let a</I> Fru$tum <I>be cut who$e Axis is V D. And as the triple of the greate$t Ba$e with the double of the mean and lea$t is to the triple of the lea$t and double of the mean and greate$t, $o is \ O to O D. It is to be proved that the Center of Gra- vity of the</I> Fru$tum <I>is in O. Let V M be the fourth part of V D. Set the Line H X by the by, equal to A D: and let K X be equal to A V: and unto H X K let X L be a third proportional, and X S a fourth. And as H S is to S X, $o let M D be to the Line taken from O towards A: which let be O N. And becau$e the greater Ba$e is in proportion to that which is mean betwixt the greater and le$$er as D A to A V; that</I> <fig> <I>is, as H X, to X K, but the $aid mean is to the lea$t as K X to X L; the greater, mean, and le$$er Ba$es $hall be in the $ame proportion as H X, X K, and X L. Wherefore as triple the greater Ba$e, with double the mean and le$$er, is to triple the lea$t with double the mean and grea- te$t; that is, as V O is to O D; $o is triple H X with double X K and X L to triple X L, with double X K and X H: And by Compo$ition and Converting the proportion, O D $hall be to V D, as H X, with double X K and triple X L, to quadruple H X, X K, and X L. There are, therefore, four proportional Lines, H X, X K, X L, and X S: And as X S is to S H, $o is the Line taken N O to 3/4 of D V, to wit, to D M; that is, to 3/4 of H K: And as H X with double X K and triple X L is to quadruple H X, X K and X L; $o is another Line taken O D to D V; that is, to H K. Therefore, by the things demon$trated, D N $hall be the fourth part of H X; that is, of A D. Wherefore the point N $hall be the Center of Gravity of the Cone or Pyramid who$e Axis is A D. Let the Center of Gra- vity of the Pyramid or Cone who$e Axis is A V be I. It is therefore manife$t that the Center of Gravity of the</I> Fru$tum <I>is in the Line I N inclining towards the part N, and in that point of it which with the point N include a Line to which I M hath the $ame proportion that the</I> Fru$tum <I>cut hath to the Pyramid or Cone who$e Axis is A V. It remaineth therefore to prove that I N hath the $ame proportion to N O, that the</I> Fru$tum <I>hath to the Cone who$e Axis is A V. But as the Cone who$e Axis is D A is to the Cone who$e Axis is A V, $o is the Cube D A to the Cube D V; that is, the Cube H X to the Cube X K: But this is the $ame proportion that H X hath to X S. Wherefore, by Divi$ion, as H S is to S X, $o $hall the</I> Fru$tum <I>who$e</I> <foot><I>Axis</I></foot> <p n=>270</p> <I>Axis is D V be to the Cone or Pyramid who$e Axis is V A. And as H S is to S X, $o al$o is M D to O N. Wherefore the</I> Fru$tum <I>is to the Pyramid who$e Axis is A V, as M D to N O. And becau$e A N is 3/4 of A D; and A I is 3/4 of A V; the remainder I N $hall be 3/4 of the remainder V D. Wherefore I N $hall be equal to M D. And it hath been demon$trated that M D is to N O, as the</I> Fru$tum <I>to the Cone A V. It is mani- fe$t, therefore, that I N hath likewi$e the $ame proportion to N O: Wherefore the Propo- $ition is manife$t.</I></P> <head><I>FINIS.</I></head> <fig> <p n=>271</p> <fig> <head>GALILEUS, HIS MECHANICKS: OF THE BENEFIT DERIVED FROM THE SCIENCE OF MECHANICKS, AND FROM ITS INSTRUMENTS.</head> <P>I judged it extreamly nece$$ary, before our de$cending to the Speculation of Mecha- nick In$truments, to con$ider how I might, as it were, $et before your eyes in a gene- ral Di$cour$e, the many benefits that are derived from the $aid In$truments: and this I have thought my $elf the more ob- liged to do, for that (if I am not mi$taken) I have $een the generality of <I>M</I>echaniti- ans deceive them$elves in going about to apply Machines to many operations of their own nature impo$$ible; by the $ucce$$e where- of they have been di$appointed, and others likewi$e fru$trate of the hope which they had conceived upon the promi$e of tho$e pre- $umptuous undertakers: of which mi$takes I think I have found the principall cau$e to be the belief and con$tant opinion the$e <foot>Artificers</foot> <p n=>272</p> Artificers had, and $till have, that they are able with a $mall force to move and rai$e great weights; (in a certain manner with their Machines cozening nature, who$e In$tinct, yea mo$t po$itive con- $titution it is, that no Re$i$tance can be overcome, but by a Force more potent then it:) which conjecture how fal$e it is, I hope by the en$uing true and nece$$ary Demon$trations to evince.</P> <P>In the mean time, $ince I have hinted, that the benefit and help derived from Machines is not, to be able with le$$e Force, by help of the Machine to move tho$e weights, which, without it, could not be moved by the $ame Force: it would not be be$ides the purpo$e to declare what the Commodities be which are derived to us from $uch like faculties, for if no profit were to be hoped for, all endeavours employed in the acqui$t thereof will be but lo$t labour.</P> <P>Proceeding therefore according to the nature of the$e Studies, let us fir$t propo$e four things to be con$idered. Fir$t, the weight to be transferred from place to place; and $econdly, the Force and Power which $hould move it; thirdly, the Di$tance between the one and the other Term of the Motion; Fourthly, the Time in which that mutation is to be made: which Time becometh the $ame thing with the Dexterity, and Velocity of the Motion; we determining that Motion to be more $wift then another, which in le$$e Time pa$$eth an equal Di$tance.</P> <P>Now, any determinate Re$i$tance and limited Force what$oever being a$$igned, and any Di$tance given, there is no doubt to be made, but that the given Force may carry the given Weight to the determinate Di$tance; for, although the Force were extream $mall, yet, by dividing the Weight into many $mall parts, none of which remain $uperiour to the Force, and by transferring them one by one, it $hall at la$t have carried the whole Weight to the a$$igned Term: and yet one cannot at the end of the Work with Rea$on $ay, that that great Weight hath been moved, and tran$- ported by a Force le$$e then it $elf, howbeit indeed it was done by a Force, that many times reiterated that Motion, and that Space, which $hall have been mea$ured but only once by the whole Weight. From whence it appears, that the Velocity of the Force hath been as many times Superiour to the Re$i$tance of the weight, as the $aid Weight was $uperiour to the Force; for that in the $ame Time that the moving Force hath many times mea$ured the intervall between the Terms of the Motion, the $aid Moveable happens to have pa$t it onely once: nor therefore ought we to affirm a great Re$i$tance to have been overcome by a $mall Force, contrary to the con$titution of Nature. Then onely may we $ay the Natural Con$titution is overcome, when the le$$er Force tran$- fers the greater Re$i$tance, with a Velocity of Motion like to that <foot>where-</foot> <p n=>273</p> wherewith it $elf doth move; which we affirm ab$olutely to be impo$$ible to be done with any Machine imaginable. But becau$e it may $ometimes come to pa$$e, that having but little Force, it is required to move a great Weight all at once, without dividing it in pieces, on this occa$ion it will be necei$ary to have recour$e to the Machine, by means whereof the propo$ed Weight may be transferred to the a$$igned Space by the Force given. But yet this doth not hinder, but that the $ame Force is to move, mea$uring that $ame Space, or another equall to it, as many $everall times as it is exceeded by the $aid Weight. So that in the end of the a- ction we $hall $ind that we have received from the Machine no other benefit tnen only that of tran$porting the $aid Weight with the given Force to the Term given, all at once. Which Weight, being divided into parts, would without any Machine have been carried by the $ame Force, in the $ame Time, through the $ame Intervall. And this ought to pa$$e for one of the benefits taken from the Mechanicks: for indeed it frequently happens, that be- ing $canted in Force but not Time, we are put upon moving great Weights unitedly or in gro$$e: but he that $hould hope, and at- tempt to do the $ame by the help of Machines without increa$e of Tardity in the Moveable, would certainly be deceived, and would declare his ignorance of the u$e of Mechanick In$truments, and the rea$on of their effects.</P> <P>Another benefit is drawn from the In$truments, which depend- eth on the place wherein the operation is to be made: for all In- $truments cannot be made u$e of in all places with equall conve- nience. And $o we $ee (to explain our $elves by an example) that for drawing of Water out of a Well, we make u$e of onely a Rope and a Bucket fitted to receive and hold Water, wherewith we draw up a determinate quantity of Water, in a certain Time, with our limited $trength: and he that $hould think he could with a Machine of what$oever Force, with the $ame $trength, and in the $ame Time, take up a great quantity of Water, is in a gro$$e Errour. And he $hall find him$elf $o much the more deceived, the more he $hall vary and multiply his Inventions: Yet never- thele$$e we $ee Water drawn up with other Engines, as with a Pump that drinks up Water in the Hold of Ships; where you mu$t note that the Pump was not imployed in tho$e Offices, for that it draws up more Water in the $ame Time, and with the $ame $trength then that which a bare Bucket would do, but becau$e in that place the u$e of the Bucket or any $uch like Ve$$el could not effect what is de$ired, namely to keep the Hold of the Ship quite dry from e- very little quantity of Water; which the Bucket cannot do, for that it cannot dimerge and dive, where there is not a con$iderable depth of Water. And thus we $ee the Holds of Ships by the <foot>Nn $aid</foot> <p n=>274</p> $aid In$trument kept dry, when Water cannot but onely oblique- ly be drawn up, which the ordinary u$e of the Bucket would not effect, which ri$eth and de$cends with its Rope perpendicu- larly.</P> <P>The third is a greater benefit, haply, then all the re$t that are derived from Mechanick In$truments, and re$pects the a$$i$tance which is borrowed of $ome Force exanimate, as of the $tream of a River, or el$e animate, but of le$$e expence by far, then that which would be nece$$ary for maintaining humane $trength: as when to turn Mills, we make u$e of the Current of a River, or the $trength of a Hor$e, to effect that, which would require the $trength of five or fix Men. And this we may al$o advantage our $elves in rai$ing Water, or making other violent Motions, which mu$t have been done by Men, if there were no other helps; becau$e with one $ole Ve$$el we may take Water, and rai$e, and empty it where occa$ion requires; but becau$e the Hor$e, or $uch other Mover wanteth Rea$on, and tho$e In$truments which are requi$ite for holding and emptying the Ve$$el in due time, returning again to fill it, and one- ly is endued with Force, therefore it's nece$$ary that the Mecha- nitian $upply the naturall defect of that Mover, furni$hing it with $uch devices and inventions, that with the $ole application of it's Force the defired effect may follow. And therein is very great advantage, not becau$e that a Wheel or other Machine can enable one to tran$port the $ame Weight with le$$e Force, and greater Dexterity, or a greater Space than an equall Force, without tho$e In$truments, but having Judgment and proper Organs, could have done; but becau$e that the $tream of a River co$teth little or nothing, and the charge of keeping of an Hor$e or other Bea$t, who$e $trength is greater then that of eight, or it may be more Men, is far le$$e then what $o many Men would be kept for.</P> <P>The$e then are the benefits that may be derived from Mecha- nick In$truments, and not tho$e which ignorant Engineers dream of, to their own di$grace, and the abu$e of $o many Princes, whil$t they undertake impo$$ible enterprizes; of which, both by the little which hath been hinted, and by the much which $hall be demon$trated in the Progre$$e of this Treati$e, we $hall come to a$$ure our $elves, if we attentively heed that which $hall be $poken.</P> <foot>DEFI-</foot> <p n=>275</p> <head>DEFINITIONS.</head> <P>That which in all Demon$trative Sciences is nece$$ary to be ob$erved, we ought al$o to follow in this Di$cour$e, that is; to propound the Definitions of the proper Terms of this Art, and the primary Suppo$itions, from which, as from $eeds full of fecundity, may of con$equence $pring and re$ult the cau$es, and true Demon$trations, of the Nature of all the Mechanick Engines which are u$ed, for the mo$t part about the Motions of Grave Matters, therefore we will determine, fir$t, what is <I>GRA- VITIE.</I></P> <P>We call <I>GRAVITIE</I> then, That propen$ion of moving naturally downwards, which is found in $olid Bodies, cau$ed by the greater or le$$e quantity of matter, whereof they are con$ti- tuted.</P> <P><I>MOMENT</I> is the propen$ion of de$cending, cau$ed not $o much by the Gravity of the moveable, as by the di$po$ure which divers Grave Bodies have in relation to one another; by means of whichMoment, we oft $ee a Body le$s Grave counterpoi$e another of greater Gravity: as in the Stiliard, a great Weight i<*> rai$ed by a very $mall counterpoi$e, not through exce$s of Gravity, but through the remotene$$e from the point whereby the Beam is up- held, which conjoyned to the Gravity of the le$$er weight adds thereunto Moment, and <I>Impetus</I> of de$cending, wherewith the Moment of the other greater Gravity may be exceeded. <I>MO- MENT</I> then is that IMPETUS of de$cending, compounded of Gravity, Po$ition, and the like, whereby that propenfion may be occa$ioned</P> <P>The <I>CENTER</I> of <I>GRAVITY</I> we define to be that point in every Grave Body, about which con$i$t parts of equall Moment: $o that, imagining $ome Grave Body to be $u$pended and $u$tain- ed by the $aid point, the parts on the right hand will Equilibrate tho$e on the left, the Anteriour, the Po$teriour, and tho$e above tho$e below; $o that be it in any what$oever fite, and po$ition, provided it be $u$pended by the $aid <I>CENTER,</I> it $hall $tand $till: and this is that point which would gladly unite with the univer$all Center of Grave Bodies, namely withthat of the Earth, if it might thorow $ome free <I>Medium</I> de$cend thither. From whence we take the$e Suppo$itions.</P> <foot>Nn 2 SUPPO</foot> <p n=>276</p> <head>SUPPOSITIONS.</head> <P>Any Grave Body, (as to what belongeth to it's proper ver- tue) moveth downwards, $o that the Center of it's Gravity never $trayeth out of that Right Line which is produced from the $aid Center placed in the fir$t Term of the Motion unto the univer$al Center of Grave Bodies. Which is a Suppo$ition very manife$t, becau$e that $ingle Center being obliged to endea- vour to unite with the common Center, it's nece$$ary, unle$$e $ome impediment intervene, that it go $eeking it by the $horte$t Line, which is the Right alone: And from hence may we $econdarily $uppo$e</P> <P>Every Grave Body putteth the greate$t $tre$$e, and weigheth mo$t on the Center of it's Gravity, and to it, as to its proper $eat, all <I>Impetus,</I> all Pondero$ity, and, in $ome, all Moment hath re- cour$e.</P> <P>We la$tly $uppo$e the Center of the Gravity of two Bodies e- qually Grave to be in the mid$t of that Right Line which conjoyns the $aid two Centers; or that two equall weights, $u$pended in equall di$tence, $hall have the point of <I>Equilibrium</I> in the common Center, or meeting of tho$e equal Di$tances. As for Example, the Di$tance C E being equall to the Di$tance E D, and there be- ing by them two equall weights $u$pended, A and B, we $uppo$e the point of <I>Equilibrium</I> to be in the point E, there being no greater rea$on for inclining to one, then to the other part. But <fig> here is to be noted, that the Di- $tances ought to be mea$ured with Perpendicular Lines, which from the point of Su$pen$ion E, fall on the Right Lines, that from the Center of the Gravity of the Weights A and B, are drawn to the common Center of things Grave; and therefore if the Di$tance E D were tran$ported into E F, the weight B would not counterpoi$e the weight A, becau$e drawing from the Centers of Gravity two Right Lines to the Cen- ter of the Earth, we $hall $ee that which cometh from the Center of the Weight I, to be nearer to the Center E, then the other produced from the Center of the weight A. Therefore our $aying that equal Weights are $u$pended by [or at] equal Di$tances, is to be under$tood to be meant when as the Right Lines that go from their Centers & to $eek out the common Center of Gravity, $hall be equidi$ta nt from that Right Line, which is produced from the $aid <foot>Term</foot> <p n=>277</p> Term of tho$e Di$tances, that is from the point of Su$pen$ion, to the $ame Center of the Earrh.</P> <P>The$e things determined and $uppo$ed, we come to the explica- tion of a Principle, the mo$t common and materiall of the greater part of Mechanick In$truments: demon$trating, that unequall Weights weigh equally when $u$pended by [or at] unequal Di$tan- ces, which have contrary proportion to that which tho$e weights are found to have, See the Demon$tration in the beginning of the $econd Dialogue of Local-Motions.</P> <head><I>Some Adverii$ements about what hath been $aid.</I></head> <P>Now being that Weights unequall come to acquire equall Moment, by being alternately $u$pended at Di$tances that have the $ame proportion with them; I think it not fit to over pa$$e with $ilence another congruicy and probability, which may confirm the $ame truth; for let the Ballance A B, be con$ide- red, as it is divided into unequal parts in the point C, and let the Weights be of the $ame propor- <fig> tion that is between the Di$tan- ces B C, and C A, alternately $u$pended by the points A, and B: It is already manife$t, that the one will counterpoi$e the other, and con$equently, that were there added to one of them a very $mall Moment of Gravity, it would preponderate, rai$ing the other, $o that an in$en$ible Weight put to the Grave B, the Ballance would move and de$cend from the point B towards E, and the other extream A would a$cend into D, and in regard that to weigh down B, every $mall Gravity is $ufficient, therefore not keeping any accompt of this in$en$ible Moment, we will put no difference between one Weights <I>$u$taining,</I> and one Weights <I>moving</I> another. Now, let us con$ider the Motion which the Weight B makes, de$cending into E, and that which the other A makes in a$cending into D, we $hall without doubt find the Space B E to be $o much greater than the Space A D, as the Di- $tance B C is greater than C A, forming in the Center C two an- gles D C A, and E C B, equall as being at the Cock, and con$e- quently two Circumferences A D and B E alike; and to have the $ame proportion to one another, as have the Semidiameters B C, and C A, by which they are de$cribed: $o that then the Velocity of the Motion of the de$cending Grave B cometh to be $o much Superiour to the Velocity of the other a$cending Moveable A, as the Gravity of this exceeds the Gravity of that; and it not being <foot>po$$ible</foot> <p n=>278</p> po$$ible that the Weight A $hould be rai$ed to D, although $low- ly, unle$$e the other Weight B do move to E $wiftly, it will not be $trange, or incon$i$tent with the Order of Nature, that the Velocity of the Motion of the Grave B, do compen$ate the greater Re$i$tance of the Weight A, $o long as it moveth $lowly to D, and the other de$cendeth $wiftly to E, and $o on the contrary, the Weight A being placed in the point D, and the other B in the point E, it will not be unrea$onable that that falling lea$urely to A, $hould be able to rai$e the other ha$tily to B, recovering by its Gravity what it had lo$t by it's Tardity of Motion. And by this Di$cour$e we may come to know how the Velocity of the Motion is able to encrea$e Moment in the Moveable, according to that $ame proportion by which the $aid Velocity of the Motion is augmented.</P> <P>There is al$o another thing, before we proceed any farther, to be confidered; and this is touching the Di$tances, whereat, or wherein Weights do hang: for it much imports how we are to under$tand Di$tances equall, and unequall; and, in $um, in what manner they ought to be mea- <fig> $ured: for that A B being the Right Line, and two equall Weights being $u$pended at the very ends thereof, the point C being taken in the mid$t of the $aid Line, there $hall be an <I>Equilibrium</I> upon the $ame: And the rea$on is for that the Di$tance C B is equal to C A. But if elevating the Line C B, moving it about the point C, it $hall be transferred into CD, $o that the Ballance $tand according to the two Lines A C, and C D, the two equall Weights hanging at the Terms A and D, $hall no longer weigh equally on that point C, becau$e the di$tance of the Weight placed in D, is made le$$e then it was when it hanged in B. For if we confider the Lines, along [or by] which the $aid Graves make their Impul$e, and would de$cend, in ca$e they were freely moved, there is no doubt but that they would make or de$cribe the Lines A G, D F, B H: Therefore the Weight hanging on the point D, maketh it's Moment and <I>Impetus</I> according to the Line D F: but when it hanged in B, it made <I>Impetus</I> in the Line B H: and becau$e the Line D F is nearer to the Fulciment C, then is the Line B H Therefore we are to under$tand that the Weights hanging on the points A and D, are not equi-di$tant from the point C, as they be when they are con$tituted according to their Right Line A C B: And la$tly, we are to take notice, that the Di$tance is to be mea$ured by <foot>Lines,</foot> <p n=>279</p> Lines, which fall at Right Angles on tho$e whereon the Weights hang, and would move, if $o be they were permitted to de$cend freely.</P> <head>Of the BALLANCE and LEAVER.</head> <P>Having under$tood by certain Demon$tration, one of the fir$t Principles, from which, as from a plenti$ul Fountain, many of the Mechanical In$truments are derived, we may take occa$ion without any difficulty to come to the knowledge of the nature of them: and fir$t $peaking of the Stiliard, an In$tru- ment of mo$t ordinary u$e, with which divers Merchandizes are weighed, $u$taining them, though very heavy, with a very $mall counterpoi$e, which is com- monly called the Roman or <fig> Plummet, we $hall prove that there is no more to be done in $uch an operation, but to re- duce into act and practice what hath been above contemplated. For if we propo$e the Bal- lance A B, who$e Fulciment or Lanquet is in the point C, by which, at the $mall Di$tance C A, hangeth the heavy Weight D, and if along the other greater C B, (which we call the Needle of the Stiliard) we $hould $uppo$e the Roman F, though of but little weight in compari$on of the Grave Body D to be $lipped to and fro, it $hall be pof$ible to place it $o remotely from the Lanquet C, that the $ame proportion may be found between the two Weights D and F, as is between the Di$tances F C, and C A: and then $hall an <I>Equilibrium</I> $ucceed; unequall Weights hanging at Di$tances alternately proportional to them.</P> <P>Nor is this In$trument different from that other called <I>Vectis,</I> <marg>If of Iron, it is called a Crow, if of wood, a Bar or Hand-$pike.</marg> and vulgarly the ^{*} Leaver, wherewith great Weights are moved by $mall Force; the application of which is according to the Fi- gure prefixed; wherein the Leaver is repre$ented by the Bar of wood or other $olid matter, <I>B</I> C D, let <fig> the heavy Weight to be rai$ed be A, and let the $teadfa$t $upport or Fulciment on which the Leaver re$ts and moves be $uppo$ed to be E, and putting one end of the Leaver under the Weight A, as may be $een in the point C, en- crea$ing the Weight or Force at the other end D, it will be able to lift up the Weight A, though not much, whenever the Force in <foot>D hath</foot> <p n=>280</p> D hath the $ame proportion to the Re$i$tance made by the Weight A, in the point C: as the Di$tance <I>B</I> C hath to the Di$tance C D, whereby it's clear, that the nearer the Fulciment E $hall approach to the Term B, encrea$ing the proportion of the Di$tance D C to the Di$tance C <I>B,</I> the more may one dimini$h the Force in D which is to rai$e the Weight A. And here it is to be noted, which I $hall al$o in its place remember you of, that the benefit drawn from all Mechanical In$truments, is not that which the vulgar Mechanitians do per$wade us, to wit, $uch, that there by Nature is overcome, and in a certain manner deluded, a $mall Force over-powring a very great Re$i$tance with help of the Leaver; for we $hall demon$trate, that without the help of the length of the Leaver, the $ame Force, in the $ame Time, $hall work the $ame effect. For taking the $ame Leaver B C D, who$e re$t or Fulci- ment is in C, let the Di$tance C D <fig> be $uppo$ed, for example, to be in quintuple proportion to the Di$tance C <I>B,</I> & the $aid Leaver to be moved till it come to I C G: In the Time that the Force $hall have pa$$ed the Space D I, the Weight $hall have been moved from B to G: and becau$e the Di$tance D C, was $uppo$ed quintuple to the other C B, it is manife$t from the things demon$trated, that the Weight placed in B may be five times greater then the moving Force $uppo$ed to be in D: but now, <marg>Or Space.</marg> if on the contrary, we take notice of the ^{*} Way pa$$ed by the Force from D unto I, whil$t the Weight is moved from B unto G, we $hall find likewi$e the Way D I, to be quintuple to the Space B G. Moreover if we take the Di$tance C L, equal to the Di$tance C B, and place the $ame Force that was in D, in the point L, and in the point B the fifth part onely of the Weight that was put there at fir$t, there is no que$tion, but that the Force in L being now equal to this Weight in B, and the Di$tances L C and C B being equall, the $aid Force $hall be able, being moved along the Space LM to transfer the Weight equall to it $elf, thorow the other equall Space B G: which five times reiterating this $ame action, $hall tran$- port all the parts of the $aid Weight to the $ame Term G: But the repeating of the Space L M, is certainly nothing more nor le$$e then the onely once mea$uring the Space D I, quintuple to the $aid L M. Therefore the transferring of the Weight from B to G, requireth no le$$e Force, nor le$$e Time, nor a $horter Way if it wee placed in D, than it would need if the $ame were applied in L: And, in $hort, the benefit that is derived from the length of the Leaver C D, is no other, $ave the enabling us to move that <foot>Body</foot> <p n=>281</p> Body all at once, which would not have been moved by the $ame Force, in the $ame Time, with an equall Motion, $ave onely in pieces, without the help of the Leaver.</P> <head><I>Of the</I> CAPSTEN <I>and of the</I> CRANE.</head> <P>The In$truments which we are now about to declare, have immediate dependence upon the Leaver, nay, are no other but a perpetual Vectis or Leaver. For if we $hall $uppo$e the Leaver B A C to be $u$tained in the point A, and the Weight G to <fig> hang at the point B, the Force be- ing placed in C; It is manife$t, that transferring the Leaver unto the points D A E, the Weight G doth alter according to the Di- $tance B D, but cannot much far- ther continue to rai$e it, $o that if it were required to elevate it yet higher, it would be nece$$ary to $tay it by $ome other Fulciment in this Po$ition, and to remit or return the Leaver to its former Po- $ition B A C, and $u$pending the Weight anew thereat, to rai$e it once again to the like height B D; and in this manner repeating the work, many times one $hall come with an interrupted Motion to effect the drawing up of the Weight, which for many re$pects will not prove very beneficial: whereupon this difficulty hath bin thought on, and remedied, by finding out a way how to unite to- gether almo$t infinite Leavers, perpetuating the operation without any interruption; and this hath been done by framing a Wheel about the Center A, according to the Semidiameter A C, and an Axis or Nave, about the $ame Center, of which let the Line A B be the Semidiameter; and all this of very tough wood, or of other $trong and $olid matter, afterwards $u$taining the whole Machine upon a Gudgeon or Pin of Iron planted in the point A, which pa$$eth quite thorow, where it is held fa$t by two fixed Fulciments, and the Rope D B G, at which the weight G hangeth, being be-laid or wound about the Axis or Barrell, and applying another Rope about the greater Wheel, at which let the other Grave I be hang- ed: It is manife$t, that the length C A having to the other A B the $elf-$ame proportion that the Weight G hath to the Weight I, it may $u$tain the Grave G, and with any little Moment more $hall move it: and becau$e the Axis turning round together with the Wheel, the Ropes that $u$tain the Weights are alwaies pendent and contingent with the extream Circumferences of that Wheel and <foot>Oo Axis,</foot> <p n=>282</p> Axis, $o that they $hall con$tantly maintain alike Site and Po$ition in re$pect of the Di$tances B A and A C, the Motion $hall be perpetuated, the Weight I de$cending, and forcing the other G to a$cend. Where we are to ob$erve the nece$$ity of be-laying or winding the Rope about the Wheel, that $o the Weight I may hang according to the Line that is tangent to the $aid Wheel: for if one $hould $u$pend the $aid Weight, $o as that it did hang by the point F, cutting the $aid Wheel, as is $een along the Line F N M, the Motion would cea$e, the Moment of the Weight M being di- mini$hed; which would weigh no more then if it did hang by the point N: becau$e the Di$tance of its Su$pen$ion from the Center A, cometh to be determined by the Line A N, which falleth per- pendicularly upon the Rope F M, and is no longer terminated by the Semidiameter of the Wheel A F, which falleth at unequall Angles upon the $aid Line F M. A violence therefore being offered in the Circumference of the Wheel by a Grave and Exanimate Body that hath no other <I>Impetus</I> then that of De$cending, it is nece$$ary that it be $u$tained by a Line that is contingent with the Wheel, and not by one that cutteth it. But if in the $ame Circumference an Animate Force were employed, that had a Mo- ment or Faculty of making an <I>Impul$e</I> on all $ides, the work might be effected in any whatever place of the $aid Circumference. And thus being placed in F, it would draw up the Weight by turning the Wheel about, pulling not according to the Line F M down- wards, but $ide-waies according to the Contingent Line F L, which maketh a Right Angle, with that which is drawn from the Center A unto the point of Contact F: $o, that if in this manner one do mea$ure the Di$tance from the Center A to the Force placed in F, according to the Line A F perpendicular to F L, along which the <I>Impetus</I> is made, a man $hall not in any part have altered the u$e of the ordinary Leaver. And we mu$t note, that the $ame would be po$$ible to be done likewi$e with an Exanimate Force, in ca$e that a way were found out to cau$e that its Moment might make Impul$e in the point F, drawing according to the Contingent Line F L: which would be done by adjoyning beneath the Line F L a turning Pulley, making the Rope wound about the Wheel to pa$$e along upon it, as it is $een to do by the Line F L X, $u$pending at the end thereof the Weight X equall to the other I, which ex- erci$ing its Force according to the Line F L, $hall alwaies keep a Di$tance from the Center A equall unto the Semidiameter of the Wheel. And from what hath been declared we will gather for a Conclu$ion, That in this In$trument the Force hath alwaies the $ame proportion to the Weight, as the Semidiameter of the Axis or Barrell hath to the Semidiameter of the Wheel.</P> <foot>From</foot> <p n=>283</p> <P>From the In$trument la$t de$cribed, the other In$trument which we call the Crane is not much different, as to form, nay, differeth nothing, $ave in the way of applying or employing it: For that the Cap$ten moveth and is con$tituted perpendicular to the Horizon, and the Crane worketh with its Moment parallel to the $ame Ho- <fig> rizon. For if upon the Circle D A E we $uppo$e an Axis to be placed Column-wi$e, turning about the Center B, and about which the Rope D H, fa$tened to the Weight that is to be drawn, is be- laid, and if the Bar F E B D be let into the $aid Axis [<I>by the Mor- tace B</I>] and the Force of a Man, of an Hor$e, or of $ome other Animal apt to draw, be applyed at its end F, which moving round, pa$$eth along the Circumference F G C, the Crane $hall be framed and fini$hed, $o that by carrying round the Bar F B D, the Barrell or Axis E A D $hall turn about, and the Rope which is twined a- bout it, $hall con$train the Weight H to go forward: And becau$e the point of the Fulciment about which the Motion is made, is the point B, and the Moment keeps at a Di$tance from it according to the Line B F, and the Re$i$tor at the Di$tance B D, the Leaver F B D is formed, by vertue of which the Force acquireth Moment equall to the Re$i$tance, if $o be, that it be in proportion to it, as the Line B D is to B F, that is, as the Semidiameter of the Axis to the Semidiameter of the Circle, along who$e Circumference the Force moveth. And both in this, and in the other In$trument we are to ob$erve that which hath been frequently mentioned, that is, That the benefit which is derived from the$e Machines, is not that which the generality of the Vulgar promi$e them$elves from the Mechanicks; namely, that being too hard for Nature, its po$$ible <foot>Oo2 with</foot> <p n=>284</p> with a Machine to overcome a Re$i$tance, though great, with a $mall Force, in regard, that we $hall manife$tly prove that the $ame Force placed in F, might in the $ame Time conveigh the $ame Weight, with the $ame Motion, unto the $ame Di$tance, without any Machine at all: For $uppo$ing, for example, that the Re$i$tance of the Grave H be ten times greater than the Force placed in F, it <fig> will be requi$ite for the mo- ving of the $aid Re$i$tance, that the Line F B be decuple to B D; and con$equently, that the Circumference of the Circle F G C be al$o decuple to the Circumference E A D: and becau$e when the Force $hall be moved once along the whole Circumference of the Circle F G C, the Barrel EAD, about which the Rope is be-laid which draweth the Weight, $hall likewi$e have given one onely turn; it is manife$t, that the Weight H $hall not have been moved more than the tenth part of that way which the Mover $hall have gone. If therefore the Force that is to move a Re$i$tance that is greater than it $elf, for $uch an a$$igned Space by help of this Machine, mu$t of nece$$ity move ten times as far, there is no doubt, but that dividing that Weight into ten parts, each of them $hall be equall to the Force, and con$equently, might have been tran$ported one at a Time, as great a Space as that which it $elf did move, $o that making ten journeys, each equal to the Circumference E A D, it $hall not have gone any farther than if it did move but once alone about the Circumference F G C; and $hall have conveighed the $ame Weight H to the $ame Di- $tance. The benefit therefore that is to be derived from the$e Machines is, that they carry all the Weight together, but not with le$$e Labour, or with greater Expedition, or a greater Way than the $ame Force might have done conveying it by parcels.</P> <head>Of PULLIES.</head> <P>The In$truments, who$e Natures are reducible unto the Bal- lance, as to their Principle and Foundation, and others little differing from them, have been already de$cribed; now for the under$tanding of that which we have to $ay touching Pullies, it is requi$ite, that we con$ider in the fir$t place another way to u$e the Leaver, which will conduce much towards the inve$tigation of the Force of Pullies, and towards the under$tanding of other Me- chanical Effects. The u$e of the Leaver above declared $uppo$ed <foot>the</foot> <p n=>285</p> the Weight to be at one extream, and the Force at the other, and the Fulciment placed in $ome point between the extreams: but we may make u$e of the Leaver another way, yet, placing, as we $ee, the Fulciment in the extream A, the Force in the other extream C, and $uppo$ing the Weight D to hang by $ome point in the mid$t, <fig> as here we $ee by the point B, in this example it's manife$t, that if the Weight did hang at a point Equi-di$tant from the two ex- treams A and C, as at the point F, the labour of $u$taining it would be equally divided betwixt the two points A and C, $o that half the Weight would be felt by the Force C, the other half being $u- $tained by the Fulciment A: but if the Grave Body $hall be hanged at another place, as at B, we $hall $hew that the Force in C is $uffi- cient to $u$tain the Weight in B, as it hath the $ame proportion to it, that the Di$tance, A B hath to the Di$tance A C. For De- mon$tration of which, let us imagine the Line B A to be continued right out unto G, and let the Di$tance B A be equall to A G, and let the Weight hanging at G, be $uppo$ed equall to D: It is ma- nife$t, that by rea$on of the equality of the Weights D and E, and of the Di$tances G A and A B, the Moment of the Weight E $hall equalize the Moment of the Weight D, and is $ufficient to $u$tain it: Therefore whatever Force $hall have Moment equall to that of the Weight E, and that $hall be able to $u$tain it, $hall be $ufficient likewi$e to $u$tain the Weight D: But for $u$taining the Weight E, let there be placed in the point C $uch a Force, who$e Moment hath that proportion to the Weight E, that the Di$tance G A hath to the Di$tance A C, it $hall be $ufficient to $u$tain it: Therefore the $ame Force $hall likewi$e be able to $u$tain the Weight D, who$e Moment is equall to the of E: But look what Proportion the Line G A hath to the Line A C; and A B al$o hath the $ame to the $aid A C, G A having been $uppo$ed equall to A B: And becau$e the Weights E and D are equall, each of them $hall have the $ame proportion to the Force placed in C: Therefore the Force in C is concluded to equall the Moment of the Weight D, as often as it hath unto it the $ame proportion that the Di$tance B A hath to the Di$tance C A. And by moving the Weight, with the Leaver u$ed in this manner, it is gathered in this al$o, as well as in the other In$truments, that what is gained in Force is lo$t in Velo- city: for the Force C rai$ing the Leaver, and transferring it to A I, the Weight is moved the Space B H, which is as much le$$er than the Space C I pa$$ed by the Force, as the Di$tance A B is le$$er <foot>than</foot> <p n=>286</p> than the Di$tance A C; that is, as the Force is le$$e than the Weight.</P> <P>The$e Principles being declared, we will pa$$e to the Contem- plation of Pullies, the compo$ition and $tructure of which, together with their u$e, $hall be de$cribed by us. And fir$t let us $uppo$e the <marg>*Called by $ome a Nut.</marg> ^{*} Little Pulley A B C, made of Mettall or hard Wood, voluble a- bout it's Axis which pa$$eth thorow it's Center D, and about this <fig> Pulley let the Rope E A B C be put, at one end of whichlet the Weight E hang, and at the other let us $uppo$e the Force F. I $ay, that the Weight being $u$tained by a Force equall to it $elf in the upper Nut or Pulley A B C, bringeth $ome benefit, as the moving or $u$taining of the $aid Weight with the Force placed in F: For if we $hall under$tand, that from the Center D, which is the place of the Fulciment, two Lines be drawn out as far as the Circumference of the Pulley in the points A and C, in which the pendent Cords touch the Circumference, we $hall have a Ballance of equal Arms which determine the Di$tance of the two Su$pen$ions from the Center and Fulciment D: Where- upon it is manife$t, that the Weight hanging at A cannot be $u$tain- ed by a le$$er Weight hanging at G, but by one equal to it; $uch is the nature of equal Weights hanging at equal Di$tances. And although in moving downwards, the Force F cometh to turn about the Pulley A B C, yet there followeth no alteration of the Alti- tude or Re$pect, that the Weight and Force have unto the two Di$tances A D and D C, nay, the Pulley encompa$$ed becometh a Ballance equal to A C, but perpetuall. Whence we may learn, how childi$hly <I>Ari$totle</I> deceiveth him$elf, who holds, that by making the $mall Pulley A B C bigger, one might draw up the Weight with a le$$er Force; he con$idering that upon the enlargement of the $aid Pulley, the Di$tance D C encrea$ed, but not con$idering that there was as great an encrea$e of the other Di$tance of the Weight, that is, the other Semidiameter D A. The benefit therefore that may be drawn from the In$trument above $aid, is nothing at all as to the diminution of the labour: and if any one $hould ask how it hap- pens, that on many occa$ions of rai$ing Weights, this means is made u$e of to help the Axis, as we $ee, for example, in drawing up the Water of Wells; it is an$wered, that that is done, becau$e that by this means the manner of employing the Force is found more commodious: for being to pull downwards, the proper Gravity of our Arms and other parts help us, whereas if we were to draw the fame Weight upwards with a meer Rope, by the $ole $trength <foot>of</foot> <p n=>287</p> of the Members and Mu$cles, and as we u$e to $ay, by Force of Armes, be$ides the extern Weight, we are to lift up the Weight of our own Armes, in which greater pains is required. Conclude we, therefore, that this upper Pulley doth not bring any Facility to the Force $imply con$idered, but onely to the manner of applying it: but if we $hall make u$e of the like Machine <fig> in another manner, as we are now about to declare; we may rai$e the Weight with di- minution of Forces: For let the Pulley B D C be voluble about the Center E placed in it's Frame B L C, at which hang the Grave G; and let the Rope A B D C F pa$$e about the Pulley; of which let the end A be fa$tned to $ome fixed $tay, and in the other F let the Force be placed; which moving to wards H $hall rai$e the Machine B L C, and con$equently the Weight G: and in this operation I $ay, that the Force in F is the half of the Weight $u$tained by it. For the $aid Weight being kept to Rights by the two ^{*} Ropes A B <marg>* Or two ends of the $ame Rope.</marg> and F C, it is manife$t, that the Labour is equally $hared betwixt the Force F and the Fulciment A: and more $ubtilly examining the nature of this In$trument, if we but continue forth the Diameter B E C, we $hall $ee a Leaver to be made, at the mid$t of which, that is at the point E, the Grave doth hang, and the Fulciment cometh to be at the end B, and the Force in the Term C: whereupon, by what hath been above demon$trated, the Force $hall have the $ame proportion to the Weight, that the Di$tance E B hath to the Di- $tance; Therefore it $hall be the half of the $aid Weight: And becau$e the Force ri$ing towards A, the Pulley turneth round, therefore that Re$pect or Con$titution which the Fulciment B and Center E, on which the Weight and Term C, in which the Force is employed do depend, $hall not change all the while; but yet in the Circuinduction the Terms B and C happen to vary in number, but not in vertue, others and others continually $ucceeding in their place, whereby the Leaver B C cometh to be perpetuated. And here (as hath been done in the other In$truments, and $hall be in tho$e that follow) we will not pa$$e without con$idering how that the journey that the Force maketh, is double to the Moment of the Weight. For in ca$e the Weight $hall be moved $o far, till that the Line B C come to arrive wi<*>h it's points B and C, at the points A and F, it is nece$$ary that the two equal Ropes be di$tended in one $ole Line F H, and con$equently, when the Weight $hall have a$cended along the Intervall B A, the Force $hall have been moved twice as far, that is, from <I>F</I> unto H. Then con$idering that the <foot><I>F</I>orce</foot> <p n=>288</p> Force in <I>F,</I> that it may rai$e the Weight, mu$t move upwards, which to exanimate Movers, as being for the mo$t part Grave Bodies, is al- <fig> together impo$$ible, or at lea$t more laborious, than the making of the $ame <I>F</I>orce down- wards: Therefore to help this inconvenience, a Remedy hath been found by adjoyning an- other Nut or Pulley above, as in the adjacent <I>F</I>igure is $een, where the Rope C E <I>F</I> hath been made to pa$s about the upper Pulley <I>F</I> G upheld by the Hook L, $o that the Rope pa$$ing to H, and thither transferring the <I>F</I>orce E, it $hall be able to move the Weight X by pulling downwards, but not that it may be le$$er than it was in E: <I>F</I>or the Motions of the <I>F</I>orce <I>F</I> H, hanging at the equal Di$tances <I>F</I> D and D G of the upper Pulley, do alwaies continue equal; nor doth that upper Pulley (as hath been $hewn above) come to produce any di- minution in the Labour. Moreover it having been nece$$ary by the addition of the upper Pulley to introduce the Appendix B, by which it is $u$tained, it will prove of $ome benefit to us to rai$e the other A, to which one end of the Rope was fa$tned, transferring it to a Ring annexed to the lower part of the <I>F</I>rame of the upper Pulley, as we $ee it done in M. Now finally, this Machine com- pounded of upper and lower Pullies, is that which the Greeks call <marg>In Latine <I>Tro- chlea.</I></marg> <G>*tpoxi/lion.</G></P> <P>We have hitherto explained, how by help of Pullies one may double the <I>F</I>orce, it remaineth that with the greate$t brevity po$- $ible, we $hew the way how to encrea$e it according to any Multi- plicity. And fir$t we will $peak of the Multiplicity according to the even numbers, and then the odde: To $hew how we may mul- tiply the <I>F</I>orce in a quadruple Proportion, we will propound the following Speculation as the Soul of all that followeth.</P> <P>Take two Leavers, A B, C D, with the <I>F</I>ulciments in the ex- <fig> treams A and C; and at the middles of each of them let the Grave G hang, $u$tained by two <I>F</I>orces of equal Mo- ment placed in B and D. I $ay, that the Moment of each of them will equal the Moment of the fourth part of the Weight G. <I>F</I>or the two <I>F</I>or- ces B and D bearing equally, it is manife$t, that the <I>F</I>orce D hath not contra$ted with more then one half of the Weight G: But if the <I>F</I>orce D do by benefit of the Leaver D C $u$tain the half of the <foot>Weight</foot> <p n=>289</p> Weight G hanging at <I>F,</I> it hath been already demon$trated, that the $aid <I>F</I>orce D hath to the Weight $o by it $u$tained, that $ame proportion which the Di$tance <I>F</I> C hath to the Di$tance C D: Which is $ubduple proportion: Therefore the Moment D is $ub- duple to the Moment of half of the Weight G $u$tained by it: Wherefore it followeth, that it is the fourth part of the Moment of the whole Weight. And in the $ame manner the $ame thing is demon$trated, of the Moment <I>B</I>; and it is but rea$onable, that the Weight G being $u$tained by the four points, A, <I>B,</I> C, D, each of them $hould feel an equall part of the Labour.</P> <P>Let us come now to apply this Con$ideration to Pullies, and let the Weight X be $uppo$ed to hang at the two Pullies A B and D E entwining about them, and about the uppermo$t Pulley G H, the Rope, as we $ee, I D E H G A B, $u$taining the whole Machine in the point K. Now I $ay, that placing the Force in L, it $hall be able to $u$tain the Weight X, if $o be, it be equal to the fourth part of it. For if we do imagine the two Diameters D E and A B, and the Weights hanging at the middle points F and C, we $hall have two Leavers like to tho$e before de$cribed, the Fulciments of which an- $wer to the points D and A. Whereupon the Force placed in B, <fig> or if you will, in L, $hall be able to $u- $tain the Weight X, being the fourth part of it: And if we adde another Pul- ley above the other two, making the Rope or Cord to pa$s along L M N, trans- ferring the Force L into N, it $hall be able to bear the $ame Weight gravitating downwards, the upper Pulley neither aug- menting or dimini$hing the Force, as hath been declared. And we will likewi$e note, that to make the: Weight a$cend the <marg>* Or four parts of the $ame Rope</marg> four Ropes B L, E H, D I, and A G ought to pa$s, whereupon the Mover will be to begin, as much as tho$e Ropes are long; and yet neverthele$s the Weight $hall move but only as much as the length of one of them: So that we may $ay by way of adverti$ement, and for confirma- tion of what hatn been many times $po- ken, namely, that look with what proportion the Labour of the <marg>* The word <I>Gy- rilla</I> $ignifieth a Shiver, Rundle, or $mall Wheel of a Pulley, tran- $lated by we $ometimes Pul- ley, $ometimes Nut or Girill.</marg> Mover is dimini$hed, the length of the Way, on the contrary, is encrea$ed with the $ame proportion</P> <P>But if we would encrea$e the Force in $excuple proportion, it will be requi$ite that we adjoyn another ^{*} $mall Pulley or Gyrill to the inferiour Pulley which that you may the better under$tand <foot>Pp we</foot> <p n=>290</p> we will $et before you the pre$ent Contemplation. Suppo$e, there- fore, that A B, C D, and E F are three Leavers; and that on the middle points of them G, H, and I the Weight K doth hang in common, $o that every one of them $hall $u$tain the third part of <fig> it: And becau$e the Power in B, $u$taining with the Leaver B A thependent Weight in G, hapneth to be the half of the $aid Weight, and it hath been already $aid, that it $u$taineth the third part of the Weight K: Therefore the Moment of the Force B is equal to half of the third part of the Weight K; that is, to the $ixth part of it: And the $ame $hall be demon$trated of the other Forces D and F: From whence we may ea$ily gather, that putting three Gyrils or Rundles into the inferiour Pulley, and two or three into the upper- <fig> mo$t, we may multiply the Force accor- <marg>* Or in Sexcuple proportion.</marg> ding to our ^{*} <I>Senarius.</I> And if we would encrea$e it according to any other even Number, the Gyrils of the Pulley below mu$t be multiplyed according to the half of that Number, according to which the Force is to be multiplyed, circumpo$ing the Rope about the Pulleys, $o as that one of the ends be fa$tned to the upper Pul- ley, and let the Force be in the other; as in this Figure adjoyning may manife$tly be gathered.</P> <P>Now pa$$ing to the Declaration of the manner how to multiply the Force ac- cording to the odd Numbers, and begin- <fig> ning at the triple proportion: fir$t, let us propo$e the pre$ent Contemplation, as that, on the under$tanding of which the knowledge of all the Work in hand doth depend. Let therefore the Leaver be A B, its Fulciment A, and from the middle of it, that is, at the point C let the Grave D be hanged; and let it be $u- $tained by two equal Forces; and let one of them be applied to the point C, and the other to the term B. I $ay, that each of tho$e Powers have Moment equal to the third part of the Weight D. For the Force in C $u$taineth a Weight equal to it $elf, being placed in the $ame Line in which the Weight D doth hang & Gravitate: But the <foot>Force</foot> <p n=>291</p> Force in B $u$taineth a part of the Weight D double to it $elf, its Di$tance from the Fulciment A, that is, the Line B A being dou- ble to the Di$tance A C at which the Grave hangeth: But becau$e the two Forces in B and C are $uppo$ed to be equal to each other: Therefore the part of the Weight D, which is $u$tained by the Force in B, is double to the part $u$tained by the Force in C. If therefore, of the Grave D two parts be made, the one double to the remainder, the greater is $u$tained by the Force in B, and the le$$er by the Force in C: But this le$$er is the third part of the Weight D: Therefore the Moment of the Force in C is equal to the Moment of the third part of the Weight D; to which, of con$equence, the Force B $hall be equal, we having $uppo$ed it equal to the other Force C: Wherefore our intention is manifell, which we were to demon$trate, how that each of the two Powers C and B is equal to the third part of the Weight D. Which be- ing demon$trated, we will pa$s forwards to the Pulleys, and will de$cribe the inferiour Gyrils of A C B, voluble about the Center G, and the Weight H hanging thereat, we will draw the other up- per one E F, winding about them both the Rope D F E A C B I, of which let the end D be fa$tned to the inferiour Pulley, and to <fig> the other I let the Force be applyed: Which, I $ay, $u$taining or moving the Weight H, $hall feele no more than the third part of the Gravity of the $ame. For con$idering the contrivance of this Ma- chine, we $hall find that the Diameter A B $upplieth the place of a Leaver, in who$e term B the Force I is applied, and in the other A the <I>F</I>uiciment is placed, at the mid- dle G the Grave H is hanged, and another <I>F</I>orce D applied at the $ame place: $o that <marg>* Or three parts of one Rope.</marg> the Weight is fa$tned to the ^{*} three Ropes I B, <I>F</I> D, and E A, which with equal Labour $u$tain the Weight. Now, by what hath already been contemplated, the two <I>F</I>orces D and B being applied, one, to the mid$t of the Leaver A B, and the other to the extream term B, it is manife$t, that each of them holdeth no more but the third part of the Weight H: Therefore the Power I, having a Moment equal to the third part of the Weight H, $hall be able to $u$tain and move it: but yet the Way of the <I>F</I>orce in I $hall be triple to the Way that the Weight $hall pa$s; the $aid Force being to di$tend it $elf according to the Length of the three Ropes I B, <I>F</I> D, and E A, of which one alone mea$ureth the Way of the Weight H.</P> <foot>Pp 2 <I>Of</I></foot> <p n=>292</p> <head><I>Of the</I> SCREW.</head> <P>Among$t the re$t of Mechanick In$truments for $undry u$es found out by the Wit of Man, the Screw doth, in my opi- nion, both for Invention and for Utility, hold the fir$t place, as that which is appo$itely accommodated, and $o contrived not only to move, but al$o to $tay and pre$s with very great Force, that taking up but little room, it worketh tho$e effects which other In$truments cannot, unle$s they were reduced to a great Machine. The Screw therefore being of mo$t ingenious and commodious contrivance, we ought de$ervedly to be at $ome pains in explaining, with all the plainne$s that is po$$ible, the Original and Nature of it. The which that we may do, we will begin at a Speculation, which, though at fir$t blu$h it may appear $omewhat remote from the con$ideration of this In$trument, yet is the <I>Ba$is</I> and Founda- tion thereof.</P> <P>No doubt, but that Natures operation in the Motions of Grave Bodies is $uch, that any whatever Body that hath a Gravity in it hath a propen$ion of moving, being at liberty, towards the Cen- <marg>* Or along.</marg> ter, and that not only ^{*} by the Right Line perpendicularly, but al- $o (when it cannot do otherwi$e) by any other Line, which ha- ving $ome inclination towards the Center goeth more and more aba$ing. And thus we $ee the Water not only to fall downwards along the Perpendicular from $ome eminent place, but al$o to run about the Surface of the Earth along Lines though very little en- clined; as we $ee in the Cour$e of Rivers, the Waters of which, if $o be that the Bed have any the lea$t declivity, go freely declining downwards. Which very effect, like as it is di$cerned in all Fluid Bodies, would appear al$o in hard Bodies, if $o be, that their Fi- gure and other Accidental and Extern Impediments did not hinder it. So that we, having a Superficies very well $moothed and poli- $hed, as for in$tance, that of a Looking-gla$s, and a Ball exactly rotund and $leek, either of Marble, or of Gla$s, or of any other Matter apt to be poli$hed, this being placed upon that Superficies $hall trundle along, in ca$e that this have any, though very $mall, inclination; and $hall lie $till only upon that Superficies which is exactly levelled and parallel to the Plane of the Horizon: as is that, for example, of a Lake or $tanding Water being frozen, up- on which the $aid Spherical Body would $tand $till, but in a con- dition of being moved by every $mall Force. For we having $up- po$ed that if that Plane did incline but an hairs breadth only, the $aid Ball would move along it $pontaneou$ly towards the part de- clining, and on the oppo$ite would have a Re$i$tance, nay, would not be able without $ome Violence to move towards the part <foot>ri$ing</foot> <p n=>293</p> ri$ing or a$cending: it of nece$$ity remaineth manife$t, that in the Superficies which is exactly equilibrated, the $aid Ball remaineth in- different and dubious between Motion and Re$t, $o that every $mall Force is $ufficient to move it, as on the contrary, every $mall Re$i- $tance, and no greater than that of the meer Air that environs it, is able to hold it $till.</P> <P>From whence we may take this Conclu$ion for indubitable, That Crave Bodies, all Extern and Adventitious Impediments being re- moved, may be moved along the Plane of the Horizon by any ne- ver $o $mall Force: but when the $ame Grave is to be thrown along an A$cending Plane, then, it beginning to $trive again$t that a$cent, having an inclination to the contrary Motion, there $hall be requi- red greater Violence, and $till greater the more Elevation that $ame Plane $hall have. As for example, the Moveable G, being po$ited upon the Line A B parallel to the Horizon, it $hall, as hath been $aid, be indifferent on it either to Motion or Re$t, $o that it may be moved by a very $mall Force: But if we $hall have the Planes Elevated, they $hall not be driven along without Violence; which <fig> Violence will be required to be greater to move it along the Line A D, than along A C; and $till greater along A E than along A D: The which hapneth, becau$e it hath greater <I>Impetus</I> of going down- wards along A E than along A D, and along A D than along A C. So that we may likewi$e conclude Grave Bodies to have greater Re$i$tance upon Planes differently Elevared, to their being moved along the $ame, according as one $hall be more or le$s elevated than the other; and, in fine, that the greate$t Re$i$tance of the $ame Grave to its being rai$ed is in the Perpendicular A F. But it will be nece$$ary to declare exactly what proportion the Force mu$t have to the Weight, that it may be able to carry it along $everal elevated Planes, before we proceed any farther, to the end that we may perfectly under$tand all that which remains to be $poken.</P> <P>Letting, therefore, Perpendiculars fall from the points C, D, and E unto the Horizontal Line A B, which let be C H, D I, and E K: it $hall be demon$trated that the $ame Weight $hall be mo- ved along the Plane A C with le$$er Force than along the Perpendi- cular A F, (where it is rai$ed by a Force equal to it $elf) accor- ding to the proportion by which the Perpendicular C H is le$s than A C: and that along the Plane A D, the Force hath the $ame pro- portion to the Weight, that the Perpendicular I D hath to D A: and, la$tly, that in the Plane A E the <I>F</I>orce to the Weight ob$er- veth the proportion of E K and E A.</P> <foot>The</foot> <p n=>294</p> <P>The pre$ent Speculation hath been attempted by <I>Pappus Alex- andrinus</I> in <I>Lib.</I> 8. <I>de Collection. Mathemat.</I> but, if I be in the right, he hath not hit the mark, and was over$een in the A$$umpti- on that he maketh, where he $uppo$eth that the Weight ought to be moved along the Horizontal Line by a <I>F</I>orce given; which is fal$e: there needing no $en$ible <I>F</I>orce (removing the Accidental Impediments, which in the Theory are not regarded) to move the given Weight along the Horizon, $o that he goeth about in vain afterwards to $eek with what <I>F</I>orce it is to be moved along the elevated Plane. It will be therefore better, the <I>F</I>orce that moveth the Weight upwards perpendicularly, (which equalizeth the Gra- vity of that Weight which is to be moved) being given, to $eek the <I>F</I>orce that moveth it along the Elevated Plane: Which we will endeavour to do in a Method different from that of <I>Pappus.</I></P> <P>Let us therefore $uppo$e the Circle A I C, and in it the Diame- ter A B C, and the Center B, and two Weights of equal Moment in the extreams B and C; $o that the Line A C being a Leaver, or Ballance moveable about the Center B, the Weight C $hall come to be $u$tained by the Weight A. But if we $hall imagine the Arm of the Ballance B C to be inclined downwards according to the Line B F, but yet in $uch a manner that the two Lines <I>A B</I> and <I>B F</I> do continue $olidly conjoyned in the point <I>B,</I> in this ca$e the Moment of the Weight C $hall not be equal to the Moment <fig> of the Weight <I>A,</I> for that the Di- $tance of the point <I>F</I> from the Line of Direction, which goeth accord- ing to B I, from the <I>F</I>ulciment B un- to the Center of the Earth, is dimi- ni$hed: But if from the point <I>F</I> we erect a Perpendicular unto B C, as is <I>F</I> K, the Moment of the Weight in <I>F</I> $hall be as if it did hang by the Line K <I>F,</I> and look how much the Di$tance K B is dimini$hed by the Di$tance B <I>A,</I> $o much is the Moment of the Weight <I>F</I> dimini$hed by the Moment of the <I>W</I>eight <I>A. A</I>nd in this fa$hion inclining the <I>W</I>eight more, as for in$tance, according to B L, its Moment $hall $till dimini$h and $hall be as if it did hang at the Di$tance <I>B</I> M, ac- cording to the <I>L</I>ine M <I>L,</I> in which point <I>L</I> it $hall be $u$tained by a <I>W</I>eight placed in <I>A,</I> $o much le$s than it $elf, by how much the Di$tance B <I>A</I> is greater than the Di$tance <I>B</I> M. See therefore that the <I>W</I>eight placed in the extream of the <I>L</I>eaver B C, in inclining downwards along the Circumference C <I>F L</I> I, cometh to dimini$h its Moment and <I>Impetus</I> of going downwards from time to time, <foot>more</foot> <p n=>295</p> more and le$s, as it is more or le$s $u$tained by the Lines B F and B L: But the con$idering that this Grave de$cending, and $u$tained by the Semidiameters B F and B L is one while le$s, and another while more con$trained to pa$s along the Circumference C F L, is no other, than if we $hould imagine the $ame Circumference C F L I to be a Super$icies $o curved, and put under the $ame Moveable: $o that bearing it $elf thereon it were con$trained to de$cend along thereby; for if in the one and other manner the Moveable de$cribeth the $ame Cour$e or Way, it will nothing im- port whether, if $u$pended at the Center B, it is $u$tained by the Semidiameter of the Circle, or el$e, whether that Fulciment being taken away, it proceed along the Circumference C F L I: So that we may confidently affirm, that the Grave de$cending downwards from the point C along the Circumference C F L I, its Moment of De$cent in the point C is total and entire, becau$e it is not in any part $u$tained by the Circumference: And there is not in that fir$t point C, any indi$po$ition to Motion different from that, which being at liberty, it would make along the Perpendicular and Con- tingent Line D C E: But if the Moveable $hall be placed in the point F, then its Gravity is in part $u$tained, and its Moment of De$cent is dimini$hed by the Circular Path or Way that is placed under it, in that proportion wherewith the <I>L</I>ine <I>B</I> K is overcome by <I>B</I> C: But if when the Moveable is in F, at the fir$t in$tant of $uch its Motion, it be as if it were in the Plane elevated according to the Contingent <I>L</I>ine G F H, for that rea$on the inclination of the Circumference in the point F differeth not from the inclination of the Contingent <I>L</I>ine F G any more $ave the in$en$ible Angle of the Contact. And in the $ame manner we $hall find the Moment of the $aid Moveable to dimini$h in the point <I>L,</I> as the <I>L</I>ine BM is dimini$hed by B C; $o that in the Plane contingent to the Circle in the point <I>L,</I> as for in$tance, according to the <I>L</I>ine N <I>L</I> O, the Moment of De$cent dimini$heth in the Moveable with the $ame proportion. If therefore ^{*} upon the Plane HG the Moment of the <marg>* Or along</marg> Moveable be dimini$hed by the total <I>Impetus</I> which it hath in its Perpendicular D C E, according to the proportion of the <I>L</I>ine K B to the <I>L</I>ine B C, and B F, being by the Solicitude of the Triangles K B F and K F H the $ame proportion betwixt the <I>L</I>ines K F and F H, as betwixt the $aid K B and <I>B</I> F, we will conclude that the proportion of the entire and ab$olute Moment, that the Moveable hath in the Perpendicular to the Horizon to that which it hath up- on the Inclined Plane H F, hath the $ame proportion that the <I>L</I>ine H F hath to the <I>L</I>ine F K; that is, that the <I>L</I>ength of the Inclined Plane hath to the Perpendicular which $hall fall from it unto the Horizon. So that pa$$ing to a more di$tinct Figure, $uch as this here pre$ent, the Moment of De$cending which the Move- <foot>able</foot> <p n=>296</p> able hath upon the inclined Plane C A hath to its total Moment wherewith it gravitates in the Perpendicular to the Horizon C P the $ame proportion that the $aid Line P C hath to C A. And if thus it be, it is manife$t, that like as the Force that $u$tai- neth the Weight in the Perpendiculation P C ought <fig> to be equal to the $ame, $o for $u$taining it in the inclined Plane C A, it will $uffice that it be $o much le$$er, by how much the $aid Perpendicular C P wan- teth of the Line C A: and becau$e, as $ometimes we $ce, it $ufficeth, that the Force for moving of the Weight do in$en$ibly $uperate that which $u$taineth it, therefore we will infer this univer$al Propo$ition, [That upon an Elevated Plane the Force hath to the Weight the $ame proportion, as the Perpendicular let fall from the Plane unto the Horizon hath to the Length of the $aid Plane.]</P> <P>Returning now to our fir$t Intention, which was to inve$tigate the Nature of the Screw, we will con$ider the Triangle A B C, of which the Line A B is Horizontal, B C perpendicular to the $aid Horizon, and A C a Plane elevated; upon which the Moveable D $hall be drawn by a Force $o much le$s than it, by how much the Line B C is $horter than C A: But to elevate or rai$e the $aid Weight along the $aid Plane A C, is as much as if the Triangle C A B $tanding $till, the Weight <fig> D be moved towards C, which is the $ame, as if the $ame Weight never removing from the Perpen- dicular A E, the Triangle did pre$s forwards towards H. For if it were in the Site F H G, the Moveable would be found to have mounted the height A I. Now, in fine, the primary Form and E$$ence of the Screw is no- thing el$e but $uch a Triangle A C B, which being forced for- wards, $hall work it $elf under the Grave Body to be rai$ed, and lifteth it up, as we $ay, by the ^{*} head and $houlders. And this was <marg><I>Levar in cape</I> <*> $ieth to lift <*> igh by force</marg> its fir$t Original: For its fir$t Inventor (whoever he was) con$i- dering how that the Triangle A B C going forwards rai$eth the Weight D, he might have framed an In$trument like to the $aid Triangle, of a very $olid Matter, which being thru$t forwards did rai$e up the propo$ed Weight: But afterwards con$idering better, how that that $ame Machine might be reduced into a much le$$er and more commodious Form, taking the $ame Triangle he twined and wound it about the Cylinder A B C D in $uch a fa$hion, that the height of the $aid Triangle, that is the Line C B, did make the Height of the Cylinder, and the A$cending Plane did beget upon <foot>the</foot> <p n=>297</p> the $aid Cylinder the Helical Line de$cribed by the Line AEFGH, which we vulgarly call the Wale of the Screw, which was produ- ced by the Line A C. And in this manner is the In$trument made, which is by the Greeks called <G>*ko/xlos,</G> and by us a Screw; which <marg>* <G>*ko/xlos,</G> in La- tine <I>Cocblea,</I> any Screw winding like the Shell of a Snail.</marg> winding about cometh to work <fig> and in$inu- ate with its Wales under the Weight, and with facility rai- $eth it. And we having demon- $trated, That up- on [<I>or along</I>] the elevated Plane the Force hath the $ame proportion to the Weight, that the perpendicular Altitude of the $aid Plane hath to its Length; $o, $uppo$ing that the Force in the Screw A B C D is multiplied according to the proportion by which the Length of the whole Wale exceedeth the Altitude C B, from hence we come to know that making the Screw with its Helix's more thick or clo$e together, it becometh $o much the more forceable, as being begot by a Plane le$s elevated, and who$e Length regards its own Per- pendicular Altitude with greater proportion. But we will not omit to adverti$e you, that de$iring to find the Force of a propo- $ed Screw, it will not be needful that we mea$ure the Length of all its Wales, and the Altitude of the whole Cylinder, but it will be enough if we $hall but examine how many times the Di- $tance betwixt two $ingle and Contiguous terms do enter into one $ole Turn of the $ame Wale, as for example, how many times the Di$tance AF is contained in the Length of the Turn AEF: For this is the $ame proportion that the Altitude CB hath to all the Wale.</P> <P>If all that be under$tood which we have hitherto $poken touch- ing the Nature of this In$trument, I do not doubt in the lea$t but that all the other circum$tances may without difficulty be compre- hended: as for in$tance, that in$teed of making the Weight to mount upon the Screw if one accommodates its Nut with the Helix incavated or made hollow, into which the Male Screw that is the Wale entring, & then being turned round it rai$eth and lifteth up the Nut or Male Screw together with the Weight which was hanged thereat. La$tly, we are not to pa$s over that Con$idera- tion with $ilence which at the beginning hath been $aid to be nece$- $ary for us to have in all Mechanick In$truments, to wit, That what is gained in Force by their a$$i$tance, is lo$t again in Time, <foot>Qq and</foot> <p n=>298</p> and in the Velocity: which peradventure, might not have $eemed to $ome $o true and manife$t in the pre$ent Contemplation; nay, rather it $eems, that in this ca$e the Force is multiplied without the Movers moving a longer way than the Moveable: In regard, that if we $hall in the Triangle A B C $uppo$e the Line A B to be the Plane of the Horizon, A C the elevated Plane, who$e Altitude is mea$ured by the Perpendicular C B, a Moveable placed upon the Plane A C, and the Cord E D <I>F</I> tyed to it, and a <I>F</I>orce or Weight applyed in <I>F</I> that hath to the Gravity of the Weight E the <fig> $ame proportion that the Line B C hath to C A; by what hath been demon$trated, the Weight <I>F</I> $hall de$cend downwards, drawing the Moveable E along the eleva- ted Plane; nor $hall the Move- able E mea$ure a greater Space when it $hall have pa$$ed the whole Line A <I>C,</I> than that which the $aid Grave <I>F</I> mea$ureth in its de$cent downwards. But here yet it mu$t be adverti$ed, that al- though the Moveable E $hall have pa$$ed the whole Line A C, in the $ame Time that the other Grave <I>F</I> $hall have been aba$ed the like Space, neverthele$s the Grave E $hall not have retired from the common Center of things Grave more than the Space of the Per- pendicular <I>C</I> B. but yet the Grave <I>F</I> de$cending Perpendicularly $hall be aba$ed a Space equal to the whole Line A <I>C.</I> And becau$e Grave Bodies make no Re$i$tance to Tran$ver$al Motions, but only $o far as they happen to recede from the <I>C</I>enter of the Earth; There- fore the Moveable E in all the Motion A <I>C</I> being rai$ed no more than the length of the Line <I>C</I>B, but the other <I>F</I> being aba$ed per- pendicularly the quantity of all the Line A <I>C</I>: Therefore we may de$ervedly affirm that Way of the <I>F</I>orce E maintaineth the $ame proportion to the <I>F</I>orce <I>F</I> that the <I>L</I>ine A <I>C</I> hath to <I>C</I> B; that is, the Weight E to the Weight <I>F.</I> It very much importeth, therefore, to con$ider by [<I>or along</I>] what <I>L</I>ines the Motions are made, e$pe- cially in exanimate Grave Bodies, the Moments of which have their total Vigour, and entire Re$i$tance in the <I>L</I>ine Perpendicular to the Horizon; and in the others tran$ver$ally Elevated and Inclined they feel the more or le$s Vigour, <I>Impetus,</I> or Re$i$tance, the more or le$s tho$e Inclinations approach unto the Perpendicular Inclina- tion.</P> <foot><I>of</I></foot> <p n=>299</p> <head><I>Of the SCREW of</I> ARCHIMEDES <I>to draw Waier.</I></head> <P>I Do not think it $it in this place to pa$s over with Silence the Invention of <I>Archimedes</I> to rai$e Wa er with the Screw, which is not only marvellous, but miraculous: for we $hall find that the Water a$cendeth in the Screw continually de$cending; and in a given Time, with a given Force doth rai$e an un$peakable quan- tity therof. But before we proceed any farther, let us declare the u$e of the Screw in making Water to ri$e: And in the en$uing Figure, let us con$ider the Line I L O P Q <fig> R S H being wrapped or twined about the Collumn M I K H, which Line you are to $uppo$e to be a Chanel thorow which the Water may run: If we $hall put the end I into the Water, making the Screw to $tand leaning, $o as the point L may be lower than the fir$t I, as the Diagram $hew- eth, and $hall turn it round about on the two Axes, T and V, the Water $hall run thorow the Cha- nel, till that in the end it $hall di$charge $orth at the mouth H. Now I $ay, that the Water, in its conveyance from the point I to the point H, doth go all the way de$cending, although the point H be higher than the point I. Which that it is $o, we will declare in this manner. We will de$cribe the Triangle A C B, which is that of which the Screw H I is generated, in $uch $ort that the Chanel of the Screw is repre$ented by the Line A C, who$e A$cent and Elevation is determined by the Angle C A B; that is to $ay, if $o be, that that Angle $hall be the third or fourth part of a Right Angle, then the Elevation of the Chanel A C $hall be ac- cording to 1/3, or 1/4 of a Right Angle. And it is manife$t; that the Ri$e of that $ame Chanel A C will be taken away deba$ing the point C as far as to B: for then the Chanel A C $hall have no Elevation. And deba$ing the point C a little below B, the Water will naturally run along the Chanel A C downwards from the point A towards C. Let us therefore conclude, that the Angle A being 1/3 of a Right Angle, the Chanel A C $hall no longer have any Ri$e, deba$ing it on the part <I>C</I> for 1/3 of a Right Angle.</P> <P>The$e things under$tood, let us infold the Triangle about the Column, and let us make the Screw B A E F G, &c. which if it $hall be placed at Right Angles with the end B in the Water, turn- ing it about, it $hall not this way draw up the Water, the Chanel about the Column being elevated, as may be $een by the part B A<*> <foot>Qq 2 But</foot> <p n=>300</p> But although the Column $tand erect at Right-Angles, yet for all that, the Ri$e along the Screw, folded about the Column, is not of a greater Elevation than of 1/3 of a Right Angle, it being generated by the Elevation of the Chanel A C: Therefore if we incline the Column but 1/3 of the <fig> $aid Right Angle, and a little more, as we $ee I K H M, there is a Tran$ition and Moti- on along the Chanel I L: Therefore the Water from the point I to the point L $hall move de$cending, and the Screw being turned about, the other parts of it $hall $ucce$$ively di$po$e or pre$ent them$elves to the Wa- ter in the $ame Po$ition as the part I L: Whereupon the Water $hall go $ucce$$ively de$cending, and in the end $hall be found to be a$cended from the point I to the point H. Which how admira- ble a thing it is, I leave $uch to judge who $hall perfectly have un- der$tood it. And by what hath been $aid, we come to know, That the Screw for rai$ing of Water ought to be inclined a little more than the quantity of the Angle of the Triangle by which the $aid Screw is de$cribed.</P> <head><I>Of the Force of the HAMMER, MALLET, or BEETLE.</I></head> <P>The Inve$tigation of the cau$e of the Force of the$e Percuti- ents is nece$$ary for many Rea$ons: and fir$t, becau$e that there appeareth in it much more matter of admiration than is ob$erved in any other Mechanick In$trument what$oever. For $triking with the Hammer upon a Nail, which is to be driven into a very tough Po$t, or with the Beetle upon a Stake that is to pene- trate into very $tiffe ground, we $ee, that by the $ole vertue of the blow of the Percutient both the one and the other is thru$t for- wards: $o that without that, only laying the Beetle upon the Nail or Stake it will not move then, nay, more, although you $hould lay upon them a Weight very much heavier than the $aid Beetle. An effect truly admirable, and $o much the more worthy of Contemplation, in that, as I conceive, none of tho$e who have <foot>hitherto</foot> <p n=>301</p> hitherto di$cour$ed upon it, have $aid any thing that hits the mark; which we may take for a certain Sign and Argument of the Ob$cu- rity and difficulty of this <I>S</I>peculation. For <I>Ari$totle,</I> or others, who would reduce the cau$e of this admirable Effect unto the length of the <I>Manubrium,</I> or Handle, may, in my judgement, be made to $ee their mi$take in the effect of tho$e In$truments, which having no Handle, yet percu$s, either in falling from on high downwards, or by being thrown with Velocity $idewaies. There- fore it is requi$ite, that we have recour$e to $ome other Principle, if we would find out the truth of this bu$ine$s; the cau$e of which, although it be of its own nature $omewhat ob$cure, and of diffi- cult con$ideration, yet neverthele$s we will attempt with the grea- te$t per$picuity po$$ible to render it clear and obvious, $hewing, for a clo$e of all, that the Principle and Original of this Effect is deri- ved from no other Fountain than this, from which the rea$ons of all other Mechanick Effects do proceed: and this we will do, by $etting before your eyes that very thing which is $een to befall in every other Mechanick Operation, <I>$cilicet,</I> That the Force, the Re$i$tance, and the Space by which the Motion is made, do go alternately with $uch proportion operating, and with $uch a rate an$wering to each other, that a Re$i$tance, equal to the Force, $hall be moved by the $aid Force along an equal Space, with Velocity equal to that with which it is moved. Likewi$e, That a Force that is le$s by half than a Re$i$tance $hall be able to move it, $o that it be moved with double Velocity, or, if you will, for a Di$tance twice as great as that which the moved Re$i$tance $hall pa$s: and, in a word, it hath been $een in all the other In$truments, that any, never $o great, Re$i$tance may be moved by every $mall Force given, provided, that the Space, along which the Re$i$tance $hall move, have the $ame proportion that is found to be betwixt the $aid great Re$i- $tance and the Force: and that this is according to the nece$$ary Order and Con$titution of Nature: So that inverting the Di$cour$e, and Arguing the contrary way, what wonder $hall it be, if that Power that $hall move a $mall Re$i$tance a great way, $hall carry one an hundred times bigger an hundredth part of that Di$tance? Certainly none at all: nay, it would be ab$urd, yea, impo$$ible that it $hould be otherwi$e. Let us therefore con$ider, what the Re$i$tance of the Beetle unto Motion may be in that point where it is to $trike, and how far, if it do not $trike, it would be carryed by the received Force beyond that point: and again, what Re$i- $tance to Motion there is in him who $triketh, and how much by that $ame Percu$$ion he is moved: and, having found that this great Re$i$tance goeth forwards by a percu$$ion $o much le$s than the Beetle driven by the <I>Impetus</I> of him that moveth it would do, by how much that $ame great Re$i$tance is greater than that of <foot>the</foot> <p n=>302</p> the Beetle; we $hall cea$e to wonder at the Effect, which doth not in the lea$t exceed the terms of Natural Con$titutions, and of what hath been $poken. Let us, for better under$tanding, give an example thereof in particular Terms. There is a Beetle, which ha- ving four degrees of Re$i$tance, is moved by $uch a Force, that being freed from it in that term where it maketh the Percu$$ion, it would, meeting with no $top, go ten Paces beyond it, and in that term a great po$t being oppo$ed to it, who$e Re$i$tance to Moti- on is as four thou$and, that is, a thou$and times greater than that of the Beetle, (but yet is not immoveable) $o that it without mea- $ure or proportion exceeds the Re$i$tance of the Beetle, yet the Percu$$ion being made on it, it $hall be driven forwards, though in- deed no more but the thou$andth part of the ten Paces which the Beetle $hall be moved: and thus in an inverted method, changing that which hath been $poken touching the other Mechanical Effects, we may inve$tigate the rea$on of the Force of the Percutient. I know that here ari$e difficulties and objections unto $ome, which they will not ea$ily be removed from, but we will freely remit them <marg>* The$e Pro- blems he here promi$eth were never yet ex- tant.</marg> to the ^{*} Problems Mechanical, which we $hall adjoyn in the end of this Di$cour$e.</P> <fig> <foot>THE</foot> <p n=>303</p> <head>THE BALLANCE OF <I>Signeur GALILEO GALILEI</I>;</head> <head>In which, in immitation of <I>Archimedes</I> in the Problem of the Crown, he $heweth how to find the proportion of the Alloy of Mixt-Metals; and how to make the $aid In$trument.</head> <P>As it is well known, by $uch who take the pains to read old Authors, that <I>Archimedes</I> detected the Cheat of the Gold$mith in the Crown of ^{*} <I>Hieron,</I> $o I think it <marg>* King of <I>Sicily,</I> and Kin$man to that Great Ma- thematician.</marg> hitherto unknown what method this Great Philo$o- pher ob$erved in that Di$covery: for the opinion, that he did per- form it by putting the Crown into the Water, having fir$t put in- <marg><I>Plutarch in Vit. Marcel.</I></marg> to it $uch another Ma$s of pure Gold, and another of Silver $eve- rally, and that from the differences in their making the Water more or le$s ri$e and run over, he came to know the Mixture or Alloy of the Gold with the Silver, of which that Crown was compounded; $eems a thing (if I may $peak it) very gro$s, and far from exactne$s. And it will $eem $o much the more dull to $uch who have read and under$tood the exqui$ite Inventions of $o Divine a Man among$t the Memorials that are extant of him; by which it is very manife$t that all other Wits are inferiour to that of <I>Archimedes.</I> Indeed I believe, that Fame divulging it abroad, that <I>Archimedes</I> had di$covered that $ame Fraud by means of the Water, $ome Writer of tho$e Times committed the memory there- of to P<*>$terity, and that this per$on, that he might add $omething to that little which he had heard by common Fame, did relate that <I>Archimedes</I> had made u$e of the Water in that manner, as $ince hath been by the generality of men believed.</P> <P>But in regard I know, that that method is altogether fallacious, and falls $hort of that exactne$s which is required in Mathematical Matters, I have often thought in what manner, by help of the Water, one might exactly find the Mixture of two Metals, and in the end, after I had diligently peru$ed that which <I>Archimedes</I> demon$trateth in his Books <I>De in$iden<*>bus aquæ,</I> and tho$e others <foot><I>De</I></foot> <p n=>304</p> <I>De æquiponder antium,</I> there came into my thoughts a Rule which exqui$itely re$olveth our Que$tion; which Rule I believe to be the $ame that <I>Archimedes</I> made u$e of, $eeing that be$ides the u$e that is to be made of the Water, the exactne$s of the Work dependeth al$o upon certain Demon$trations found by the $aid <I>Archimedes.</I></P> <P>The way is by help of a Ballance, who$e Con$truction and U$e $hall be $hewn by and by, after we $hall have declared what is nece$$ary for the knowledge thereof. You mu$t know there- fore, that the Solid Bodies that $ink in the Water weigh $o much le$s in the Water than in the Air, as a Ma$s of Water equal to the $aid Solid doth weigh in the Air: which hath been demon- $trated by <I>Archimedes.</I> But, in regard his Demon$tration is very mediate, becau$e I would not be over long, laying it a$ide, I $hall declare the $ame another way. Let us con$ider, therefore, that putting into the Water <I>v. g.</I> a Ma$s of Gold, if that Ma$s were of Water it would have no weight at all: For the Water moveth neither upwards, nor downwards in the Water: It remains, therefore, that the Ma$s of Gold weigheth in the Water only $o much as the Gravity of the Gold exceeds the Gravity of the Wa- ter. And the like is to be under$tood of other Metals. And be- cau$e the Metals are different from each other in Gravity, their Gravity in the Water $hall dimini$h according to $everal proporti- ons. As for example: Let us $uppo$e that Gold weigheth twenty times more than Water, it is manife$t by that which hath been $poken, that the Gold will weigh le$s in the Water than in the Air by a twentieth part of its whole weight. Now, let us $uppo$e that Silver, as being le$s Grave than Gold, weigheth 12 times more than Water: this then, being weighed in the Water, $hall di- mini$h in Gravity the twelfth part of its whole weight. Therefore the Gravity of Gold in the Water decrea$eth le$s than that of Silver; for that dimini$heth a twentieth part, and this a twelfth. If therefore in an exqui$ite Ballance we $hall hang a Metal at the one Arm, and at the other a Counterpoi$e that weigheth equally with the $aid Metal in the Water, leaving the Counterpoi$e in the Air, to the end that it may equivalate and compen$ate the Me- tal, it will be nece$$ary to hang it nearer the Perpendicular or Cook. As for example, Let the Ballance be A B, its Perpendicu- <fig> lar C, and let a Ma$s of $ome Metal be $u- $pended at B, counterpoi$edby the Weight D: putting the Weight B into the Water, the Weight D in A would weigh more: therefore that they may <foot>weigh</foot> <p n=>305</p> weigh equally it would be nece$$ary to hang it nearer to the Perpendicular C, as <I>v. gr.</I> in E: and look how many times the Di- $tance C A $hall contain A E, $o many times $hall the Metal weigh more than the Water. Let us therefore $uppo$e that the Weight in B be Gold, and that weighed in the Water it with- draws the Counterpoi$e D into E; and then doing the $ame with pure Silver, let us $uppo$e that its Counterpoi$e, when afterwards it is weighed in the Water, returneth to F: which point $hall be nearer to the point C, as Experience $heweth, becau$e the Silver is le$s grave than the Gold: And the Di$tance that is between A and F $hall have the $ame Difference with the Di$tance A E, that the Gravity of the Gold hath with that of the Silver. But if we have a Mixture of Gold and Silver, it is clear, that by rea$on it participates of Silver, it $hall weigh le$s than the pure Gold, and by rea$on it participates of Gold, it $hall weigh more than the pure Silver: and therefore being weighed in the Air, and de$iring that the $ame Counterpoi$e $hould counterpoi$e it, when that Mixture $hall be put into the Water it will be nece$$ary to draw the $aid Counterpoi$e more towards the Perpendicular C, than the point E is, which is the term of the Gold; and more from C than F is, which is the term of the pure Silver; Therefore it $hall fall between the points E and F: And the proportion into which the Di$tance EF $hall be divided, $hall exactly give the proportion of the two Metals which compound that Mixture. As for exam- ple: Let us $uppo$e the Mixture of Gold and Silver to be in B, <fig> counterpoi$ed in the Air by D, which Counter- poi$e when the Compound Me- tal is put into the Water returneth into G: I $ay now, that the Gold and the Silver which compound this Mixture are to one ano- ther in the $ame proportion, as the Di$tance F G is to the Di$tance G E. But you mu$t know that the Di$tance G F terminated in the mark of the Silver, $hall denote unto us the quantity of the Gold, and the Di$tance G E, terminated in the mark of the Gold, $hall $hew us the quantity of the Silver: in$omuch that if F G $hall prove double to G E, then that Mixture $hall be two parts Gold, and one part Silver: and in the $ame method proceeding i<*> the examination of other Mixtures, one $hall exactly find the quantity of the $imple Metals.</P> <P>To compo$e the Ballance, therefore, take a Rod at lea$t a yard long, (and the longer it is, the exacter the In$trument $hall be) and divide it in the mid$t, where place the Perpendicular: then adju$t the Arms that they may $tand in <I>Equilibrium,</I> by filing or <foot>Rr $having</foot> <p n=>306</p> $having that le$s which weigheth mo$t; and upon one of the Arms note the terms to which the Counterpoi$es of $imple Metals return when they $hall be weighed in the Water: taking care to weigh the pure$t Metals that can be found. This being done, it remaineth that we find out a way, how we may with facility di$cover the proportion, according to which, the Di$tances between the terms of the $imple and pure Metals are divided by the Marks of the Mixt Metals: Which $hall be effected in this manner.</P> <P>We are to have two very $mall Wires drawn thorow the $ame drawing-Iron, one of Steel, the other of Bra$s, and above the terms of the $imple Metals we mu$t wind the Steel Wyer; as for example: above the point E, the term of the pure Gold, we are to wind the Steel Wyer, and under it the other Bra$s Wyre, and having made ten folds of the Steel Wyer, we mu$t make ten more with that of Bra$s, and thus we are to continue to do with ten of Steel, and ten of Bra$s, until that the whole Space be- tween the points E and F, the terms of the pure Metals, be full; cau$ing tho$e two terms to be alwaies vi$ible and per$picuous: and thus the Di$tance E F $hall be divided into many equal parts, and numbred by ten and ten. And if at any time we would know the proportion that is between F G and G E, we mu$t count the Wyers F G, and the Wyers G E: and finding the Wyers F G to be, for example, 40, and the Wyers G E, 21: we will $ay that there is in the mixt Metal 40 parts of Gold, and 21 of Silver. But here you mu$t note, that there is $ome difficulty in the counting, for tho$e Wyers being very $mall, as it is requi$ite for exactne$s $ake, it is not po$$ible with the eye to tell them, becau$e the $malne$s of the Spaces dazleth & confoundeth the Sight. Therefore to number them with facility, take a Bodkin as $harp as a Needle and $et it into an handle, or a very fine pointed Pen-knife, with which we may ea$ily run over all the $aid Wyers, and this way partly by help of hearing, partly by the impediments the hand $hall feel at every Wyer, tho$e Wyers $hall be counted; the number of which, as I $aid before, $hall give us the exact quantity of the $unple Metals, of which the Mixt-Metal is com- pounded: taking notice that the Simple an$wer alternately to the Di$tances. As for example, in a Mixture of Gold and Silver, the Wyers that $hall be towards the term of Gold $hall $hew us the quantity of the Silver: And the $ame is to be under$tood of other Metals.</P> <foot>Annotations</foot> <p n=>307</p> <head>Annotations of <I>Dominico Mantovani</I> upon the Bal- lance of <I>Signore Galileo Galilei.</I></head> <P>Fir$t, I conceive that the difficulty of Numbring the Wyres is removed by wrapping about the Ballance ten of Steel, and then ten of Bra$s, which being divided by tens, there only remains that tenth part to be numbred, in which the term of the Mixt Metal falleth. For although <I>Signore Galileo,</I> who is Author of this Invention, makes mention of two Wyres, one of Steel, the other of Bra$s, yet he doth not $ay, that we are to take ^{*} ten of the one, and ten of the other: which it may be <marg>* <I>Galileus</I> $aith it expre$ly in this Copy which I fol- low, but might omit it in the Co- py which came to the hands of <I>Man- tovani.</I></marg> hapneth by the negligence of him that hath tran$cribed it; al- though I mu$t confe$s that the Copy which came to my hands was of his own writing.</P> <P>Secondly, it is $uppo$ed in this Problem that the Compo$ition of two Metals do retain the $ame proportion of Ma$s in the Mixture as the two Simple Metals, of which it is compounded, had at fir$t. I mean, that the Simple Metals retain and keep in the Compo$ition (after that they are incorporated and commix- ed) the $ame proportion in Ma$s that the Simple Metals had when they were $eparated: Which in the Ca$e of <I>Signore Gali- leo,</I> touching the Commixtion of Gold and Silver, I do neither deny, nor particularly confe$s. But if one would, for example, unite 101 pounds of Copper with 21 pounds of Tin, to make thereof 120 pounds of Bell-Metal, (I abate two pounds, $uppo$ed to be wa$ted in the Melting) I do think that 120 pounds of Compound Metal will have a le$s Bulk than the 100 pounds of pure Copper, and the 20 pounds of Tin unmixt, that is, before they were incorporated and melted into one Ma$s, and that the Compo$ition is more grave <I>in Specie</I> than the $ingle Cop- per, and the $ingle Bra$s: and in the Ca$e of <I>Signore Galileo</I> the Compo$ition of Gold and Silver is $uppo$ed to be lighter <I>in Specie</I> than the pure Gold, and heavier <I>in Specie</I> than the pure Silver. Of which it would be ea$ie to make $ome $uch like experiment, melt- ing together, <I>v. gr.</I> 10 pounds of Lead with 5 pounds of Tin, and ob$erving whether tho$e 15 pounds, or whatever the Mixture maketh, do give the difference betwixt the weight in the Water to the weight in the Air, in the proportion that the 15 pounds of the two Metals di$-united gave before: I do not $ay, the $ame diffe- rence, becau$e I pre $uppo$e that they will wa$te in melting down, and that the Compound will be le$s than 15 pounds, therefore I $ay in proportion.</P> <P>Thirdly, He doth al$o $uppo$e, that one ought to take the <foot>Rr 2 Simple</foot> <p n=>308</p> Simple Metals, that is, the Gold and the Silver, each of the $ame weight as the Mixture, although he doth not $ay $o; which may be collected in that he marketh the ballance only betwixt the Terms of the Gold and the Silver, which is the cau$e of the great facility in re$olving the Problem by only counting the Wyers.</P> <P>One might take the pure Gold, and pure Silver of the $ame weight, in re$pect of one another, but yet different from the weight of the Mixture, that is, either more or le$s grave than the Mixt Metal: and being equal in weight to one another they might $hew the proportion of the Ma$s of the Gold to that of the Silver; but yet with this difference, that the more grave will $hew the $aid proportion more exactly than the $mall and le$s grave. But the Simple and pure Metals not being of the $ame weight as the Compound, it will be nece$$ary, having found the proportion of the Ma$s of the Gold to that of the Silver; to find by numbers proportionally the exact quantity of each of the two Metals com- pounding the Mixture.</P> <P>A man may likewi$e u$e the quantity of the $imple Metals ac- cording to nece$$ity and convenience, although of different Weights, both as to each other, and to the Mixture, provided that each of them be pure in its kind: but then we mu$t after- wards by numbers find the proportion of the Ma$$es of the two Simple ones of equal weight (which is $oon done, taking them of equal weight as was $aid before) and then according to this pro- portion to find, by means of the Weight, and of the Ma$s of the Compound Metal, the di$tinct quantity of each of the two Sim- ple ones that make the Compo$ition: of each of which Ca$es examples might be given. But to conclude, if the pure Gold, and pure Silver, and the Mixt Metal $hould be of equal Ma$s, they would be unequal in Weight, and it would not need to weigh them in the Water, for being of equal Bulk, the differen- ces of their Weights in the Air and in the Water would be al$o equal: for the difference of the weight of any Body in the Air to its weight in the Water, is alwaies equal to the Weight of $o much Water as equalleth the $ame Body in Ma$s, by <I>Archimedes</I> his fifth Propo$ition, <I>De ijs quæ vehuntur in aqua.</I></P> <P>And la$t of all, the Simple and pure Metals may have the $ame proportion in Gravity, mutually or reciprocally, as their Bodies have in Bulk: In which ca$e, as well the Ma$s, found by help of the weight in Water, or by any other meanes, as their Weight in the Air $hall $hew the proportion of their Specifical Gravities; as their Weights in the Water do when their Weights in the Air are equal; but yet alternately weighed: that is to $ay, the Spe- cifical Gravity of the Gold $hall have $uch proportion to the <foot>Specifical</foot> <p n=>309</p> Specifical Gravity of the Silver, as the Ma$s of the Silver hath to the Ma$s of the Gold; that is, as the difference betwixt the Weight in Water and Weight in Air of the Silver, hath to the difference betwixt the Weight in Water and Weight in Air of the Gold.</P> <P>With this $ame Ballance one may with facility mea$ure the Ma$s or Magnitude of any Body, in any manner what$oever Irre- gular in manner following, namely:</P> <P>We will have at hand a Solid Body of a $ub$tance more grave <I>in Specie</I> than the Water; as for in$tance of Lead; or if it were of Wood, or other matter more light <I>in Specie</I> than the Water, it may be made heavier by fa$tning unto it Lead, or $ome other thing that makes it $ink in the Water, and let us take $ome known Mea$ure, and with it mea$ure the Irregular Solid; as for in$tance, the Roman Palm, the Geometrical Foot, or any other known mea$ure, or part of the $ame, as the half Foot, the quar- ter of a Foot, or any $uch like part known; then let it be weighed in the Air, and $uppo$e that it weigh 10 pounds; let the $ame Mea$ure be weighed in the Air, and $uppo$e that it weigh 8 pounds: and $ub$tract 8 pounds, the Weight in the Water, from 10 pounds, the Weight in the Air, and there remaineth 2 pounds for the Weight of a Body of Water equal in Magnitude to the Mea$ure known. Now, if we would mea$ure a Statue of Mar- ble, let it be weighed fir$t in the Air, and then in the Water, and $ub$tract the Weight in the <I>W</I>ater from the <I>W</I>eight in the Air, and the remainder $hall be the weight of $o much <I>W</I>ater as equalleth the Statue in Ma$s; which being divided by the difference betwixt the <I>W</I>eight in <I>W</I>ater and the <I>W</I>eight in Air of the Mea$ure known, the Quotient will give how many times the Statue containeth the $ame given Mea$ure. As for example; if the Statue in Air weigh 100 pounds, and in the <I>W</I>ater 80 pounds, 80 pounds being $ub- $tracted from 100 there re$teth 20 pounds for the <I>W</I>eight of $o much <I>W</I>ater in Ma$s as equalleth the Statue. But becau$e the difference betwixt the <I>W</I>eight in <I>W</I>ater, and the <I>W</I>eight in Air equal in Magnitude to the Mea$ure known, was $uppo$ed to be 2 pounds; divide 18 pounds by two pounds, and the Quotient is 9, for the number of times that the propo$ed Statue containeth the given Mea$ure. The $ame Method may be ob$erved, if it were required, to mea$ure a Statue, or other Ma$s of any kind of Metal: only it mu$t be adverti$ed, that all the holes mu$t be $topt, that the <I>W</I>ater may not enter into the Body of the Statue: but he that de$ireth only the Solid content of the Metal of the $aid Statue mu$t open the holes, and with Tunnels fill the whole cavity of the Statue with <I>W</I>ater. And if the Statue were of a Sub$tance lighter <I>in Specie</I> than the <I>W</I>ater; as, for example, of <foot><I>W</I>ax,</foot> <p n=>310</p> Wax, it will be requi$ite to add unto the Statue $ome Counter- poi$e, that maketh it $ink in the <I>W</I>ater, and then to mea$ure the Counterpoi$e, as above, and to $ub$tract its mea$ure from the Compound Body, and there will remain the Mea$ure of the Statue of <I>W</I>ax. And la$tly, to make u$e of the $aid Ballance, in$tead of $eeking the numbers of the pounds of the Differences of the <I>W</I>eights of the Mea$ure known, and of the Solid to be mea$ured in <I>W</I>ater, and in Air, we may count the <I>W</I>yers of the Arm of the Ballance, which being very $mall will give the Mea$ure exactly.</P> <head><I>FINIS.</I></head> <fig> <p n=>311</p> <head>DISCOURSES OF THE MECHANICKS: A MANVSCRIPT of Mon$ieur Des-Cartes.</head> <head>The Explication.</head> <P><I>Of Engines, by help of which we may rai$e a very great weight with $mall $trength.</I></P> <P>The Invention of all the$e Engines de- pends upon one $ole Principle, which is, That the $ame Force that can lift up a Weight, for example, of 100 pounds to the height of one foot, can life up one of 200 pounds to the height of half a foot, or one of 400 pounds to the height of a fourth part of a foot, and $o of the re$t, be there never $o much applyed to it: and this Principle cannot be denied if we con$ider, that the Effect ought to be proportioned to the Action that is nece$$ary for the production of it; $o that, if it be nece$$ary to employ an Action by which we may rai$e a Weight of 100 pounds to the height of two foot, for to rai$e one $uch to the height of one foot only this $ame ought to weigh 200 pounds: for its the $ame thing to rai$e 100 pounds to the height of one foot, and again yet another 100 pounds to the height of one foot, as to rai$e one of 200 pounds to the height of one foot, and the $ame, al$o, as to rai$e 100 pounds to the height of two feet.</P> <P>Now, the Engines which $erve to make this Application of a Force which acteth at a great Space upon a <I>W</I>eight which it cau- <foot>$eth</foot> <p n=>312</p> $eth to be rai$ed by a le$$er, are the Pulley, the Inclined Plane, the Wedg, the Cap$ten, or Wheel, the Screw, the Leaver, and $ome others, for if we will not apply or compare them one to another, we cannot well number more, and if we will apply them we need not in$tance in $o many.</P> <head>The PVLLEY, <I>Trochlea.</I></head> <P>Let A B C be a Chord put about the Pulley D, to which let the Weight E be fa$tned; and fir$t, $uppo$ing that two men $u$tain or pull up equally each of them one of the <fig> ends of the $aid Chord: it is manife$t, that if the Weight weigheth 200 pounds, each of tho$e men $hal employ but the half thereof, that is to $ay, the Force that is requi$ite for $u$taining or rai$ing of 100 pounds, for each of them $hal bear but the half of it.</P> <P>Afterwards, let us $up- po$e that A, one of the ends of this Chord, being made fa$t to $ome Nail, the other C be again $u- $tained by a Man; and it is manife$t, that this Man in C, needs not (no more than before) for the $u$taining the Weight E, more Force than is requi$ite for the $u$taining of 100 pounds: becau$e the Nail at A doth the $ame Office as the Man which we $uppo$ed there before. In fine, let us $uppo$e that this Man in C do pull the Chord to make the Weight E to ri$e, and it is manife$t, that if he there employeth the Force which is requi$ite for the rai$ing of 100 pounds to the height of two feet, he $hall rai$e this Weight E of 200 pounds to the height of one foot: for the Chord A B C being doubled, as it is, it mu$t be pull'd two feet by the end C, to make the Weight E ri$e as much, as if two men did draw it, the one by the end A, and the other by the end C, each of them the length of one foot only.</P> <P>There is alwaies one thing that hinders the exactne$s of the Cal- culation, that is the pondero$ity of the Chord or Pulley, and the difficulty that we meet with in making the Chord to $lip, and in bearing it: but this is very $mall in compari$on of that which <foot>rai$eth</foot> <p n=>313</p> rai$eth it, and cannot be e$timated $ave wthin a $mall matter.</P> <P>Moreover, it is nece$$ary to ob$erve, that it is nothing but the redoubling of the Chord, and not the Pulley, that cau$eth this Force: for if we fa$ten yet another Pulley towards A, about which we pa$s the Chord A B C H, there will be required no le$s Force to draw H towards K, and $o to lift up the Weight E, than there was before to draw C towards G. But if to the$e two Pul- leys we add yet another towards D, to which we fa$ten the Weight, and in which we make the Chord to run or $lip, ju$t as we did in the fir$t, then we $hall need no more Force to lift up this Weight of 200 pounds than to lift up 50 pounds without the Pulley: be- cau$e that in drawing four feet of Chord we lift it up but one foot. And $o in multiplying of the Pulleys one may rai$e the great- e$t Weights with the lea$t Forces. It is requi$ite al$o to ob$erve, that a little more Force is alwaies nece$$ary for the rai$ing of a Weight than for the $u$taining of it: which is the rea$on why I have $poken here di$tinctly of the one and of the other.</P> <head><I>The Inclined</I> PLANE.</head> <P>If not having more Force than $ufficeth to rai$e 100 pounds, one would neverthele$s rai$e this Body F, that weigheth 200 pounds, to the height of the Line B A, there needs no more but to draw or rowl it along the Inclined Plane C A, which I $uppo$e to be twice as long as the Line <fig> A B, for by this means, for to make it arrive at the point A, we mu$t there employ the Force that is nece$$ary for the rai$ing 100 pounds twice as high, and the more inclined this Plane $hall be made, $o much the le$s Force $hall there need to rai$e the Weight F. But yet there is to be rebated from this Calculation the difficulty that there is in moving the Body F, along the Plane A C, if that Plane were laid down upon the Line B C, all the parts of which I $uppo$e to be equidi$tant from the Center of the Earth.</P> <P>It is true, that this impediment being $o much le$s as the Plane is more united, more hard, more even, and more polite; it cannot likewi$e be e$timated but by gue$s, and it is not very con$ide- rable.</P> <P>We need not neither much to regard that the Line B C being a part of a Circle that hath the $ame Center with the Earth, the Plane A C ought to be (though but very little) curved, and to have the Figure of part of a Spiral, de$cribed between two Circles, <foot>S$ which</foot> <p n=>314</p> which likewi$e have for their Center that of the Earth, for that it is not any way $en$ible.</P> <head><I>The</I> WEDGE, <I>Cuneus.</I></head> <P>The Force of the Wedge A B C D is ea$ily under$tood after that which hath been $poken above of the Inclined Plane, for the Force wherewith we $trike downwards acts as if it were to make it move according to the Line B D; and the Wood, or other thing and Body that it cleaveth, openeth not, or the Weight that it rai$eth doth not ri$e, $ave only according to the <fig> Line A C, in$omuch that the Force, wherewith one driveth or $triketh this Wedge, ought to have the $ame Pro- portion to the Re$i$tance of this Wood or Weight, that A C hath to A B. Or el$e again, to be exact, it would be convenient that B D were a part of a Circle, and A D and C D two portions of Spirals that had the $ame Center with the Earth, and that the Wedge were of a Matter $o perfectly hard and polite, and of $o $mall weight, as that any little Force would $uffice to move it.</P> <head><I>The</I> CRANE, <I>or the</I> CAPSTEN, <I>Axis in Peritrochio.</I></head> <P>We $ee al$o very ea$ily, that the Force wherewith the Wheel A or Cogg B is turned, which make the Axis or Cylinder C to move, about which a Chord is rolled, to which the Weight D, which we would rai$e, is fa$tned, ought to have the <fig> $ame proportion to the $aid Weight, as the Circumference of the Cylinder hath to the Cir- cumference of a Circle which that Force de$cribeth, or that the Diameter of the one hath unto the Diameter of the other; for that the Circumferences have the $ame proportion as the Diame- ters: in$omuch that the Cylinder C, having no more but one foot in Diameter, if the Wheel AB be $ix feet in its Diameter, and the Weight D do weigh 600 pounds, it $hall $uffice that the Force in B $hall be capable to rai$e 100 pounds, and $o of others. One may <foot>al$o</foot> <p n=>315</p> al$o in$tead of the Chord that rolleth about the Cylinder C, place there a $mall Wheel with teeth or Coggs, that may turn another greater, and by that means multiply the power of the Force as much as one $hall plea$e, without having any thing to deduct of the $ame, $ave only the difficulty of moving the Machine, as in the others.</P> <head><I>The</I> SCREW, <I>Cochlea.</I></head> <P>When once the Force of the Cap$ten and of the In- clined Plane is under$tood, that of the Screw is ea$ie to be computed, for it is compo$ed only of a Plane much inclined, which windeth about a Cylinder: and if this Plane be in $uch manner Inclined, as that the Cylinder ought to make <I>v. gr.</I> ten turns to advance forwards the length of a foot in the Screw, and that the bigne$s of the Circumference of the Circle <fig> which the Force that turneth it about doth de$cribe be of ten feet; fora$much as ten times ten are one hundred, one Man alone $hall be able to pre$s as $trongly with this In$trument, or Screw, as one hundred without it, provided alwaies, that we rebate the Force that is required to the turning of it.</P> <P>Now I $peak here of Pre$$ing rather than of Rai$ing, or Remo- ving, in regard that it is about this mo$t commonly that the Screw is employed, but when we would make u$e of it for the rai$ing of Weights, in$tead of making it to advance into a Female Screw, we joyn or apply unto it a Wheel of many Coggs, in $uch $ort made, that if <I>v. gr.</I> this <I>W</I>heel have thirty Coggs, whil$t the Screw maketh one entire turn, it $hall not cau$e the <I>W</I>heel to make more than the thirtieth part of a turn, and if the <I>W</I>eight be fa$tned to a Chord that rowling about the Axis of this <I>W</I>heel $hall rai$e it but one foot in the time that the <I>W</I>heel makes one entire revolution, and that the greatne$s of the Circumference of the Circle that is de$cribed by the Force that turneth the Screw about be al$o of ten $eet, by rea$on that 10 times 30 make 300, one $ingle Man $hall be able to rai$e a <I>W</I>eight of that bigne$s with this In$trument, which is called the Perpetual Screw, as would require 300 men with- out it.</P> <P>Provided, as before, that we thence deduct the difficulty that we meet with in turning of it, which is not properly cau$ed by the Pondero$ity of the <I>W</I>eight, but by the Force or Matter of the In- <foot>S$ <*> $trument</foot> <p n=>316</p> $trument: which difficulty is more $en$ible in it than in tho$e afore- going, fora$much as it hath greater Force.</P> <head>The LEAVER, <I>Vectis.</I></head> <P>I Have deferred to $peak of the Leaver until the la$t, in regard that it is of all Engines for rai$ing of Weights, the mo$t diffi- cult to be explained.</P> <P>Let us $uppo$e that C H is a Leaver, in $uch manner $upported at the point O, (by means of an Iron Pin that pa$$eth thorow it acro$s, or otherwi$e) that it may turn about on this point O, its part C de$cribing the Semicircle A B C D E, and its part H the <fig> Semicircle F G H I K; and that the Weight which we would rai$e by help of it were in H, and the Force in C, the Line C O being $uppo$ed triple of O H. Then let us con$ider that in the Time whil$t the Force that moveth this Leaver de$cri- beth the whole Semicircle A B C D E, and acteth accord- ing to the Line A B C D E, al- though that the Weight de$cri- beth likewi$e the Semicircle F G H I K, yet it is not rai$ed to the length of this curved Line F G H I K, but only to that of the Line F O K; in$omuch that the Proportion that the Force which moveth this Weight ought to have to its Pondero$ity ought not to be mea$ured by that which is between the two Diameters of the$e Circles, or between their two Circumferences, as it hath been $aid above of the Wheel, but ra- ther by that which is betwixt the Circumference of the greater, and the Diameter of the le$$er. Furthermore let us con$ider, that there is a nece$$ity that this Force needeth not to be $o great, at $uch time as it is near to A, or near to E, for the turning of the Leaver, as then when it is near to B, or to D; nor $o great when it is near to B or D, as then when it is near to C: of which the rea- $on is, that the Weights do there mount le$s: as it is ea$ie to un- der$tand, if having $uppo$ed that the Line C O H is parallel to the Horizon, and that A O F cutteth it at Right Angles, we take the point G equidi$tant from the points F and H, and the point B equi- di$tant from A and C; and that having drawn G S perpendicular to F O, we ob$erve that the Line F S (which $heweth how much the Weight mounteth in the Time that the Force operates along <foot>the</foot> <p n=>317</p> the Line A B) is much le$$er than the Line S O, which $heweth how much it mounteth in the Time that the Force opperates along the Line B C.</P> <P>And to mea$ure exactly what his Force ought to be in each Point of the curved Line A B C D E, it is requi$ite to know that it ope- rates there ju$t in the $ame manner as if it drew the Weight along a Plane Circularly Inclined, and that the Inclination of each of the Points of this circular Plane were to be mea$ured by that of the right Line that toucheth the Circle in this Point. As for example, when the Force is at the Point B, for to find the proportion that it ought to have with the pondero$ity of the Weight which is at that time at the Point G, it is nece$$ary to draw the Contingent Line G M, and to account that the pondero$ity of the Weight is to the Force which is required to draw it along this Plane, and con$e- quently to rai$e it, according to the Circle F G H, as the Line G M is to SM Again, for as much as B O is triple of O G, the Force in B needs to be to the Weight in G but as the third part of the Line SM is unto the whole Line G M. In the $elf $ame manner, when the Force is at the Point D, to know how much the Weight weigheth at I, it is nece$$ary to draw the Contingent Line betwixt I and P, and the right Line I N perpendicular upon the Horizon, and from the Point P taken at di$cretion in the Line I P, provided that it be below the Point I, you mu$t draw P N parallel to the $ame Horizon, to the end you may have the proportion that is be- twixt the Line I P and the third part of the Line I N, for that which betwixt the pondero$ity of the Weight, and the Force that ought to be at the Point D for the moving of it: and $o of others. Where, neverthele$s, you mu$t except the Point H, at which the Contin- gent Line being perpendicular upon the Horizon, the Weight can be no other than triple the Force which ought to be in C for the moving of it: in the Points F and K, at which the Contingent Line being parallel unto the Horizon it $elf, the lea$t Force that one can a$$ign is $ufficient to move the Weight. Moreover, that you may be perfectly exact, you mu$t ob$erve that the Lines S G and P N ought to be parts of a Circle that have for their Center that of the Earth; and GM and I P parts of Spirals drawn between two $uch Circles; and, la$tly, that the right Lines S M and I N both tending towards the Center of the Earth are not exactly Paral- lels: and furthermore, that the Point H where I $uppo$e the Contingent Line to be perpendicular unto the Horizon ought to be $ome $mall matter nearer to the Point F than to K, at the which F and K the Contingent Lines are Parallels unto the $aid Horizon.</P> <P>This done, we may ea$ily re$olve all the difficulties of the Ba- lance, and $hew, That then when it is mo$t exact, and for in$tance, <foot>$uppo$ing</foot> <p n=>318</p> $uppo$ing it's Centre at O by which it is $u$tained to be no more but an indivi$ible Point, like as I have $uppo$ed here for the Leaver, if the Armes be declined one way or the other, that which $hall be the lowermo$t ought evermore to be adjudged the heavier; $o that the Centre of Gravity is not $ixed and immoveable in each $everal Body, as the Ancients have $uppo$ed, which no per$on, that I know of, hath hitherto ob$erved.</P> <P>But the$e la$t Con$iderations are of no moment in Practice, and it would be good for tho$e who $et them$elves to invent new Machines, that they knew nothing more of this bu$i- ne$$e than this little which I have now writ thereof, for then they would not be in danger of decei- ving them$elves in their Computation, as they frequently do in $uppo$ing other Principles.</P> <head><I>FINIS.</I></head> <fig> <p n=>319</p> <head>A LETTER OF Mon$ieur Des-Cartes TO THE REVEREND FATHER <I>MARIN MERSENNE.</I></head> <P><I>Reverend Father,</I></P> <P>I Did think to have deferred writing unto you yet eight or fifteen dayes, to the end I might not trouble you too often with my Letters, but I have received yours of the fir$t of <I>Sept.</I> which giveth me to under$tand that it is an hard matter to admit the Principle which I have $uppo$ed in my Examination of the Geo$tatick Que$tion, and in regard that if it be not true, all the re$t that I have inferred from it would be yet le$$e true: I would not one onely day defer $ending you a more particular Explication. It is requi$ite above all things to con$ider that I did $peak of the Force that $erveth to rai$e a Weight to $ome heighth, the which Force hath evermore two Dimen$ions, and not of that which $erveth in each point to $u$tain it, which hath never more than one Dimen$ion, in$omuch that the$e two Forces differ as much the one from the other, as a Superficies differs from a Line: for the $ame Force which a Nail ought to have for the $u$taining of a Weight of 100 pound one moment of time, doth al$o $uffice for to $u$tain it the $pace of a year, provided that it do not dimini$h, but the $ame Quantity of this Force which $erveth to rai$e the Weight to the heighth of one foot, $ufficeth not <I>(eadem numero)</I> to rai$e it two feet; and it is not more manife$t that two and two make four, than it's manife$t that we are to employ double as much therein.</P> <P>Now, fora$much as that this is nothing but the $ame thing that I have $uppo$ed for a Principle, I cannot gue$$e on what the Scruple $hould be grounded that men make of receiving it; but I $hall in <foot>this</foot> <p n=>320</p> this place $peak of all $uch as I $u$pect, which for the mo$t part ari$e onely from this, that men are before-hand over-knowing in the Mechanicks; that is to $ay, that they are pre-occupied with Principles that others prove touching the$e matters, which not being ab$olutely true, they deceive the more, the more true they $eem to be.</P> <P>The fir$t thing wherewith a man may be pre-occupied in this bu$ine$$e, is, that they many times confound the Con$ideration of <fig> Space, with that of Time, or of the Ve- locity, $o that, for Example, in the <I>L</I>eaver, or (which is the $ame) the Ba- llance A B C D having $uppo$ed that the Arm A B is double to B C, and the Weight in C double to the Weight in A, and al$o that they are in <I>Equilibrium,</I> in$tead of $aying, that that which cau$eth this <I>Equilibrium</I> is, that if the Weight C did $u$tain, or was rai$ed up by the Weight A, it did not pa$$e more than half $o much Space as it, they $ay that it did move $lower by the half: which is a fault $o much the more prejudicial, in that it is very difficult to be known: for it is not the difference of <fig> the Velocity that is the cau$e why the$e Weights are to be one double to the other, but the difference of the Space, as appeareth by this, that to rai$e, for Example, the Weight F with the hand unto G, it is not nece$$ary to employ a Force that is preci$ely double to that which one $hould have therein employed the fir$t bout, to rai$e it twice as quick- ly, but it is requi$ite to employ therein either more or le$s than the double, according to the different proportion that this Velocity may have unto the Cau$es that re$i$t it.</P> <P>In$tead of requiring a Force ju$t double for the rai$ing of it with the $ame Velocity twice as high, unto H, I $ay that it is ju$t dou- ble in counting (as two and two make four) that one and one make two, for it is requi$ite to employ a certain quantity of this Force to rai$e the Weight from F to G, and again al$o, as much more of the $ame Force to rai$e it from G to H.</P> <P>For if I had had a mind to have joyned the Con$ideration of the Velocity with that of the Space, it had been nece$$ary to have a$$igned three Dimen$ions to the Force, whereas I have a$$igned it no more but two, on purpo$e to exclude it. And if I have te$tified that there is $o little of worth in any part of this $mall Tract of the Staticks, yet I de $ire that men $hould know, that there is more in this alone than in all the re$t: for it's impo$$ible to $ay any thing that is good and $olid touching Velocity, without having rightly explained what we are to under$tand by Gravity, as al$o the whole Sy$teme of the World. Now becau$e I would not under take it, <foot>I have</foot> <p n=>321</p> I have thought good to omit this Con$ideration, and in this manner to $ingle out the$e others that I could explain without it: for though there be no Motion but hath $ome Velocity, neverthele$s it is onely the Augmentations and Diminutions of this Velocity that are con$iderable. And now that $peaking of the Motion of a Body, we $uppo$e that it is made according to the Velocity which is mo$t naturall to it, which is the $ame as if we did not con$ider it at all.</P> <P>The other rea$on that may have hindred men from rightly un- der$tanding my Principle is, that they have thought that they could demon$trate without it $ome of tho$e things which I demon$trate not without it: As, for example, touching the Pulley A B C, they have thought that it was enough to know that the Nail in A did <fig> $u$tain the half of the Weight B; to conclude that the Hand in C had need but of half $o much Force to $u$tain or rai$e the Weight, thus wound about the Pulley, as it would need for to $u$tain or rai$e it without it. But howbeit that this ex- plaineth very well, how the application of the Force at C is made unto a Weight double to that which it could rai$e without a Pulley, and that I my $elf did make u$e thereof, yet I deny that this is $imply, becau$e that that the Nail A $u- $taineth one part of the Weight B, that the Force in C, which $u$taineth it, might be le$s than if it had been $o $u$tained. For if that had been true, the Rope C E be- ing wound about the Pulley D, the Force in E might by the $ame rea$on be le$s than the Force in C: for that the Nail A doth not $u$tain the Weight le$s than it did before, and that there is al$o another Nail that $u$tains it, to wit, that to wich the Pulley D is fa$tned. Thus therefore, that we may not be mi$taken in this, that the Nail A $u$taineth the half of the Weight B, we ought to con- clude no more but this, that by this application the one of the Di- men$ions of the Force that ought to be in C <fig> to rai$e up this Weight is dimini$hed the one half; and that the other, of con$equence, be- cometh double, in $uch $ort that if the Line F G repre$ent the Force that is required for the $u$taining the Weight B in a point, with- out the help of any Machine, and the Quadrangle G H that which is required for the rai$ing of it to the height of a foot, the $upport of the Nail A dimini$heth the Di- men$ion which is repre$ented by the Line F G the one half, and the redoubling of the Rope A B C maketh the other Dimen$ion to <foot>Tt double</foot> <p n=>322</p> double, which is repre$ented by the Line FH; and $o the Force that ought to be in C for the rai$ing of the Weight B to the height of one foot is repre$ented by the Quadrangle IK; and, as we know in Geometry, that a Line being added to, or taken from a Superfi- cies, neither augmenteth, nor dimini$heth it in the lea$t, $o the Force where with the Nail A $u$tains the Weight B, having but one $ole Dimen$ion, cannot cau$e that the Force in C, con$idered ac- cording to its two Dimen$ions, ought to be le$s for the rai$ing in like manner the Weight E, than for the rai$ing it without any Pulley.</P> <P>The third thing which may make men imagine $ome Ob$curity in my Principle is, that they, it may be, have not had regard to all the words by which I explain it; for I do not $ay $imply that the Force that can rai$e a Weight of 50 pounds to the height of four feet can rai$e one of 200 pounds to the height of one foot; but I $ay that it may do it, if $o be that it be applyed to it: now it is impo$$ible to apply the $ame thereto, but by the means of $ome Ma- chine, or other Invention that $hall cau$e this Weight to a$cend but one, in the time whil$t the Force pa$$eth the whole length of four feet, and $o that it do transform the Quandrangle, by which the Force is repre$ented that is required to rai$e this Weight of 400 pounds to the height of one foot into another that is equall and like to that which repre$ents the Force that is required for to rai$e a Weight of 50 pounds to the height of four feet.</P> <P>In fine, it may be that men may have thought the wor$e of my Principle, becau$e they have imagined that I have alledged the Ex- amples of the Pulley, of the Inclined Plane, and of the Leaver, to the end that I might better per$warde the truth thereof, as if it had been dubious, or el$e that I had $o ill di$cour$ed as to offer to a$$ume from thence a Principle, which ought of it felf to be $o clear, as not to need any proof by things that are $o difficult to comprehend as that; it may be, they have never been well demon$trated by any man: but neither have I made u$e of them, $ave only with a de$ign to $hew that this Principle extends it $elf to all matters of which one treateth in the Staticks: or, rather, I have made u$e of this oc- ca$ion for to in$ert them into my Treati$e, for that I conceived that it would have been too dry and barren if I had therein $po- ken of nothing el$e but of this Que$tion, that is of no u$e, as of that of the Geo$taticks, which I purpo$ed to examine.</P> <P>Now one may perceive, by what hath already been $aid, how the Forces of the Leaver and Pulley are demon$trated by my Principle $o well, that there only remains the Inclined Plane, of which you $hall clearly $ee the Demon$tration by this Figure; in which G F repre$ents the fir$t Dimen$ion of the Force that the <foot>Rectangle</foot> <p n=>323</p> Rectangle F H de$cribeth whil$t it draweth the Weight D along the Plane B A, by the means of a Chord parallel to this Plane, and pa$$ing about the Pulley E, in $uch $ort, that H G, that is the height of this Rectangle, is equal to B A, along which the Weight D is to move, whil$t it mounteth to the height of the Line C A. And N O repre$ents the fir$t Dimen$ion of $uch another Force, that is de- $cribed by the Rectan- gle N P, in the time that <fig> it is rai$ing the Weight L to M. And I $uppo$e that L M is equal to B A, or double to C A; and that N O is to F G, as O P is to G H. This done, I con$ider that at $uch time as the Weight D is moved from B to- wards A, one may ima- gine its Motion to be compo$ed of two others, of which the one carrieth it from B R to- wards C A, (to which operation there is no Force required, as all tho$e $uppo$e who treat of the Mechanicks) and the other rai$eth it from B C towards R A, for which alone the Force is required: in$omuch that it needs neither more nor le$s Force to move it along the Inclined Plane B A, than along the Perpendicular C A. For I $uppo$e that the unevenne$$es, <I>&c.</I> of the Plane do not at all hinder it, like as it is alwaies $uppo$ed in treating of this matter.</P> <P>So then the whole Force F H is employed only about the rai$ing of D to the height of C A: and fora$much as it is exactly equal to the Force N P, that is required for the rai$ing of L to the Height of L M, double to C A, I conclude by my Principle that the Weight D is double to the Weight L. For in regard that it is nece$$ary to employ as much Force for the one as for the other, there is as much to be rai$ed in the one as in the other; and no more knowledge is required than to count unto two for the knowing that it is alike facile to rai$e 200 pounds from C to A, as to rai$e 100 pounds from L to M: $ince that L M is double to C A.</P> <P>You tell me, moreover, that I ought more particularly to ex- plain the nature of the Spiral Line that repre$enteth the Plane equally enclined, which hath many qualities that render it $uffi- ciently knowable.</P> <foot>Tt 2 For</foot> <p n=>324</p> <P>For if A be the Center of the Earth, <fig> and A N B C D the Spiral Line, having drawn the Right Lines A B, A D, and the like, there is the $ame proportion betwixt the Curved Line A N B and the Right Line AB, as is betwixt the Curved Line A N B C, and the Right Line A C; or betwixt A N B C D and A D: and $o of the re$t.</P> <P>And if one draw the Tangents D E, C F, and B G, the Angles A D E, A C F, A B G, &c. $hall be equal. As for the re$t I will, &c.----</P> <P>Reverend Father,</P> <P>Your very humble Servant</P> <P><I>DES-CARTES.</I></P> <fig> <p n=>325</p> <head>A LETTER OF Mon$ieur de Robberval TO Mon$ieur de Fermates, Coun$ellour of <I>THOULOUSE,</I> Containing certain Propo$itions in the MECHANICKS.</head> <P>MONSIEUR,</P> <P>I have, according to my promi$e, $ent you the Demon$tration of the Fundamental Propo$i- tion of our Mechanicks, in which I follow the common method of explaining, in the fir$t place, the Definitions and Principles of which we make u$e.</P> <P>We in general call that Quality a Force or Power, by means of which any thing whatever doth tend or a$pire into another place than that in which it is, be it downwards, upwards, or $ide waies, whether this Quality naturally belongeth to the Body, or be communicated to it from without. From which definition it followeth, that all Weights are a $pecies of Force, in regard that it is a Quality, by means whereof Bodies do tend downwards. We often al$o a$$ign the name of Force to that very thing to which the Force belongeth, as a ponderous Bo- dy is called a Weight, but with this pre-caution, that this is in re- ference to the true Force, the which augmenting or dimini$hing $hall be called a greater or le$$er Force, albeit that the thing to which it belongeth do remain alwaies the $ame.</P> <P>If a Force be $u$pended or fa$tned to a Flexible Line that is without Gravity, and that is made fa$t by one end unto $ome <I>Ful- ciment</I> or $tay, in $uch $ort as that it $u$tain the Force, drawing <foot>without</foot> <p n=>326</p> without impediment by this Line, the Force and the Line $hall take $ome certain po$ition in which they $hall re$t, and the Line $hall of nece$$ity be $treight, let that Line be termed <I>the Pendant,</I> or <I>Line of Direction of the Force.</I> And let the Point by which it is fa$tned to the Fulciment be called <I>the Point of Su$pen$ion</I>: which may $ometimes be the Arm of a Leaver or Ballance; and then let the Line drawn from the Center of the Fulciment of the Leaver or Ballance to the Point of Su$pen$ion be named <I>the Di$tance</I> or <I>the Arm of the Force</I>: which we $uppo$e to be a Line fixed, and con$idered without Gravity. Moreover, let the Angle comprehen- ded betwixt the Arm of the Force and the Line of Direction be termed <I>the Angle of the Direction of the Force.</I></P> <head>AXIOM I.</head> <P>After the$e Definitions we lay down for a Principle, that in the Leaver, and in the Ballance, Equal Forces drawing by Arms that are equal, and at equall Angles of Direction, do draw equal- ly. And if in this Po$ition they draw one again$t the other they $hall make an <I>Equilibrium</I>: but if they draw together, or towards the $ame part, the Effect $hall be double.</P> <P>If the Forces being equal, and the Augles of Direction al$o equal, the Arms be unequal, the Force that $hall be $u$pended at the greater Arm $hall work the greater Effect.</P> <P>As in this Figure, the Center of the Ballance or Leaver being A, <fig> if the Arms A B and A C are equal, as al$o the Angles A B D, and A C E, the equal Forces D and E $hall draw equally, and make an <I>Equili- brium.</I> So likewi$e the Arm A F be- ing equal to A B, the Angle A F G to the Angle A B D, and the Force G to D, the$e two Forces ^{*} G and D <marg>* In the M. S. Copy it is <I>C and D.</I></marg> draw equally; and in regard that they draw both one way, the Effect $hall be double.</P> <P>In the $ame manner the Forces G and E $hall make an <I>Equilibri- um</I>; as al$o I and L $hall counterpoi$e, if (being equal) the Arms A K and A H, and the Angles A H T, and A K L be equal.</P> <P>The $ame $hall befall in the Forces P and R, if all things be di$po$ed as before. And in this ca$e we make no other di$tinction betwixt Weights and other Forces $ave only this, that Weights all tend towards the Center of Grave Bodies, and Forces may be un- der$tood to tend all towards all parts of the Univer$e, with $o much greater or le$$er <I>Impetus</I> than Weights. So that Weights and <foot>their</foot> <p n=>327</p> their parts do draw by Lines of Direction, which all concur in one and the $ame Point; and Forces and their parts may be under$tood to draw in $uch $ort that all the Lines of Direction are parallel to each other.</P> <head>AXIOM II.</head> <P>In the $econd place, we $uppo$e that a Force and its Line of Di- rection abiding alwaies in the $ame po$ition, as al$o the Center of the Ballance or Leaver, be the Arm what it will that is drawn from the Center of the Ballance to the Line of Direction, the Force drawing alwaies in the $ame fa$hion, will alwaies produce the $ame Effect.</P> <P>As, in this $econd Figure, the Center of the Ballance being A, the Force B, and the Line of Direction <fig> B <I>F</I> prolonged, as occa$ion $hall re- quire, in which the Arms A G, A C, and A <I>F</I> do determine, in this po$ition let the Line B <I>F</I> be fa$tned to the Arm A <I>F,</I> or A C, or to another Arm drawn from the Center A to the Line of Di- rection ^{*} B <I>F</I>: we $uppo$e that this <marg>* In the Original it is writ, but by the mi$take of the Tran$criber, <I>a la ligue de</I> di- rection A F.</marg> <I>F</I>orce B $hall alwaies work the $ame Effect upon the Ballance. And if drawing by the Arm A C it make an <I>Equilibrium</I> with the <I>F</I>orce <I>D</I> drawing by the <I>A</I>rm <I>A</I> E, when ever it $hall draw by the <I>A</I>rms <I>A F</I> or <I>A</I> G, it $hall likewi$e make an <I>Equilibrium</I> with the <I>F</I>orce D drawing by the <I>A</I>rm <I>A E.</I> This Principle although it be not expre$ly found in <I>A</I>uthors, yet it is tacitly $uppo$ed by all tho$e that have writ on this <I>A</I>rgument, and Experience con$tantly confirmeth it.</P> <head>AXIOM III.</head> <P>I<I>f</I> the Arms of a Ballance or Leaver are directly placed the one to the other, and that being equal they $u$tain equal <I>F</I>orces, of which the Angles of Direction are Right An- <fig> gles, the$e <I>F</I>orces do alwaies weigh equally upon the Center of the Bal- lance, whether that they be near to the $ame Center, or far di$tant, or both conjoyned in the Center it $elf; as in this <I>F</I>igure the Ballance being E D, the Center A, the equal Arms A D and <I>A</I> E, let us $u$tain equal <I>F</I>orces H and I, of which the <I>A</I>ngles <foot>of</foot> <p n=>328</p> of Direction <I>A</I> D H and <I>A</I> E I are Right <I>A</I>ngles, we $uppo$e that the$e two <I>F</I>orces I and H weigh alike upon the Center <I>A</I> as if they were nearer to the Center, at the equal Di$tances <I>A</I> B and A C, and we al$o $uppo$e the $ame if the$e very <I>F</I>orces were $u$pended both together in <I>A,</I> the <I>A</I>ngles of Directions being $till Right <I>A</I>ngles.</P> <head>PROPOSITION I.</head> <P>The$e Principles agreed upon, we will ea$ily demon$trate, in Imitation of <I>Archimedes,</I> that upon a $traight Balance the <I>F</I>orces, of which and of all their parts the Lines of Dire- ction are parallel to one another, and perpendicular to the Balance, $hall couuterpoi$e and make an <I>Equilibrium,</I> when the $aid <I>F</I>orces $hall be to one another in Reciprocal proportion of their Arms, which we think to be $o manife$t to you, that we thence $hall de- rive the Demon$tration of this Univer$al Propo$ition to which we ha$ten.</P> <head>PROPOS. II.</head> <P>In every Balance or Leaver, if the proportion of the <I>F</I>orces is reciprocal to that of the Perpendicular Lines drawn from the Center or Point of the <I>F</I>ulciment unto the Lines of Direction of the <I>F</I>orces, drawing the one again$t the other, they $hall make an <I>Equilibrium,</I> and drawing on one and the $ame $ide, they $hall have a like Effect, that is to $ay, that they $hall have as much <I>F</I>orce the one as the other, to move the Balance.</P> <P>In this <I>F</I>igure let the Center of the Balance be <I>A,</I> the <I>A</I>rm <I>A</I> B, bigger than <I>A</I> C, and fir$t let the <I>L</I>ines of Direction B D, and E C be perpendicular to the <I>A</I>rms <I>A</I> B and <I>A</I> C, by which Lines the <I>F</I>orces D and E (which may be made of Weights if one will) do draw; and that there is the $ame rate <fig> of the <I>F</I>orce D to the Force E as there is betwixt the <I>A</I>rm <I>A</I> C to the Arm <I>A</I> B: the Forces drawing one again$t the other, I $ay, that they will make an <I>Equilibrium</I> upon the Balance <I>C</I> A B. For let the <I>A</I>rm C <I>A</I> be prolonged unto F, $o as that <I>A</I>F may be equal to <I>A</I> B: and let C <I>A</I> F be con$idered as a $treight Balance, of which let the Center be <I>A</I>: and let there be $uppo$ed two Forces G and H, of which and of all their parts the Lines of Direction are parallel to the Line C E, and that the Force G be equal to the Force D, and H to E, the one, to wit G, <foot>drawing</foot> <p n=>329</p> drawing upon the Arm A <I>F,</I> and the other, to wit H, upon the Arm A C: now, by the fir$t Propo$ition, G and H $hall make an <I>Equili- brium</I> upon the Balance C A F: But, by the fir$t Principle, the Force D upon the Arm A B worketh the $ame effect as the Force G on the Arm A F: Therefore the Force D upon the Arm A B maketh an <I>Equilibrium</I> with the Force H upon A C: And the Force H drawing in the $ame manner upon the Arm <I>A</I> C as the Force E, by the $ame fir$t <I>A</I>xiom, the Force D upon the Arm <I>A</I> B $hall make an <I>Equilibrium</I> with the Force E upon the Arm <I>A</I> C.</P> <P>Now, in the following Figure, let the Center of the Balance be <I>A,</I> the Arms A B and A C, the Lines of Direction B D and C E which are not Perpendicular to the Arms, and the Forces D and E drawing likewi$e by the Lines of Direction, upon which Perpen- diculars are erected unto the Center A, that is A F upon B D, and A G upon E C, and that as A F is to A G, $o is the Force E to the Force D: which Forces draw one <fig> again$t the other: I $ay, that they will make an <I>Equilibrium</I> upon the Balance C A B: For let the Lines A F and A G be under$tood to be the two Arms of a Balance G A F, upon which the For- ces D and E do draw by the Lines of Direction F D and G E: The$e Forces $hall make an <I>Equilibrium,</I> by the fir$t part of this $econd Propo$ition; but, by the $econd Axiom, the Force D upon the Arm A F hath the $ame Effect as upon the Arm A B: Therefore the Force D upon the Arm A B maketh an <I>Equilibrium</I> with the Force E upon the Arm A C.</P> <P>There are many Ca$es, according to the Series of Perpendicu- lars, but it will be ea$ie for you to $ee that they have all but one and the $ame Demon$tration.</P> <P>It is al$o ea$ie to demon$trate, that if the Forces draw both on one $ide they $hall make the $ame Effect one as another, and that the Effect of two together $hall be double to that of one alone.</P> <head>OF THE GEOSTATICKS.</head> <P>The Principle which you demand for the <I>Geo$taticks</I> is, That if two equal Weights are conjoyned by a right Line fixed and void of Gravity, and that being $o di- $po$ed they may de$cend freely, they will never re$t till that the middle of the Line, that is the Center of Gravitation of the Ancients, unites it $elf to the common Center of Grave Bodies.</P> <foot>Uu This</foot> <p n=>330</p> <P>This Principle $eems at the fir$t very plau$ible, but when the Que$tion concerneth a Principle, you know what Conditions are required to it, that it may be received, the principal of which are wanting in the Principle now in controver$ie<I>: $cil.</I> that we do not know what is the radical Cau$e why Grave Bodies de$cend; and whence the Original of this Gravity ari$eth: as al$o that we are to- tally ignorant of that which would arrive at the Center whither Grave Bodies do tend, nor to other places without the Surface of the Earth, of which, in regard we inhabit upon it, we have $ome Expe- riments upon which we ground our Principles.</P> <P>For it may be, that Gravity is a Quality that re$ides in the Body it $elf that falleth; it may be that it is in another that attracteth that which de$cends, as in the Earth: It may be, and it is very likely that it is a Natural Attraction, or a Natural De$ire of two Bodies to unite together, as in the Iron and Load$tone, which are $uch, that if the Load$tone be $taid, the Iron, if nothing hinder it, will go find it out; and if the Iron be $taid the Load$tone will go towards it; and if they be both at liberty, they will reciprocally approach one another, yet after $uch a fa$hion, that the $tronge$t of the two will move the lea$t way.</P> <P>If the fir$t be true, according to the common opinion, we $ee not how your Principle can $ub$i$t, for Common Sen$e tells us, that in whatever place a Weight is, it alwaies weigheth alike, having ever- more the $ame Quality that maketh it to weigh, and that then a Bo- dy will repo$e at the Common Center of things Grave when the parts of the Body which $hall be on each part of the $aid Center $hall be of equal Pondero$ity to counterpoi$e one another, without having any regard whether they be little or much removed from the Center. Since therefore that of the$e three po$$ible Cau$es of Gra- vitation, we know not which is the right, nay, that we are not cer- tain that it is any of them, it being po$$ibly that there is a fourth from which one may draw Conclu$ions very different, it $eemeth to me impo$$ible for us to lay down other Principles in this bufine$s than tho$e of which we are a$$ured by a continual Experience, and a $ound Judgment. As for our parts, we call tho$e Bodies equally or unequally Grave which have an equal or unequal Force of mo- ving towards the Common Center: and a Body is $aid to have the $ame Weight when it alwaies hath this $ame Force: but if this Force augmenteth or dimini$heth, then, although it be the $ame Bo- dy, we con$ider it no longer as the $ame Weight: Now $ince that this hapneth to Bodies that recede or approach to the Common Center, this is it which we de$ire to know, but finding nothing that giveth me content upon this Subject, I will leave the Que$tion un- determined and unde$cribed.</P> <foot>FINIS.</foot> <pb> <head>ARCHIMEDES HIS TRACT De Incidentibus Humido, OR OF THE NATATION OF BODIES VPON, OR SVBMERSION IN, THE WATER OR OTHER LIQUIDS.</head> <head>IN TWO BOOKS.</head> <head>Tran$lated from the Original Greek,</head> <head>Fir$t into Latine, and afterwards into Italian, by <I>NICOLO TARTAGLIA,</I> and by him familiarly demon- $trated by way of Dialogue, with <I>Richard Wentworth,</I> a Noble Engli$h Gentleman, and his Friend.</head> <head>Together with the Learned Commentaries of <I>Federico Commandino,</I> who hath Re$tored $uch of the Demon$trations as, thorow the Injury of Time, were obliterated.</head> <head>Now compared with the ORIGINAL, and Engli$hed By <I>THOMAS SALVSBVRY,</I> E$q.</head> <head><I>LONDON,</I> Printed by <I>W. Leybourn,</I> 1662.</head> <p n=>335</p> <head>ARCHIMEDES HIS TRACT <I>De INCIDENTIBUS HUMIDO,</I> OR OF The Natation of Bodies upon, or Submer$ion in, the Water, or other Liquids.</head> <head>BOOK I.</head> <head>RICARDO.</head> <P><I>Dear Companion,</I> I have peru$ed your <I>Indu$trious Invention,</I> in which I find not any thing that will not certainly hold true; but, truth is, there are many of your Conclu$ions of which I under$tand uot the Cau$e, and therefore, if it be not a trouble to you, I would de$ire you to declare them to me, for, indeed, nothing plea$eth me, if the Cau$e thereof be hid from me.</P> <P>NICOLO. My obligations unto you are $o many and great, <I>Honoured Campanion,</I> that no reque$t of yours ought to be trouble$ome to me, and therefore tell me what tho$e Perticulars are of which you know not the Cau$e, for I $hall endeavour with the utmo$t of my power and under$tanding to $atisfie you in all your demands.</P> <P>RIC. In the fir$t <I>Direction</I> of the fir$t Book of that your <I>Indu$trious Invention</I> you conclude, That it is impo$$ible that the Water $hould wholly receive into it any material Solid Body that is lighter than it $eif (as to <I>$peciæ</I>) nay, you $ay, That there will alwaies a part of the Body $tay or remain above the Waters Surface (that is uncovered by it;) and, That as the whole Solid Body put into the Water is in proportion to that part of it that $hall be immerged, or received, into the Wa- ter, $o $hall the Gravity of the Water be to the Gravity <I>(in $peciæ)</I> of that $ame material Body: And that tho$e Solid Bodies, that are by nature more Grave than the Water, being put into the Water, $hall pre$ently make the $aid Water give place; and, That they do not only wholly enter or $ubmerge in the $ame, but go continu- ally de$cending untill they arrive at <I>t</I>he Bottom; and, That they $ink to the Bot- tom $o much fa$ter, by how much they are more Grave than the Water. And, again, That tho$e which are preci$ely of the $ame Gravity with the Water, being put into the $ame, are of nece$$ity wholly received into, or immerged by it, but yet retained in the Surface of the $aid Water, and much le$s will the Water con- $ent that it do de$cend to the Bottom: and, now, albeit that all the$e things are manife$t to Sen$e and Experience, yet neverthele$s would I be very glad, if it be po$$ible, that you would demon$trate to me the mo$t apt and proper Cau$e of the$e Effects.</P> <foot>NIC. The</foot> <p n=>334</p> <P>NIC. The Cau$e of all the$e Effects is a$$igned by <I>Archimedes,</I> the <I>Siracu$an,</I> in <marg>* <I>Aquæ,</I> tan$lated by me <I>Humido,</I> as the more Compre- hen$ive word, for his Doctrine holds true in all Liquids as well as in Wa- ter, <I>$oil.</I> in Wine, Oyl, Milk, <I>&c.</I></marg> that Book <I>De Incidentibus (^{*}) Aquæ,</I> by me publi$hed in Latine, and dedicated to your $elf, as I al$o $aid in the beginning of that my <I>Indu$trions Invention.</I></P> <P>RIC. I have $een that $ame <I>Archimedes,</I> and have very well under$tood tho$e two Books in which he treateth <I>De Centro Gravitatis æque<*>e<*>entibus,</I> or of the Center of Gravity in Figures plain, or parallel to the Horizon; and likewi$e tho$e <I>De Quadratura Parabolæ,</I> or, of Squaring the Parabola; but ^{*}<I>that</I> in which he treat- eth of Solids that Swim upon, or $ink in Liquids, is $o ob$cure, that, to $peak the truth, there are many things in <I>it</I> which I do not under$tand, and therefore before <marg>* He $peaks of but one Book, <I>Tartag- li<*></I> having tran$la- ted no more.</marg> we proceed any farther, I $hould take it for a favour if you would declare it to me in your Vulgar Tongue, beginning with his fir$t <I>Suppo$ition,</I> which $peaketh in this manner.</P> <head>SVPPOSITION I.</head> <P><I>It is $uppo$ed that the Liquid is of $uch a nature, that its parts being equi-jacent and contiguous, the le$s pre$$ed are repul$ed by the more pre$$ed. And that each of its parts is pre$$ed or repul$ed by the Liquor that lyeth over it, perpendicularly, if the Liquid be de$cending into any place, or pre$$ed any whither by another.</I></P> <P>NIC. Every Science, Art, or Doctrine (as you know, <I>Honoured Companion,</I>) hath its fir$t undemon$trable Principles, by which (they being granted or $uppo$ed) the $aid Science is proved, maintained, or de- mon$trated. And of the$e Principles, $ome are called <I>Petitions,</I> and others <I>Demands,</I> or <I>Suppo$itions.</I> I $ay, therefore, that the Science or Doctrine of tho$e Material Solids that Swim or Sink in Liquids, hath only two undemon- $trable <I>Suppo$itions,</I> one of which is that above alledged, the which in compliance with your de$ire I have $et down in our Vulgar Tongue.</P> <P>RIC. Before you proceed any farther tell me, how we are to under$tand the parts of a Liquid to be <I>Equijacent.</I></P> <P>NIC. When they are equidi$tant from the Center of the World, or of the Earth (which is the $ame, although ^{*} $ome hold that the Centers of the Earth and Worldare different.)</P> <P>RIC. I under$tand you not unle$s you give me $ome Example thereof in Figure.</P> <marg>* The Coperni- cans.</marg> <P>NIC. To exemplifie this particular, Let us $uppo$e a quantity of Liquor (as for in$tance of Water) to be upon the Earth; then let us with the Imagination cut the whole Earth together with that Water into two equal parts, in $uch a manner as that the $aid Section may pa$s ^{*} by the Center of the Earth: And let us $uppo$e that one part of the Superficies of that Section, as well of the Water as of the Earth, be the Superficies A B, and that the Center of the Earth be the point K. This being done, let us in our Imagination de$cribe a Circle upon the <marg>* Or through.</marg> $aid Center K, of $uch a bigne$s as that the Circumference may pa$s by the Super- ficies of the Section of the Water: Now let this Circumference be E F G: and let many Lines be drawn from the point K to the $aid Circumference, cutting the $ame, as KE, KHO, KFQ KLP, KM. Now I $ay, that all the$e parts of the $aid Water, terminated in that Circumference, are Equijacent, as being all <foot>equi-</foot> <p n=>335</p> equidi$tant from the point K, the Center of the World, which parts are G M, M L, L F, F H, H E.</P> <P>RIC. I under$tand you very well, as to this particular: But tell me a little; he $aith that each of the parts of the Liquid is pre$$ed or repul$ed by the Liquid that is above it, according to the Perpendicular: I know not what that Liquid is that lieth upon a part of another Perpendicularly.</P> <P>NIC. Imagining a Line that cometh from the Center of the Earth penetrating thorow $ome Water, each part of the Water that is in that Line he $uppo$eth to be pre$$ed or repul$ed by the Water that lieth above it in that $ame Line, and that that repul$e is made according to the $ame Line, (that is, directly towards the Center of the World) which Line is called a Perpendicular; becau$e every Right-Line that departeth from any point, and goeth directly towards the Worlds Center is called a Perpendicular. And that you may the better under$tand me, let <fig> us imagine the Line KHO, and in that let us imagine $everal parts, as $uppo$e RS, S T, T V, V H, H O. I $ay, that he $up- po$eth that the part V H is pre$$ed by that placed a- bove it, H O, according to the Line OK; the which O K, as hath been $aid above, is called the Perpendicular pa$$ing thorow tho$e two parts. In like manner, I $ay that the part T V is expul$ed by the part V H, ac- cording to the $aid Line O K: and $o the part S T to be pre$$ed by T V, according to the $aid Perpendicular O K, and R S by S T. And this you are to under$tand in all the other Lines that were protracted from the $aid Point K, penetrating the $aid Water, As for Example, in <I>K</I> G, <I>K</I> M, <I>K</I> L, <I>K</I> F, <I>K</I> E, and infinite others of the like kind.</P> <P>RIC. Indeed, <I>Dear Companion,</I> this your Explanation hath given megreat $a- tisfaction; for, in my Judgment, it $eemeth that all the difficulty of this Suppo$ition con$i$ts in the$e two particulars which you have declared to me.</P> <P>NIC. It doth $o; for having under$tood that the parts E H, H F, F L, L M, and MG, determining in the Circumference of the $aid Circle are equijacent, it is an ea$ie matter to under$tand the fore$aid <I>Suppo$ition</I> in Order, which $aith, <I>That it is $uppo$ed that the Liquid is of $uch a nature, that the part thereof le$s pre$$ed or thrust is re- pul$ed by the more thru$t or pre$$ed.</I> As for example, if the part E H were by chance more thru$t, crowded, or pre$$ed from above downwards by the Liquid, or $ome other matter that was over it, than the part H F, contiguous to it, it is $uppo$ed that the $aid part H F, le$s pre$$ed, would be repul$ed by the $aid part E H. And thus we ought to under$tand of the other parts equijacent, in ca$e that they be contiguous, and not $evered. That each of the parts thereof is pre$$ed and repul. $ed by the <I>L</I>iquid that lieth over it Perpendicularly, is manife$t by that which was $aid above, to wit, that it $hould be repul$ed, in ca$e the <I>L</I>iquid be de$cending into any place, and thru$t, or driven any whither by another.</P> <P>RIC. I under$tand this Suppo$ition very well, but yet me thinks that before the Suppo$ition, the Author ought to have defined tho$e two particulars, which you fir$t declared to me, that is, how we are to under$tand the parts of the <I>L</I>iquid equijacent, and likewi$e the Perpendicular.</P> <foot>NIC. You</foot> <p n=>336</p> <P>NIC. You $ay truth.</P> <P>RIC. I have another que$tion to aske you, which is this, Why the Author u$eth the word <I>L</I>iquid, or Humid, in$tead of Water.</P> <P>NIC. It may be for two of the$e two Cau$es; the one is, that Water being the principal of all <I>L</I>iquids, therefore $aying <I>Humidum</I> he is to be under$tood to mean the chief Liquid, that is Water: The other, becau$e that all the Propo$itions of this Book of his, do not only hold true in Water, but al$o in every other <I>L</I>iquid, as in Wine, Oyl, and the like: and therefore the Author might have u$ed the word <I>Humidum,</I> as being a word more general than <I>Aqua.</I></P> <P>RIC. This I under$tand, therefore let us come to the fir$t <I>Propo$ition,</I> which, as you know, in the Original $peaks in this manner.</P> <head>PROP. I. THEOR. I.</head> <P><I>If any Superficies $hall be cut by a Plane thorough any Point, and the Section be alwaies the Circumference of a Circle, who$e Center is the $aid Point: that Su- perficies $hall be Spherical.</I></P> <P>Let any Superficies be cut at plea$ure by a Plane thorow the Point K; and let the Section alwaies de$cribe the Circumfe- rence of a Circle that hath for its Center the Point K: I $ay, that that $ame Superficies is Sphærical. For were it po$$ible that the $aid Superficies were not Sphærical, then all the Lines drawn through the $aid Point K unto that Superficies would not be equal, Let therefore A and B be two Points in the $aid Superficies, $o that <fig> drawing the two Lines K A and K B, let them, if po$$ible, be une- qual: Then by the$e two Lines let a Plane be drawn cutting the $aid Superficies, and let the Section in the Superficies make the Line D A B G: Now this Line D A B G is, by our pre-$uppo$al, a Circle, and the Center thereof is the Point K, for $uch the $aid Superficies was $uppo$ed to be. Therefore the two Lines K A and K B are equal: But they were al$o $uppo$ed to be unequal; which is impo$$ible: It followeth therefore, of nece$$ity, that the $aid Superficies be Sphærical, that is, the Superficies of a Sphære.</P> <P>RIC. I under$tand you very well; now let us proceed to the $econd <I>Propo$ition,</I> which, you know, runs thus.</P> <foot>PROP.</foot> <p n=>337</p> <head>PROP. II. THEOR. II.</head> <P><I>The Superficies of every Liquid that is con$i$tant and $etled $hall be of a Sphærical Figure, which Figure $hall have the $ame Center with the Earth.</I></P> <P>Let us $uppo$e a Liquid that is of $uch a con$i$tance as that it is not moved, and that its Superficies be cut by a Plane along by the Center of the Earth, and let the Center of the Earth be the Point K: and let the Section of the Superficies be the Line A B G D. I $ay that the Line A B G D is the Circumference of a <fig> Circle, and that the Center thereof is the Point K And if it be po$$ible that it may not be the Circumference of a Circle, the Right- <marg>* O: through.</marg> Lines drawn ^{*} by the Point K to the $aid Line A B G D $hall not be equal. There- fore let a Right-Line be taken greater than $ome of tho$e produced from the Point K unto the $aid Line A B G D, and le$$er than $ome other; and upon the Point K let a Circle be de$cribed at the length of that Line, Now the Circumference of this Circle $hall fall part without the $aid Line A B G D, and part within: it having been pre$uppo$ed that its Semidiameter is greater than $ome of tho$e Lines that may be drawn from the $aid Point K unto the $aid Line A B G D, and le$$er than $ome other. Let the Circumference of the de$cribed Circle be R B G H, and from B to K draw the Right-Line B K: and drawn al$o the two Lines K R, and K E L which make a Right- Angle in the Point K: and upon the Center K de$cribe the Circum- ference X O P in the Plane and in the Liquid. The parts, there- fore, of the Liquid that are ^{*} according to the Circumference <marg>* <I>i.e.</I> Parallel.</marg> X O P, for the rea$ons alledged upon the fir$t <I>Suppo$ition,</I> are equi- jacent, or equipo$ited, and contiguous to each other; and both the$e parts are pre$t or thru$t, according to the $econd part of the <I>Suppo$ition,</I> by the Liquor which is above them. And becau$e the two Angles E K B and B K R are $uppo$ed equal [<I>by the</I> 26. <I>of</I> 3. <I>of Euclid,</I>] the two Circumferences or Arches B E and B R $hall be equal (fora$much as R B G H was a Circle de$cribed for $atis- faction of the Oponent, and K its Center:) And in like manner the whole Triangle B E K $hall be equal to the whole Triangle B R K. And becau$e al$o the Triangle O P K for the $ame rea$on <foot>Xx $hall</foot> <p n=>338</p> $hall be equal to the Triangle O X K; Therefore (by common Notion) $ub$tracting tho$e two $mall Triangles O P K and O X K from the two others B E K and B R K, the two Remainders $hall be equal: one of which Remainders $hall be the Quadrangle B E O P, and the other B R X O. And becau$e the whole Quadran- gle B E O P is full of Liquor, and of the Quadrangle B R X O, the part B A X O only is full, and the re$idue B R A is wholly void of Water: It followeth, therefore, that the Quadrangle B E O P is more ponderous than the Quadrangle B R X O. And if the $aid Quadrangle B E O P be more Grave than the Quadrangle B R X O, much more $hall the Quadrangle B L O P exceed in Gra- vity the $aid Quadrangle B R X O: whence it followeth, that the part O P is more pre$$ed than the part O X. But, by the fir$t part of the Suppo$ition, the part le$s pre$$ed $hould be repul$ed by the part more pre$$ed: Therefore the part O X mu$t be repul$ed by the part O P: But it was pre$uppo$ed that the Liquid did not move: Wherefore it would follow that the le$s pre$$ed would not be repul$ed by the more pre$$ed: And therefore it followeth of nece$$ity that the Line A <I>B</I> G D is the Circumference of a Circle, and that the Center of it is the point K. And in like manner $hall it be demon$trated, if the Surface of the Liquid be cut by a Plane thorow the Center of the Earth, that the Section $hall be the Cir- cumference of a Circle, and that the Center of the $ame $hall be that very Point which is Center of the Earth. It is therefore mani- fe$t that the Superficies of a Liquid that is con$i$tant and $etled $hall have the Figure of a Sphære, the Center of which $hall be the $ame with that of the Earth, by the fir$t <I>Propo$ition</I>; for it is $uch that being ever cut thorow the $ame Point, the Section or Di- vi$ion de$cribes the Circumference of a Circle which hath for Cen- ter the $elf-$ame Point that is Center of the Earth: Which was to be demon$trated.</P> <P>RIC. I do thorowly under$tand the$e your Rea$ons, and $ince there is in them no umbrage of Doubting, let us proceed to his third <I>Propo$ition.</I></P> <head>PROP. III. THEOR. III.</head> <P><I>Solid Magnitudes that being of equal Ma$s with the Liquid are al$o equal to it in Gravity, being demit-</I> <marg>* I add the word $etled, as nece$$ary in making the Ex- periment.</marg> <I>ted into the [^{*} $etled] Liquid do $o $ubmerge in the $ame as that they lie or appear not at all above the Surface of the Liquid, nor yet do they $ink to the Bottom.</I></P> <foot>NIC. In</foot> <p n=>339</p> <P>NIC. In this <I>Propo$ition</I> it is affirmed that tho$e Solid Magnitules that hap- pen to be equal in $pecifical Gravity with the Liquid being lefeat liber- ty in the $aid Liquid do $o $ubmerge in the $ame, as that they lie or ap- pear not at all above the Surface of the Liquid, nor yet do they go or $ink to the Bottom.</P> <P>For $uppo$ing, on the contrary, that it were po$$ible for one of tho$e Solids being placed in the Liquid to lie in part without the Liquid, that is above its Surface, (alwaies provided that the $aid Liquid be $etled and undi$turbed,) let us imagine any Plane pro- duced thorow the Center of the Earth, thorow the Liquid, and thorow that Solid Body: and let us imagine that the Section of the Liquid is the Superficies A B G D, and the Section of the Solid Body that is within it the Super$icies E Z H T, and let us $uppo$e the Center of the Earth to be the Point K: and let the part of the $aid Solid $ubmerged in the Liquid be B G H T, and let that above be B E Z G: and let the Solid Body be $uppo$ed to be comprized in a Pyramid that hath its Parallelogram Ba$e in the upper Surface of the Liquid, and its Summity or Vertex in the Center of the Earth: which Pyramid let us al$o $uppo$e to be cut or divided by the $ame Plane in which is the Circumference A B G D, and let the Sections <fig> of the Planes of the $aid Pyramid be K L and K M: and in the Liquid about the Center K let there be de$cribed a Su- perficies of another Sphære below E Z H T, which let be X O P; and let this be cut by the Superficies of the Plane: And let there be another Pyramid ta- ken or $uppo$ed equal and like to that which compri$eth the $aid Solid Body, and contiguous and conjunct with the $ame; and let the Sections of its Superficies be K M and K N: and let us $uppo$e another Solid to be taken or imagined, of Liquor, contained in that $ame Pyramid, which let be R S C Y, equal and like to the partial Solid B H G T, which is immerged in the $aid Liquid: But the part of the Liquid which in the fir$t Pyramid is under the Super- ficies X O, and that, which in the other Pyramid is under the Su- perficies O P, are equijacent or equipo$ited and contiguous, but are not pre$$ed equally; for that which is under the Superficies X O is pre$$ed by the Solid T H E Z, and by the Liquor that is contained between the two Spherical Superficies X O and L M and the Planes of the Pyramid, but that which proceeds accord- ing to F O is pre$$ed by the Solid R S C Y, and by the Liquid <foot>Xx 2 contained</foot> <p n=>340</p> contained between the Sphærical Superficies that proceed accord- ing to P O and M N and the Planes of the Pyramid; and the Gra- vity of the Liquid, which is according to M N O P, $hall be le$$er than that which is according to L M X O; becau$e that Solid of Liquor which proceeds according to R S C Y is le$s than the Solid E Z H T (having been $uppo$ed to be equal in quantity to only the part H B G T of that:) And the $aid Solid E Z H T hath been $uppo$ed to be equally grave with the Liquid: Therefore the Gra- vity of the <I>L</I>iquid compri$ed betwixt the two Sphærical Superfi- cies L M and <I>X</I> O, and betwixt the $ides L <I>X</I> and M O of the <fig> Pyramid, together with the whole Solid EZHT, $hall exceed the Gravity of the Liquid compri- $ed betwixt the other two Sphærical Superfi- cies M N and O P, and the Sides M O and N P of the Pyramid, toge- ther with the Solid of Liquor R S C Y by the quantity of the Gra- vity of the part E B Z G, $uppo$ed to remain above the Surface of the Liquid: And therefore it is manife$t that the part which pro- ceedeth according to the Circumference O P is pre$$ed, driven, and repul$ed, according to the <I>Suppo$ition,</I> by that which proceeds ac- cording to the Circumference X O, by which means the Liquid would not be $etled and $till: But we did pre$uppo$e that it was $etled, namely $o, as to be without motion: It followeth, therefore, that the $aid Solid cannot in any part of it exceed or lie above the Superficies of the Liquid: And al$o that being dimerged in the Li- quid it cannot de$cend to the Bottom, for that all the parts of the Liquid equijacent, or di$po$ed equally, are equally pre$$ed, becau$e the Solid is equally grave with the Liquid, by what we pre$uppo$ed.</P> <P>RIC. I do under$tand your Argumentation, but I under$tand not that Phra$e <I>Solid Magnitudes.</I></P> <P>NIC. I will declare this Term unto you. <I>Magnitude</I> is a general Word that re$pecteth all the Species of Continual Quantity; and the Species of Continual Quantity are three, that is, the <I>L</I>ine, the Superficies, and the Body; which Body is al$o called a Solid, as having in it $elf <I>L</I>ength, Breadth, and Thickne$s, or Depth: and therefore that none might equivocate or take that Term <I>Magnitudes</I> to be meant of <I>L</I>ines, or Superficies, but only of Solid <I>Magnitudes,</I> that is, Bodies, he did $pecifie it by that manner of expre$$ion, as was $aid. The truth is, that he might have expre$t that <I>Propo$ition</I> in this manner: <I>Solids (or Bodies) which being of equal Gravity with an equal Ma$s of the Liquid,</I> &c. And this <I>Propo$ition</I> would have been more cleer and intelligible, for it is as $ignificant to $ay, a <I>Solid,</I> or, a <I>Body,</I> as to $ay, a <I>Solid Magnitude:</I> therefore wonder not if for the future I u$e the$e three kinds of words indifferently.</P> <P>RIC. You have $ufficiently $atisfied me, wherefore that we may lo$e no time let us go forwards to the fourth <I>Propo$ition.</I></P> <foot>PROP.</foot> <p n=>341</p> <head>PROP. IV. THEOR. IV.</head> <P><I>Solid Magnitudes that are lighter than the Liquid, being demitted into the $etled Liquid, will not total- ly $ubmerge in the $ame, but $ome part thereof will lie or $tay above the Surface of the Liquid.</I></P> <P>NIC. In this fourth <I>Propo$ition</I> it is concluded, that every Body or Solid that is lighter (as to Specifical Gravity) than the <I>L</I>iquid, being put into the <I>L</I>iquid, will not totally $ubmerge in the $ame, but that $ome part of it will $tay and appear without the <I>L</I>iquid, that is above its Surface.</P> <P>For $uppo$ing, on the contrary, that it were po$$ible for a Solid more light than the Liquid, being demitted in the Liquid to $ub- merge totally in the $ame, that is, $o as that no part thereof re- maineth above, or without the $aid Liquid, (evermore $uppo$ing that the Liquid be $o con$tituted as that it be not moved,) let us imagine any Plane produced thorow the Center of the Earth, tho- row the Liquid, and thorow that Solid Body: and that the Surface of the Liquid is cut by this Plane according to the Circumference A <I>B</I> G, and the Solid <I>B</I>ody according to the Figure R; and let the Center of the Earth be K. And let there be imagined a Pyramid <fig> that compri$eth the Figure R, as was done in the pre. cedent, that hath its Ver- tex in the Point K, and let the Superficies of that Pyramid be cut by the Superficies of the Plane A <I>B</I> G, according to A K and K <I>B</I>. And let us ima- gine another Pyramid equal and like to this, and let its Superficies be cut by the Superficies A <I>B</I> G according to K <I>B</I> and K <I>G</I>; and let the Superficies of another Sphære be de$cribed in the Liquid, upon the Center K, and beneath the Solid R; and let that be cut by the $ame Plane according to <I>X</I> O P. And, la$tly, let us $uppo$e ano- ther Solid taken ^{*} from the Liquid, in this $econd Pyramid, which <marg>* That is a Ma$s of the Liquid.</marg> let be H, equal to the Solid R. Now the parts of the Liquid, name- ly, that which is under the Spherical Superficies that proceeds ac- cording to the Superficies or Circumference <I>X</I> O, in the fir$t Py- ramid, and that which is under the Spherical Superficies that pro- ceeds according to the Circumference O P, in the $econd Pyramid, are equijacent, and contiguous, but are not pre$$ed equally; for <foot>that</foot> <p n=>342</p> that of the fir$t Pyramid is pre$$ed by the Solid R, and by the Liquid which that containeth, that is, that which is in the place of the Py- ramid according to A B O X: but that part which, in the other Py- ramid, is pre$$ed by the Solid H, $uppo$ed to be of the $ame Li- quid, and by the Liquid which that containeth, that is, that which is in the place of the $aid Pyramid according to P O B G: and the Gravity of the Solid R is le$s than the Gravity of the Liquid H, for that the$e two Magnitudes were $uppo$ed to be equal in Ma$s, and the Solid R was $uppo$ed to be lighter than the Liquid: and the Ma$$es of the two Pyramids of Liquor that containeth the$e <marg>* For that the Py- ramids were $uppo- $ed equal.</marg> two Solids R and H are equal ^{*} by what was pre$uppo$ed: There- fore the part of the Liquid that is under the Superficies that pro- ceeds according to the Circumference O P is more pre$$ed; and, therefore, by the <I>Suppo$ition,</I> it $hall repul$e that part which is le$s pre$$ed, whereby the $aid Liquid will not be $etled: But it was be- fore $uppo$ed that it was $etled: Therefore that Solid R $hall not totally $ubmerge, but $ome part thereof will remain without the Liquid, that is, above its Surface, Which was the <I>Propo$ition.</I></P> <P>RIC. I have very well under$tood you, therefore let us come to the fifth <I>Pro- po$ition,</I> which, as you know, doth thus $peak.</P> <head>PROP. V. THEOR. V.</head> <P><I>Solid Magnitudes that are lighter than the Liquid, being demitted in the ($etled) Liquid, will $o far $ubmerge, till that a Ma$s of Liquor, equal to the Part $ubmerged, doth in Gravity equalize the whole Magnitude.</I></P> <P>NIC. It having, in the precedent, been demon$trared that Solids lighter than the Liquid, being demitted in the <I>L</I>iquid, alwaies a part of them remains without the <I>L</I>iquid, that is above its Surface; In this fifth <I>Propo$ition</I> it is a$$erted, that $o much of $uch a Solid $hall $ubmerge, as that a Ma$s of the <I>L</I>iquid equal to the part $ubmerged, $hall have equal Gravity with the whole Solid.</P> <P>And to demon$trate this, let us a$$ume all the $ame Schemes as before, in <I>Propo$ition</I> 3. and likewi$e let the Liquid be $et- led, and let the Solid E Z H T be lighter than the Liquid. Now if the $aid Liquid be $etled, the parts of it that are equija- cent are equally pre$$ed: Therefore the Liquid that is beneath <foot>the</foot> <p n=>343</p> the Superficies that proceed according to the Circumferences X O and P O are equally pre$$ed; whereby the Gravity pre$$ed is equal. <fig> But the Gravity of the Liquid which is in the <marg>* <I>Without, i.e.</I> that being deducted.</marg> fir$t Pyramid ^{*} without the Solid B H T G, is equal to the Gravity of the Liquid which is in the other Pyramid with- out the Liquid R S C Y: It is manife$t, therefore, that the Gravity of the Solid E Z H T, is equal to the Gravity of the Liquid R S C Y: Therefore it is manife$t that a Ma$s of Liquor equal in Ma$s to the part of the Solid $ubmerged is equal in Gra- vity to the whole Solid.</P> <P>RIC. This was a pretty Demon$tration, and becau$e I very well under$tand it, let us lo$e no time, but proceed to the $ixth <I>Propo$ition,</I> $peaking thus.</P> <head>PROP. VI. THEOR. VI.</head> <P><I>Solid Magnitudes lighter than the Liquid being thru$t into the Liquid, are repul$ed upwards with a Force as great as is the exce$s of the Gravity of a Ma$s of Liquor equal to the Magnitude above the Gra- vity of the $aid Magnitude.</I></P> <P>NIC. This $ixth <I>Propo$ition</I> $aith, that the Solids lighter than the Liquid demitted, thru$t, or trodden by Force underneath the Liquids Sur- face, are returned or driven upwards with $o much Force, by how much a quantity of the Liquid equal to the. Solid $hall exceed the $aid Solid in Gravity.</P> <P>And to delucidate this <I>Propo$ition,</I> let the Solid A be lighter than the <I>L</I>iquid, and let us $uppo$e that the Gravity of the $aid Solid A is B: and let the Gravity of a <I>L</I>iquid, equal in Ma$s to A, be B G. I $ay, that the Solid A depre$$ed or demitted with Force into the $aid <I>L</I>iquid, $hall be returned and repul$ed upwards with a Force equal to the Gravity G. And to demon$trate this <I>Propo- $ition,</I> take the Solid D, equal in Gravity to the $aid G. Now the Solid compounded of the two Solids A and D will be lighter than the <I>L</I>iquid: for the Gravity of the Solid compounded of them both is BG, and the Gravity of as much Liquor as equal- leth in greatne$s the Solid A, is greater than the $aid Gravity BG, <foot>for</foot> <p n=>344</p> for that B G is the Gravity of the Liquid equal in Ma$s unto it: Therefore the Solid compounded of tho$e two Solids A and D being dimerged, it $hall, by the precedent, $o much of it $ubmerge, as that a quantity of the Liquid equal to the $aid $ubmerged part $hall have equal Gravity with the $aid compounded Solid. And <fig> for an example of that <I>Propo$ition</I> let the Su- perficies of any Liquid be that which pro- ceedeth according to the Circumference A B G D: Becau$e now a Ma$s or quantity of Liquor as big as the Ma$s A hath equal Gravity with the whole compounded Solid A D: It is manife$t that the $ubmerged part thereof $hall be the Ma$s A: and the remain- der, namely, the part D, $hall be wholly a- top, that is, above the Surface of the Liquid. It is therefore evident, that the part A hath $o much virtue or Force to return upwards, that is, to ri$e from below above the Li- quid, as that which is upon it, to wit, the part D, hath to pre$s it downwards, for that neither part is repul$ed by the other: But D pre$$eth downwards with a Gravity equal to G, it having been $up- po$ed that the Gravity of that part D was equal to G: Therefore that is manife$t which was to be demon$trated.</P> <P>RIC. This was a fine Demon$tration, and from this I perceive that you colle- cted your <I>Indu$trious Invention</I>; and e$pecially that part of it which you in$ert in the fir$t Book for the recovering of a Ship $unk: and, indeed, I have many Que- $tions to ask you about that, but I will not now interrupt the Di$cour$e in hand, but de$ire that we may go on to the $eventh <I>Propo$ition,</I> the purport whereof is this.</P> <head>PROP. VII. THEOR. VII.</head> <P><I>Solid Magnitudes beavier than the Liquid, being de- mitted into the [$etled] Liquid, are boren down- wards as far as they can de$cend: and $hall be lighter in the Liquid by the Gravity of a Liquid Ma$s of the $ame bigne$s with the Solid Magnitude.</I></P> <P>NIC. This $eventh <I>Propo$ition</I> hath two parts to be demon$trated.</P> <P>The fir$t is, That all Solids heavier than the Liquid, being demit- ted into the <I>L</I>iquid, are boren by their Gravities downwards as far as they can de$cend, that is untill they arrive at the Bottom. Which fir$t part is manife$t, becau$e the Parts of the <I>L</I>iquid, which $till lie under that Solid, are more pre$$ed than the others equijacent, becau$e that that Solid is $uppo$ed more grave than the Liquid. <foot>But</foot> <p n=>345</p> But now that that Solid is lighter in the Liquid than out of it, as is affirmed in the $econd part, $hall be demon$trated in this man- ner. Take a Solid, as $uppo$e A, that is more grave than the Li- quid, and $uppo$e the Gravity of that $ame Solid A to be BG. And of a Ma$s of <I>L</I>iquor of the $ame bigne$s with the Solid A, $up- po$e the Gravity to be B: It is to be demon$trated that the Solid A, immerged in the Liquid, $hall have a Gravity equal to G. And to demon$trate this, let us imagine another Solid, as $uppo$e D, more light than the Liquid, but of $uch a quality as that its Gravi- ty is equal to B: and let this D be of $uch a Magnitude, that a Ma$s of <I>L</I>iquor equal to it hath its Gravity equal to the Gravity B G. Now the$e two Solids D and A being compounded toge- ther, all that Solid compounded of the$e two $hall be equally Grave with the Water: becau$e the Gravity of the$e two Solids together $hall be equal to the$e two Gravities, that is, to B G, and <fig> to B; and the Gravity of a Liquid that hath its Ma$s equal to the$e two Solids A and D, $hall be equal to the$e two Gravities B G and B. <I>L</I>et the$e two Solids, therefore, be put in the <I>L</I>iquid, <marg>* Or, according to <I>Commandine,</I> $hall be equall in Gravi- ty to the Liquid, neither moving up- wards or down- wards.</marg> and they $hall ^{*} remain in the Surface of that <I>L</I>i- quid, (that is, they $hall not be drawn or driven upwards, nor yet downwards:) For if the Solid A be more grave than the Liquid, it $hall be drawn or born by its Gravity downwards to- wards the Bottom, with as much Force as by the Solid D it is thru$t upwards: And becau$e the Solid D is lighter than the <I>L</I>iquid, it $hall rai$e it upward with a Force as great as the Gravity G: Be- cau$e it hath been demon$trated, in the $ixth <I>Propo$ition,</I> That So- lid Magnitudes that are lighter than the Water, being demitted in the $ame, are repul$ed or driven upwards with a Force $o much the greater by how much a <I>L</I>iquid of equal Ma$s with the Solid is more Grave than the $aid Solid: But the <I>L</I>iquid which is equal in Ma$s with the Solid D, is more grave than the $aid Solid D, by the Gra- vity G: Therefore it is manife$t, that the Solid A is pre$$ed or born downwards towards the Centre of the World, with a Force as great as the Gravity G: Which was to be demon$trated.</P> <P>RIC. This hath been an ingenuous Demon$tration; and in regard I do $uffici- ently under$tand it, that we may lo$e no time, we will proceed to the $econd <I>Suppo- $ition,</I> which, as I need not tell you, $peaks thus.</P> <foot>Yy SUP.</foot> <p n=>346</p> <head>SVPPOSITION II.</head> <P><I>It is $uppo$ed that tho$e Solids which are moved up- wards, do all a$cend according to the Perpendicular which is produced thorow their Centre of Gravity.</I></P> <head>COMMANDINE.</head> <P><I>And tho$e which are moved downwards, de$cend, likewi$e, according to the Perpendicular that is produced thorow their Centre of Gravity, which he pretermitted either as known, or as to be collected from what went before.</I></P> <P>NIC. For under$tanding of this $econd <I>Suppo$ition,</I> it is requi$ite to take notice that every Solid that is lighter than the Liquid being by violence, or by $ome other occa$ion, $ubmerged in the Liquid, and then left at liberty, it $hall, by that which hath been proved in the $ixth <I>Propo$ition,</I> be thru$t or born up wards by the Liquid, and that impul$e or thru$ting is $uppo$ed to be directly according to the Perpendi- cular that is produced thorow the Centre of Gravity of that Solid; which Per- pendicular, if you well remember, is that which is drawn in the Imagination from the Centre of the World, or of the Earth, unto the Centre of Gravity of that Body, or Solid.</P> <P>RIC. How may one find the Centre of Gravity of a Solid?</P> <P>NIC. This he $heweth in that Book, intituled <I>De Centris Gravium, vel de Æqui- ponderantibus</I>; and therefore repair thither and you $hall be $atisfied, for to declare it to you in this place would cau$e very great confu$ion.</P> <P>RIC. I under$tand you: $ome other time we will talk of this, becau$e I have a mind at pre$ent to proceed to the la$t <I>Propo$ition,</I> the Expo$ition of which $eemeth to me very confu$ed, and, as I conceive, the Author hath not therein $hewn all the Subject of that <I>Propo$ition</I> in general, but only a part: which Propo$ition $peaketh, as you know, in this form.</P> <head>PROP. VIII. THEOR. VIII.</head> <marg>A</marg> <P><I>If any Solid Magnitude, lighter than the Liquid, that hath the Figure of a Portion of a Sphære, $hall be</I> <marg>B</marg> <I>demitted into the Liquid in $uch a manner as that the Ba$e of the Portion touch not the Liquid, the Figure $hall $tand erectly, $o, as that the Axis of the $aid Portion $hall be according to the Perpen- dicular. And if the Figure $hall be inclined to any $ide, $o, as that the Ba$e of the Portion touch the Liquid, it $hall not continue $o inclined as it was de- mitted, but $hall return to its uprightne$s.</I></P> <foot>For</foot> <p n=>347</p> <P>For the declaration of this <I>Propo$ition,</I> let a Solid Magnitude that hath the Figure of a portion of a Sphære, as hath been $aid, be imagined to be de- <fig> mitted into the Liquid; and al$o, let a Plain be $uppo$ed to be produced thorow the Axis of that portion, and thorow the Center of the Earth: and let the Section of the Surface of the Liquid be the Circumference A B C D, and of the Figure, the Circumference E F H, & let E H be a right line, and F T the Axis of the Portion. If now it were po$$ible, for $atisfact- ion of the Adver$ary, Let it be $uppo$ed that the $aid Axis were not according to the <I>(a)</I> Per- <marg>(a) <I>Perpendicular is taken kere, as in all other places, by this Author for the Line K L drawn thorow the Centre and Cir- cumference of the Earth.</I></marg> pendicular; we are then to demon$trate, that the Figure will not continue as it was con$tituted by the Adver$ary, but that it will re- turn, as hath been $aid, unto its former po$ition, that is, that the Axis F T $hall be according to the Perpendicular. It is manife$t, by the <I>Corollary</I> of the 1. of 3. <I>Euclide,</I> that the Center of the Sphære is in the Line F T, fora$much as that is the Axis of that Figure. And in regard that the Por- <fig> tion of a Sphære, may be greater or le$$er than an He- mi$phære, and may al$o be an Hemi$phære, let the Cen- tre of the Sphære, in the He- mi$phære, be the Point T, and in the le$$er Portion the Point P, and in the greater, the Point K, and let the Cen- tre of the Earth be the Point L. And $peaking, fir$t, of that greater Portion which hath its Ba$e out of, or a- bove, the Liquid, thorew the Points K and L, draw the Line KL cutting the Circumference E F H in the Point N, Now, becau$e <marg>C</marg> every Portion of a Sphære, hath its Axis in the Line, that from the Centre of the Sphære is drawn perpendicular unto its Ba$e, and hath its Centre of Gravity in the Axis; therefore that Portion of the Fi- gure which is within the Liquid, which is compounded of two Por- <foot>Y y 2 tions</foot> <p n=>348</p> tions of a Sphære, $hall have its Axis in the Perpendicular, that is drawn through the point K; and its Centre of Gravity, for the $ame rea$on, $hall be in the Line N K: let us $uppo$e it to be the Point R: <marg>D</marg> But the Centre of Gravity of the whole Portion is in the Line F T, betwixt the Point R and <fig> the Point F; let us $uppo$e it to be the Point <I>X</I>: The re- mainder, therefore, of that <marg>E</marg> Figure elivated above the Surface of the Liquid, hath its Centre of Gravity in the Line R X produced or continued right out in the Part towards X, taken $o, that the part prolonged may have the $ame proportion to X R, that the Gravity of that Portion that is demer- ged in the Liquid hath to the Gravity of that Figure which is above the Liquid; let us $uppo$e <marg>* <I>i. e,</I> The Center of Gravity.</marg> that ^{*} that Centre of the $aid Figure be the Point S: and thorow that <marg>F</marg> $ame Centre S draw the Perpendicular L S. Now the Gravity of the Fi- gure that is above the Liquid $hall pre$$e from above downwards ac- cording to the Perpendicular S L; & the Gravity of the Portion that is $ubmerged in the Liquid, $hall pre$$e from below upwards, accor- ding to the Perpendicular R L. Therefore that Figure will not conti- nue according to our Adver$aries Propo$all, but tho$e parts of the $aid Figure which are towards E, $hall be born or drawn downwards, & tho$e which are towards H $hall be born or driven upwards, and this $hall be $o long untill that the Axis F T comes to be according to the Perpendicular.</P> <P>And this $ame Demon$tration is in the $ame manner verified in the other <I>P</I>ortions. As, fir$t, in the Hæmi$phere that lieth with its whole Ba$e above or without the <I>L</I>iquid, the Centre of the Sphære hath been $uppo$ed to be the <I>P</I>oint T; and therefore, imagining T to be in the place, in which, in the other above mentioned, the <I>P</I>oint R was, arguing in all things el$e as you did in that, you $hall find that the Figure which is above the <I>L</I>iquid $hall pre$s from above downwards according to the <I>P</I>erpendicular S <I>L</I>; and the <I>P</I>ortion that is $ubmerged in the <I>L</I>iquid $hall pre$s from below up- wards according to the <I>P</I>erpendicular R <I>L.</I> And therefore it $hall follow, as in the other, namely, that the parts of the whole Figure which are towards E, $hall be born or pre$$ed downwards, and tho$e <marg>* Or according to the Perpendi- cular.</marg> that are towards H, $hall be born or driven upwards: and this $hall be $o long untill that the Axis F T come to $tand ^{*} <I>P</I>erpendicular- <foot>ly</foot> <p n=>349</p> ly. The like $hall al$o hold true in the <I>P</I>ortion of the Sphære le$s than an Hemi$phere that lieth with its whole Ba$e above the Liquid.</P> <head>COMMANDINE.</head> <P><I>The Demon$tration of this Propo$ition is defaced by the Injury of Time, which we have re- $tored, $o far as by the Figures that remain, one may collect the Meaning of</I> Archimedes, <I>for we thought it not good to alter them: and what was wanting to their declaration and ex- planation we have $upplyed in our Commentaries, as we have al$o determined to do in the $e- cond Propo$ition of the $econd Book.</I></P> <P>If any Solid Magnitude lighter than the Liquid.] <I>The$e words, light-</I> <marg>A</marg> <I>er than the Liquid, are added by us, and are not to be found in the Tran$iation; for of the$e kind of Magnitudes doth</I> Archimedes <I>$peak in this Propo$ition.</I></P> <P>Shall be demitted into the Liquid in $uch a manner as that the <marg>B</marg> Ba$e of the Portion touch not the Liquid.] <I>That is, $hall be $o demitted into the Liquid as that the Ba$e $hall be upwards, and the</I> Vertex <I>downwards, which he oppo$eth to that which he $aith in the Propo$ition following</I>; Be demitted into the Liquid, $o, as that its Ba$e be wholly within the Liquid; <I>For the$e words $ignifie the Portion demit- ted the contrary way, as namely, with the</I> Vertex <I>upwards and the Ba$e downwards. The $ame manner of $peech is frequently u$ed in the $econd Book; which treateth of the Portions of Rectangle Conoids.</I></P> <P>Now becau$e every Portion of a Sphære hath its Axis in the Line <marg>C</marg> that from the Center of the Sphære is drawn perpendicular to its Ba$e.] <I>For draw a Line from B to C, and let K L cut the Circumference A B C D in the Point G, and the Right Line B C in M</I>: <fig> <I>and becau$e the two Circles A B C D, and E F H do cut one another in the Points B and C, the Right Line that conjoyneth their Centers, namely, K L, doth cut the Line B C in two equall parts, and at Right Angles; as in our Commentaries upon</I> Prolomeys <I>Plani$phære we do prove: But of the Portion of the Circle B N C the Diameter is M N; and of the Portion B G C the Diameter is M G;</I> <marg><I>(a)</I> By 29. of the fir$t of <I>Encl.</I></marg> <I>for the</I> (a) <I>Right Lines which are drawn on both $ides parallel to B C do make</I> <marg><I>(b)</I> By 3. of the third.</marg> <I>Right Angles with N G; and</I> (b) <I>for that cau$e are thereby cut in two equall parts: Therefore the Axis of the Portion of the Sphære B N C is N M; and the Axis of the Portion B G C is M G: from whence it followeth that the Axis of the Portion demerged in the Liquid is in the Line K L, namely N G. And $ince the Center of Gravity of any Portion of a Sphære is in the Axis, as we have demonstrated in our Book</I> De Centro Gravitatis Solidorum, <I>the Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is, of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra- vity of tho$e Portions of Sphæres. For $uppo$e, if po$$ible, that it be out of the Line N G, as in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V Q. Becau$e therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha- ving the $ame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the Portion B G C $hall, by the 8 of the fir$t Book of</I> Archimedes, De Centro Gravitatis <foot>Plano-</foot> <p n=>350</p> Planotum, <I>be in the Line V Q prolonged: But that is impo$$ible; for it is in the Axis G: It followeth, therefore, that the Center of Gravity of the Portion demerged in Liquid be in the Line N K: which we propounded to be proved.</I></P> <P>But the Centre of Gravity of the whole Portion is in the Line <marg>D</marg> T, betwixt the Point R and the Point F; let us $uppo$e it to <*> the Point X.] <I>Let the Sphære becompleated, $o as that there be added of that Porti<*> the Axis T Y, and the Center of Gravity Z. And becau$e that from the whole Sphæ<*> who$e Centre of Gravity is K, as we have al$o demon$trated in the (c) Book before named, the is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind</I> <marg>(c) <I>By 8 of the fir$t</I> of Archimedes.</marg> <I>of the Portion E F H $hall be in the Line Z K prolonged: And therefore it mu$t of nece$$<*> fall betwixt K and F.</I></P> <marg>E</marg> <P>The remainder, therefore, of the Figure, elevated above the Su<*> face of the Liquid, hath its Center of Gravity in the <I>L</I>ine R <*> prolonged.] <I>By the $ame 8 of the fir$t Book of</I> Archimedes, de Centro Gravit<*> tis Planorum.</P> <P>Now the Gravity of the Figure that is above the <I>L</I>iquid $hal<*> <marg>F</marg> pre$s from above downwards according to S L; and the Gravit of the Portion that is $ubmerged in the <I>L</I>iquid $hall pre$s from be low upwards, according to the Perpendicular R L.] <I>By the $econd Su<*> po$ition of this. For the Magnitude that is demerged in the Liquid is moved upwards with <*> much Force along R L, as that which is above the Liquid is moved downwards along S L; <*> may be $hewn by Propo$ition 6. of this. And becau$e they are moved along $everall other Line <*> neither cau$eth the others being le$s moved; the which it continually doth when the Porti<*> is $et according to the Perpendicular: For then the Centers of Gravity of both the Magnitud<*> do concur in one and the $ame Perpendicular, namely, in the Axis of the Portion: and loo<*> with what force or</I> Impetus <I>that which is in the Lipuid tendeth upwards, and with the lik<*> doth that which is above or without the Liquid tend downwards along the $ame Line: An<*></I> <marg>* <I>Or overcome.</I></marg> <I>therefore, in regard that the one doth not ^{*} exceed the other, the Portion $hall no longer move but $hall $tay and re$t allwayes in one and the $ame Po$ition, unle$s $ome extrin$ick Cau$<*> chance to intervene.</I></P> <head>PROP. IX. THEOR. IX.</head> <marg>* In $ome Greek Coppies this is no di$tinct Propo$i- tion, but all Commentators, do divide it from the Prece- dent, as having a di$tinct demon- $tration in the Originall.</marg> <P>^{*} <I>But if the Figure, lighter than the Liquid, be demit- ted into the Liquid, $o, as that its Ba$e be wholl<*> within the $aid Liquid, it $hall continue in $uch manner erect, as that its Axis $hall $tand according to the Perpendicular.</I></P> <P>For $uppo$e, $uch a Magnitude as that aforenamed to be de mitted into the Liquid; and imagine a Plane to be produce<*> thorow the Axis of the Portion, and thorow the Center of the Earth: And let the <I>S</I>ection of the Surface of the Liquid, be the Cir<*> cumference A B C D, and of the Figure the Circumference E F <I>H</I> And let E H be a Right Line, and F T the Axis of the Portion. I<*> now it were po$$ible, for $atisfaction of the Adver$ary, let it be $uppo$ed that the $aid Axis were not according to the Perpendicu- lar: we are now to demon$trate that the Figure will not $o conti- <foot>nue,</foot> <p n=>351</p> nue, but will return to be according to the <fig> Perpendieular. It is manife$t that the Gen- tre of the Sphære is in the Line F T. And again, fora$much as the Portion of a Sphære may be greater or le$$er than an Hemi$- phære, and may al$o be an Hemi$phære, let the Centre of the Sphære in the Hemi$- phære be the Point T, & in the le$$er Por- tion the Point P, and in the Greater the <marg>A</marg> Point R. And $peaking fir$t of that greater Portion which hath its Ba$e within the Liquid, thorow R and L, the Earths Cen- <fig> tre, draw the line RL. The Portion that is above the Liquid, hath its Axis in the Per- pendicular pa$$ing thorow R; and by what hath been $aid before, its Centre of Gravity $hall be in the Line N R; let it be the Point R: But the Centre of Gra- vity of the whole Portion is in the line F T, betwixt R and F; let it be X: The re- mainder therefore of that Figure, which is within the Liquid $hall have its Centre in the Right Line R <I>X</I> prolonged in the part <fig> towards <I>X,</I> taken $o, that the part pro- longed may have the $ame Proportion to X R, that the Gravity of the Portion that is above the Liquid hath to the Gravity of the Figure that is within the Liquid. Let O be the Centre of that $ame Figure: and thorow O draw the Perpendicular L O. Now the Gravity of the Portion that is above the Liquid $hall pre$s according to the Right Line R L downwards; and the Gravity of the Figure that is in the Liquid according to the Right Line O L upwards: There the Figure $hall not continue; but the parts of it towards H $hall move down- wards, and tho$e towards E upwards: & <fig> this $hall ever be, $o long as F T is accord- ing to the Perpendicular.</P> <head>COMMANDINE.</head> <P>The Portion that is above the Liquid <marg>A</marg> hath its Axis in the Perpendicular pa$$ing thorow K.] <I>For draw B C cutting the Line N K in M; and let N K out the Circumference</I> A B <I>C D in G. In the $ame manner as before me will demon$trate, that the Axis</I> <foot><I>of</I></foot> <p n=>352</p> <I>of the Portion of the Sphære is N M; and of the Portion B G C the Axis is G M: Wherefore the Centre of Gravity of them both $hall be in the Line N M: And becau$e that from the Por- tion B N C the Portion B G C, not having the $ame Centre of Gravity, is cut off, the Centre of Gravity of the remainder of the Magnitude that is above the Surface of the Liquid $hall be in the Line N K; namely, in the Line which conjoyneth the Centres of Gravity of the $aid Portions by the fore$aid 8 of</I> Archimedis de Centro Gravitatis Planorum.</P> <P>NIC. Truth is, that in $ome of the$e Figures C is put for X, and $o it was in the Greek Copy that I followed.</P> <P>RIC. This Demo$tration is very difficult, to my thinking; but I believe that it is becau$e I have not in memory the Propo$itions of that Book entituled <I>De Cen- tris Gravium.</I></P> <P>NIC. It is $o.</P> <P>RIC. We will take a more convenient time to di$cour$e of that, and now return <marg>A</marg> to $peak of the two la$t Propo$itions. And I $ay that the Figures incerted in the demon$tration would in my opinion, have been better and more intelligble unto me, drawing the Axis according to its proper Po$ition; that is in the half Arch of the$e Figures, and then, to $econd the Objection of the Adver$ary, to $uppo$e that the $aid Figures $tood $omewhat Obliquely, to the end that the $aid Axis, if it were po$$ible, did not $tand according to the Perpendicular $o often mentioned, which doing, the Propo$ition would be proved in the $ame manner as before: and this way would be more naturall and clear.</P> <marg>B</marg> <P>NIC. You are in the right, but becau$e thus they were in the Greek Copy, I thought not fit to alter them, although unto the better.</P> <P>RIC. Companion, you have thorowly $atisfied me in all that in the beginning of our Di$cour$e I asked of you, to morrow, God permitting, we will treat of $ome other ingenious Novelties.</P> <head>THE TRANSLATOR.</head> <P>I $ay that the Figures, &c. would have been more intelligible to <marg>A</marg> me, drawing the Axis Z T according to its proper Po$ition, that is in the half Arch of the$e Figures.] <I>And in this con$ideration I have followed the Schemes of</I> Commandine, <I>who being the Re$torer of the Demon$trations of the$e two la$t Propo$itions, hath well con$idered what</I> Ricardo <I>here propo$eth, and therefore hath drawn the $aid Axis (which in the Manu$cripts that he had by him is lettered F T, and not as in that of</I> Tartaylia <I>Z T,) according to that its proper Po$ition.</I></P> <P>But becau$e thus they were in the Greek Copy, I thought not <marg>B</marg> fit to alter them although unto the better.] <I>The Schemes of tho$e Manu-</I> <fig> <I>$cripts that</I> Tartaylia <I>had $een were more imperfect then tho$e in Commandines Copies; but for variety $ake, take here one of</I> Tartaylia, <I>it being that of the Portion of a Sphære, equall to an Hemi$phære, with its Axis oblique, and its Ba$e dimitted into the Liquid, and Lettered as in this Edition.</I></P> <P><I>Now Courteous Readers, I hope that you may, amid$t the great Ob$curity of the Originall in the Demon$trations of the$e two la$t Propo$itions, be able from the joynt light of the$e two Famous Commentators of our more famous Author, to di$cern the truth of the Doctrine affirmed, namely, That Solids of the Figure of Portions of Sphæres demitted into the Liquid with their Ba$es upwards $hall $tand erectly, that is, with their Axis according to the Perpendicular drawn from the Centre of the Earth unto its Circumference: And that if the $aid Portions be demitted with their Ba$es oblique and touching the Liquid in one Point, they $hall not rest in that Obliquity, but $hall return to Rectitude: And that la$tly, if the$e Portions be demitted with their Ba$es downwards, they $hall continue erect with their Axis according to the Perpendicular afore$aid: $o that no more remains to be done, but that we$et before you the 2 Books of this our Admirable Author.</I></P> <foot>AR-</foot> <p n=>353</p> <head>ARCHIMEDES, HIS TRACT <I>DE INSIDENTIBUS HUMIDO,</I> OR, Of the NATATION of BODIES Upon, or Submer$ion In the WATER, or other LIQUIDS.</head> <head><I>BOOK</I> II.</head> <head>PROP. I. THEOR. I.</head> <P><I>If any Magnitude lighter than the Liquid be demitted into the $aid Liquid, it $hall have the $ame proporti- on in Gravity to a Liquid of equal Ma$$e, that the part of the Magnitude demerged hath unto the whole Magnitude.</I></P> <P>For let any Solid Magnitude, as for in- $tance F A, lighter than the Liquid, be de- merged in the Liquid, which let be F A: And let the part thereof immerged be A, and the part above the Liquid F, It is to be demon$trated that the Magnitude F A hath the $ame proportion in Gravity to a Liquid of Equall Ma$$e that A hath to F A. Take any Liquid Magnitude, as $up- po$e N I, of equall Ma$$e with F A; and let F be equall to N, and A to I: and let the Gravity of the whole Magnitude F A be B, and let that of the Magnitude N I be O, and let that of I be R. Now the <fig> Magnitude F A hath the $ame pro- portion unto N I that the Gravity B hath to the Gravity O R: But for a$much as the Magnitude F A demit- ted into the Liquid is lighter than the $aid Liquid, it is manife$t that a Ma$$e of the Liquid, I, equall to the part of the Magnitude demerged, A, hath equall Gravity <marg>(a) <I>By 5. of the fir$t of this.</I></marg> with the whole Magnitnde, F A: For this was <I>(a)</I> above demon- $trated: But B is the Gravity of the Magnitude F A, and R of I: <foot>Zz Therefore</foot> <p n=>354</p> Therefore B and R are equall. And becau$e that of the Magni- tude FA the <I>G</I>ravity is B: Therefore of the Liquid Body <I>N</I> I the Gravity is O R. As F A is to N I, $o is B to O R, or, $o is R to O R: But as R is to O R, $o is I to N I, and A to F A: Therefore <marg>(b) <I>By 11. of the fifth of</I> Eucl.</marg> I is to N I, as F A to N I: And as I to N I $o is <I>(b)</I> A to F A. Therefore F A is to N I, as A is to F A: Which was to be demon- $trated.</P> <head>PROP. II. THEOR. II.</head> <marg>A</marg> <P>^{*} <I>The Right Portion of a Right angled Conoide, when it $hall have its Axis le$$e than</I> $e$quialter ejus quæ ad Axem (<I>or of its</I> Semi-parameter) <I>having any what ever proportion to the Liquid in Gravity, being de- mitted into the Liquid $o as that its Ba$e touch not the $aid Liquid, and being $et $tooping, it $hall not remain $tooping, but $hall be restored to uprightne$$e. I $ay that the $aid Portion $hall $tand upright when the Plane that cuts it $hall be parallel unto the Sur- face of the Liquid.</I></P> <P>Let there be a Portion of a Rightangled Conoid, as hath been $aid; and let it lye $tooping or inclining: It is to be demon- $trated that it will not $o continue but $hall be re$tored to re- ctitude. For let it be cut through the Axis by a plane erect upon the Surface of the Liquid, and let the Section of the Portion be A PO L, the Section of a Rightangled Cone, and let the Axis <fig> of the Portion and Diameter of the Section be N O: And let the Sect- ion of the Surface of the Liquid be I S. If now the Portion be not erect, then neither $hall A L be Pa- rallel to I S: Wherefore N O will not be at Right Angles with I S. <marg>* <I>Supplied by</I> Fe- derico Comman- dino.</marg> Draw therefore K <G>w,</G> touching the Section of the Cone I, in the Point P [that is parallel to I S: and from the Point P unto I S <marg>B</marg> draw P F parallel unto O N, ^{*} which $hall be the Diameter of the Section I P O S, and the Axis of the Portion demerged in the <I>L</I>i- <marg>C</marg> quid. In the next place take the Centres of Gravity: ^{*} and of the Solid Magnitude A P O L, let the Centre of Gravity be R; and <marg>D</marg> of I P O S let the Centre be B: ^{*} and draw a Line from B to R prolonged unto G; which let be the Centre of Gravity of the <foot>remaining</foot> <p n=>355</p> remaining Figure I S L A. Becau$e now that N O is <I>Se$quialter</I> of R O, but le$s than <I>Se$quialter ejus quæ u$que ad Axem</I> (or of its <I>Semi-parameter</I>;) ^{*} R O $hall be le$$e than <I>quæ u$que ad Axem</I> (or <marg>E</marg> than the <I>Semi-parameter</I>;) ^{*} whereupon the Angle R P <G>w</G> $hall be <marg>F</marg> acute. For $ince the Line <I>quæ u$que ad Axem</I> (or <I>Semi-parameter</I>) is greater than R O, that Line which is drawn from the Point R, and perpendicular to K <G>w,</G> namely RT, meeteth with the line F P without the Section, and for that cau$e mu$t of nece$$ity fall be- tween the Points <I>P</I> and <G>w;</G> Therefore if <I>L</I>ines be drawn through B and G, parallel unto R T, they $hall contain Right Angles with the Surface of the Liquid: ^{*} and the part that is within the Li- <marg>G</marg> quid $hall move upwards according to the Perpendicular that is drawn thorow B, parallel to R T, and the part that is above the Li- quid $hall move downwards according to that which is drawn tho- row G; and the Solid A P O L $hall not abide in this Po$ition; for that the parts towards A will move upwards, and tho$e towards B downwards; Wherefore N O $hall be con$tituted according to the Perpendicular.]</P> <head>COMMANDINE.</head> <P><I>The Demon$tration of this propo$ition hath been much de$ired; which we have (in like man- ner as the 8 Prop. of the fir$t Book) re$tored according to</I> Archimedes <I>his own Schemes, and illustrated it with Commentaries.</I></P> <P>The Right Portion of a Rightangled Conoid, when it $hall <marg>A</marg> have its Axis le$$e than <I>Se$quialter ejus quæ u$que ad Axem</I> (or of its <I>Semi-parameter] In the Tran$lation of</I> Nicolo Tartaglia <I>it is fal$lyread</I> great- er then Se$quialter, <I>and $o its rendered in the following Propo$ition; but it is the Right Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we $ay that Conoids are then con$tituted erect when the cutting Plane, that is to $ay, the Plane of the Ba$e, $hall be parallel to the Surface of the Liquid.</I></P> <P>Which $hall be the Diameter of the Section I P O S, and the <marg>B</marg> Axis of the Portion demerged in the Liquid.] <I>By the 46 of the fir$t of the Conicks of</I> Apollonious, <I>or by the Corol- lary of the 51 of the $ame.</I></P> <fig> <P>And of the Solid Magnitude A P <marg>C</marg> O L, let the Centre of Gravity be R; and of I P O S let the Centre be B.] <I>For the Centre of Gravity of the Portion of a Right- angled Conoid is in its Axis, which it $o divideth as that the part thereof terminating in the vertex, be double to the other part terminating in the Ba$e; as in our Book</I> De Centro Gravitatis Solidorum Propo. 29. <I>we have demon$trated. And $ince the Centre of Gravity of the Portion A P O L is R, O R $hall be double to RN and there- fore N O $hall be Se$quialter of O R. And for the $ame rea$on, B the Centre of Gravity of the Por- tion I P O S is in the Axis P F, $o dividing it as that P B is double to B F;</I></P> <P>And draw a Line from B to R prolonged unto G; which let <marg>D</marg> be the Centre of Gravity of the remaining Eigure I S L A.] <foot>Zz 2 <I>For</I></foot> <p n=>356</p> <I>For if, the Line B R being prolonged unto G, G R hath the $ame proportion to R B as the Por- tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the Liquid, the Toine G $hall be its Centre of Gravity; by the 8 of the $econd of</I> Archimedes de Centro Gravitatis Planorum, vel de <I>Æ</I>quiponderantibus.</P> <marg>E</marg> <P>R O $hall be le$s than <I>quæ u$que ad Axem</I> (or than the Semi- parameter.] <I>By the 10 Propofit. of</I> Euclids <I>fifth Book of Elements. The Line</I> quæ u$que ad Axem, <I>(or the Semi-parameter) according to</I> Archimedes, <I>is the half of that</I> juxta quam po$$unt, quæ á Sectione ducuntur, (<I>or of the Parameter;) as appeareth by the 4 Propo$it of his Book</I> De Conoidibus & Shpæroidibus: <I>and for what rea$on it is $o called, we have declared in the Commentaries upon him by us publi$hed.</I></P> <marg>F</marg> <P>Whereupon the Angle R P <G>w</G> $hall be acute.] <I>Let the Line N O be continued out to H, that $o RH may be equall to the Semi-parameter. If now from the Point H</I> <fig> <I>a Line be drawn at Right Angles to N H, it $hall meet with FP without the Section; for being drawn thorow O parallel to A L, it $hall fall without the Section, by the 17 of our $irst Book of</I> Conicks; <I>Therefore let it meet in V: and becau$e F P is parallel to the Diameter, and H V perpendicular to the $ame Diameter, and R H equall to the Semi-parameter, the Line drawn from the Point R to V $hall make Right Angles with that Line which the Section toucheth in the Point P: that is with K</I> <G>w,</G> <I>as $hall anon be demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and</I> <G>w;</G> <I>and the Argle R</I> P <G>w</G> <I>$hall be an Acute Angle.</I></P> <P>Let A B C be the Section of a Rightangled Cone, or a Parabola, and its Diameter B D; and let the Line E F touch the $ame in the Point G: and in the Diameter B D take the Line H K equall to the Semi-parameter: and thorow G, G L be- ing drawn parallel to the Diameter, draw KM from the <I>P</I>oint K at Right Angles to B D cutting G L in M: I $ay that the Line prolonged thorow Hand Mis perpendicular to E F, which it cutteth in N.</P> <P><I>For from the Point G draw the Line G O at Right Angles to E F cutting the Diameter in O: and again from the $ame Point draw G P perpendicular to the Diameter: and let the $aid Diameter prolonged cut the Line E F in Q. P B $hall be equall to B Q, by the 35 of</I> <marg>(a) <I>By Cor. of 8. of 6. of</I> Euclide.</marg> <I>our fir$t Book of</I> Conick <I>Sections,</I> (a) <I>and G</I> <fig> <I>P a Mean-proportion all betmixt Q P<*>and PO</I>; <marg>(b) <I>By 17. of the</I> 6.</marg> (b) <I>and therefore the Square of G P $hall be e- quall to the Rectangle of O P Q: But it is al$o equall to the Rectangle comprehended under P B and the Line</I> juxta quam po$$unt, <I>or the Par- ameter, by the 11 of our fir$t Book of</I> Conicks: <marg>(c) <I>By 14. of the</I> 6.</marg> (c) <I>Therefore, look what proportion Q P hath to P B, and the $ame hath the Parameter unto P O: But Q P is double unto</I> P B, <I>for that</I> P B <I>and B Q are equall, as hath been $aid: And therefore the Parameter $hall be double to the $aid P O: and by the $ame Rea$on P O is equall to that which we call the Semi-parameter, that is, to K H</I>: <marg>(d) <I>By 33. of the</I> 1.</marg> <I>But</I> (d) <I>P G is equall to K M, and</I> (e) <I>the Angle O P G to the Angle H K M; for they are both</I> <marg>(e) <I>By 4. of the</I> 1.</marg> <I>Right Angles: And therefore O G al$o is equall to H M, and the Angle P O G unto the</I> <foot><I>Angle</I></foot> <p n=>357</p> <fig> <I>Angle K H M: Therefore</I> (f) O G <I>and H N are parallel,</I> <marg>(f) <I>By 28. of the</I> 1.</marg> <I>and the</I> (g) <I>Angle H N F equall to the Angle O G F; for that G O being Perpendicular to E F, H N $hall al$o be per-</I> <marg>(g) <I>By 29. of th</I> 1</marg> <I>pandicnlar to the $ame: Which was to be demon $trated.</I></P> <P>And the part which is within the Liquid <marg>G</marg> doth move upwards according to the Per- pendicular that is drawn thorow B parallel to R T.] <I>The rea$on why this moveth upwards, and that other downwards, along the Perpendicular Line, hath been $hewn above in the 8 of the fir$t Book of this; $o that we have judged it needle$$e to repeat it either in this, or in the re$t that follow.</I></P> <head>THE TRANSLATOR.</head> <P><I>In the</I> Antient <I>Parabola (namely that a$$umed in a Rightangled Cone) the Line</I> juxta quam Po$$unt quæ in Sectione ordinatim du- cuntur <I>(which I, following</I> Mydorgius, <I>do call the</I> Parameter<I>) is</I> (a) <marg>(a) Rîvalt. <I>in</I> Ar- chimed. <I>de Cunoid & Sphæroid.</I> Prop. 3. Lem. 1.</marg> <I>double to that</I> quæ ducta e$t à Vertice Sectionis u$que ad Axem, <I>or in</I> Archimedes <I>phra$e,</I> <G>ta_s us/xri tou_ a)/con<34>;</G> <I>which I for that cau$e, and for want of a better word, name the</I> Semiparameter: <I>but in</I> Modern <I>Parabola's it is greater or le$$er then double. Now that throughout this Book</I> Archimedes <I>$peaketh of the Parabola in a Rectangled Cone, is mani- fe$t both by the fir$t words of each Propo$ition, & by this that no Parabola hath its Parameter double to the Line</I> quæ e$t a Sectione ad Axem, <I>$ave that which is taken in a Rightangled Cone. And in any other Parabola, for the Line</I> <G>ta_s ms/xritou_ a)/eon<34></G> <I>or</I> quæ u$que ad Axem <I>to u$urpe the Word</I> Se- miparameter <I>would be neither proper nor true: but in this ca$e it may pa$s</I></P> <head>PROP. III. THEOR. III.</head> <P><I>The Right Portion of a Rightangled Conoid, when it $hall have its Axis le$$e than $e$quialter of the Se- mi-parameter, the Axis having any what ever pro- portion to the Liquid in Gravity, being demitted into the Liquid $o as that its Ba$e be wholly within the $aid Liquid, and being $et inclining, it $hall not re- main inclined, but $hall be $o re$tored, as that its Ax- is do $tand upright, or according to the Perpendicular.</I></P> <P>Let any Portion be demitted into the Liquid, as was $aid; and let its Ba$e be in the <I>L</I>iquid; <fig> and let it be cut thorow the Axis, by a Plain erect upon the Sur- face of the Liquid, and let the Se- ction be A P O <I>L,</I> the Section of a Right angled Cone: and let the Axis of the Portion and Diameter of the <foot>Section</foot> <p n=>356</p> Section of the Portion be A P O L, the Section of a Rightangled Cone; and let the Axis of the Portion and Diameter of the Section be N O, and the Section of the Surface of the Liquid I S. If now the Portion be not erect, then N O $hall not be at equall Angles with I S. Draw R <G>w</G> touching the Section of the Rightangled Conoid in P, and parallel to I S: and from the Point P and parall to O N draw <I>P</I> F: and take the Centers of Gravity; and of the Solid A <I>P</I> O L let the Centre be R; and of that which lyeth within the Liquid let the Centre be B; and draw a Line from B to R pro- longing it to G, that G may be the Centre of Gravity of the Solid that is above the Liquid. And becau$e N O is $e$quialter of R O, and is greater than $e$quialter of the Semi-Parameter; it is ma- <marg>(a) <I>By 10. of the fifth.</I></marg> nife$t that <I>(a)</I> R O is greater than the <fig> Semi-parameter. ^{*}Let therefore R <marg>A</marg> H be equall to the Semi-Parameter, <marg>B</marg> ^{*} and O <I>H</I> double to H M. Fora$- much therefore as N O is $e$quialter <marg>(b) <I>By 19. of the fifth.</I></marg> of R O, and M O of O H, <I>(b)</I> the Remainder N M $hall be $e$quialter of the Remainder R H: Therefore the Axis is greater than $e$quialter of the Semi parameter by the quan- tity of the Line M O. And let it be $uppo$ed that the Portion hath not le$$e proportion in Gravity unto the Liquid of equall Ma$$e, than the Square that is made of the Exce$$e by which the Axis is greater than $e$quialter of the Semi- parameter hath to the Square made of the Axis: It is therefore ma- nife$t that the Portion hath not le$$e proportion in Gravity to the Liquid than the Square of the Line M O hath to the Square of N O: But look what proportion the <I>P</I>ortion hath to the Liquid in Gravity, the $ame hath the <I>P</I>ortion $ubmerged to the whole Solid: for this hath been demon$trated <I>(c)</I> above: ^{*}And look what pro- <marg>C</marg> portion the $ubmerged Portion hath to the whole <I>P</I>ortion, the <marg>(c) <I>By 1. of this $econd Book.</I></marg> $ame hath the Square of <I>P</I> F unto the Square of N O: For it hath been demon$trated in <I>(d) Lib. de Conoidibus,</I> that if from a Right- <marg>(d) <I>By</I> 6. De Co- noilibus & <I>S</I>phæ- roidibus <I>of</I> Archi- medes.</marg> angled Conoid two <I>P</I>ortions be cut by Planes in any fa$hion pro- duced, the$e <I>P</I>ortions $hall have the $ame Proportion to each other as the Squares of their Axes: The Square of P F, therefore, hath not le$$e proportion to the Square of N O than the Square of M O hath to the Square of N O: ^{*}Wherefore P F is not le$$e than <marg>D</marg> M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn <marg>E</marg> from H at Right Angles unto N O, it $hall meet with B <I>P,</I> and $hall <marg>F</marg> fall betwixt B and P; let it fall in T: <I>(e)</I> And becau$e <I>P</I> F is <marg>(e) <I>By 2. of this $econd Book.</I></marg> parallel to the Diameter, and H T is perpendicular unto the $ame Diameter, and R H equall to the Semi-parameter; a Line drawn from R to T and prolonged, maketh Right Angles with the Line <foot>contingent</foot> <p n=>360</p> contingent unto the Section in the Point P: Wherefore it al$o maketh Right Angles with the Surface of the Liquid: and that part of the Conoidall Solid which is within the Liquid $hall move upwards according to the Perpendicular drawn thorow B parallel to R T; and that part which is above the Liquid $hall move down- wards according to that drawn thorow G, parallel to the $aid R T: And thus it $hall continue to do $o long untill that the Conoid be re$tored to uprightne$$e, or to $tand according to the Perpendicular.</P> <head>COMMANDINE.</head> <marg>A</marg> <P>Let therefore R H be equall to the Semi-parameter.] <I>So it is to be read, and not R M, as</I> Tartaglia's <I>Tran$lation hath is; which may be made appear from that which followeth.</I></P> <marg>B</marg> <P>And O H double to H M.] <I>In the Tran$lation aforenamed it is fal$ly render- ed,</I> O N <I>double to</I> R M.</P> <marg>C</marg> <P>And look what proportion the Submerged Portion hath to the whole Portion, the $ame hath the Square of P F unto the Square of N O.] <I>This place we have re$tored in our Tran$lation, at the reque$t of $ome friends: But it is demon- $trated by</I> Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. 26.</P> <marg>D</marg> <P>Wherefore P F is not le$$e than M O.] <I>For by 10 of the fifth it followeth that the Square of P F is not le$$e than the Square of M O: and therefore neither $hall the Line P F be leße than the Line M O, by 22 of the</I> <fig> <marg>E</marg> <I>$ixth.</I></P> <marg>(a) <I>By 14. of the $ixth.</I></marg> <P>Nor B P than H O,] <I>For as P F is to P B, $o is M O to H O: and, by Permutation, as</I> <marg>F</marg> <I>P F is to M O, $o is B P to H O; But P F is not le$$e than M O as hath bin proved; (a) Therefore neither $hall B P be le$$e than H O.</I></P> <P>If therefore a Right Line be drawn from H at Right Angles unto N O, it $hall meet with B P, and $hall fall be- twixt B and P.] <I>This Place was corrupt in the Tran$lation of</I> Tartaglia<I>: But it is thus demonstra- ted. In regard that P F is not le$$e than O M, nor P B than O H, if we $uppo$e P F equall to O M, P B $hall be likewi$e equall to O H: Wherefore the Line drawn thorow O, parallel to A L $hall fall without the Section, by 17 of the fir$t of our Treati$e of Conicks; And in regard that B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth al$o meet with the $ame beneath B, and it doth of nece$$ity fall betwixt B and P: But the $ame is much more to follow, if we $uppo$e P F to be greater than O M.</I></P> <foot>PROP.</foot> <p n=>361</p> <head>PROP. V. THEOR. V.</head> <P><I>The Right Portion of a Right-Angled Conoid lighter than the Liquid, when it $hall have its Axis great- er than</I> Se$quialter <I>of the Semi-parameter, if it have not greater proportion in Gravity to the Liquid [of equal Ma$s] than the Exce$$e by which the Square made of the Axis is greater than the Square made of the Exce$$e by which the Axis is greater than</I> $e$quialter <I>of the Semi-Parameter hath to the Square made of the Axis being demitted into the Li- quid, $o as that its Ba$e be wholly within the Liquid, and being $et inclining, it $hall not remain $o inclined, but $hall turn about till that its Axis $hall be accor- ding to the Perpendicular.</I></P> <P>For let any Portion be demitted into the Liquid, as hath been $aid; and let its Ba$e be wholly within the Liquid, And being cut thorow its Axis by a Plain erect upon the Surface of the Liquid; its Section $hall be the Section <fig> of a Rightangled Cone: Let it be A P O L, and let the Axis of the Por- tion and Diameter of the Section be N O; and the Section of the Surface of the Liquid I S. And becau$e the Axis is not according to the Perpendicu- lar, N O will not be at equall angles with I S. Draw K <G>w</G> touching the Se- ction A P O L in P, and parallel unto I S: and thorow P, draw P F parallel unto N O: and take the Centres of Gravity; and of the Solid A P O L let the Centre be R; and of that which lyeth above the Liquid let the Centre be B; and draw a Line from B to R, prolonging it to G; which let be the Centre of Gravity of the Solid demerged within the Liquid: and moreover, take R H equall to the Semi-parameter, and let O H be double to H M; and do in the re$t as hath been $aid <I>(a)</I> above. <marg>(a) <I>In <*>. Prop. of this.</I></marg> Now fora$much as it was $uppo$ed that the Portion hath not greater proportion in Gravity to the Liquid, than the Exce$$e by which the Square N O is greater than the Square M O, hath to the $aid Square N O: And in regard that whatever proportion in Gravity <foot>Aaa the</foot> <p n=>362</p> the Portion hath to the Liquid of equall Ma$$e, the $ame hath the Magnitude of the Portion $ubmerged unto the whole Portion; as hath been demon$trated in the fir$t Propo$ition; The Magnitude $ubmerged, therefore, $hall not have greater proportion to the <marg>(a) <I>By 11. of the fifth.</I></marg> whole <I>(b)</I> Portion, than that which hath been mentioned: ^{*}And therefore the whole Portion hath not greater proportion unto that <marg>A</marg> which is above the Liquid, than the Square N O hath to the Square <marg>(b) <I>By 26. of the Book</I> De Conoid. & Sphæroid.</marg> M O: But the <I>(c)</I> whole Portion hath the $ame proportion unto that which is above the Liquid that the Square N O hath to the Square P F: Therefore the Square N O hath not greater propor- <marg>B</marg> tion unto the Square P F, than it hath unto the Square M O: ^{*}And hence it followeth that P F is not le$$e than O M, nor P B than O <marg>C</marg> H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O $hall meet with B P betwixt P and B: Let it be in T: And be- cau$e that in the Section of the Rectangled Cone P F is parallel unto the Diameter N O; and H T perpendicular unto the $aid Diame- ter; and R H equall to the Semi-parameter: It is manife$t that R T prolonged doth make Right Angles with K P <G>w</G>: And there- fore doth al$o make Right Angles with I S: Therefore R T is per- pendicular unto the Surface of the Liquid; And if thorow the Points B and G Lines be drawn parallel unto R T, they $hall be perpendicular unto the Liquids Surface. The Portion, therefore, which is above the Liquid $hall move downwards in the Liquid ac- cording to the Perpendicular drawn thorow B; and that part which is within the Liquid $hall move upwards according to the Perpendicular drawn thorow G; and the Solid Portion A P O L $hall not continue $o inclined, [<I>as it was at its demer$ion</I>], but $hall move within the Liquid untill $uch time that N O do $tand accor- ding to the Perpendicular.</P> <head>COMMANDINE.</head> <marg>A</marg> <P>And therefore the whole Portion hath not greater proportion unto that which is above the Liquid, than the Square N O hath to the Square M O.] <I>For in regard that the Magnitude of the Portion demerged within the Liquid hath not greater proportion unto the whole Portion than the Exce$$e by which the Square N O is greater than the Square M O hath to the $aid Square N O; Converting of the Proportion, by the 26. of the fifth of</I> Euclid, <I>of</I> Campanus <I>his Tran$lation, the whole Portion $hall not have le$$er proportion unto the Magnitude $ubmerged, than the Square N O hath unto the Exce$$e by which N O is greater than the Square M O. Let a Portion be taken; and let that part of it which is above the Liquid be the fir$t Magnitude; the part of it which is $ubmerged the $econd: and let the third Magnitude be the Square M O; and let the Exce$$e by which the Square N O is greater than the Square M O be the fourth. Now of the$e Mag- nitudes, the proportion of the fir$t and $econd, unto the $econd, is not le$$e than that of the third & fourth unto the fourth: For the Square M O together with the Exce$$e by which the Square N O exceedeth the Square M O is equall unto the $aid Square N O: Wherefore, by Conver$i- on of Proportion, by 30 of the $aid fifth Book, the proportion of the fir$t and $econd unto the fir$t, $hall not be greater than that of the third and fourth unto the third: And, for the $ame</I> <foot><I>cau$e,</I></foot> <p n=>363</p> <I>the proportion of the whole Portion unto that part thereof which is above the Liquid $hall not be greater than that of the Square N O unto the Square M O: Which was to be demon$trated.</I></P> <P>And hence it followeth that P F is not le$$e than O M, nor P B <marg>B</marg> than O H.] <I>This followeth by the 10 and 14 of the fifth, and by the 22 of the $ixth of</I> Euclid, <I>as hath been $aid above.</I></P> <P>A <I>L</I>ine, therefore, drawn from Hat Right Angles unto N O $hall <marg>C</marg> meet with P B betwixt P and B.] <I>Why this $o falleth out, we will $hew in the next.</I></P> <head>PROP. VI. THEOR. VI.</head> <P><I>The Right Portion of a Rightangled Conoid lighter than the Liquid, when it $hall have its Axis greater than $e$quialter of the Semi-parameter, but le$$e than to be unto the Semi-parameter in proportion as fifteen to fower, being demitted into the Liquid $o as that its Ba$e do touch the Liquid, it $hall never stand $o enclined as that its Ba$e toucheth the Liquid in one Point only.</I></P> <P>Let there be a Portion, as was $aid; and demit it into the Li- quid in $uch fa$hion as that its Ba$e do touch the Liquid in one only Point: It is to be demon$trated that the $aid Portion <marg>A</marg> $hall not continue $o, but $hall turn about in $uch manner as that its Ba$e do in no wi$e touch the Surface of the Liquid. For let it be cut thorow its Axis by a Plane erect <fig> upon the <I>L</I>iquids Surface: and let the Section of the Superficies of the Portion be A P O L, the Section of a Rightangled Cone; and the Sect- ion of the Surface of the <I>L</I>iquid be A S; and the Axis of the Portion and Diameter of the Section N O: and let it be cut in F, $o as that O F be double to F N; and in <G>w</G> $o, as that N O may be to F <G>w</G> in the $ame proportion as fifteen to four; and at Right Angles to N O draw <G>w</G> <I>N</I>ow becau$e N O hath greater proportion unto F <G>w</G> than unto the Semi-parameter, let the Semi-parameter be equall to F B: <marg>B</marg> and draw P C parallel unto A S, and touching the Section A P O L in P; and P I parallel unto <I>N O</I>; and fir$t let P I cut K<G>w</G> in H. For- <marg>C</marg> a$much, therefore, as in the Portion A P O L, contained betwixt the Right <I>L</I>ine and the Section of the Rightangled Cone, K <G>w</G> is parallel to A L, and P I parallel unto the Diameter, and cut by the <foot>Aaa 2 $aid</foot> <p n=>364</p> $aid K <G>w</G> in H, and A S is parallel unto the <I>L</I>ine that toucheth in P; It is nece$$ary that P I hath unto P H either the $ame proportion that <I>N</I> <G>w</G> hath to <G>w</G> O, or greater; for this hath already been de- mon$trated: But <I>N</I> <G>w</G> is $e$quialter of <G>w</G> O; and P I, therefore, is either Se$quialter of H P, or more than $e$quialter: Wherefore <marg>D</marg> P H is to H I either double, or le$$e than double. <I>L</I>et P T be double to T I: the Centre of Gravity of the part which is within the <I>L</I>iquid $hall be the Point T. Therefore draw a <I>L</I>ine from T to F prolonging it; and let the Centre of <fig> Gravity of the part which is above the <I>L</I>iquid be G: and from the Point B at Right Angles unto <I>N O</I> draw B R. And $eeing that P I is parallel unto the Diameter <I>N O,</I> and B R perpendicular unto the $aid Diameter, and F B equall to the Semi-parameter; It is mani- fe$t that the <I>L</I>ine drawn thorow the Points F and R being prolonged, maketh equall Angles with that which toucheth the Section A P O L in the Point P: and therefore doth al$o make Right An- gles with A S, and with the Surface of the <I>L</I>iquid: and the <I>L</I>ines drawn thorow T and G parallel unto F R $hall be al$o perpendicu- lar to the Surface of the <I>L</I>iquid: and of the Solid Magnitude A P O L, the part which is within the <I>L</I>iquid moveth upwards according to the Perpendicular drawn thorow T; and the part which is above the <I>L</I>iquid moveth downwards according to that drawn thorow G: <marg>E</marg> The Solid A <I>P</I> O L, therefore, $hall turn about, and its Ba$e $hall not in the lea$t touch the Surface of the <I>L</I>iquid, And if <I>P</I> I do not cut the <I>L</I>ine K <G>w,</G> as in the $econd Figure, it is manife$t that the <I>P</I>oint T, which is the Centre of Gravity of the $ubmerged <I>P</I>ortion, falleth betwixt <I>P</I> and I: And for the other particulars remaining, they are demon$trated like as before.</P> <head>COMMANDINE.</head> <marg>A</marg> <P>It is to be demon$trated that the $aid <I>P</I>ortion $hall not continue $o, but $hall turn about in $uch manner as that its Ba$e do in no wi$e touch the Surface of the Liquid.] <I>The$e words are added by us, as having been omitted by</I> Tartaglia.</P> <P><I>N</I>ow becau$e N O hath greater proportion to F <G>w</G> than unto <marg>B</marg> the Semi parameter.] <I>For the Diameter of the Portion N O hath unto F</I> <G>w</G> <I>the $ame proportion as fifteen to fower: But it was $uppo$ed to have le$$e proportion unto the Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F</I> <G>w</G> <I>than unto the Semi-parameter: And therefore</I> (a) <I>the Semi-parameter $hall be greater</I> <marg>(a) <I>By 10. of the fifth.</I></marg> <I>than the $aid F</I> <G>w.</G></P> <P>Fora$much, therefore, as in the <I>P</I>ortion <I>A P O L,</I> contained, be- <marg>C</marg> twixt the Right <I>L</I>ine and the Section of the Rightangled Cone K <G>w</G> is parallel to A L, and <I>P I</I> parallel unto the Diameter, and cut by <foot>the</foot> <p n=>365</p> the $aid K <G>w</G> in H, and A S is parallel unto the Line that toucheth in P; It is nece$$ary that P I hath unto P H either the $ame propor- tion that N <G>w</G> hath to <G>w</G> O, or greater; for this hath already been demon$trated.] <I>Where this is demon$trated either by</I> Archimedes <I>him$elf, or by any other, doth not appear; touching which we will here in$ert a Demon$tration, after that we have explained $ome things that pertaine thereto.</I></P> <head>LEMMA I.</head> <P>Let the Lines A B and A C contain the Angle B A C; and from the point D, taken in the Line A C, draw D E and D F at plea$ure unto A B: and in the $ame Line any Points G and L being taken, draw G H & L M parallel to D E, & G K and L N parallel unto F D: Then from the Points D & G as farre as to the Line M L draw D O P, cutting G H in O, and G Q parallel unto B A. I $ay that the Lines that lye betwixt the Pa- rallels unto F D have unto tho$e that lye betwixt the Par- allels unto D E (namely K N to G Q or to O P; F K to D O; and F N to D P) the $ame mutuall proportion: that is to $ay, the $ame that A F hath to A E.</P> <P><I>For in regard that the Triangles A F D, A K G, and A N L</I> <fig> <I>are alike, and E F D, H K G, and M N L are al$o alike: There-</I> <marg>(a) <I>By 4. of the $ixth.</I></marg> <I>fore,</I> (a) <I>as A F is to F D, $o $hall A K be to K G; and as F D is to F E, $o $hall K G be to K H: Wherefore,</I> ex equali, <I>as A F is to F E, $o $hall A K be to K H: And, by Conver$ion of proportion, as A F is to A E, $o $hall A K be to K H. It is in the $ame manner proved that, as A F is to A E, $o $hall A N be to A M. Now A</I> <marg>(b) <I>By 5. of the fifth.</I></marg> <I>N being to A M, as A K is to A H; The</I> (b) <I>Remainder K N $hall be unto the Remainder H M, that is unto G Q, or unto O P, as A N is to A M; that is, as A F is to A E: Again, A K is to A H, as A F is to A E; Therefore the Remainder F K $hall be to the Remainder E H, namely to D O, as A F is to A E. We might in like manner demonstrate that $o is F N to D P: Which is that that was required to be demonstrated.</I></P> <head>LEMMA II.</head> <P>In the $ame Line A B let there be two Points R and S, $o di$po- $ed, that A S may have the $ame Proportion to A R that A F hath to A E; and thorow R draw R T parallel to E D, and thorow S draw S T parallel to F D, $o, as that it may meet with R T in the Point T. I $ay that the Point T fall- eth in the Line A C.</P> <foot><I>For</I></foot> <p n=>366</p> <fig> <P><I>For if it be po$$ible, let it fall $hort of it: and let R T be pro- longed as farre as to A C in V: and then thorow V draw V X pa- rallel to F D. Now, by the thing <*> we have last demon$trated, A X $hall have the $ame proportion unto A R, as A F hath to A E. But A S hath al$o the $ame proportion to A R: Wherefore</I> (a) <marg>(a) <I>By 9. of the fifth.</I></marg> A S <I>is equall to A X, the part to the whole, which is impo$$i- ble. The $ame ab$urdity will follow if we $uppo$e the Toint T to fall beyond the Line A C: It is therefore nece$$ary that it do fall in the $aid A C. Which we propounded to be demonstrated.</I></P> <head>LEMMA III.</head> <P>Let there be a Parabola, who$e Diameter <marg>* Or touch it.</marg> let be A B; and let the Right Lines A C and B D be ^{*} con- tingent to it, A C in the Point C, and B D in B: And two Lines being drawn thorow C, the one C E, parallel unto the Diameter; the other C F, parallel to B D; take any Point in the Diameter, as G; and as F B is to B G, $o let B G be to B H: and thorow G and H draw G K L, and H E M, parallel unto B D; and thorow M draw M N O parallel to <I>A C,</I> and cutting the Diameter in O: and the Line <I>N P</I> being drawn thorow <I>N</I> unto the Diameter let it be parallel to B D. I $ay that H O is double to G B.</P> <P><I>For the Line M N O cutteth the Diameter either in G, or in other Points: and if it do cut it in G, one and the $ame Point $hall be noted by the two letters G and O. Therfore F C, P N, and H E M being Parallels, and A C being Parallels to M N O, they $hall make the</I> <fig> <I>Triangles A F C, O P N and O H M like to</I> <marg>(a) <I>By 4. of the $ixth.</I></marg> <I>each other: Wherefore</I> (a) <I>O H $hall be to H M, as A F to FC: and</I> ^{*} Permutando, <marg>* Or permitting.</marg> <I>O H $hall be to A F, as H M to F C: But the Square H M is to the Square G L as the Line H B is to the Line B G, by 20. of our fir$t Book of</I> Conicks; <I>and the Square G L is unto the Square F C, as the Line G B is to the Line B F: and the Lines H B, B G and B F are thereupon</I> <marg>(b) <I>By 22. of the $ixth.</I></marg> <I>Proportionals: Therefore the</I> (b) <I>Squares H M, G L and F C and there Sides, $hall al$o be Proportionals: And, therefore, as the (c) Square H M is to the Square G L, $o is the Line</I> <marg>(c) <I>By</I> Cor. <I>of 20. of the $ixth.</I></marg> <I>H M to the Line F C: But as H M is to F C, $o is O H to A F; and as the Square H M is to the Square G L, $o is the Line H B to B G; that is, B G to B F: From whence it followeth that O H is to A F, as B G to B F: And</I> Permu- tando, <I>O H is to B G, as A F to F B; But A F is double to F B: Therefore A B and B F are equall, by 35. of our fir$t Book of</I> Conicks: <I>And therefore N O is double to G B: Which was to be demon$trated.</I></P> <foot>LEMMA</foot> <p n=>367</p> <head>LEMMA IV.</head> <P>The $ame things a$$umed again, and M Q being drawn from the Point M unto the Diameter, let it touch the Section in the Point M. I $ay that H Q hath to Q O, the $ame proportion that G H hath to C N.</P> <P><I>For make H R equall to G F; and $eeing that</I> <fig> <I>the Triangles A F C and O P N are alike, and P N equall to F C, we might in like manner de- mon$trate P O and F A to be equall to each other: Wherefore P O $hall be double to F B: But H O is double to G B: Therefore the Remainder P H is al$o double to the Remainder F G; that is, to R H: And therefore is followeth that P R, R H and F G are equall to one another; as al$o that R G and P F are equall: For P G is common to both R P and G F. Since therefore, that H B is to B G, as G B is to B F, by Conver$ion of Pro- portion, B H $hall be to H G, as B G is to G F: But Q H is to H B, as H O to B G. For by 35 of our fir$t Book of</I> Conicks, <I>in regard that Q M toucheth the Section in the Point M, H B and B Q $hall be equall, and Q H double to H B: Therefore,</I> ex æquali, <I>Q H $hall be to H G, as H O to G F; that is, to H R: and,</I> Permu- tando, <I>Q H $hall be to H O, as H G to H R: again, by Conver$ion, H Q $hall be to Q O, as H G to G R; that is, to P F; and, by the $ame rea$on, to C N: Whichwas to be de- mon$trated.</I></P> <P>The$e things therefore being explained, we come now to that which was propounded. I $ay, therefore, fir$t that <I>N C</I> hath to C K the $ame proportion that H G hath to G B.</P> <P><I>For $ince that H Q is to Q O, as H G to C N</I>; <fig> <I>that is, to A O, equall to the $aid C N: The Re- mainder G Q $hall be to the Remainder Q A, as H Q to Q O: and, for the $ame cau$e, the Lines A C and G L prolonged, by the things that wee have above demonstrated, $hall inter$ect or meet in the Line Q M. Again, G Q is to Q A, as H Q to Q O: that is, as H G to F P; as</I> <marg>(a) <I>By</I> 2. Lemma.</marg> (a) <I>was bnt now demonstrated, But unto</I> (b) <I>G</I> <marg>(b) <I>By</I> 4. Lemma.</marg> <I>Q two Lines taken together, Q B that is H B, and B G are equall: and to Q A H F is equall; for if from the equall Magnitudes H B and B Q there be taken the equall Magnitudes F B and B A, the Re mainder $hall be equall; Therefore taking H G from the two Lines H B and B G, there $hall re- main a Magnitude double to B G; that is, O H: and P F taken from F H, the Remainder is H P: Wherefore</I> (c) <I>O H is to H P, as G Q to Q A:</I> <marg>(b) <I>By 19. of the fifth.</I></marg> <I>But as G Q is to Q A, $o is H Q to Q O;</I> <foot><I>that</I></foot> <p n=>368</p> <marg>(d) <I>By 15. of the fifth.</I></marg> <I>that is, H G to N C: and as</I> (d) <I>O H is to H P, $o is G B to C K; For O H is double to G B, and H P al$o double to G F; that is, to C K; Therefore H G hath the $ame propor- tion to N C, that G B hath to C K: And</I> Permutando, <I>N C hath to C K the $ame proportion that H G hath to G B.</I></P> <P>Then take $ome other Point at plea$ure in the Section, which let be S: and thorow S draw two Lines, the one S T paral- lel to D B, and cutting the Diameter in the Point T; the other S V parallel to A C, and cutting C E in V. I $ay that V C hath greater proportion to C K, than T G hath to G B.</P> <P><I>For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the Diameter parallel to B D: G T $hall be le$$e than G Y, in regard that V S is leße than V X: And, by the fir$t Lemma, Y G $hall be to V C, as H G to N C; that is, as G B to C K, which was demon$trated but now: And,</I> Permutando, <I>Y G $hall be to G B, as V C to C K: But T G, for that it is le$$e than Y G, hath le$$e proportion to G B, than Y G hath to the $ame; Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de- mon$trated. Therefore a Po$ition given G K, there $hall be in the Section one only Point, to wit M, from which two Lines M E H and M N O being drawn, N C $hall have the $ame pro- portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall- eth betwixt A C, and the Line parallel unto it $hall alwayes have greater proportion to C K, than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there- fore, is manife$t which was affirmed by</I> Archimedes, <I>to wit, that the Line P I hath unto P H, either the $ame proportion that N</I> <G>w</G> <I>hath to</I> <G>w</G> <I>O, or greater.</I></P> <marg>D</marg> <P>Wherefore P H is to H I either double, or le$$e than double.] <I>If le$$e than double, let P T be double to T I: The Centre of Gravity of that part of the Portion that is within the Liquid $hall be the</I> <fig> <I>Point T: But if P H be double to H I, H $hall be the Centre of Gravity; And draw H F, and prolong it unto the Centre of that part of the Por- tion which is above the Liquid, namely, unto G, and the re$t is demon$trated as before. And the $ame is to be under$tood in the Propo$ition that followeth.</I></P> <P>The Solid A P O L, therefore, $hall turn about, and its Ba$e $hall not in the lea$t touch the Surface of the Liquid.] <I>In</I> Tartaglia's <I>Tran$lation it is rendered</I> ut Ba$is ip$ius non tangent $uperficiem humidi $ecundum unum $ignum; <I>but we have cho$en to read</I> ut Ba$is ip$ius nullo modo humidi $uperficiem contingent, <I>both here, and in the following Propo$itions, becau$e the Greekes frequently u$e</I> <G>w(de\ei)=s, w(de\<*></G> <I>pro</I> <G>w)dei\s<*> & ou)di\n</G>: <I>$o that</I> <G>ou)de)/sinoudei/s,</G> nullus e$t; <G>ou)d<*>u(p)e(ro\s</G> à nullo, <I>and $o of others of the like nature.</I></P> <foot>PROP.</foot> <p n=>369</p> <head>PROP. VII. THE OR. VII.</head> <P><I>The Right Portion of a Rightangled Conoid lighter than the Liquid, when it $hall have its Axis greater than Se$quialter of the Semi-parameter, but le$$e than to be unto the $aid Semi-parameter in proportion as fi$teen to fower, being demitted into the Liquid $o as that its Ba$e be wholly within the Liquid, it $hall never $tand $o as that its Ba$e do touch the Surface of the Liquid, but $o, that it be wholly within the Liquid, and $hall not in the lea$t touch its Surface.</I></P> <P>Let there be a Portion as hath been $aid; and let it be de- mitted into the Liquid, as we have $uppo$ed, $o as that its Ba$e do touch the Surface in one Point only: It is to be de- mon$trated that the $ame $hall not $o <fig> continue, but $hall turn about in $uch manner as that its Ba$e do in no wi$e touch the Surface of the Liquid. For let it be cut thorow its Axis by a Plane erect upon the Liquids Sur- face: and let the Section be A P O L, the Section of a Rightangled Cone; the Section of the Liquids Surface S L; and the Axis of the Portion and Diameter of the Section P F: and let P F be cut in R, $o, as that R P may be double to R F, and in <*> $o as that P F may be to R <G>w</G> as fifteen to fower: and draw <G>w</G> K at Right Angles <marg>(a) <I>By 10 of the fifth.</I></marg> to P F: <I>(a)</I> R <G>w</G> $hall be le$$e than the Semi-parameter. There- fore let R H be $uppo$ed equall to the Semi-parameter: and draw C O touching the Section in O and parallel unto S L; and let N O be parallel unto P F; and fir$t let N O cut K <G>w</G> in the Point I, as in the former Schemes: It $hall be demon$trated that N O is to O I either $e$quialter, or greater than $e$quialter. Let O I be le$$e than double to I N; and let O B be double to B N: and let them be di$po$ed like as before. We might likewi$e demon$trate that if a Line be drawn thorow R and T it will make Right Angles with the Line C O, and with the Surface of the Liquid: Where- fore Lines being drawn from the Points B and G parallels unto R T, they al$o $hall be Perpendiculars to the Surface of the Liquid: The Portion therefore which is above the Liquid $hall move down- <foot>Bbb wards</foot> <p n=>370</p> <fig> wards according to that $ame Perpendicular which pa$$eth thorow B; and the Portion which is within the Liquid $hall move up- wards acording to that pa$$ing thorow G: From whence it is manife$t that the Solid $hall turn about in $uch manner, as that its Ba$e $hall in no wi$e touch the Surface of the Liquid; for that now when it touch- eth but in one Point only, it moveth down- wards on the part towards L. And though N O $hould not cut <*> K, yet $hall the $ame hold true.</P> <head>PROP. VIII. THE OR. VIII.</head> <P><I>The Right Portion of a Rightangled Conoid, when it $hall have its Axis greater than $e$quialter of the Se- mi-parameter, but le$$e than to be unto the $aid Semi- parameter, in proportion as fifteen to fower, if it have a le$$er proportion in Gravity to the Liquid, than the Square made of the Exce$$e by which the Axis is greater than Se$quialter of the Semi-parameter hath to the Square made of the Axis, being demitted into the Liquid, $o as that its Ba$e touch not the Liquid, it $hall neither return to Perpendicularity, nor conti- nue inclined, $ave only when the Axis makes an Angle with the Surface of the Liquid, equall to that which we $hall pre$ently $peak of.</I></P> <P>Let there be a Portion as hath been $aid; and let B D be equall to the Axis: and let B K be double to K D; and R K equall <marg>A</marg> to the Semi-parameter: and let C B be Se$quialter of B R: C D $hall be al$o Sefquialter of K R. And as the Portion is to the Liquid in Gravity, $o let the Square F Q be to the Square D B; and let F be double to Q: It is manife$t, therefore, that F Q hath to D B, le$s proportion than C B hath to B D; For C B is the Exc<*>$s by which the Axis is greater than Se$quialter of the Semi- <marg>B</marg> parameter: And, therefore, F Q is le$s than B C; and, for the <marg>C</marg> $ame rea$on, F is le$s than B R. Let R <G>y</G> be equall to F; and draw <G>y</G> E perpendicular to B D; which let be in power or contence the half of that which the Lines K R and <G>y</G> B containeth; and draw a Line from B to E: It is to be demon$trated, that the <foot>Portion</foot> <p n=>371</p> Portion demitted into the Liquid, like as hath been $aid, $hall $tand enclined $o as that its Axis do make an Angle with the Surface of the Liquid equall unto the Angle E B <G>*y.</G> For demit any Portion into the Liquid $o as that its Ba$e <fig> touch not the Liquids Surface; and, if it can be done, let the Axis not make an Angle with the Liquids Surface equall to the Angle E B <G>*y</G>; but fir$t, let it be greater: and the Portion being cut thorow the Axis by a Plane e- rect unto [<I>or upon</I>] the Surface of the Liquid, let the Section be A P O L the Section of a Rightangled Cone; the Section of the Surface of the Liquid X S; and let the Axis of the Portion and Diameter of the Section be N O: and draw P Y parallel to X S, and touching the Section A P O L in P; and P M parallel to N O; and P I perpendicular to N O: and moreover, let B R be equall to O <G>w,</G> and R K to T <G>w;</G> and let <G>w</G> H be perpendicular to the Axis. Now becau$e it hath been $uppo$ed <marg>D</marg> that the Axis of the Portion doth make an Angle with the Surface of the Liquid greater than the Angle B, the Angle P Y I $hall be greater than the Angle B: Therefore the Square P I hath greater <marg>E</marg> proportion to the Square Y I, than the Square E <G>*y</G> hath to the Square <G>*y</G> B: But as the Square P I is to the Square Y I, $o is the <marg>F</marg> Line K R unto the Line I Y; and as the Square E <G>*y</G> is to the Square <marg>G</marg> <G>*y</G> B, $o is half of the Line K R unto the Line <G>*y</G> B: Wherefore <I>(a)</I> K R hath greater proportion to I Y, than the half of K R hath <marg>(a) <I>By 13. of the fifth.</I></marg> to <G>*y</G> B: And, con$equently, I Y isle$$e than the double of <G>*y</G> B, and is the double of O I: Therefore O I is le$$e than <G>*y</G> B; and I <G>w</G> <marg>H</marg> greater than <G>*y</G> R: but <G>*y</G> R is equall to F: Therefore I <G>w</G> is greater <marg>K</marg> than F. And becau$e that the Portion is $uppo$ed to be in Gra- vity unto the Liquid, as the Square F Q is to the Square B D; and $ince that as the Portion is to the Liquid in Gravity, $o is the part thereof $ubmerged unto the whole Portion; and in regard that as the part thereof $ubmerged is to the whole, $o is the Square P M to the Square O N; It followeth, that the Square P M is to the Square N O, as the Square F Q is to the Square B D: And therefore F <marg>L</marg> Q is equall to P M: But it hath been demon$trated that P H is <marg>M</marg> greater than F: It is manife$t, therefore, that P M is le$$e than $e$quialter of P H: And con$equently that P H is greater than the double of H M. Let P Z be double to Z M: T $hall be the Cen- tre of Gravity of the whole Solid; the Centre of that part of it which is within the Liquid, the Point Z; and of the remaining <marg>N</marg> part the Centre $hall be in the Line Z T prolonged unto G. In <foot>Bbb 2 the</foot> <p n=>372</p> the $ame manner we might demon- <fig> $trate the <I>L</I>ine T H to be perpendi- cular unto the Surface of the Liquid: and that the Portion demerged with- in the <I>L</I>iquid moveth or a$cend- eth out of the <I>L</I>iquid according to the Perpendicular that $hall be drawn thorow Z unto the Surface of the Liquid; and that the part that is above the Liquid de$cendeth into the Liquid according to that drawn thorow G: therefore the Portion will not continue $o inclined as was $uppo$ed: But neither $hall it return to Rectitude or Per- pendicularity; For that of the Perpendiculars drawn thorow Z and G, that pa$$ing thorow Z doth fall on tho$e parts which are to- wards L; and that that pa$$eth thorow G on tho$e towards A: Wherefore it followeth that the Centre Z do move upwards, and G downwards: Therefore the parts of the whole Solid which are towards A $hall move downwards, and tho$e towards L up- wards. Again let the Propo$ition run in other termes; and let the Axis of the Portion make an Angle with the Surface of the <marg>O</marg> Liquid le$$e than that which is at B. Therefore the Square P I hath le$$er Proportion unto the Square <fig> I Y, than the Square E <G>*y</G> hath to the Square <G>*y</G> B: Wherefore K R hath le$$er proportion to I Y, than the half of K R hath to <G>*y</G> B: And, for the $ame rea$on, I Y is greater than dou- ble of <G>*y</G> B: but it is double of O I: Therefore O I $hall be greater than <G>*y</G> B: But the Totall O <G>w</G> is equall to R B, and the Remainder <G>w</G> I le$$e than <G>y</G> R: Wherefore P H $hall al$o be le$$e than F. And, in regard that M P is equall to F Q, it is manife$t that P M is greater than $e$qui- alter of P H; and that P H is le$$e than double of <I>H</I> M. <I>L</I>et P Z be double to Z M. The Centre of Gravity of the whole Solid $hall again be T; that of the part which is within the Liquid Z; and drawing a Line from Z to T, the Centre of Gravity of that which is above the Liquid $hall be found in that Line portracted, that is in G: Therefore, Perpendiculars being drawn thorow Z and G <marg>P</marg> unto the Surface of the Liquid that are parallel to T H, it followeth that the $aid Portion $hall not $tay, but $hall turn about till that its Axis do make an Angle with the Waters Surface greater than that which it now maketh. And becau$e that when before we <foot>did</foot> <p n=>373</p> did $uppo$e that it made an Angle greater than the Angle B, the Poriton did not re$t then neither; It is manife$t that it $hall $tay <marg>Q</marg> or re$t when it $hall make an Angle eqnall to B. For $o $hall I O be equall to <G>*y</G> <I>B</I>; and <G>w</G> I equall to <fig> <G>*y</G> R; and P H equall to F: There- fore <I>M P</I> $hall be $e$quialter of <I>P H,</I> and <I>P H</I> double of H M: And there- fore $ince H is the Centre of Gravity of that part of it which is within the Liquid, it $hall move upwards along the $ame <I>P</I>erpendicular according to which the whole <I>P</I>ortion moveth; and along the $ame al$o $hall the part which is above move downwards: The <I>P</I>ortion therefore $hall re$t; for- a$much as the parts are not repul$ed by each other.</P> <head>COMMANDINE.</head> <P>And let <I>C B</I> be $e$quialter of <I>B R</I>: C D $hall al$o be $e$quialter <marg>A</marg> of K R.] <I>In the Tran$lation it is read thus:</I> Sit autem & CB quidem hemeolia ip$ius B R: C D autem ip$ius K R. <I>But we at the reading of this pa$$age have thought fit thus to correctit; for it is not $uppo$ed $o to be, but from the things $uppo$ed is proved to be $o. For if B</I> <G>y</G> <I>be double of</I> <G>y</G> <I>D, D B $hall be $e$quialter of B</I> <G>y.</G> <I>And becau$e E B is $e$quialter of B R, it followeth that the</I> (a) <I>Remainder C D is $e$quialter of</I> <G>y</G> <I>R; that is, of</I> <marg>(a) <I>By 19. of the fifth.</I></marg> <I>the Semi-parameter: Wherefore B C $hall be the Exce$$e by which the Axis is greater than $e$quialter of the Semi-parameter.</I></P> <P>And therefore F Q is le$$e than <I>B C.] For in regard that the Portion hath</I> <marg>B</marg> <I>the $ame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B; and hath le$$er proportion than the Square made of the Exce$$e by which the Axis is greater than Se$quialter of the Semi parameter, hath to the Square made of the Axis; that is, leßer than the Square C B hath to the Square B D; for the Line B D was $uppo$ed to be equall unto the Axis: Therefore the Square F Q $hall have to the Square D B le$$er proporti- on than the Sqnare C B to the $ame Square B D: And therefore the Square</I> (b) <I>F Q $hall be</I> <marg>(b) <I>By 8 of the fifth.</I></marg> <I>leße than the Square C B: And, for that rea$on, the Line F Q $hall be leße than B C.</I></P> <P>And, for the $ame rea$on, F is le$$e than <I>B R.] For C B being $e$qui-</I> <marg>C</marg> <I>alter of B R, and F Q $e$quialter of F</I>: (c) F <I>Q $hall be likewi$e le$$e than B C; and F</I> <marg>(c) <I>By 14 of the fifth.</I></marg> <I>leße than B R.</I></P> <P>Now becau$e it hath been $uppo$ed that the Axis of the <I>P</I>ortion <marg>D</marg> doth make an Angle with the Surface of the Liquid greater than the Angle <I>B,</I> the Angle <I>P Y I</I> $hall be greater than the Angle <I>B.] For the Line P Y being parallel to the Surface of the Liquid, that is, to XS</I>; (d) <I>the Angle</I> <marg>(d) <I>By 29 of the fir$t.</I></marg> <I>P Y I $hall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the Line X S: And therefore $hall be greater than the Angle B.</I></P> <P>Therefore the Square <I>P I</I> hath greater proportion to the Square <marg>E</marg> Y I, than the Square E <G>*y</G> hath to the Square <G>*y</G> <I>B] Let the Triangles P I Y and E</I> <G>y</G> <I>B, be de$cribed apart: And $eeing that the Angle P Y I is greater than the Angle E B</I> <G>y,</G> <I>unto the Line I Y, and at the Point Y a$$igned in</I> <fig> <I>the $ame, make the Angle V Y I equall to the Angle E B</I> <G>y</G>; <I>But the Right Angle at I, is equall unto the Right Angle at</I> <G>y;</G> <I>therefore the</I> <foot><I>Remaining</I></foot> <p n=>374</p> <I>Remaining Angle Y V I is equall to the Remaining Angle B E</I> <G>y.</G> <I>And therefore the</I> <marg>(e) <I>By 4. of the $ixth.</I></marg> (e) <I>Line V I hath to the Line I Y the $ame proportion that the Line E</I> <G>y</G> <I>hath to</I> <G>y</G> <I>B: But the</I> (f) <I>Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-</I> <marg>(f) <I>By 8. of the fifth.</I></marg> <I>to the $ame: Therefore</I> (g) <I>T I $hall have greater proportion unto I Y, than E</I> <G>y</G> <I>hath to</I> <G>y</G> <I>B: And, by the $ame rea$on, the Square T I $hall have greater proportion to the Square I Y, than</I> <marg>(g) <I>By 13 of the fifth.</I></marg> <I>the Square E</I> <G>y</G> <I>hath to the Square</I> <G>y</G> <I>B.</I></P> <marg>F</marg> <P>But as the Square P I is to the Square Y I, $o is the Line K R unto the Line I Y] <I>For by 11. of the fir$t of our</I> Conicks, <I>the Square P I is equall to the Rectangle contained under the Line I O, and under the Parameter; which we $uppo$ed to be eqnall to the Semi-parameter; that is, the double of K R</I>: <marg>(h) <I>By 26. of the $ixth.</I></marg> <I>But I Y is double of I O, by 33 of the $ame: And, therefore, the</I> (h) <I>Rectangle made of K R and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;</I> <marg>(i) <I>By</I> Lem. 22 <I>of the tenth.</I></marg> <I>that is, to the Square P I: But as the</I> (i) <I>Rectangle compounded of K R and I Y is to the Square I Y, $o is the Line K R unto the Line I Y: Therefore the Line K R $hall have unto I Y, the $ame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I hath to the Square I Y.</I></P> <marg>G</marg> <P>And as the Square E <G>*y</G> is to the Square <G>*y</G> B, $o is half of the Line K R unto the Line <G>y</G> B.] <I>For the Square E</I> <G>y</G> <I>having been $uppo$ed equall to half the Rectangle contained under the Line K R and</I> <G>y</G> <I>B; that is, to that contained under the half of K R and the Line</I> <G>y</G> <I>B; and $eeing that as the</I> (k) <I>Rectangle made of half K R</I> <marg>(k) <I>By</I> Lem. 22 <I>of the tenth.</I></marg> <I>and of B</I> <G>y</G> <I>is to the Square</I> <G>y</G> <I>B, $o is half K R unto the Line</I> <G>y</G> <I>B; the half of K R $hall have the $ame proportion to</I> <G>y</G> <I>B, as the Square E</I> <G>y</G> <I>hath to the Square</I> <G>y</G> <I>B.</I></P> <marg>H</marg> <P>And, con$equently, I Y is le$$e than the double of <G>y</G> B.] <I>For, as half K R is to</I> <G>y</G> <I>B, $o is K R to another Line: it $hall be</I> (1) <I>greater than I Y; that</I> <marg>(l) <I>By 10 of the fifth.</I></marg> <I>is, than that to which K R hath le$$er proportion; and it $hall be double of</I> <G>y</G> <I>B: Therefore I Y is le$$e than the double of</I> <G>y</G> <I>B.</I></P> <marg>K</marg> <P>And I <G>w</G> greater than <G>y</G> R.] <I>For O having been $uppo$ed equall to B R, if from B R,</I> <G>y</G> <I>B be taken, and from O</I> <G>w,</G> <I>O I, which is le$$er than B, be taken; the Remainder I</I> <G>w</G> <I>$hall be greater than the Remainder</I> <G>*y</G> <I>R.</I></P> <marg>L</marg> <P>And, therefore, F Q is equall to P M.] <I>By the fourteenth of the fifth of</I> Euclids <I>Elements: For the Line O N is equall to B D.</I></P> <marg>M</marg> <P>But it hath been demon$trated that P H is greater than F.] <I>For it was demon$trated that I</I> <G>w</G> <I>is greater than F: And P H is equall to I</I> <G>w.</G></P> <marg>N</marg> <P>In the $ame manner we might demon$trate the Line T H to be Perpendicular unto the Surface of the Liquid.] <I>For T</I> <G>a</G> <I>is equall to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated, the Line T H $hall be drawn Perpendicular unto the Liquids Surface.</I></P> <marg>O</marg> <P>Therefore, the Square P I hath le$$er proportion unto the Square I Y, than the Square E <G><*>y</G> hath to the Square <G>y</G> B.] <I>The$e, and other particulars of the like nature, that follow both in this and the following Propo$itions, $hall be demon$trated by us no otherwi$e than we have done above.</I></P> <marg>P</marg> <P>Therefore Perpendiculars being drawn thorow Z and G, unto the Surface of the Liquid, that are parallel to T H, it followeth that the $aid Portion $hall not $tay, but $hall turn about till that its Axis do make an Angle with the Waters Surface greater than that which it now maketh.] <I>For in that the Line drawn thorow G, doth fall perpendicu- larly towards tho$e parts which are next to L; but that thorow Z, towards tho$e next to A; It is nece$$ary that the Centre G do move downwards, and Z upwards: and, therefore, the parts of the Solid next to L $hall move downwards, and tho$e towards A upwards, that the Axis may makea greater Angle with the Surface of the Liquid.</I></P> <marg>Q</marg> <P>For $o $hall I O be equall to <G>y</G> B; and <G>w</G> I equall to I R; and P H equall to F.] <I>This plainly appeareth in the third Figure, which is added by us.</I></P> <foot>PROP.</foot> <p n=>375</p> <head>PROP. IX. THE OR. IX.</head> <P><I>The Right Portion of a Rightangled Conoid, when it $hall have its Axis greater than Se$quialter of the Semi-parameter, but le$$er than to be unto the $aid Semi-parameter in proportion as fifteen to four, and hath greater proportion in Gravity to the Liquid, than the exce$s by which the Square made of the Axis is greater than the Square made of the Exce$s, by which the Axis is greater than Se$quialter of the Semi- parameter, hath to the Square made of the Axis, being demitted into the Liquid, $o as that its Ba$e be wholly within the Liquid, and being $et inclining<*> it $hall neither turn about, $o as that its Axis $tand according to the Perpendicular, nor remain inclined, $ave only when the Axis makes an Angle with the Surface of the Liquid, equall to that aßigned as before.</I></P> <P>Let there be a Portion as was $aid; and $uppo$e D B equall to the Axis of the <I>P</I>ortion: and let B K be double to K D; and K R equall to the Semi-parameter: and C B Se$quialter of B R. And as the Portion is to the Liquid in Gravity, $o let the Ex- ce$$e by which the Square B D exceeds the Square F Q be to the Square B D: and let F be double to Q: It is manife$t, therefore, that the Exce$$e by which the <fig> Square B D is greater than the Square B C hath le$ser proportion to the Square B D, than the Exce$s by which the Square B D is greater than the Square F Q hath to the Square B D; for B C is the Exce$s by which the Axis of the Portion is greater than Se$quialter of the Semi-parameter: And, therefore, <marg>A</marg> the Square B D doth more exceed the Square F Q, than doth the Square B C: And, con$equently, the Line F Q is le$s than B C; <foot>and</foot> <p n=>376</p> and F le$s than B R. Let R <G>*y</G> be equall to F; and draw <G>*y</G> E perpendicular to B D; which let be in power the half of that which the Lines K R and <G>*y</G> B containeth; and draw a Line from B to E: I $ay that the Portion demitted into the Liquid, $o as that its Ba$e be wholly within the Liquid, $hall $o $tand, as that its Axis do make an Angle with the Liquids Surface, equall to the Angle B. For let the <I>P</I>ortion be demitted into the Liquid, as hath been $aid; and let the Axis not make an Angle with the Liquids Surface, equall to B, but fir$t a greater: and the $ame being cut thorow the Axis by a Plane erect unto the Surface of the Liquid, let the Section of the Portion be A P O L, the Section of a Rightangled Cone; the Section of the Surface of the Liquid <G>*g</G> I; and the Axis of the Portion and Diameter of the Section N O; which let be cut in the Points <G>w</G> and T, as before: and draw Y P, parallelto <G>*g</G> I, and touching the Section in P, and MP parallel to N O, and P S perpen- dicular to the Axis. And becau$e now that the Axis of the Portion maketh an <I>A</I>ngle with the Liquids Surface greater than the Angle B, the Angle S Y P $hall al$o be greater than the Angle B: And, therefore, the Square P S hath greater proportion to the Square <marg>B</marg> S Y, than the Square <G>*y</G> E hath to the Square <G>*y</G> B: And, for that cau$e, K R hath greater proportion to S Y, than the half of K R hath to <G>*y</G> B: Therefore, S Y is le$s than the double of <G>*y</G> B; and <marg>C</marg> S O le$s than <G>y</G> B: <I>A</I>nd, therefore, S <G>w</G> is greater than R <G>y</G>; and <marg>D</marg> P H greater than F. <I>A</I>nd, becau$e that the <I>P</I>ortion hath the $ame proportion in Gravity unto the Liquid, that the Exce$s by which the Square B D, is greater than the Square F Q, hath unto the Square B D; and that as the Portion is in proportion to the Liquid in Gravity, $o is the part thereof $ubmerged unto the whole Portion; It followeth that the part $ubmerged, hath the $ame proportion to the whole <I>P</I>ortion, that the Exce$s by which the Square B D is greater than the Square F Q hath unto the Square B D: <I>A</I>nd, therefore, the whole <I>P</I>ortion $hall have the $ame propor- <marg>E</marg> tion to that part which is above the <fig> Liquid, that the Square B D hath to the Square F Q: But as the whole Portion is to that part which is above the Liquid, $o is the Square N O unto the Square P M: Therefore, P M $hall be equall to F Q: But it hath been demon$trated, that P H is greater than F. And, therefore, MH $hall be le$s than Q; and P H greater than double of H M. Let therefore, P Z be double to Z M: <foot>and</foot> <p n=>377</p> and drawing a Line from Z to T pro- <fig> long it unto G. The Centre of Gravity of the whole Portion $hall be T; of that part which is above the Liquid Z; and of the Remain- der which is within the Liquid, the Centre $hall be in the Line Z T pro- longed; let it be in G: It $hall be demon$trated, as before, that T H is perpendicular to the Surface of the Liquid, and that the Lines drawn thorow Z and G parallel to the $aid T H, are al$o perpen- diculars unto the $ame: Therefore, the Part which is above the Liquid $hall move downwards, along that which pa$seth thorow Z; and that which is within it, $hall move upwards, along that which pa$seth thorow G: And, therefore, the Portion $hall not remain $o inclined, nor $hall $o turn about, as that its Axis be perpendicular <marg>F</marg> unto the Surface of the Liquid; for the parts towards L $hall move downwards, and tho$e towards <I>A</I> upwards; as may appear by the things already demon$trated. And, if the Axis $hould make an Angle with the Surface of the Liquid, le$s than the Angle B; it $hall in like manner be demon$trated, that the Portion will not <marg>G</marg> re$t, but incline untill that its Axis do make an Angle with the Surface of the Liquid, equall to the Angle B.</P> <head>COMMANDINE.</head> <P>And, therefore, the Square B D doth more exceed the Square <marg>A</marg> F Q, than doth the Square B C: And, con$equently, the Line F Q, is le$s than B C; and F le$s than B R.] <I>Becau$e the Exce$s by which the Square B D exceedeth the Square B C; having le$s proportion unto the Square B D, than the Exce$s by which the Square B D exceedeth the Square F Q, hath to the $aid Square</I>; (a) <I>the Exce$s by which the Square B D exceedeth the Square B C $hall be le$s than the Exce$s</I> <marg>(a) <I>By 8. of the fifth.</I></marg> <I>by which it exceedeth the Square F Q: Therefore, the Square F Q is le$s than the Square B C: and, con$quently, the Line F Q le$s than the Line BC: But F Q hath the $ameproportion to F, that B C hath to B R; for the Antecedents are each Se$quialter of their con$equents: And</I> (b) <I>F Q being le$s than B C, F $hall al$o be le$s than B R.</I></P> <marg>(b) <I>By 14. of the fifth.</I></marg> <P>And, for that cau$e, K R hath greater proportion to S Y, than the half of K R hath to <G>y</G> B.] <I>For K R is to S Y, as the Square P S is to the Square</I> <marg>B</marg> <I>S Y: and the half of the Line K R is to the Line</I> <G>y</G> <I>B, as the Square E</I> <G>y</G> <I>is to the Square</I> <G>y</G> <I>B.</I></P> <P>And S O le$s than <G>y</G> B.] <I>For S Y is double of S O.</I></P> <marg>C</marg> <P>And P H greater than F.] <I>For P H is equall to S</I> <G>w,</G> <I>and R</I> <G>y</G> <I>equall to F.</I></P> <marg>D</marg> <P>And, therefore, the whole Portion $hall have the $ame propor- <marg>E</marg> tion to that part which is above the Liquid, that the Square B D hath to the Square F Q] <I>Becau$e that the part $ubmerged, being to the whole Portion as the Exce$s by which the Square B D is greater than the Square F Q, is to the Square B D; the whole Portion, Converting, $hall be to the part thereof $ubmerged, as the Square B D is to</I> <foot><I>Ccc the</I></foot> <p n=>378</p> <I>the Exce$s by which it exceedeth the Square F Q: And, therefore, by Conver$ion of Proportion, the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square, F Q; for the Square B D is $o much greater than the Exce$s by which it exceedeth the Squar, F Q as is the $aid Square F Q.</I></P> <marg>F</marg> <P>For the parts towards L $hall move downwards, and tho$e to- wards A upwards.] <I>We thus carrect the$e words, for in</I> Tartaglia's <I>Tran$lation it is fal$ly, as I conceive, read</I> Quoniam quæ ex parte L ad $uperiora ferentur, <I>becau$e the Line thàt pa$$eth thorow Z falls perpendicularly on the parts towards L, and that thor<*> G falleth perpendicularly on the parts towards A: Whereupon the Centre Z, together with tho$e parts which are towards L $hall move downwards; and the Centre G, together with the parts which are towards A upwards.</I></P> <marg>G</marg> <P>It $hall in like manner be demon$trated that the Portion $hall not re$t, but incline untill that its Axis do make an Angle with the Surface of the Liquid, equall to the Angle B.] <I>This may be ea$ily demon- $tratred, as nell from what hath been $aid in the precedent Propo$ition, as al$o from the two latter Figures, by us in$erted</I></P> <head>PROP. X. THEOR. X.</head> <P><I>The Right Portion of a Rightangled Conoid, lighter than the Liquid, when it $hall have its Axis greater than to be unto the Semiparameter, in proportion as fifteen to four, being demitted into the Liquid, $o as</I> <marg>A</marg> <I>that its Ba$e touch not the $ame, it $hall $ometimes</I> <marg>B</marg> <I>$tand perpendicular; $ometimes inclined; and $ome- times $o inclined, as that its Ba$e touch the Surface of the Liquid in one Point only, and that in two Po-</I> <marg>C</marg> <I>$itions; $ometimes $o that its Ba$e be more $ubmer-</I> <marg>D</marg> <I>ged in the Liquid; and $ometimes $o as that it doth not in the lea$t touch the Surface of the Liquid;</I> <marg>E</marg> <I>according to the proportion that it hath to the Liquid in Gravity. Every one of which Ca$es $hall be anon demon$trated.</I></P> <P>Let there be a Portion, as hath been $aid; and it being cut thorow its Axis, by a Plane erect unto the Superficies of the Liquid, let the Section be A P O L, the Section of a Right angled Cone; and the Axis of the Portion and Diameter of the Section B D: and let B D be cut in the Point K, $o as that B K be double of K D; and in C, $o as that B D may have the $ame <marg>F</marg> proportion to K C, as fifteen to four: It is manife$t, therefore, <marg>G</marg> that K C is greater than the Semi-parameter: Let the Semi- <foot>parameter</foot> <p n=>379</p> parameter be equall to K R: and <fig> <marg>H</marg> let D S be Se$quialter of K R: but S B is al$o Se$quialter of B R: Therefore, draw a Line from A to B; and thorow C draw C E Per- pendicular to B D, cutting the Line A B in the Point E; and thorow E draw E Z parallel unto B D. Again, A B being divided into two equall parts in T, draw T H parallel to the $ame B D: and let Sections of Rightangled Cones be de$cribed, A E I about the Diameter E Z; and A T D about the Diameter T H; and let them be like to the <marg>K</marg> Portion A B L: Now the Section of the Cone A E I, $hall pa$s <marg>L</marg> thorow K; and the Line drawn from R perpendicular unto B D, $hall cut the $aid A E I; let it cut it in the Points Y G: and thorow Y and G draw P Y Q and O G N parallels unto B D, and cutting A T D in the Points F and X: la$tly, draw P <G>*f</G> and O X touching the Section A P O L in the <I>P</I>oints P and O. In regard, <marg>M</marg> therefore, that the three <I>P</I>ortions A P O L, A E I, and A T D are contained betwixt Right Lines, and the Sections of Rightangled Cones, and are right alike and unequall, touching one another, upon one and the $ame Ba$e; and N X G O being drawn from the <I>P</I>oint N upwards, and Q F Y P from Q: O G $hall have to G X a proportion compounded of the proportion, that I L hath to L A, and of the proportion that A D hath to DI: But I L is to L A, as two to five: And C B is to B D, as $ix to fifteen; that is, as two <marg>N</marg> to five: And as C B is to B D, $o is <I>E B to B A</I>; and D Z to <marg>O</marg> D A: And of D Z and D A, L I and L A are double: and A D <marg>P</marg> is to D I, as five to one: <I>B</I>ut the proportion compounded of the proportion of two to five, and of the proportion of five to one, is <marg>Q</marg> the $ame with that of two to one: and two is to one, in double proportion: Therefore, O G is double of GX: and, in the $ame manner is P Y proved to be double of Y F: Therefore, $ince that D S is Se$quialter of K R; <I>B S</I> $hall be the Exce$s by which the Axis is greater than Se$quialter of the Semi-parameter. If there- fore, the <I>P</I>ortion have the $ame proportion in Gravity unto the Liquid, as the Square made of the Line <I>B S,</I> hath to the Square made of <I>B D,</I> or greater, being demitted into the Liquid, $o as hat its <I>B</I>a$e touch not the Liquid, it $hall $tand erect, or perpendicular: For it hath been demon$trated above, that the <I>P</I>ortion who$e <marg>R</marg> Axis is greater than Se$quialter of the Semi-parameter, if it have not le$ser proportion in Gravity unto the Liquid, than the Square <foot>Ccc <*> made</foot> <p n=>380</p> made of the Exce$s by which the Axis is greater than Se$quialter of the Semi-parameter, hath to the Square made of the Axis, being demitted into the Liquid, $o as hath been $aid, it $hall $tand erect, or <I>P</I>erpendicular.</P> <head>COMMANDINE.</head> <P><I>The particulars contained in this Tenth Propo$ition, are divided by</I> Archimedes <I>into five Parts and Conclu$ions, each of which he proveth by a di$tinct Demon$tration.</I></P> <marg>A</marg> <P>It $hall $ometimes $tand perpendicular.] <I>This is the fir$t Conclu$ion, the Demonstration of which he hath $ubjoyned to the Propo$ition.</I></P> <marg>B</marg> <P>And $ometimes $o inclined, as that its Ba$e touch the Surface of the Liquid, in one Point only.] <I>This is demon$trated in the third Con- clu$ion.</I></P> <P>Sometimes, $o that its Ba$e be mo$t $ubmerged in the Liquid.] <marg>C</marg> <I>This pertaineth unto the fourth Conclu$ion.</I></P> <P><I>A</I>nd, $ometimes, $o as that it doth not in the lea$t touch the Sur- <marg>D</marg> face of the Liquid.] <I>This it doth hold true two wayes, one of which is explained is the $econd, and the other in the fifth Conclu$ion.</I></P> <P>According to the proportion, that it hath to the Liquid in Gra- <marg>E</marg> vity. Every one of which Ca$es $hall be anon demon$trated.] <I>In</I> Tartaglia's <I>Ver$ion it is rendered, to the confu$ion of the $ence,</I> Quam autem pro- portionem habeant ad humidum in Gravitate fingula horum demon$trabuntur.</P> <P>It is manife$t, therefore, that K C is greater than the Semi- <marg>F</marg> parameter] <I>For, $ince B D hath to K C the $ame proportion, as fifteen to four, and hath unto the Semi-parameter greater proportion; (a) the Semi-parameter $hall be le$s</I> <marg>(a) <I>By 10. of the fifth.</I></marg> <I>than K C.</I></P> <P>Let the Semi-parameter be equall to KR.] <I>We have added the$e words,</I> <marg>G</marg> <I>which are not to be found in</I> Tartaglia.</P> <P>But S B is al$o Se$quialter of BR.] <I>For, D B is $uppo$ed Se$quialter of</I> <marg>H</marg> <I>B K; and D S al$o is Se$quialter of K R: Wherefore as</I> (b) <I>the whole D B, is to the whole B K, $o is the part D S to the part K R. Therefore, the Remainder S B, is al$o to the</I> <marg>(b) <I>By 19 of the fifth.</I></marg> <I>Remainder B R, as D B is to B K.</I></P> <P><I>A</I>nd let them be like to the <I>P</I>ortion <I>A B L.</I>] Apollonius <I>thus defineth</I> <marg>K</marg> <I>like Portions of the Sections of a Cone, in</I> Lib. 6. Conicornm, <I>as</I> Eutocius <I>writeth</I> ^{*}; <marg>* <I>Upon prop. 3 lib.</I> 2 Archim. <I>Æqui- pond.</I></marg> <G>o)/n oi(_s a)x deisw_n o)/n e(xa/sw| warallh/lwn th_ <35>a\sei, i(/swn to\ plh_o<34>, ai( para/llhlos, kai\ a(i <35>a/seis wro\s ta/s apotrm<*> nome/nas a)po\ <*> diame/tswn tw_s korufai_s e)n toi_s a)ntoi_s lo/gois ei)si, kai\ ai( a)potemno/menai wro\s ta\s a) temnomi/nas<*></G> <I>that is,</I> In both of which an equall number of Lines being drawn parallel to the Ba$e; the parallel and the Ba$es have to the parts of the Diameters, cut off from the Vertex, the $ameproportion: as al$o, the parts cut off, to the parts cut off. <I>Now the Lines parallel to the Ba$es are drawn, as I $uppo$e, by making a Rectilineall Figure (cal-</I> <marg><I>Vide</I> Archim, <I>ante prop. 2. lib. 2. Æquipond.</I></marg> <I>led)</I> Signally in$cribed [<G>xh_ma giwri/mws e)gn\<36>ro/menon</G>] <I>in both portions, having an equall num- ber of Sides in both. Therefore, like Portions are cut off from like Sections of a Cone; and their Diameters, whether they be perpendicular to their Ba$es, or making equall Angles with their Ba$es, have the $ame proportion unto their Ba$es.</I></P> <marg>L</marg> <P>Now the Section of the Cone <I>A E I</I> $hall pa$s thorow K.] <I>For, if it be po$$ible, let it not pa$s thorow K, but thorow $ome other Point of the Line D B, as thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, who$e Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth both A E and A I; A E in B, and A I in D; D B $hall have to B V, the $ame proportion</I> <foot><I>that</I></foot> <p n=>381</p> <I>that A Z hath to Z D; by the fourth Propo$ition of</I> Archimedes, De quadratura Para- bolæ: <I>But A Z is Se$quialter of Z D; for it is as three to two, as we $hallanon demon-</I> <marg>(c) <I>By 9 of the fifth,</I></marg> <I>$trate: Therefore D B is Se$quialter of B V; but D B and B K are Se$quialter: And, therefore, the Lines</I> (c) <I>B V and B K are equall: Which is impos$ible: Therefore the Section of the Right-angled Cone A E I, $hall pa$s thorow the Point K; which we would demonstrate.</I></P> <P>In regard, therefore, that the three <I>P</I>ortions A P O L, A E I <marg>M</marg> and A T D are contained betwixt Right Lines and the Sections of Right-angled Cones, and are Right, alike and unequall, touching one another, upon one and the $ame Ba$e.] <I>After the$e words,</I> upon one and the $ame Ba$e, <I>we may $ee that $omething is obliterated, that is to be de$ired: and for the Demon$tration of the$e particulars, it is requi$ite in this place to premi$e $ome things: which will al$o be nece$$ary unto the things that follow.</I></P> <head>LEMMA. I.</head> <P>Let there be a Right <I>L</I>ine A B; and let it be cut by two <I>L</I>ines, parallel to one another, A C and D E, $o, that as <I>A B</I> is to B D. $o <I>A C</I> may be to D E. I $ay that the Line that con- joyneth the Points C and B $hall likewi$e pa$s by E.</P> <fig> <P><I>For, if po$$ible, let it not pa$s by E, but either above or below it. Let it first pa$s below it, as by F. The Triangles A B C and D B F $hall be alike: And, therefore, as</I> (a) <I>A B is to B D,</I> <marg>(a) <I>By 4. of the $ixth.</I></marg> <I>$o is A C to D F: But as A B is to B D, $o was A C to D E: Therefore</I> (b) <I>D F $hall be equall to</I> <marg>(b) <I>By 9. of the fifth.</I></marg> <I>D E: that is, the part to the whole: Which is ab$urd. The $ame ab$urditie will follow, if the Line C B be $uppo$ed to pa$s above the Point E: And, therefore, C B mu$t of neces$ity pa$s thorow E: Which was required to be demon$trated.</I></P> <head>LEMMA. II.</head> <P>Let there be two like <I>P</I>ortions, contained betwixt Right Lines, and the Sections of Right-angled Cones; A B C the great- er, who$e Diameter let be B D; and E F C the le$ser, who$e Diameter let be F G: and, let them be $o applyed to one another, that the greater include the le$ser; and let their Ba$es A C and E C be in the $ame Right Line, that the $ame Point C, may be the term or bound of them both: And, then in the Section A B C, take any Point, as H; and draw a Line from H to C. I $ay, that the Line H C, hath to that part of it $elf, that lyeth betwixt C and the Section E F C, the $ame proportion that A C hath to C E.</P> <P><I>Draw B C, which $hall pa$s thorow F, For, in regard, that the Portions are alike, the Diameters with the Ba$es contain equall Angles: And, therefore, B D and F G are parallel to one another: and B D is to A C, as F G it to E C: and,</I> Permutando, <I>B D is to F G, as A C is to C E; that is,</I> (a) <I>as their halfes D C to C G; therefore, it followeth, by the</I> <marg>(a) <I>By 15. of the fifth.</I></marg> <I>preceding Lemma, that the Line B C $hall pa$s by the Point F. Moreover, from the Point H unto the Diameter B D, draw the Line H K, parallel to the Ba$e A C: and, draw a Line</I> <foot><I>from</I></foot> <p n=>382</p> <fig> <I>from K to C, cutting the Diameter F G in L: and, thorow L, unto the Section E F. G, on the part E, draw the Line L M, parallel unto the $ame Ba$e A C. And, of the Section A B C, let the Line B N be the Parameter; and, of the Section E F C, let F O be the Parameter. And, becau$e the Triangles C B D and C F G are alike</I>; (b) <I>therefore, as B C is to C F, $o $hall D C be</I> <marg>(b) <I>By 4. of the $ixth.</I></marg> <I>to C G, and B D to F G. Again, becau$e the Triangles C K B and C L F, are al$o alike to one another; therefore, as B C is to C F, that is, as B D is to F G, $o $hall K C be to C L, and B K to F L: Wherefore, K C to C L, and,</I> <marg>(c) <I>By 15. of the fifth.</I></marg> <I>B K to F L, are as D C to C G; that is,</I> (c) <I>as their duplicates A C and C E: But as B D is to F G, $o is D C to C G; that is, A D to E G: And,</I> Permutando, <I>as B D is to A D, $o is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11 of our fir$t of</I> Conicks: <I>Therefore, the</I> (d) <I>three Lines B D, A D and B N are</I> <marg>(d) <I>By 17. of the $ixth.</I></marg> <I>Proportionalls. By the $ame rea$on, likewi$e, the Square E G being equall to the Rectangle G F O, the three other Lines F G, E G and F O, $hall be al$o Proportionals: And, as B D is to A D, $o is F G to E G: And, therefore, as A D is to B N, $o is E G to F O:</I> Ex equali, <I>therefore, as D B is to B N, $o is G F to F O: And,</I> Permutando, <I>as D B is to G F, $o is B N to F O: But as D B is to G F, $o is B K to F L: Therefore, B K is to F L, as B N is to F O: And,</I> Permutando, <I>as B K is to B N, $o is F L to F O. Again, becau$e the</I> (e) <I>Square H K is equall to the Rectangle B N; and the Square M L, equall</I> <marg>(e) <I>By 11 of our fir$t of</I> Conicks.</marg> <I>to the Rectangle L F O, therefore, the three Lines B K, K H and B N $hall be Proportionals: and F L, L M, and F O $hall al$o be Proportionals: And, therefore,</I> (f) <I>as the Line</I> <marg>(f) <I>By</I> Cor. <I>of 20. of the $ixth.</I></marg> <I>B K is to the Line B N, $o $hall the Square B K, be to the Square H K: And, as the Line F L is to the Line F O, $o $hall the Square F L be to the Square L M: Therefore, becau$e that as B K is to B N, $o is F L to F O; as the Square</I> <marg>(g) <I>By 23. of the $ixth.</I></marg> <I>B K is to the Square K H, $o $hall the Square F L be to the Square L M: Therefore,</I> (g) <I>as the Line B K is to the Line K H, $o is the Line F L to L M: And,</I> Permutando, <I>as B K is to F L, $o is K H to L M: But B K was to F L, as K C to C L: Therefore, K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manife$t that the Line H C al$o $hall pa$s thorow the Point M: As K C, therefore, is to C L, that is, as A C to C E, $o is H C to C M; that is, to the $ame part of it $elf, that lyeth betwixt C and the Section E F C. And, in like manner might we demon$trate, that the $ame happeneth in other Lines, that are produced from the Point C, and the Sections E B C. And, that B C hath the $ame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G; that is, as their Duplicates A C to C E.</I></P> <P>From whence it is manife$t, that all Lines $o drawn, $hall be cut by the $aid Section in the $ame proportion. For, by Divi$ion and Conver$ion, C M is to M H, and C F to F B, as C E to E A.</P> <head>LEMMA. III.</head> <P>And, hence it may al$o be proved, that the Lines which are drawn in like Portions, $o, as that with the Ba$es, they con- tain equall Angles, $hall al$o cut off like Portions; that is, as in the foregoing Figure, the Portions H B C and M F C, which the Lines C H and C M do cut off, are al$o alike to each other.</P> <P><I>For let C H and C M be divided in the midst in the Points P and Q; and thorow tho$e Points draw the Lines R P S and T Q V parallel to the Diameters. Of the Portion H S C the Diameter $hall be P S, and of the Portion M V C the Diameter $hall be</I> <foot><I>QV.</I></foot> <p n=>383</p> <I>Q V. And, $uppo$e that as the Square C R is to the Square C P, $o is the Line B N unto another Line; which let be S X: And, as the Square C T is to the Square C Q $o let F O be to V Y. Now it is manife$t, by the things which we have demon$trated, in our Commentaries, upon the fourth Propo$ition of</I> Archimedes, De Conoidibus & Spheæroidibus, <I>that the Square C P is equall to the Rectangle P S X; and al$o, that the Square C Q is equall to the Rectangle Q V Y; that is, the Lines S X and V Y, are the Parameters of the Sections H S C and M V C: But $ince the Triangles C P R and C Q T are alike; C R $hall have to C P, the $ame Proportion that C T hath to C Q: And, therefore, the</I> (a) <I>Square C R $hall have</I> <marg>(a) <I>By 22. of the $ixth.</I></marg> <I>to the Square C P, the $ame proportion that the</I> <fig> <I>Square C T hath to the Square C Q: There- fore, al$o, the Line B N $hall be to the Line S X, as the Line F O is to V Y: But H C was to C M, as A C to C E: And, therefore, al$o, their halves C P and C Q, are al$o to one another, as A D and E G: And.</I> Permu- tando, <I>C P is to A D, as C Q is to E G: But it hath been proved, that A D is to B N, as E G to F O; and B N to S X, as F O to V Y: Therefore,</I> exæquali, <I>C P $hall be to S X, as C Q is to V Y. And, $ince the Square C P is equall to the Rectangle P S X, and the Square C Q to the Rectangle Q V Y, the three Lines S P, PC and S X $hall be proportionalls, and V Q, Q C and V Y $hal be Proportionalls al$o: And therefore al$o S P $hall be to P C as V Q to Q C And as P C is to C H, $o $hall Q C. be to C M: Therefore,</I> ex æquali, <I>as S P the Diameter of the Portion H S C is to its Ba$e C H, $o is V Q the Diameter of the portion M V S the Ba$e C M; and the Angles which the Diameter with the Ba$es do contain, are equall; and the Lines S P and V Q are parallel: Therefore the Portions, al$o, H S C and M V C $hall be alike: Which was propo$ed to be demon$trated</I></P> <head>LEMMA. IV.</head> <P><I>L</I>et there be two <I>L</I>ines A <I>B</I> and C D; and let them be cut in the Points E and F, $o that as A E is to E B, C F may be to F D: and let them be cut again in two other Points G and H; and let C H be to H D, as A G is to G B. I $ay that C F $hall be to F H as A E is E G.</P> <P><I>For in regard that as A E is to E B, $o is C F to F D; it followeth that, by Compounding, as A B is to E B, $o $hall C D be to F D. Again, $ince that as A G is to G B, $o is C H, to H D; it followeth that, by Compounding and Converting, as G B is to A B, $o $hall H D be</I> <fig> <I>C D: Therefore,</I> ex æquali, <I>and Converting as E B is to G B, $o $hall F D be to H D; And, by Conver- $ion of Propo$ition, as E B is to E G, $o $hall F D be to F H: But as A E is to E B, $o is C F to F D:</I> Ex æquali, <I>therefore, as A E is to E G, $o $hall CF be to F H.</I> Again, another way. <I>Let the Lines A B and C D be applyed to one another, $o as that they doe make an Angle at the parts A and C; and let A and C be in one and the $ame Point: then draw Lines from D to B, from H to G, and from F to E. And $ince that as A E is to E B, $o is C F, that is A F to F D; therefore F E $hall be parallel to D B</I>; (a) <I>and likewi$e</I> <marg>(a) <I>By 2. of the $ixth.</I></marg> <I>H G $hall be parallel to D B; for that A H is to H D, as A G to G B</I>: (b) <I>Therefore F E and H G are parallel to each other: And con$equently, as A E is to E G, $o is A H, that is,</I> <marg>(b) <I>By 30 of the fir$t.</I></marg> <I>C F to F H: Which was to be demon$trated.</I></P> <foot><I>L</I>EMMA</foot> <p n=>384</p> <head>LEMMA. V.</head> <P>Again, let there be two like Portions, contained betwixt Right Lines and the Sections of Right-angled Cones, as in the fore- going figure, A B C, who$e Diameter is B D; and E F C, who$e Diameter is F G; and from the Point E, draw the Line E H parallel to the Diameters B D and F G; and let it cut the Section A B C in K: and from the Point C draw C H touching the Section A B C in C, and meeting with the Line E H in H; which al$o toucheth the Section E F C in the $ame Point C, as $hall be demon$trated: I $ay that the Line drawn from C <I>H</I> unto the Section E F C $o as that it be parallel to the Line E H, $hall be divided in the $ame proportion by the Section A B C, in which the <I>L</I>ine C A is divided by the Section E F C; and the part of the <I>L</I>ine C A which is betwixt the two Sections, $hall an$wer in proportion to the part of the Line drawn, which al$o falleth betwixt the $ame Sections: that is, as in the foregoing Figure, if D B be produced untill it meet with C H in L, that it may inter$ect the Section E F C in the Point M, the <I>L</I>ine <I>L</I> B $hall have to B M the $ame proportion that C E hath to E A.</P> <P><I>For let G F be prolonged untill it meet the $ame Line C H in N, cutting the Section A B C in O; and drawing a Line from B to C, which $hall pa$$e by F, as hath been $hewn, the</I> <fig> <I>Triangles C G F and C D B $hall be alike; as al$o the Triangles C F N and C B L: Wherefore</I> (a) <I>as G F is to D B, $o $hall C F b to C B:</I> <marg>(a) <I>By 4. of the $ixth.</I></marg> <I>And as</I> (b) <I>C F is to C B, $o $hall F N be to B L: Therefore G F $hall be to D B, as F N</I> <marg>(b) <I>By 11 of the fifth,</I></marg> <I>to B L: And,</I> Permutando, <I>G F $hall be to F N, as D B to B L: But D B is equall to B L, by 35 of our Fir$t Book of</I> Conicks: <I>Therefore</I> (c) <I>G F al$o $hall be equall to F N:</I> <marg>(c) <I>By 14 of the fifth.</I></marg> <I>And by 33 of the $ame, the Line C H touch- eth the Section E F C in the $ame Point. There- fore, drawing a Line from C to M, prolong it untill it meet with the Section A B C in P; and from P unto A C draw P Q parallel to B D. Becau$e, now, that the Line C H toucheth the Section E F C in the Point C; L M $hall have the $ame proportion to M D that C D hath to D E, by the Fifth Propo$ition of</I> Archimedes <I>in his Book</I> De Quadratura Patabolæ: <I>And by rea$on of the Similitude of the Triangles C M D and C P Q, as C M is to C D, $o $hall C P be to C Q: And,</I> Permutando, <I>as C M is to C P, $o $hall C D be to C Q: But as C M is to C P, $o is C E to C A,; as we have but even now demon$trated: And therefore, as C E is to C A, $o is C D to C Q; that is as the whole is to the whole, $o is the part to the part: The remainder, therefore, D E is to the Remainder Q A, as C E is to C A; that is, as C D is to C Q: And,</I> Permutando, <I>C D is to D E, as C Q is to Q A: And L M is al$o to M D, as C D to D E: Therefore L M is</I> <foot><I>to</I></foot> <p n=>385</p> <I>to M D, as C Q to Q A: But L B is to B D, by 5 of</I> Archimedes, <I>before recited, as C D to D A: It is manife$t therefore, by the precedent Lemma, that C D is to D Q, as L B is to B M: But as C D is to D Q, $o is C M to M P: Therefore L B is to B M, as C M to M P:</I> <marg><I>By 2. of the $ixth</I></marg> <I>And it haveing been demon$trated, that C M is to M P, as C E to E A; L B $hall be to B M<*> as C E to E A. And in like manner it $hall be demonstrated that $o is N O to O F; as al$o the Remainders. And that al$o H K is to K E, as C E to E A, doth plainly appeare by the $ame</I> 5. <I>of</I> Archimedes<I>: Which is that that we propounded to be demon$trated.</I></P> <head>LEMMA. VI.</head> <P>And, therefore, let the things $tand as above; and de$cribe yet another like Portion, contained betwixt a Right Line, and the Section of the Rightangled Cone D R C, who$e Diameter is R S, that it may cut the Line F G in T; and prolong S R unto the <I>L</I>ine C H in V, which meeteth the Section A B C in X, and E F C in Y. I $ay, that B M hath to M D, a propor- tion compounded of the proportion that E A hath to A C; and of that which C D hath to D E.</P> <P><I>For, we $hall fir$t demon$trate, that the Line C H toucheth the Section D R C in the Point C; and that L M is to M D, as al$o N F to F T, and V Y to Y R, as C D is to E D. And, becau$e now that L B is to B M, as C E is to E A; therefore, Compounding and Conver- ting, B M $hall be to L M, as E A to A C: And, as L M is to M D, $o $hall C D be to D E: The proportion, therefore, of B M to M D, is compounded of the proportion that B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion of B M to M D, $hall al$o be compounded of the proportion that E A hath to A C, and of that which C D hath to D E. In the $ame manner it $hal be demon$trated, that O F hath to F T, and al$o X Y to Y R, a proportion compounded of tho$e $ame proportions; and $o in the re$t: Which was to be demonstrated.</I></P> <P>By which it appeareth that the <I>L</I>ines $o drawn; which fall betwixt the Sections A B C and D R C, $hall be divided by the Section E F C in the $ame Proportion.</P> <P>And C B is to B D, as $ix to fifteen.] <I>For we have $uppo$ed that B K is</I> <marg>N</marg> <I>double of K D: Wherefore, by Compo$ition B D $hall be to K D as three to one; that is, as fifteen to five: But B D was to K C as fifteen to four; Therefore B D is to D C as fifteen to nine: And, by Conver$ion of proportion and Convert- ing, C B is to B D, as $ix to $ifteen.</I></P> <fig> <P>And as C B is to B D, $o is <marg>O</marg> E B to B A; and D Z to D A.] <I>For the Triangles C B E and D B A being alike; As C B is to B E, $o $hall D B be to B A: And,</I> Permutando, <I>as C B is to B D, $o $hall E B be to B A: Againe, as B C is to C E $o $hall B D be to D A, And,</I> Permutando, <I>as C B is to B D, $o $hall C E, that is, D Z equall to it, be to D A.</I></P> <P>And of D Z and D A, L I and <marg>P</marg> L A are double.] <I>That the Line L A is double of D A, is manife$t, for that B D is the Diameter of the Portion. And that L I is dovble to D Z $hall be thus demon$trated. For as much as ZD is to D A, as two to five: therefore, Converting and Dividing, A Z, that is, I Z, $hall be to Z D, as three to two:</I> <foot><I>Dd Again,</I></foot> <p n=>386</p> <I>Again, by dividing, I D $hall be to D Z, as one to two: But Z D was to D A, that is, to D L, as two to five: Therefore,</I> ex equali, <I>and Converting, L D is to D I, as five to one: and, by Conver$ion of Proportion, D L is to D I, as five to four: But D Z was to D L, as two to five: Therefore, again,</I> ex equali, <I>D Z is to L I, as two to four: Therefort L I is double of D Z: Which was to be demon$trated.</I></P> <marg>Q</marg> <P>And, A D is to D I, as five to one.] <I>This we have but ju$t now demon- $trated.</I></P> <marg>R</marg> <P>For it hath been demon$trated, above, that the Portion who$e Axis is greater than Se$quialter of the Semi-parameter, if it have not le$$er proportion in Gravity to the Liquid, &c.] <I>He hath demonstra- ted this in the fourth Propo$ition of this Book.</I></P> <head>CONCLVSION II.</head> <P><I>If the Portion have le$$er proportion in Gravity to the</I> <marg>A</marg> <I>Liquid, than the Square S B hath to the Square B D, but greater than the Square X O hath to the Square B D, being demitted into the Liquid, $o in- clined, as that its Ba$e touch not the Liquid, it $hall continue inclined, $o, as that its Ba$e $hall not in the lea$t touch the Surface of the Liquid, and its Axis $hall make an Angle with the Liquids Surface, greater than the Angle X.</I></P> <P>Therfore repeating the fir$t figure, let the Portion have unto the Liquid in Gravitie a proportion greater than the Square X O hath to the $quare B D, but le$$er than the Square made of the Exce$$e by which the Axis is greater than Se$quialter of the Semi- <fig> Parameter, that is, of S B, hath to the Square B D: and as the Portion is to the Liquid in Gravity, $o let the Square made of the Line <G>y</G> be to the Square B D: <G>y</G> $hall be great- <marg>B</marg> er than X O, but le$$er than the Exce$$e by which the Axis is grea- ter than Se$quialter of the Semi- parameter, that is, than S B. Let a Right Line M N be applyed to fall between the Conick-Sections A M Q L and A <I>X</I> D, [<I>parallel to B D falling betwixt O X and B D,</I>] and equall to the Line <G>y</G>: and let it cut the remaining Conick Section A H I in the point H, and the <marg>C</marg> Right Line R G in V. It $hall be demon$trated that M H is double to H N, like as it was demon$trated that O G is double to G X. <foot>And,</foot> <p n=>387</p> <fig> And from the Point M draw M Y touching the Section A M Q L in M; and M C perpendicular to B D: and la$tly, having drawn A N & prolong- ed it to Q, the Lines A N & N Q $hall be equall to each other. For in regard that in the Like Portions <marg>D</marg> A M Q L and A <I>X</I> D the Lines A Q and A N are drawn from the Ba$es unto the Portions, which Lines contain equall Angles with the $aid Ba$es, Q A $hall have the $ame proportion to A M that L A hath to A D: Therefore A N is equall to N Q, and A Q parallel to M Y. <marg>E F</marg> It is to be demon$trated that the Portion being demitted into the Liquid, and $o inclined as that its Ba$e touch not the Liquid, it $hall continue inclined $o as that its Ba$e $hall not in the lea$t touch the Surface of the Liquid, and its Axis $hall make an Angle with the Liquids Surface greater than the Angle X. Let it be demitted into the Liquid, and let it $tand, $o, as that its Ba$e do touch the Surface of the Liquid in one Point only; and let the Portion be cut thorow the Axis by a Plane erect unto the Surface of the Liquid, <fig> and Let the Section of the Super- ficies of the Portion be A P O L, the Section of a Rightangled Cone, and let the Section of the Liquids Surface be A O; And let the Axis of the Portion and Diameter of the Section be <I>B</I> D: and let B D be <marg>G</marg> cut in the Points K and R as hath been $aid; al$o draw P G Parallel to A O and touching the Section A P O L in P; and from that Point draw P T Parallel to B D, and P S perpendicular to the $ame B D. Now, fora$much as the Portion is unto the Liquid in Gravity, as the Square made of the Line <G>y</G> is to the Square B D; and $ince that as the portion is unto the Liquid in Gravitie, $o is the part thereof $ubmerged unto the whole Portion; and that as the part $ubmerged is to the whole, $o is the Square T P to the Square B D; It follow- eth that the Line <G>y</G> $hall be equall to T P: And therefore the Lines M N and P T, as al$o the Portions A M Q and A P O $hall like- wi$e be equall to each other. And $eeing that in the Equall and Like Portions A P O L and A M Q L the Lines A O and A Q <marg>H</marg> are drawn from the extremites of their Ba$es, $o, as that the Portions cut off do make Equall Angles with their Diameters; as al$o the <foot>Ddd 2 Angle</foot> <p n=>388</p> Angles at Y and G being equall; therefore the Lines Y B and G B, and B C and B S $hall al$o be equall: And therefore C R and S R, and M V and P Z, and V N and Z T, $hall be equall likewi$e. <marg>K</marg> Since therefore M V is Le$$er than double of V N, it is manife$t that P Z is le$$er than double of Z T. <I>L</I>et P <G>w</G> be double of <G>w</G> T; and drawing a <I>L</I>ine from <G>w</G> to K, prolong it to E. Now the Centre of Gravity of the whole Portion $hall be the point K; and the Centre of that part which is in the Liquid $hall be <G>w,</G> and of that which is above the Liquid $hall be in the <I>L</I>ine K E, which let be E: But the <I>L</I>ine K Z $hall be perpendicular unto the Surface of the <I>L</I>iquid: And therefore al$o the Lines drawn thorow the Points E and <G>w</G> parall- <marg>L</marg> lell unto K Z, $hall be perpendicular sunto the $ame: Therefore the Portion $hall not abide, but $hall turn about $o, as that its <I>B</I>a$e do not in the lea$t touch the Surface of the <I>L</I>iquid; in regard that now when it toucheth in but one Point only, it moveth upwards, on <marg>M</marg> the part towards A: It is therefore per$picuous, that the Portion $hall con$i$t $o, as that its Axis $hall make an Angle with the <I>L</I>iquids Surface greater than the Angle X.</P> <head>COMMANDINE.</head> <marg>A</marg> <P>If the Portion have le$$er proportion in Gravity to the Liquid, than the Square S B hath to the Square B D, but greater than the Square X O hath to the Square B D.] <I>This is the $econd part of the Tenth propo$ition; and the other pat is with their Demon$trations, $hall hereafter follow in the $ame Order.</I></P> <P><G>*y</G> $hall be greater than <I>X</I> O, but le$$er than the Exce$s by <marg>B</marg> which the Axis is greater than Se$quialter of the Semi-parameter, that is than S B.] <I>This followeth from the 10 of the fifth Book of</I> Euclids <I>Elements.</I></P> <marg>C</marg> <P>It $hall be demon$trated, that M H is double to H N, like as it was demon$trated, that O G is double to G X.] <I>As in the fir$t Conclu$ion of this Propo$ition, and from what we have but even now written, thereupon appeareth:</I></P> <marg>D</marg> <P>For in regard that in the like Portions A M Q L and A X D, the Lines A Q and A N are drawn from the Ba$es unto the Portions, which Lines contain equall Angles with the $aid Ba$es, Q A $hall have the $ame proportion to A N, that L A hath to A D.] <I>This we have demonstrated above.</I></P> <marg>E</marg> <P>Therefore A N is equall to N Q] <I>For $ince that Q A is to A N, as L A to A D; Dividing and Converting, A N $hall be to N Q as A D to D L: But A D is equall to D L; for that D B is $uppo$ed to be the Diameter of the Portion: Therefore</I> <marg>(a) <I>By 14 of the fifth.</I></marg> <I>al$o</I> (a) <I>A N is equall to N Q.</I></P> <P>And A Q parallel to M Y.] <I>By the fifth of the $econd Book of</I> Apollonius <I>his Conicks<*></I></P> <marg>F</marg> <P>And let B D be cut in the Points K and R as hath been $aid.] <marg>G</marg> <I>In the fir$t Conciu$ion of this Propo$ition: And let it be cut in K, $o, as that B K be double to K D, and in R $o, as that K R may be equall to the Semi-parameter.</I></P> <P>And, $eeing that in the Equall and Like Portions A P O L and <marg>H</marg> A <I>M</I> Q L, the Lines A O and A Q are drawn from the Extremities of their Ba$es, $o, as that the Portions cut off, do make equall Angles <foot>with</foot> <p n=>389</p> with their Diameters; as al$o, the Angles at Y and G being equall; Therefore, the Lines Y B and G B, & B C & B S, $hall al$o be equall.] <I>Let the Line A Q cut the Diameter D B in</I> <G>g,</G> <I>and let it cut A O in</I> <G>d.</G> <I>Now becau$e that in</I> <fig> <I>the equall and like Portions A P O L & A M Q L, from the Extremities of their Ba$es, A O and A Q are drawn, that contain equall Angles with tho$e Ba$es; and $ince the Angles at D, are both Right; Therefore, the Remaining Angles A</I> <G>d</G> <I>D and A</I> <G>g</G> D <I>$hall be equall to one another: But the Line P G is parallel unto the Line A O; al$o M Y is parallel to A Q; and P S and M C to A D: Therefore the Triangles P G S and M Y C, as al$o the Triangles A</I> <G>d</G> <I>D and A</I> <G>g</G> <I>D, are all alike to each other</I>: (b) <I>And as A D is to A</I> <G>d,</G> <marg>(b) <I>By 4. of the $ixth.</I></marg> <I>$o is A D to A</I> <G>g</G><I>: and,</I> Permutando, <I>the Lines A D and A D are equall to each other: Therefore, A</I> <G>d</G> <I>and A</I> <G>g</G> <I>are al$o equall: But A O and A Q are equall to each other; as al$o their halves A T and A N: Therefore the Remainders T</I> <G>d</G> <I>and N</I> <G>g</G><I>; that is, TG and MY, are al$o</I> <marg>(c) <I>By 34 of the fir$t,</I></marg> <fig> <I>equall. And, as</I> (c) <I>P G is to G S, $o is M Y to Y C: and</I> Permutando, <I>as P G is to M Y, $o is G S to Y C: And, therefore, G S and Y C are equall; as al$o their halves B S and B C: From whence it followeth, that the Remainders S R and C R are al$o equall: And, con$equently, that P Z and M V, and V N and Z T, are lkiewi$e equall to one another.</I></P> <P>Since, therefore, that N V is le$$er <marg>K</marg> than double of V N.] <I>For M H is double of H N, and M V is le$$er than M H: Therefore, M V is le$$er than double of H N, and much le$$er than double of V N.</I></P> <P>Therefore, the Portion $hall not abide, but $hall turn about, <marg>L</marg> $o, as that its Ba$e do not in the lea$t touch the Surface of the Liquid; in regard that now when it toucheth in but one Point only, it moveth upwards on the part towards A.] Tartaglia's <I>his Tran$la- tion hath it thus,</I> Non ergo manet Portio $ed inclinabitur ut Ba$is ip$ius, nec $ecundum unum tangat Superficiem Humidi, quon am nunc $ecundum unum tacta ip$a reclina- tur<I>: Which we have thought fit in this manner to correct, from other Places of</I> Archimedes, <I>that the $en$e might be the more per$picuous. For in the $ixth Propo$ition of this, he thus writeth (as we al$o have it in the Tran$lation,)</I> The Solid A P O L, therefore, $hall turn about, and its Ba$e $hall not in the lea$t touch the Surface of the Liquid. <I>Again, in the $eventh Propo$ition</I>; From whence it is manife$t, that its Ba$e $hall turn about in $uch manner, a that its Ba$e doth in no wi$e touch the Surface of the Liquid; For that now when it toucheth but in one Point only, it moveth downwards on the part towards L. <I>And that the Portion moveth upwards, on the part towards A, doth plainly ap- pear: For $ince that the Perpendiculars unto the Surface of the Liquid, that pa$s thorow <*>, de fall on the part towards A, and tho$e that pa$s thorow E, on the part towards L; it is nece$$ary that the Centre</I> <G>w</G> <I>do move upwards, and the Centre E downwards.</I></P> <P>It is therefore per$picuous, that the Portion $hall con$i$t, $o, as that its Axis $hall make an Angle with the Liquids Surface greater than the Angle <I>X.] For dræwing a Line from A to X, prolong it nntill it do cut the Diamter</I> <foot><I>B D</I></foot> <p n=>390</p> <fig> <I>B D in</I> <G>l</G><I>; and from the Point O, and parallel to A</I> <G>l,</G> <I>draw O X; and let it touch the Section in O, as in the first Figure: And the</I> (d) <I>Angle at X,</I> <marg>(d) <I>By 29 of the fir$t.</I></marg> <I>$hall be equall al$o to the angle</I> <G>l</G><I>: But the angle at Y is equall to the Angle at</I> <G>g;</G> <I>and the</I> (e) <I>Angle</I> <marg>(e) <I>By 16. of the fir$t.</I></marg> A <G>*g</G> D <I>greater than the Angle A</I> <G>l</G> <I>D, which falleth without it: Therefore the Angle at Y $hall be great- er than that at X. And becau$e now the Portion turneth about, $o, as that the Ba$e doth not touch the Liquid, the Axis $hall make an Angle with its Surface greater than the Angle G; that is, than the Angle Y: And, for that rea$on, much greater than the Angle X.</I></P> <head>CONCLUSION III.</head> <P><I>If the Portion have the $ame proportion in Gravity to the Liquid, that the Square X O hath to the Square</I> BD, <I>being demitted into the Liquid, $o inclined, as that its Ba$e touch not the Liquid, it $hall $tand and continue inclined, $o, as that its Ba$e touch the Sur- face of the Liquid, in one Point only, and its Axis $hall make an Angle with the Liquids Surface equall to the Angle X. And, if the Portion have the $ame proportion in Gravity to the Liquid, that the Square P F hath to the Square B D, being demitted into the Liquid, & $et $o inclined, as that its Ba$e touch not the Liquid, it $hall $tand inclined, $o, as that its Ba$e touch the Surface of the Liquid in one Point only, & its Axis $hall make an Angle with it, equall to the Angle</I> <G>*f.</G></P> <P>Let the Portion have the $ame proportion in Gravity to tho Liquid that the Square <I>X</I>O hath to the Square B D; and let it be demitted into the Liquid $o inclined, as that its Ba$e touch <fig> not the Liquid. And cutting it by a Plane thorow the Axis, erect unto the Surface of the Liquid, let the Section of the Solid, be the Section of a Right-angled Cone, A P M L; let the Section of the Surface of the Liquid be I M; and the Axis of the Portion and Diameter of the Section B D; and let B D be divided as be- fore; and draw PN parallel to IM <p n=>391</p> and touching the Section in P, and T P parallel to B D; and P S perpen- dicular unto B D. It is to be demon$trated that the Portion $hall <fig> not $tand $o, but $hall encline until that the Ba$e touch the Surface of the Liquid, in one Point only, for let the $uperior figure $tand as it was, and draw O C, Perpendicular to B D; and drawing a <I>L</I>ine from A to <I>X,</I> prolong it to Q: A X $halbe equall to <I>X</I> Q. Then draw O X parallel to A Q. And becau$e the Portion is $uppo$ed to have the $ame pro- portion in Gravity to the Liquid that the $quare X O hath to the Square B D; the part thereof $ubmerged $hall al$o have the $ame proportion to the whole; that is, the Square T P to the Square <marg>A</marg> B D; and $o T P $hall be equal to <I>X</I> O: And $ince that of the <I>P</I>ortions I P M and A O Q the Diameters are equall, the portions $hall al$o be <marg>B</marg> equall. <I>A</I>gain, becau$e that in the Equall and <I>L</I>ike <I>P</I>ortions A O Q L <marg>C</marg> and AP ML the Lines A Q and I M, which cut off equall <I>P</I>or- tions, are drawn, that, from the Extremity of the <I>B</I>a$e, and this not from the Extremity; it appeareth that that which is drawn from the end or Extremity of the <I>B</I>a$e, $hall make the Acute Angle with the Diameter of the whole <I>P</I>ortion le$set. <I>A</I>nd the Angle at <I>X</I> <marg>D</marg> being le$$e than the Angle at N, B C $hall be greater than B S; and C R le$$er than S R: <I>A</I>nd, therfore O G $hall be le$$er than P Z; and G <I>X</I> greater than Z T: Therfore P Z is greater than double of Z T; being that O G is double of G X. Let P H be double to H T; and drawing a Line from H to K, prolong it to <G>w.</G> The Center of Gravity of the whole Portion $hall be K; the Center of the part which is within the Liquid H, and that of the part which is above the Liquid in the Line K <G>w</G>; which $uppo$ed to be <G>w.</G> Therefore it $hall be demon$trated, both, that K H is perpendicular to the Surface of the Liquid, and tho$e Lines al$o that are drawn thorow the Points Hand <G>w</G> parallel to K H: And therfore the Portion $hall not re$t, but $hall encline untill that its Ba$e do touch the Surface of the Liquid in one Point; and $o it $hall continue. For in the Equall Portions A O Q L and A P M L, the <fig> Lines A Q and A M, that cut off equall Portions, $hall be dawn from the Ends or Terms of the Ba$es; and A O Q and A P M $hall be demon$trated, as in the former, to <marg>E</marg> be equall: Therfore A Q and A M, do make equall Acute Angles with the Diameters of the Portions; and <foot>the</foot> <p n=>392</p> the Angles at X and N are equall. And, therefore, if drawing HK, it be prolonged to <G>w,</G> the Centre of Gravity of the whole Portion $hall be K; of the part which is within the Liquid H; and of the part which is above the Liquid in K <G>w)</G> as $uppo$e in <G>w;</G> and H K perpendicular to <fig> the Surface of the Liquid. Therfore along the $ame Right Lines $hall the part which is within the Liquid move upwards, and the part above it down- wards: And therfore the Portion $hall re$t with one of its Points touching the Surface of the Liquid, and its Axis $hall make with the <marg>F</marg> $ame an Angle equall to X. It is to be demon$trated in the $ame manner that the Portion that hath the $ame proportion in Gravity to the Liquid, that the Square P F hath to the Square B D, being demitted into the Liquid, $o, as that its Ba$e touch not the Liquid, it $hall $tand inclined, $o, as that its Ba$e touch the Surface of the Liquid in one Point only; and its Axis $hall make therwith an Angle equall to the Angle <G>f.</G></P> <head>COMMANDINE.</head> <marg>A</marg> <P>That is the Square T P to the Square B D.] <I>By the twenty $ixth of the Book</I> <marg>(a) <I>By 9 of the fifth.</I></marg> <I>of</I> Archimedes, De Conoidibus & Sphæroidibus: <I>Therefore, (a) the Square T P $hall be equall to the Square X O: And for that rea$on, the Line T P equall to the Line X O.</I></P> <marg>B</marg> <P>The Portions $hall al$o be equall.] <I>By the twenty fifth of the $ame Book.</I></P> <marg>C</marg> <P>Again, becau$e that in the Equall and Like Portions, A O Q L and A P M L.] <I>For, in the Portion A P M L, de$cribe the Portion A O Q equall to the Portion I P M: The Point Q falleth beneath M; for otherwi$e, the Whole would be equall to the Part. Then draw I V parallel to A Q, and cutting the Diameter is</I> <G>y;</G> <I>and let I M cut the $ame</I> <G>s;</G> <I>and A Q in</I> <G>s.</G> <I>I $ay that the Angle A</I> <G>u</G> <I>D, is le$$er than the Angle</I> <fig> <I>I</I> <G>s</G> <I>D. For the Angle I</I> <G>y</G> <I>D is equall to the Angle A</I> <G>u</G> <I>D: (b) But the interiour Angle</I> <marg>(b) <I>By 29 of the fir$t.</I></marg> <I>I</I> <G>y</G> <I>D is le$$er than the exteriour I</I> <G>s</G> <I>D: There-</I> <marg><I>(c) By 16 of the fir$t.</I></marg> <I>fore, (c) A</I> <G>u</G> <I>D $hall al$o be lefter than I</I> <G>s</G> <I>D.</I></P> <marg>D</marg> <P>And the Angle at X, being le$$e than the Angle at N.] <I>Thorow O draw twe Lines, O C perpendicular to the Diameter B D, and O X touching the Section in the Point O, and cutting</I> <marg><I>(d) By 5 of our<*> $e- cond of</I> Conicks.</marg> <I>the Diameter in X: (d) O X $hall be parallel to A Q; and the</I> (e) <I>Angle at X, $hall be equall to</I> <marg>(e) <I>By 29 of the fir$t.</I></marg> <I>that at</I> <G>u</G>: <I>Therefore, the</I> (f) <I>Angle at X,</I> <marg>(f) <I>By 39 of our fir$t of</I> Conicks.</marg> <I>$hall be le$$er than the Angle at</I> <G>s;</G> <I>that is, to that at N: And, con$equently, X $hall fall beneath N: Therefore, the Line X B is greater than N B. And, $ince B C is equall to X B, and B S equall to N B; B C $hall be greater than B S.</I></P> <foot>Therefore,</foot> <p n=>397</p> <P>Therefore, A Q and A M do make equall Acute Angles with <marg>E</marg> the Diameters of the Portions.] <I>We demon$trate this as in the Commentaries upon the $econd Conclu$ion.</I></P> <P>It is to be demon$trated in the $ame manner, that the Portion <marg>F</marg> that hath the $ame proportion in Gravity to the Liquid, that the Square P F hath to the Square B D, being demitted into the Liquid, $o, <fig> as that its Ba$e touch not the Li- quid, it $hall $tand inclined, $o, as that its Ba$e touch the Surface of the Liquid in one point only; and its Axis $hall make therewith an angle equall to the Angle <G>f.</G>] <I>Let the Portion be to the Liquid in Gravity, as the Square P F to the Square B D: and being demitted into the Liquid, $o inclined, as that its Ba$e touch not the Liquid, let it be cut thorow the Axis by a Plane erect to the Surface of the Liquid, that that the Section may be A M O L, the Section of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit of the Portion and Diameter of the Section B D; which let be cut into the $ame parts as we $aid before, and draw M N parallel to I O, that it may touch the Section in the Point M; and M T parallel to B D, and P M S perpe ndicular to the $ame. It is to be demon- strated, that the Portion $hall not re$t, but $hall incline, $o, as that it touch the Liquids Surface, in one Point of its Ba$e only. For,</I> <fig> <I>draw P C perpendicular to B D; and drawing a Line from A to F, prolong it till it meet with the Section in Q; and thorow P draw P</I> <G>f</G> <I>pa- rallel to A Q: Now, by the things allready de- mon$trated by us, A F and F Q $hall be equall to one another. And being that the Portion hath the $ame proportion in Gravity unto the Liquid, that the Square P F hath to the Square B D; and $eeing that the part $ubmerge<*>, hath the $ame pro-</I> <marg>(g) <I>By 9 of t fifth.</I></marg> <I>partion to the whole Portion; that is, the Squàre M T to the Square B D; (g) the Square M T $hall be equall to the Square P F; and, by the $ame rea$on, the Line M T equall to the Line P F. So that there being drawn in the equall & like portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the fir$t from the Extreme term of the Ba$e, the la$t not from the Extremity; it followeth, that A Q drawn from the Extremity, containeth a le$$er Acute Angle with the Diameter of the Portion, than I O: But the Line P</I> <G>f</G> <I>is parallel to the Line A Q, and M N to I O: There- fore, the Angle at</I> <G>f</G> <I>$hall be le$$er than the Angle at N; but the Line B C greater than B S; and S R, that is, M X, greater than C R, that is, than P Y: and, by the $ame rea$on, X T le$$er than Y F. And, $ince P Y is double to Y F, M X $hall be greater than double to Y F, and much greater than double of X T. Let M H be double to H T, and draw a Line from H to K, prolonging it. Now, the Centre of Gravity of the whole Fortion $hall be the Point K; of the part within the Liquid H; and of the Remaining part above the Liquid in the Line H K produced, as $uppo$e in</I> <G><*></G> <I>It $hall be demon$trated in the $ame manner, as before, that both the Line K H and tho$e that are drawn thorow the Points H and</I> <G>w</G> <I>parallel to the $aid K H, are perpendicular to the Surface of the Liquid: The Portion therefore, $hall not re$t; but when it $hall be enclined $o far as to touch the Sur- face of the Liquid in one Point and no more, then it $hall $tay. For the Angle at N</I> <foot><I>Eee $hall</I></foot> <p n=>398</p> <fig> <I>$hall be equall to the Angle at</I> <G>f;</G> <I>and the Line B S equall to the Line B C; and S R to C R: Where- fore, M H $hall be likewi$e equall to P Y. There- fore, having drawn HK and prolonged it; the Centre of Gravity of the whole Portion $hall be K; of that which is in the Liquid H; and of that which is above it, the Centre $hall be in the Line prolonged: let it be in</I> <G><*>.</G> <I>There- fore, along that $ame Line K H, which is per- pendicular to the Surface of the Liquid, $hall the part which is within the Liquid move up- wards, and that which is above the Liquld downwards: And, for this cau$e, the Portion, $hall be no longer moved, but $hall $tay, and re$t, $o, as that its Ba$e do touch the Liquids Surface in but one Point; and its Axis maketh an Angle therewith equall to the Angle</I> <G>f</G><I>; And, this is that which we were to demon$trate.</I></P> <head>CONCLVSION IV.</head> <P><I>If the Portion have greater proportion in Gravity to the Liquid, than the Square F P to the Square B D, but le$$er than that of the Square X O to the Square B D, being demitted into the Liquid, and inclined, $o, as that its Ba$e touch not the Liquid, it $hall $tand and re$t, $o, as that its Ba$e $hall be more $ubmerged in the Liquid.</I></P> <P>Again, let the Portion have greater proportion in Gravity to the Liquid, than the Square F P to the Square B D, but le$$er than that of the Square X O to the Square B D; and as the Portion is in Gravity to the Liquid, $o let the Square made of the Line <G>y</G> be to the Square B D. <G>*y</G> $hall be greater than F P, and le$$er than X O. Apply, therefore, the right Line I V to fall betwixt the Portions A V Q L and A X D; and let it be equall to <G>y,</G> and parallel to B D; and let it meet the Remaining Section in Y: V Y $hall al$o be proved double to Y I, like as it hath been demon$trated, that O G is double off G X. And, draw from V, the Line V <G>w,</G> touching the Section A V Q L in V; and drawing a Line from A to I, prolong it unto Q. We prove in the $ame manner, that the Line A I is equall to I Q; and that A Q is parallel to V <G>w.</G> It is to be demon$trated, that the Portion being demitted into the Liquid, and $o inclined, as that its Ba$e touch not the Liquid, $hall $tand, $o, that its Ba$e $hall be more $ubmerged in the Liquid, than to touch it Surface in <foot>but</foot> <p n=>399</p> but one Point only. For let it be de- <fig> mitted into the Liquid, as hath been $aid; and let it fir$t be $o inclined, as that its Ba$e do not in the lea$t touch the Surface of the Liquid. And then it being cut thorow the Axis, by a Plane erect unto the Surface of the Liquid, let the Section of the Portion be A N Z G; that of the Liquids Surface E Z; the Axis of the Portion and Diameter of the Section B D; and let B D be cut in the Points K and R, as before; and draw N L parallel to E Z, and touching the Section A N Z G in N, and N S perpendicular to <fig> B D. Now, $eeing that the Por- tion is in Gravity unto the Liquid, as the Square made of the Line is to the Square B D; <G>y</G> $hall be equall to N T: Which is to be demon$trated as above: And, therefore, N T is al$o equall to V I: The Portions, therefore, A V Q and E N Z are equall to one another. And, $ince that in the Equall and like Portions A V Q L and A N Z G, there are drawn A Q and E Z, cutting off equall Portions, that from the <fig> Extremity of the Ba$e, this not from the Extreme, that which is drawn from the Extremity of the Ba$e, $hall make the Acute Angle with the Diameter of the Portion le$$er: and in the Triangles N L S and V <G>w</G> C, the Angle at L is greater than the Angle at <G>w</G>: Therefore, B S $hall be le$$er than B C; and S R le$$er than C R: and, con$equently, N X greater than V H; and X T le$$er than H I. Seeing, therefore, that V Y is double to Y I; It is manife$t, that N X is greater than double to X T. Let N M be double to M T: It is manife$t, from what hath been $aid, that the Portion $hall not re$t, but will incline, untill that its Bafe do touch the Surface of the Liquid: and it toucheth it in one Point only, as appeareth in the Figure: And other things <foot>Eee 2 $tanding</foot> <p n=>400</p> <fig> $tanding as before, we will again demon$trate, that N T is equall to V I; and that the Portions A V Q and A N Z are equall to each other. Therefore, in regard, that in the Equall and Like Portions A V Q L and A N Z G, there are drawn A Q and A Z cutting off equall Por- tions, they $hall with the Diameters of the Portions, contain equall Angles. Therefore, in the Triangles N L S and V <G>w</G> C, the Angles at the Points <I>L</I> and <G>w</G> are equall; and the Right Line B S equall to B C; S R to C R; N X to V H; and X T to H I: And, $ince V Y is double to Y I, N X $hall be greater than double of X T. Let therefore, N M be double to M T. It is hence again manife$t, that the Portion will not remain, but $hall incline on the part towards A: But it was $uppo$ed, that the $aid Portion did touch the Surface of the Liquid in one $ole Point: Therefore, its Ba$e mu$t of nece$$ity $ubmerge farther into the Liquid.</P> <head>CONCLVSION V.</head> <P><I>If the Portion have le$$er proportion in Gravity to the Liquid, than the Square F P to the Square B D, being demitted into the Liquid, and in- clined, $o, as that its Ba$e touch not the Liquid, it $hall $tand $o inclined, as that its Axis $hall make an Angle with the Surface of the Liquid, le$$e than the Angle</I> <G>y;</G> <I>And its Ba$e $hall not in the lea$t touch the Liquids Surface.</I></P> <P>Finally, let the Portion have le$$er proportion to the Liquid in Gravity, than the Square F P hath to the Square B D; and as the Portion is in Gravity to the Liquid, $o let the Square made of the Line <G>y</G> be to the Square B D. <G>y</G> $hall be le$$er than P F. Again, apply any Right Line as G I, falling betwixt the Sections A G Q L and A X D, and parallel to B D; and let it cut the Middle Conick Section in the Point H, and <foot>the</foot> <p n=>401</p> the Right Line R Y in Y. We <fig> $hall demon$trate G H to be double to H I, as it hathbeen demon$tra- ted, that O G is double to G X. Then draw G <G>w</G> touching the Section A G Q L in G; and G C perpen di- cular to B D; and drawing a Line from A to I, prolong it to Q. Now A I $hall be equall to I Q; and A Q parallel to G <G>w.</G> It is to be demon$trated, that the Portion being demitted into the Liquid, and inclined, $o, as that its Ba$e touch the Liquid, it $hall $tand $o incli- <fig> ned, as that its Axis $hall make an Angle with the Surface of the Liquid le$$e than the Angle <G>f;</G> and its Ba$e $hall not in the lea$t touch the Liquids Surface. For let it be demitted into the Liquid, and let it $tand, $o, as that its Ba$e do touch the Surface of the Liquid in one Point only: and the Portion being cut thorow the Axis by a Plane erect unto the Surface of the Liquid, let the Section of <fig> the Portion be A N Z L, the Section of a Rightangled Cone; that of the Surface of the Liquid A Z; and the Axis of the Portion and Dia- meter of the Section B D; and let B D be cut in the Points K and R as hath been $aid above; and draw N F parallel to A Z, and touching the Section of the Cone in the Point N; and N T parallel to B D; and N S perpendicular to the $ame. Be- cau$e, now, that the Portion is in Gravity to the Liquid, as the Square made of <G>y</G> is to the Square B D; and $ince that as the Portion is to the Liquid in Gravity, $o is the Square N T to the Square B D, by the things that have been $aid; it is plain, that N T is equall to the Line <G>y</G>: And, therefore, al$o, the Portions A N Z and A G Q are equall. And, $eeing that in the Equall and Like Portions A G Q L and A N Z L; there are drawn from the Extremities of their Ba$es, A Q and A Z which cut off equall Porti- ons: It is obvious, that with the Diameters of the Portions they <foot>make</foot> <p n=>402</p> make equall Angles; and that in the Triangles N F S and G <G>w</G> C the Angles at F and <G>w</G> are equall; as al$o, that S B and B C, an<*> S R and C R are equall to one another: And, therefore, N X an<*> G Y are al$o equall; and X T and Y I. And $ince G H is doubl<*> to H I, N X $hall be le$$er than double of X T. Let N M therefor<*> be double to M T; and drawing a Line from M to K, prolong i<*> unto E. Now the Centre of Gravity of the whole $hall be th<*> Point K; of the part which is in the Liquid the Point M; an<*> that of the part which is above the Liquid in the Line prolonged as $uppo$e in E. Therefore, by what was even now demon$trated it is manife$t that the Portion $hall not $tay thus, but $hall incline, $o as that its Ba$e do in no wi$e touch the Surface of the Liquid And that the Portion will $tand, $o, as to make an Angle with th<*> Surface of the Liquid le$$er tha<*> <fig> the Angle <G>f,</G> $hall thus be demon $trated. Let it, if po$$ible, $tan<*> $o, as that it do not make an Angl<*> le$$er than the Angle <G>f;</G> and di$po<*> all things el$e in the $ame manner a before; as is done in the pre$et Figure. We are to demon$trat in the $ame method, that N T is <*> quall to <G>y;</G> and by the $ame rea$or equall al$o to G I. And $ince that i<*> the Triangles P <G>f</G> C and N F S, the Angle F is not le$$er than th<*> Angle <G>f,</G> B F $hall not be greater than B C: And, therefore, neithe<*> $hall S R be le$$er than C R; nor N X than P Y: But $ince P F <*> greater than N T, let P F be Se$quialter of P Y: N T $hall be le$$e<*> than Se$quialter of N X: And, therefore, N X $hall be greate than double of X T. Let N M be double of M T; and drawing Line from M to K prolong it. It is manife$t, now, by what hat<*> been $aid, that the Portion $hall not continue in this po$ition, but $hal<*> turn about, $o, as that its Axis do make an Angle with the Surfac<*> of the Liquid, le$$er than the Angle <G>f.</G></P> <foot><I>FINIS.</I></foot> <pb> <head>A DISCOURSE <I>PRESENTED</I> TO THE MOST SERENE Don Co$imo II. GREAT DUKE <I>OF</I> TUSCANY, CONCERNING The <I>NATATION</I> of BODIES Vpon, And <I>SUBMERSION</I> In, THE WATER.</head> <head>By GALILEUS GALILEI: Philo$opher and Mathematician, unto His mo$t Serene Highne$$e.</head> <head>Engli$hed from the Second Edition of the ITALIAN, compared with the Manu$cript Copies, and reduced into PROPOSITIONS: By <I>THOMAS SALUSBURY,</I> E$q;</head> <head><I>LONDON</I>: Printed by WILLIAM LEYBOURN: <I>M D C LXIII.</I></head> <p n=>401</p> <head>A DISCOVRSE Pre$ented to the Mo$t Serene DON COSIMO II. GREATDUKE of <I>TUSC ANY:</I> CONCERNING</head> <head><I>The Natation of BODIES Upon, or Submer$ion In, the WATER.</I></head> <P>Con$idering (Mo$t Serene Prince) that the publi$hing this pre$ent Treati$e, of $o different an Argument from that which <marg>His Nuncio Sl- derio.</marg> many expect, and which according to the intentions I propo$ed in my ^{*} A$tronomi- call <I>Advi$o,</I> I $hould before this time have put forth, might peradventure make $ome thinke, either that I had wholly relinqui$hed my farther imployment about the new Cele$tiall Ob$ervations, or that, at lea$t, I handled them very remi$$ely; I have judged fit to render an account, a$well of my deferring that, as of my writing, and publi$hing this treati$e.</P> <P>As to the fir$t, the la$t di$coveries of <I>Saturn</I> to be tricorporeall, and of the mutations of Figure in <I>Venus,</I> like to tho$e that are $een in the Moon, together with the Con$equents depending thereupon, have not $o much occa$ioned the demur, as the inve$tigation of the times of the Conver$ions of each of the Four Medicean Planets about <I>Ju- piter,</I> which I lighted upon in <I>April</I> the year pa$t, 1611, at my being in <I>Rome</I>; where, in the end, I a$$ertained my $elfe, that the fir$t and neere$t to <I>Jupiter,</I> moved about 8 <I>gr.</I> & 29 <I>m.</I> of its Sphere in an houre, make- ing its whole revolution in one naturall day, and 18 hours, and almo$t an halfe. The $econd moves in its Orbe 14 <I>gr. 13 min.</I> or very neer, in an hour, and its compleat conver$ion is con$ummate in 3 dayes, 13 hours, and one third, or thereabouts. The third pa$$eth in an hour, 2 <I>gr. 6 min.</I> little more or le$s of its Circle, and mea$ures it all in 7 dayes, 4 hours, or very neer. The fourth, and more remote than the re$t, goes in one houre, o <I>gr 54 min.</I> and almo$t an halfe of its Sphere, and fini$heth it all in 16 dayes, and very neer 18 hours. But be- cau$e the exce$$ive velocity of their returns or re$titutions, requires a mo$t $crupulous preci$ene$$e to calculate their places, in times pa$t <foot>Fff and</foot> <p n=>402</p> and future, e$pecially if the time be for many Moneths or Years; I am therefore forced, with other Ob$ervations, and more exact than the former, and in times more remote from one another, to correct the Tables of $uch Motions, and limit them even to the $horte$t mo- ment: for $uch exactne$$e my fir$t Ob$ervations $uffice not; not only in regard of the $hort intervals of Time, but becau$e I had not as then found out a way to mea$ure the di$tances between the $aid Planets by any In$trument: I Ob$erved $uch Intervals with $imple relation to the Diameter of the Body of <I>Jupiter</I>; taken, as we have $aid, by the eye, the which, though they admit not errors of above a Minute, yet they $uffice not for the determination of the exact greatne$s of the Spheres of tho$e Stars. But now that I have hit upon a way of ta- king $uch mea$ures without failing, $carce in a very few Seconds, I will continue the ob$ervation to the very occultation of <I>JVPITER,</I> which $hall $erve to bring us to the perfect knowledge of the Moti- ons, and Magnitudes of the Orbes of the $aid Planets, together <marg>The Authors Ob$ervations of the Solar Spots.</marg> al$o with $ome other con$equences thence ari$ing. I adde to the$e things the ob$ervation of $ome ob$cure Spots, which are di$cover- ed in the Solar Body, which changing, po$ition in that, propounds to our con$ideration a great argument either that the Sun revolves in it $elfe, or that perhaps other Starts, in like manner as <I>Venus</I> and <I>Mercury,</I> revolve about it, invi$ible in other times, by rea$on of their $mall digre$$ions, le$$e than that of <I>Mercury,</I> and only vi$ible when they interpo$e between the Sun and our eye, or el$e hint the truth of both this and that; the certainty of which things ought not to be contemned, nor omitted.</P> <P><I>Continuall ob$ervation hath at la$t a$$ured me that the$e Spots are matters contiguous to the Body of the Sun, there continually produced in great number, and afterwards di$$olved, $ome in a $horter, $ome in a longer time, and to be by the Conver$ion or Revolution of the Sun in it $elfe, which in a Lunar Moneth, or thereabouts, fini$heth its Period, caried about in a Circle, an accident great of it $elfe, and greater for its Con$equences.</I></P> <marg>The occa$ion in- ducing the Au- thor to write this Treati$e.</marg> <P>As to the other particular in the next place. ^{*} Many cau$es have moved me to write the pre$ent Tract, the $ubject whereof, is the Di$pute which I held $ome dayes $ince, with $ome learned men of this City, about which, as your Highne$$e knows, have followed many Di$cour$es: The principall of which Cau$es hath been the Intimation of your Highne$$e, having commended to me Writing, as a $ingular means to make true known from fal$e, reall from appa- rent Rea$ons, farr better than by Di$puting vocally, where the one or the other, or very often both the Di$putants, through too <foot>great</foot> <p n=>403</p> greate heate, or exalting of the voyce, either are not under$tood, or el$e being tran$ported by o$tentation of not yeilding to one ano- ther, farr from the fir$t Propo$ition, with the novelty, of the various Propo$als, confound both them$elves and their Auditors.</P> <P>Moreover, it $eemed to me convenient to informe your High- ne$$e of all the $equell, concerning the Controver$ie of which I treat, as it hath been adverti$ed often already by others: and becau$e the Doctrine which I follow, in the di$cu$$ion of the point in hand, is different from that of <I>Ari$totle</I>; and interferes with his Principles, I have con$idered that again$t the Authority of that mo$t famous Man, which among$t many makes all $u$pected that comes not from the Schooles of the Peripateticks, its farr better to give ones Rea$ons by the Pen than by word of mouth and therfore I re$olved to write the pre$ent di$cour$e: in which yet I hope to demon$trate that it was not out of capritiou$ne$$e, or for that I had not read or under$tood <I>Ari$totle,</I> that I $ometimes $werve from his opinion, but becau$e $everall Rea$ons per$wade me to it, and the $ame <I>Ari$totle</I> hath <marg><I>Ari$totle</I> prefers Rea$on to the Authority ofan Author.</marg> tought me to fix my judgment on that which is grounded upon Rea$on, and not on the bare Authority of the Ma$ter; and it is mo$t certaine according to the $entence of <I>Alcinoos,</I> that philo$opha- <marg>The benefit of this Argument.</marg> ting $hould be free. Nor is the re$olution of our Que$tion in my judgment without $ome benefit to the Univer$all, fora$much as treating whether the figure of Solids operates, or not, in their going, or not going to the bottome in Water, in occurrences of building Bridges or other Fabricks on the Water, which happen commonly in affairs of grand import, it may be of great availe to know the truth.</P> <P>I $ay therfore, that being the la$t Summer in company with certain <marg>Conden$ation the Propriety of Cold, according to the Peripate- ticks.</marg> Learned men, it was $aid in the argumentation; That Conden$ation was the propriety of Cold, and there was alledged for in$tance, the example of Ice: now I at that time $aid, that, in my judgment, the Ice $hould be rather Water rarified than conden$ed, and my <marg>Ice rather water rarified, than conden$ed, and why:</marg> rea$on was, becau$e Conden$ation begets diminution of Ma$s, and augmentation of gravity, and Rarifaction cau$eth greater Lightne$s, and augmentarion of Ma$$e: and Water in freezing, encrea$eth in Ma$$e, and the Ice made thereby is lighter than the Water on which it $wimmeth.</P> <P><I>What I $ay, is manife$t, becau$e, the medium $ubtracting from the whole Gravity of Sollids the weight of $uch another Ma$$e of the $aid</I> <marg>In lib: 1. of Na- tation of Bodies Prop. 7.</marg> <I>Medium; was</I> Archimedes <I>proves in his ^{*} Fir$t Booke</I> De In$identibus Humido; <I>when ever the Ma$$e of the $aid Solid encrea$eth by Di$traction, the more $hall the</I> Medium <I>detract from its entire Gravity; and le$$e, when by Compre$$ion it $hall be conden$ed and reduced to a le$$e Ma$$e.</I></P> <foot><I>Fff</I> 2 It</foot> <p n=>404</p> <marg>Figure operates not in the Nata- tion of Sollids.</marg> <P>It was an$wered me, that that proceeded not from the greater Levity; but from the Figure, large and flat, which not being able to pene- trate the Re$i$tance of the Water, is the cau$e that it $ubmergeth not. I replied, that any piece of Ice, of what$oever Figure, $wims upon the Water, a manife$t $igne, that its being never $o flat and broad, hath not any part in its floating: and added, that it was a manife$t proofe hereof to $ee a piece of Ice of very broad Figure being thru$t to the botome of the Water, $uddenly return to flote atoppe, which had it been more grave, and had its $wimming proceeded from its Forme, unable to penetrate the Re$i$tance of the <I>Medium,</I> that would be altogether impo$$ible; I concluded therefore, that the Figure was in $ort a Cau$e of the Natation or Submer$ion of Bodies, but the greater or le$$e Gravity in re$pect of the Water: and there- fore all Bodyes heavier than it of what Figure $oever they be, indiffe- rently go to the bottome, and the lighter, though of any figure, float indifferently on the top: and I $uppo$e that tho$e which hold other- wi$e, were induced to that beliefe, by $eeing how that diver$ity of Formes or Figures, greatly altereth the Velo$ity, and Tardity of Motion; $o that Bodies of Figure broad and thin, de$cend far more lea$urely into the Water, than tho$e of a more compacted Figure, though both made of the $ame Matter: by which $ome might be induced to believe that the Dilatation of the Figure might reduce it to $uch amplene$$e that it $hould not only retard but wholly impede and take away the Motion, which I hold to be fal$e. Upon this Conclu$ion, in many dayes di$cour$e, was $poken much, and many things, and divers Experiments produced, of which your Highne$$e heard, and $aw $ome, and in this di$cour$e $hall have all that which hath been produced again$t my A$$ertion, and what hath been $ugge$ted to my thoughts on this matter, and for con- firmation of my Conclu$ion: which if it $hall $uffice to remove that (as I e$teem hitherto fal$e) Opinion, I $hall thinke I have not unprofitably $pent my paynes and time. and although that come not to pa$$e, yet ought I to promi$e another benefit to my $elfe, namely, of attaining the knowledge of the truth, by hearing my Fallacyes confuted, and true demon$trations produced by tho$e of the contrary opinion.</P> <P>And to proceed with the greate$t plainne$s and per$picuity that I can po$$ible, it is, I conceive, nece$$ary, fir$t of all to declare what is the true, intrin$ecall, and totall Cau$e, of the a$cending of $ome Sollid Bodyes in the Water, and therein floating; or on the contrary, of their $inking. and $o much the rather in a$much as I cannot $atisfie my $elfe in that which <I>Ari$totle</I> hath left written on this Subject.</P> <marg>The cau$e of the Natation & $ub-</marg> <P>I $ay then the Cau$e why $ome Sollid Bodyes de$cend to the <foot>Bottom</foot> <p n=>405</p> Bottom of Water, is the exce$$e of their Gravity, above the <marg>mer$ion of Sol- ids in the Wa- ter.</marg> Gravity of the Water; and on the contrary, the exce$s of the Waters Gravity above the Gravity of tho$e, is the Cau$e that others do not de$cend, rather that they ri$e from the Bottom, and a$cend to the Surface. This was $ubtilly demon$trated by <I>Archimedes</I> in his Book Of the NATATION of BODIES: Conferred afterwards by a very grave Author, but, if I erre not invi$ibly, as below for defence of him, I $hall endeavour to prove.</P> <P>I, with a different Method, and by other meanes, will endeavour to demon$trate the $ame, reducing the Cau$es of $uch Effects to more intrin$ecall and immediate Principles, in which al$o are di$co- vered the Cau$es of $ome admirable and almo$t incredible Acci- dents, as that would be, that a very little quantity of Water, $hould be able, with its $mall weight, to rai$e and $u$tain a Solid Body, an hundred or a thou$and times heavier than it.</P> <P>And becau$e demon$trative Order $o requires, I $hall define cer- tain Termes, and afterwards explain $ome Propo$itions, of which, as of things true and obvious, I may make u$e of to my pre$ent pur- po$e.</P> <head>DEFINITION I.</head> <P><I>I then call equally Grave</I> in $pecie, <I>tho$e Matters of which equall Ma$$es weigh equally.</I></P> <P>As if for example, two Balls, one of Wax, and the other of $ome Wood of equall Ma$$e, were al$o equall in Weight, we $ay, that $uch Wood, and the Wax are <I>in $pecie</I> equally grave.</P> <head>DEFINITION II.</head> <P><I>But equally grave in Ab$olute Gravity, we call two Sollids, weighing equally, though of Ma$s they be unequall.</I></P> <P>As for example, a Ma$s of Lead, and another of Wood, that weigh each ten pounds, I call equall in Ab$olute Gravity, though the Ma$s of the Wood be much greater then that of the Lead.</P> <P><I>And, con$equently, le$s Grave</I> in $pecie.</P> <head>DEFINITION III.</head> <P><I>I call a Matter more Grave</I> in $pecie <I>than another, of which a Ma$s, equall to a Ma$s of the other, $hall weigh more.</I></P> <foot>And</foot> <p n=>406</p> <P>And $o I $ay, that Lead is more grave <I>in $pecie</I> than Tinn, becau$e if you take of them two equall Ma$$es, that of the Lead weigheth more.</P> <head>DEFINITION IV.</head> <P><I>But I call that Body more grave ab$olutely than this, if that weigh more than this, without any re$pect had to the Ma$$es.</I></P> <P>And thus a great piece of Wood is $aid to weigh more than a little lump of Lead, though the Lead be <I>in $pecie</I> more heavy than the Wood. And the $ame is to be under$tood of the le$s grave <I>in $pecie,</I> and the le$s grave ab$olutely.</P> <P>The$e Termes defined, I take from the Mechanicks two Princi- ples: the fir$t is, that</P> <head>AXIOME. I.</head> <P><I>Weights ab$olutely equall, moved with equall Velocity, are of equall Force and Moment in their operations.</I></P> <head><I>DEFINITION V.</I></head> <P>Moment, among$t Mechanicians, $igrifieth that Vertue, that Force, or that Efficacy, with which the Mover moves, and the Moveable re$i$ts.</P> <P><I>Which Vertue dependes not only on the $imple Gravity, but on the Velocity of the Motion, and on the diver$e Inclinations of the Spaces along which the Motion is made: For a de$cending Weight makes a greater</I> Impetus <I>in a Space much declining, than in one le$s declining; and in $umme, what ever is the occa$ion of $uch Vertue, it ever retaines the name of</I> Moment; <I>nor in my Judgement, is this $ence new in our Idiome, for, if I mistake not, I think we often $ay; This is a weighty bu$ine$$e, but the other is of $mall moment: and we con$ider lighter mat- ters and let pa$s tho$e of Moment; a Metaphor, I $uppo$e, taken from the Mechanicks.</I></P> <P>As for example, two weights equall in ab$olute Gravity, being put into a Ballance of equall Arms, they $tand in <I>Equilibrium,</I> nei- ther one going down, nor the other up: becau$e the equality of the Di$tances of both, from the Centre on which the Ballance is $uppor- ted, and about which it moves, cau$eth that tho$e weights, the $aid Ballance moving, $hall in the $ame Time move equall Spaces, that is, $hall move with equall Velocity, $o that there is no rea$on for which <foot>this</foot> <p n=>407</p> this Weight $hould de$cend more than that, or that more than this; and therefore they make an <I>Equilibrium,</I> and their Moments continue of $emblable and equall Vertue.</P> <P>The $econd Principle is; That</P> <head>AXIOME II.</head> <P><I>The Moment and Force of the Gravity, is encrea$ed by the Velocity of the Motion.</I></P> <P>So that Weights ab$olutely equall, but conjoyned with Velocity unequall, are of Force, Moment and Vertue unequall: and the more potent, the more $wift, according to the proportion of the Ve- locity of the one, to the Velocity of the other. Of this we have a very pertinent example in the Balance or Stiliard of unequall Arms, at which Weights ab$olutely equall being $u$pended, they do not weigh down, and gravitate equally, but that which is at a greater di$tance from the Centre, about which the Beam moves, de$cends, rai$ing the other, and the Motion of this which a$cends is $low, and the other $wift: and $uch is the Force and Vertue, which from the Velocity of the Mover, is conferred on the Moveable, which receives it, that it can exqui$itely compen$ate, as much more Weight added to the other $lower Moveable: $o that if of the Arms of the Balance, one were ten times as long as the other, whereupon in the Beames moving about the Centre, the end of that would go ten times as far as the end of this, a Weight $u$pended at the greater di$tance, may $u$tain and poy$e another ten times more grave ab$olutely than it: and that becau$e the Stiliard moving, the le$$er Weight $hall move ten times fa$ter than the bigger. It ought alwayes therefore to be under$tood, that Motions are according to the $ame Inclinations, namely, that if one of the Moveables move perpendicularly to the Horizon, then the other makes its Motion by the like Perpendicular; and if the Motion of one were to be made Horizontally; that then the other is made along the $ame Horizontall plain: and in $umme, alwayes both in like Inclinations. This proportion between the Gravity and Velocity is found in all Mechanicall In$truments: and is con$idered by <I>Ari$totle,</I> as a Principle in his <I>Mechanicall Que$tions</I>; whereupon we al$o may take it for a true A$$umption, That</P> <head>AXIOME III.</head> <P><I>Weights ab$olutely unequall, do alternately counterpoy$e and become of equall Moments, as oft as their Gravi- ties, with contrary proportion, an$wer to the Velocity of their Motions.</I></P> <foot>That</foot> <p n=>408</p> <P>That is to $ay, that by how much the one is le$s grave than the other, by $o much is it in a con$titution of moving more $wiftly than that.</P> <P>Having prefatically explicated the$e things, we may begin to en- quire, what Bodyes tho$e are which totally $ubmerge in Water, and go to the Bottom, and which tho$e that by con$traint float on the top, $o that being thru$t by violence under Water, they return to $wim, with one part of their Ma$s vi$ible above the Surface of the Water: and this we will do by con$idering the re$pective operati- on of the $aid Solids, and of Water: Which operation followes the Submer$ion and $inking; and this it is, That in the Submer$ion <marg>How the $ub- mer$ion of So- lids in the Wa- ter, is effected.</marg> that the Solid maketh, being depre$$ed downwards by its proper Gravity, it comes to drive away the water from the place where it $ucce$$ively $ubenters, and the water repul$ed ri$eth and a$cends above its fir$t levell, to which A$cent on the other $ide it, as being a grave Body of its own nature, re$i$ts: And becau$e the de$cending Solid more and more immerging, greater and greater quantity of Water a$cends, till the whole Sollid be $ubmerged; its nece$$ary to compare the Moments of the Re$i$tance of the water to A$cen$ion, with the Moments of the pre$$ive Gravity of the Solid: And if the Moments of the Re$i$tance of the water, $hall equalize the Moments <marg>What Solids $hall float on the Water.</marg> of the Solid, before its totall Immer$ion; in this ca$e doubtle$s there $hall be made an <I>Equilibrium,</I> nor $hall the Body $ink any farther. But if the Moment of the Solid, $hall alwayes exceed the Moments <marg>What Solids $hall $inke to the botome.</marg> wherewith the repul$ed water $ucce$$ively makes Re$i$tance, that Solid $hall not only wholly $ubmerge under water, but $hall de$cend to the Bottom. But if, la$tly, in the in$tant of totall Submer$ion, <marg>What Solids $hall re$t in all places of the Wa- ter.</marg> the equality $hall be made between the Moments of the prement Solid, and the re$i$ting Water; then $hall re$t en$ue, and the $aid Solid $hall be able to re$t indifferently, in what$oever part of the water. By this time is manife$t the nece$$ity of comparing the <marg>The Gravitie of the Water and <I>S</I>olid mu$t be compared in all Problems, of Na- tation of Bodies.</marg> Gravity of the water, and of the Solid; and this compari$on might at fir$t $ight $eem $ufficient to conclude and determine which are the Solids that float a-top, and which tho$e that $ink to the Bottom in the water, a$$erting that tho$e $hall float which are le$$e grave <I>in $pecie</I> than the water, and tho$e $ubmerge, which are <I>in $pecie</I> more grave. For it $eems in appearance, that the Sollid in $inking continually, rai$eth $o much Water in Ma$s, as an$wers to the parts of its own Bulk $ubmerged: whereupon it is impo$$ible, that a Solid le$s grave <I>in $pecie,</I> than water, $hould wholly $ink, as being unable to rai$e a weight greater than its own, and $uch would a Ma$s of water equall to its own Ma$s be. And likewi$e it $eems nece$$ary, that the graver Solids do go to the Bottom, as being of a Force more than $ufficient for the rai$ing a Ma$$e of water, equall to its own, though inferiour in weight. Neverthele$s the bu$ine$s $ucceeds otherwi$e: and <foot>though</foot> <p n=>409</p> though the Conclu$ions are true, yet are the Cau$es thus a$$igned deficient, nor is it true, that the Solid in $ubmerging, rai$eth and repul$eth Ma$$es of Water, equall to the parts of it $elf $ubmerged; but the Water repul$ed, is alwayes le$s than the parts of the Solid <marg>The water re- pul$ed is ever le$s than the parts of the Sollid $ub- merged.</marg> $ubmerged: and $o much the more by how much the Ve$$ell in which the Water is contained is narrower: in $uch manner that it hinders not, but that a Solid may $ubmerge all under Water, with- out rai$ing $o much Water in Ma$s, as would equall the tenth or twentieth part of its own Bulk: like as on the contrary, a very <marg><I>A</I> $mall quantity of water, may float a very great Solid Ma$s.</marg> $mall quantity of Water, may rai$e a very great Solid Ma$s, though $uch Solid $hould weigh ab$olutely a hundred times as much, or more, than the $aid Water, if $o be that the Matter of that $ame Solid be <I>in $pecie</I> le$s grave than the Water. And thus a great Beam, as $uppo$e of a 1000 weight, may be rai$ed and born afloat by Water, which weighs not 50: and this happens when the Mo- ment of the Water is compen$ated by the Velocity of its Motion.</P> <P>But becau$e $uch things, propounded thus in ab$tract, are $ome- what difficult to be comprehended, it would be good to demon$trate them by particular examples; and for facility of demon$tration, we will $uppo$e the Ve$$els in which we are to put the Water, and place the Solids, to be inviron'd and included with $ides erected perpendi- cular to the Plane of the Horizon, and the Solid that is to be put into $uch ve$$ell to be either a $treight Cylinder, or el$e an upright Pri$me</P> <P><I>The which propo$ed and declared, I proceed to demonstrate the truth of what hath been hinted, forming the en$uing Theoreme.</I></P> <head><I>THEOREME I.</I></head> <P>The Ma$s of the Water whicha$cends in the $ub- <marg><I>T</I>he Proportion of the water rai- $ed to the <I>S</I>olid $ubmerged.</marg> merging of a Solid, Pri$me or Cylinder, or that aba$eth in taking it out, is le$s than the Ma$s of the $aid Solid, $o depre$$ed or advanced: and hath to it the $ame proportion, that the Surface of the Water circumfu$ing the Solid, hath to the $ame circumfu$ed Surface, together with the Ba$e of the Solid.</P> <P><I>Let the Ve$$ell be A B C D, and in it the Water rai$ed up to the Levell E F G, before the Solid Pri$me H I K be therein immerged; but after that it is depre$$ed under Water, let the Water be rai$ed as high as the Levell L M, the Solid H I K $hall then be all under Water, and the Ma$s of the elevated Water $hall be L G, which is le$s than the</I> <foot><I>Ggg Ma$s</I></foot> <p n=>410</p> <fig> <I>Ma$$e of the Solid depre$$ed, namely of H I K, being equall to the only part E I K, which is contained under the fir$t Levell E F G. Which is manife$t, becau$e if the Solid H I K be taken out, the Water I G $hall return into the place occupied by the Ma$s E I K, where it was continuate be- fore the $ubmer$ion of the Pri$me. And the Ma$s L G being equall to the Ma$s E K: adde thereto the Ma$s E N, and it $hall be the whole Ma$s E M, compo$ed of the parts of the Pri$me E N, and of the Water N F, equall to the whole Solid H I K: And, there- fore, the Ma$s L G $hall have the $ame proportion to E M, as to the Ma$s H I K: But the Ma$s L G hath the $ame proportion to the Ma$s E M, as the Surface L M hath to the Surface M H: Therefore it is ma- nife$t, that the Ma$s of Water repul$ed L G, is in proportion to the Ma$s of the Solid $ubmerged H I K; as the Surface L M, namely, that of the Water ambient about the Sollid, to the whole Surface H M, compounded of the $aid ambient water, and the Ba$e of the Pri$me H N. But if we $uppo$e the fir$t Levell of the Water the according to the Surface H M, and the Pri$me allready $ubmerged H I K; and after to be taken out and rai$ed to E A O, and the Water to be faln from the fir$t Levell H L M as low as E F G; It is manife$t, that the Pri$me E A O being the $ame with H I K, its $uperiour part H O, $hall be equall to the inferiour E I K: and remove the common part E N, and, con$equently, the Ma$s of the Water L G is equall to the Ma$s H O; and, therefore, le$s than the Solid, which is without the Water, namely, the whole Pri$me E A O, to which likewi$e, the $aid Ma$s of Water abated L G, hath the $ame propor- tion, that the Surface of the Waters circumfu$ed L M hath to the $ame circumfu$ed Surface, together with the Ba$e of the Pri$me A O: which hath the $ame demon$tration with the former ca$e above.</I></P> <P><I>And from hence is inferred, that the Ma$s of the Water, that ri$eth in the immer$ion of the Solid, or that ebbeth in elevating it, is not equall to all the Ma$s of the Solid, which is $ubmerged or elevated, but to that part only, which in the immer$ion is under the fir$t Levell of the Water, and in the elevation remaines above the fir$t Levell: Which is that which was to be demon$trated. We will now pur$ue the things that remain.</I></P> <P>And fir$t we will demon$trate that,</P> <foot>THEO-</foot> <p n=>411</p> <head>THEOREME II.</head> <P><I>When in one of the above $aid Ve$$els, of what ever</I> <marg>The proportion of the water aba- ted, to the Solid rai$ed.</marg> <I>breadth, whether wide or narrow, there is placed $uch a Pri$me or Cylinder, inviron'd with Water, if we ele- vate that Solid perpendicularly, the Water circumfu- $ed $hall abate, and the Abatement of the Water, $hall have the $ame proportion to the Elevation of the Pri$me, as one of the Ba$es of the Pri$me, hath to the Surface of the Water Circumfu$ed.</I></P> <P>Imagine in the Ve$$ell, as is afore$aid, the <fig> Pri$me A C D B to be placed, and in the re$t of the Space the Water to be dif- fu$ed as far as the Levell E A: and rai- $ing the Solid, let it be transferred to G M, and let the Water be aba$ed from E A to N O: I $ay, that the de$cent of the Water, mea$ured by the Line A O, hath the $ame proportion to the ri$e of the Pri$me, mea$ured by the Line G A, as the Ba$e of the Solid G H hath to the Surface of the Water N O. The which is manife$t: becau$e the Ma$s of the Solid G A B H, rai$ed above the fir$t Levell E A B, is equall to the Ma$s of Water that is aba$ed E N O A. Therefore, E N O A and G A B H are two equall Pri$mes; for of equall Pri$mes, the Ba$es an$wer contrarily to their heights: There- fore, as the Altitude A O is to the Altitude A G, $o is the Superfi- cies or Ba$e G H to the Surface of the Water N O. If therefore, for example, a Pillar were erected in a wa$te Pond full of Water, or el$e in a Well, capable of little more then the Ma$s of the $aid Pillar, in elevating the $aid Pillar, and taking it out of the Water, according as it ri$eth, the Water that invirons it will gradually abate, and the aba$ement of the Water at the in$tant of lifting out the Pillar, $hall have the $ame proportion, that the thickne$s of the Pillar hath to the exce$s of the breadth of the $aid Pond or Well, above the thickne$s of the $aid Pillar: $o that if the breadth of the Well were an eighth part larger than the thickne$s of the Pillar, and the <marg>Why a Solid le$s grave <I>in $pe- cie</I> than water, $tayeth not un- der water, in ve- ry $mall depthst.</marg> breadth of the Pond twenty five times as great as the $aid thickne$s, in the Pillars a$cending one foot, the water in the Well $hall de$cend $even foot, and that in the Pond only 1/25 of a foot.</P> <P>This Demon$trated, it will not be difficult to $hew the true cau$e, how it comes to pa$s, that,</P> <foot>Ggg 2 THEO-</foot> <p n=>412</p> <head>THEOREME III.</head> <P><I>A Pri$me or regular Cylinder, of a $ub$tance $pecifically le$s grave than Water, if it $hould be totally $ubmerged in Water, $tayes not underneath, but ri$eth, though the Water circumfu$ed be very little, and in ab$olute Gravity, never $o much inferiour to the Gravity of the $aid Pri$me.</I></P> <P>Let then the Pri$me A E F B, be put into the Ve$$ell C D F B, the $ame being le$s grave <I>in $pecie</I> than the Water: and let the Water infu$ed ri$e to the height of the Pri$me: I $ay, that the Pri$me left at liberty, it $hall ri$e, being born up by the Water circumfu$ed C D E A. For the <fig> Water C E being $pecifically more grave than the Solid A F, the ab$olute weight of the water C E, $hall have greater proportion to the ab$o- lute weight of the Pri$me A F, than the Ma$s C E hath to the Ma$s A F (in regard the Ma$s hath the $ame proportion to the Ma$s, that the weight ab$olute hath to the weight ab$olute, in ca$e the Ma$$es are of the $ame Gravity <I>in $pecie.</I>) But the Ma$s C E is to the Ma$s A F, as the Surface of the water A C, is to the Superficies, or Ba$e of the Pri$me A B; which is the $ame pro- portion as the a$cent of the Pri$me when it ri$eth, hath to the de$cent of the water circumfu$ed C E.</P> <P>Therefore, the ab$olute Gravity of the water C E, hath greater proportion to the ab$olute Gravity of the Pri$me A F; than the A$cent of the Pri$me A F, hath to the de$cent of the $aid water C E. The Moment, therefore, compounded of the ab$olute Gravity of the water C E, and of the Velocity of its de$cent, whil$t it forceably repul$eth and rai$eth the Solid A F, is greater than the Moment compounded of the ab$olute Gravity of the Pri$me A F, and of the Tardity of its a$cent, with which Moment it contra$ts and re- fi$ts the repul$e and violence done it by the Moment of the water: Therefore, the Pri$me $hall be rai$ed.</P> <marg>The Proportion according to which the Sub- mer$ion & Na tation of Solids is made.</marg> <P>It followes, now, that we proceed forward to demon$trate more particularly, how much $uch Solids $hall be inferiour in Gravity to the water elevated; namely, what part of them $hall re$t $ubmerged, and what $hall be vi$ible above the Surface of the water: but fir$t it is nece$$ary to demon$trate the $ub$equent Lemma.</P> <foot>LEMM</foot> <p n=>413</p> <head>LEMMA I.</head> <P><I>The ab$olute Gravities of Solids, have a proportion com-</I> <marg>The ab$olute Gravity of So- lids, are in a pro- portion com- pounded of their Specifick Gravi- ties, and of their Ma$$es.</marg> <I>pounded of the proportions of their $pecificall Gravities, and of their Ma$$es.</I></P> <P>Let A and B be two Solids. I $ay, that the Ab$olute Gravity of A, hath to the Ab$olute Gravity of B, a proportion com- pounded of the proportions of the $pecificall Gravity of A, to the Specificall Gravity of B, and of the Ma$s A to the Ma$s B. Let the Line D have the <fig> $ame proportion to E, that the $pecifick Gravity of A, hath to the $pecifick Gravity of B; and let E be to F, as the Ma$s A to the Ma$s B: It is manife$t, that the proportion of D to F, is compounded of the proportions D and E; and E and F. It is requi$ite, therefore, to demon$trate, that as D is to F, $o the ab$olute Gravity of A, is to the ab$olute Gravity of B. Take the Solid C, equall in Ma$s to the Solid A, and of the $ame Gravity <I>in $pecie</I> with the Solid B. Becau$e, therefore, A and C are equall in Ma$s, the ab$olute Gravity of A, $hall have to the ab$olute Gravity of C, the $ame pro- portion, as the $pecificall Gravity of A, hath to the $pecificall Gravity of C, or of B, which is the $ame <I>in $pecie</I>; that is, as D is to E. And, be- cau$e, C and B are of the $ame Gravity <I>in $pecie,</I> it $hall be, that as the ab$olute weight of C, is to the ab$olute weight of B, $o the Ma$s C, or the Ma$s A, is to the Ma$s B; that is, as the Line E to the Line F. As therefore, the ab$olute Gravity of A, is to the ab$olute Gravity of C, $o is the Line D to the Line E: and, as the ab$olute Gravity of C, is to the ab$olute Gravity of B, $o is the Line E to the Line F: Therefore, by Equality of proportion, the ab$olute Gra- vity of A, is to the ab$olute Gravity of B, as the Line D to the Line F: which was to be demon$trated. I proceed now to demon- $trate, how that,</P> <foot>THEO-</foot> <p n=>414</p> <head>THEOREME IV.</head> <marg>The proportion of water requi- $ite to make a Solid $wim.</marg> <P><I>If a Solid, Cylinder, or Pri$me, le$$e grave $pecifically than the Water, being put into a Ve$$el, as above, of what$oever greatne$$e, and the Water, be afterwards infu$ed, the Solid $hall re$t in the bottom, unrai$ed, till the Water arrive to that part of the Altitude, of the $aid Pri$me, to which its whole Altitude hath the $ame proportion, that the Specificall Gravity of the Water, hath to the Specificall Gravity of the $aid Solid: but infu$ing more Water, the Solid $hall a$cend.</I></P> <P>Let the Ve$$ell be M L G N of any bigne$s, and let there be pla- ced in it the Solid Pri$me D F G E, le$s grave <I>in $pecie</I> than the water; and look what proportion the <I>S</I>pecificall Gravity of the water, hath to that of the Pri$me, $uch let the Altitude D F, have to the Altitude F B. I $ay, that infu$ing water to the Altitude F B, the Solid D G $hall not float, but $hall $tand in <I>Equilibrium,</I> $o, that that every little quantity of water, that is infu$ed, $hall rai$e it. Let the water, therefore, be infu$ed to the Levell A B C, and, becau$e the Specifick Gravity of the Solid D G, is to the Specifick Gravity of the water, as the altitude B F is to the altitude F D; that is, as the Ma$s B G to the Ma$s G D; as the proportion of the Ma$s B G is to the Ma$s G D, as the proportion of the Ma$s G D is to the Ma$s A F, they compo$e the Proportion of the Ma$s B G to the Ma$s A F. Therefore, the Ma$s B G is to the Ma$s A F, in a proportion compounded of the proportions of the Specifick Gravity of the Solid G D, to the Speci- fick Gravity of the water, and of the Ma$s G D to the Ma$s A F: But the $ame proportions <fig> of the Specifick Gravity of G D, to the Specifick Gravity of the water, and of the Ma$s G D to the Ma$s A F, do al$o by the precedent <I>Lemma,</I> compound the proportion of the ab$olute Gra- vity of the Solid D G, to the ab$olute Gravity of the Ma$s of the water A F: Therefore, as the Ma$s B G is to the Ma$s A F, $o is the Ab$olute Gravity of the Solid D G, to the Ab- $olute Gravity of the Ma$s of the water A F. But as the Ma$s B G is to the Ma$s A F; $o is the Ba$e of the Pri$me D E, to the Surface of the water AB; and $o is the de$cent of the water A B, to the Elevation of the Pri$me D G; Therefore, the de$cent of the <foot>water</foot> <p n=>415</p> water is to the elevation of the Pri$me, as the ab$olute Gravity of the Pri$me, is to the ab$olute Gravity of the water: Therefore, the Moment re$ulting from the ab$olute Gravity of the water A F, and the Velocity of the Motion of declination, with which Moment it forceth the Pri$me D G, to ri$e and a$cend, is equall to the Moment that re$ults from the ab$olute Gravity of the Pri$me D G, and from the Velocity of the Motion, wherewith being rai$ed, it would a$cend: with which Moment it re$i$ts its being rai$ed: becau$e, therefore, $uch Moments are equall, there $hall be an <I>Equilibrium</I> between the water and the Solid. And, it is manife$t, that putting a little more water unto the other A F, it will increa$e the Gravity and Moment, whereupon the Pri$me D G, $hall be overcome, and elevated till that the only part B F remaines $ubmerged. Which is that that was to be demon$trated.</P> <head>COROLLARY I.</head> <P><I>By what hath been demon$trated, it is manife$t, that Solids le$s grave</I> <marg><I>H</I>ow far Solids le$s grave <I>in $pe- cie</I> than water, do $ubmerge.</marg> in $pecie <I>than the water, $ubmerge only $o far, that as much water in Ma$s, as is the part of the Solid $ubmerged, doth weigh ab$olutely as much as the whole Solid.</I></P> <P>For, it being $uppo$ed, that the Specificall Gravity of the water, is to the Specificall Gravity of the Pri$me D G, as the Altitude D F, is to the Altitude F B; that is, as the Solid D G is to the Solid B G; we might ea$ily demon$trate, that as much water in Ma$s as is equall to the Solid B G, doth weigh ab$olutely as much as the whole Solid D G; For, by the <I>Lemma</I> foregoing, the Ab$olute Gravity of a Ma$s of water, equall to the Ma$s B G, hath to the Ab- $olute Gravity of the Pri$me D G, a proportion compounded of the proportions, of the Ma$s B G to the Ma$s G D, and of the Specifick Gravit 7 of the water, to the Specifick Gravity of the Pri$me: But the Gravity <I>in $pecie</I> of the water, to the Gravity <I>in $pecie</I> of the Pri$me, is $uppo$ed to be as the Ma$s G D to the Ma$s G B. There- fore, the Ab$olute Gravity of a Ma$s of water, equall to the Ma$s B G, is to the Ab$olute Gravity of the Solid D G, in a proportion compounded of the proportions, of the Ma$s B G to the Ma$s G D, and of the Ma$s D G to the Ma$s G B; which is a proportion of equalitie. The Ab$olute Gravity, therefore, of a Ma$s of Water equall to the part of the Ma$s of the Pri$me B G, is equall to the Ab- $olute Gravity of the whole Solid D G.</P> <foot>COROL-</foot> <p n=>416</p> <head>COROLLARY II.</head> <marg><I>A</I> Rule to equi- librate <I>S</I>olids in the water.</marg> <P><I>It followes, moreover, that a Solid le$s grave than the water, being put into a Ve$$ell of any imaginable greatne$s, and water being circumfu$ed about it to $uch a height, that as much water in Ma$s, as is the part of the Solid $ubmerged, do<*> weigh ab$olutely as much as the whole Solid; it $hall by that water be ju$tly $u$tained, be the circumfu$ed Water in quantity greater or le$$er.</I></P> <P>For, if the Cylinder or Pri$me M, le$s grave than the water, <I>v. gra.</I> in Sub$equiteriall proportion, $hall be put into the capaci- ous Ve$$ell A B C D, and the water rai$ed about it, to three quarters of its height, namely, to its Levell A D: it $hall be $u$tained and exactly poy$ed in <I>Equi- librium.</I> The $ame will hap- pen, if the Ve$$ell E N S F <fig> were very $mall, $o, that be- tween the Ve$$ell and the So- lid M, there were but a very narrow $pace, and only capable of $o much water, as the hundredth part of the Ma$s M, by which it $hould be likewi$e rai$ed and erected, as before it had been elevated to three fourths of the height of the Solid: which to many at the fir$t $ight, may $eem a notable Paradox, and beget a conceit, that the Demon$tration of the$e effects, were $ophi$ticall and fallacious: but, for tho$e who $o repute it, the Ex- periment is a means that may fully $atisfie them. But he that $hall but comprehend of what Importance Velocity of Motion is, and how it exactly compen$ates the defect and want of Gravity, will cea$e to wonder, in con$idering that at the elevation of the Solid M, the great Ma$s of water A B C D abateth very little, but the little Ma$s of water E N S F decrea$eth very much, and in an in$tant, as the Solid M before did li$e, howbeit for a very $hort $pace: Whereupon the Moment, compounded of the $mall Ab$olute Gravity of the water E N S F, and of its great Velocity in ebbing, equalizeth the Force and and Moment, that re$ults from the compo$icion of the immen$e Gra- vity of the water A B C D, with its great $lowne$$e of ebbing; $ince that in the Elevation of the Sollid M, the aba$ement of the le$- <marg><I>T</I>he proportion according to which water ri- $eth and falls in different Ve$$els at the Immer$i- on and Elevati- on of <I>s</I>olids.</marg> $er water E S, is performed ju$t $o much more $wiftly than the great Ma$s of water A C, as this is more in Ma$s than that which we thus demon$trate.</P> <P>In the ri$ing of the Solid M, its elevation hath the $ame proportion to the circumfu$ed water E N S F, that the Surface of the $aid water, hath to the Superficies or Ba$e of the $aid Solid M; which Ba$e hath the $ame proportion to the Surface of the water A D, that the aba$e- <foot>ment</foot> <p n=>417</p> ment or ebbing of the water A C, hath to the ri$e or elevation of the $aid Solid M. Therefore, by Perturbation of proportion, in the a$cent of the $aid Solid M, the aba$ement of the water A B C D, to the aba$ement of the water E N S F, hath the $ame proportion, that the Surface of the water E F, hath to the Surface of the water A D; that is, that the whole Ma$s of the water E N S F, hath to the whole Ma$s A B C D, being equally high: It is manife$t, therefore, that in the expul$ion and elevation of the Solid M, the water E N S F $hall exceed in Velocity of <I>M</I>otion the water A B C D, a$much as it on the other $ide is exceeded by that in quantity: whereupon their Moments in $uch operations, are mutually equall.</P> <P><I>And, for ampler confirmation, and clearer explication of this, let us con$ider the pre$ent Figure, (which if I be not deceived, may $erve to detect the errors of $ome Practick Mechanitians, who upon a fal$e founda- tion $ome times attempt impo$$ible enterprizes,) in which, unto the large Ve$$ell E I D F, the narrow Funnell or Pipe I C A B is continued, and $up- po$e water infu$ed into them, unto the Levell L G H, which water $hall re$t in this po$ition, not without admiration in $ome, who cannot conceive</I> <fig> <I>how it can be, that the heavie charge of the great Ma$s of water G D, pre$$ing downwards, $hould not elevate and repul$e the little quantity of the other, contained in the Funnell or Pipe C L, by which the de$cent of it is re$isted and hindered: But $uch wonder $hall cea$e, if we begin to $uppo$e the water G D to be aba$ed only to Q D, and $hall afterwards con$ider, what the water C L hath done, which to give place to the other, which is de$cended from the Levell G H, to the Levell Q O, $hall of nece$$ity have a$cended in the $ame time, from the Levell Lunto A B. And the a$cent L B, $hall be $o much greater than the de- $cent G Q, by how much the breadth of the Ve$$ell G D, is greater than that of the Funnell I C; which, in $umme, is as much as the water G D, is more than the water L C: but in regard that the Moment of the Velocity of the Motion, in one Moveable, compen$ates that of the Gravity of ano- ther, what wonder is it, if the $wift a$cent of the le$$er Water C L, $hall re$i$t the $low de$cent of the greater G D</I>?</P> <P>The $ame, therefore, happens in this operation, as in the Stilliard, in which a weight of two pounds counterpoy$eth an other of 200, asoften as that $hall move in the $ame time, a $pace 100 times great- er than this: which falleth out when one Arme of the Beam is an <foot>Hhh hundred</foot> <p n=>418</p> hundred times as long as the other. Let the erroneous opinion o <marg>A $hip flotes as well in ten Tun of water as in an Ocean.</marg> tho$e therefore cea$e, who hold that a Ship is better, and ea$ter born up in a great abundance of water, then in a le$$er quantity, (<I>this was believed by</I> Ari$totle <I>in his Problems, Sect. 23, Probl.</I> 2.) it being or the contrary true, that its po$$ible, that a Ship may as well float in ten Tun of water, as in an Ocean.</P> <marg>A Solid $peci- fiaclly graver than the water, cannot be born up by any quan- tity of it.</marg> <P>But following our matter, I $ay, that by what hath been hitherto demon$trated, we may under$tand how, that</P> <head>COROLL-ARY III.</head> <P><I>One of the above named Solids, when more grave</I> in $pecie <I>than the water, can never be $u$tained, by any whatever quantity of it.</I></P> <P>For having $een how that the Moment wherewith $uch a Solid as grave <I>in $pecie</I> as the water, contra$ts with the Moment of any Ma$s of water what$oever, is able to retain it, even to its totall Submer$ion: without its ever a$cending; it remaineth, manife$t, that the water is far le$s able to rai$e it up, when it exceeds the $ame <I>in $pecie</I>: $o<*> that though you infu$e water till its totall Submer$ion, it $hall $till $tay at the Bottome, and with $uch Gravity, and Re$i$tance to Eleva- tion, as is the exce$s of its Ab$olute Gravity, above the Ab$olute Gra- vity of a Ma$s equall to it, made of water, or of a Matter <I>in $pecie</I> equally grave with the water: and, though you $hould moreover adde never $o much water above the Levell of that which equalizeth the Altitude of the Solid, it $hall not, for all that, encrea$e the Pre$$ion or Gravitation, of the parts circumfu$ed about the $aid Solid, by which greater pre$$ion, it might come to be repul$ed, becau$e, the Re$i$tance is not made, but only by tho$e parts of the water, which at the Motion of the $aid Solid do al$o move, and the$e are tho$e only, which are comprehended by the two Superficies equidi$tant to the Horizon, and their parallels, that comprehend the Altitude of the Solid immerged in the water.</P> <P>I conceive, I have by this time $ufficiently declared and opened the way to the contemplation of the true, intrin$ecall and proper Cau$es of diver$e Motions, and of the Re$t of many Solid Bodies i<*> diver$e <I>Mediums,</I> and particularly in the water, $hewing how all i<*> effect, depend on the mutuall exce$$es of the Gravity of the Movea- bles and of the <I>Mediums</I>: and, that which did highly import, re- moving the Objection, which peradventure would have begotter much doubting, and $cruple in $ome, about the verity of my Con- clu$ion, namely, how that notwith$tanding, that the exce$s of the Gravity of the water, above the Gravity of the Solid, demitted into it, be the cau$e of its floating and ri$ing from the Bottom to the Sur- face, yet a quantity of water, that weighs not ten pounds, can rai$e <foot>Solid</foot> <p n=>419</p> Solid that weighs above 100 pounds: in that we have demon$tra- ted, That it $ufficeth, that $uch difference be found between the Specificall Gravities of the <I>Mediums</I> and Moveables, let the particular and ab$olute Gravities be what they will: in$omuch, that a Solid, provided that it be Specifically le$s grave than the water, although its ab$olute weight were 1000 pounds, yet may it be born up and elevated by ten pounds of water, and le$s: and on the contrary, a- nother Solid, $o that it be Specifically more grave than the water, though in ab$olute Gravity it were not above a pound, yet all the water in the Sea, cannot rai$e it from the Bottom, or float it. This $ufficeth me, for my pre$ent occa$ion, to have, by the above declared Examples, di$covered and demon$trated, without extending $uch matters farther, and, as I might have done, into a long Treati$e: yea, but that there was a nece$$ity of re$olving the above propo$ed doubt, I $hould have contented my $elf with that only, which is demon$trated by <I>Archimedes,</I> in his fir$t Book <I>De In$identibus hu- mido</I>: where in generall termes he infers and confirms the $ame <marg><I>Of Natation</I></marg> <marg>(a) <I>Lib. 1. Prop.</I> 4.</marg> Conclu$ions, namely, that Solids (<I>a</I>) le$s grave than water, $wim or <marg>(b) <I>Id. Lib. 1. Prop.</I> 3.</marg> float upon it, the (<I>b</I>) more grave go to the Bottom, and the (<I>c</I>) e- <marg>(c) <I>Id. Lib. 1. Prop.</I> 3.</marg> qually grave re$t indifferently in all places, yea, though they $hould be wholly under water.</P> <P>But, becau$e that this Doctrine of <I>Archimedes,</I> peru$ed, tran$cri- <marg>The <I>Authors</I> defence of <I>Ar- chimedes</I> his Do- ctrine, again$t the oppo$itions of <I>Buonamico.</I></marg> bed and examined by <I>Signor France$co Buonamico,</I> in his <I>fifth Book of Motion, Chap.</I> 29, and afterwards by him confuted, might by the Authority of $o renowned, and famous a Philo$opher, be rendered dubious, and $u$pected of fal$ity; I have judged it nece$$ary to de- fend it, if I am able $o to do, and to clear <I>Archimedes,</I> from tho$e cen$ures, with which he appeareth to be charged. <I>Buonamico</I> re- <marg>His fir$t Objecti- on again$t the Doctrine of <I>Ar- chimedes.</I></marg> jecteth the Doctrine of <I>Archimedes,</I> fir$t, as not con$entaneous with the Opinion of <I>Aristotle,</I> adding, that it was a $trange thing to him, <marg>His Second Ob- jection.</marg> that the Water $hould exceed the Earth in Gravity, $eeing on the contrary, that the Gravity of water, increa$eth, by means of the parti- <marg>His third Obje- ction.</marg> cipation of Earth. And he $ubjoyns pre$ently after, that he was not $atisfied with the Rea$ons of <I>Archimedes,</I> as not being able with that Doctrine, to a$$ign the cau$e whence it comes, that a Boat and a Ve$$ell, which otherwi$e, floats above the water, doth $ink to the Bottom, if once it be filled with water; that by rea$on of the e- quality of Gravity, between the water within it, and the other water without, it $hould $tay a top; but yet, neverthele$s, we $ee it to go to the Bottom.</P> <marg>His $ourth Ob- jection.</marg> <P>He farther addes, that <I>Ari$totle</I> had clearly confuted the Ancients, who $aid, that light Bodies moved upwards, driven by the impul$e <marg>The <I>Ancients</I> denved <I>Ao$olute</I> Levity.</marg> of the more grave Ambient: which if it were $o, it $hould $eem of nece$$ity to follow, that all naturall Bodies are by nature heavy, <foot>Hhh2 and</foot> <p n=>420</p> and none light: For that the $ame would befall the Fire and Air, if put in the Bottom of the water. And, howbeit, <I>Ari$totle</I> grants a Pul$ion in the Elements, by which the Earth is reduced into a Sphe- ricall Figure, yet neverthele$s, in his judgement, it is not $uch that it can remove grave Bodies from their naturall places, but rather, that it $end them toward the Centre, to which (as he $omewhat ob$curely continues to $ay,) the water principally moves, if it in the interim meet not with $omething that re$i$ts it, and, by its Gravity, thru$ts it out of its place: in which ca$e, if it cannot directly, yet at lea$t as well as it can, it tends to the Centre: but it happens, that light Bodies by $uch Impul$ion, do all a$cend upward: but this properly they have by nature, as al$o, that other of $wimming. He concludes, <marg>The cau$es of Natation & Sub- mer$ion, accord- ing to the Peri- pateticks.</marg> la$tly, that he concurs with <I>Archimedes</I> in his Conclu$ions; but not in the Cau$es, which he would referre to the facile and difficult Sepa- ration of the <I>Medium,</I> and to the predominance of the Elements, $o that when the Moveable $uperates the power of the <I>Medium</I>; as for example, Lead doth the Continuity of water, it $hall move thorow it, el$e not.</P> <P>This is all that I have been able to collect, as produced again$t <I>Archimedes</I> by <I>Signor Buonamico</I>: who hath not well ob$erved the Principles and Suppo$itions of <I>Archimedes</I>; which yet mu$t be fal$e, if the Doctrine be fal$e, which depends upon them; but is contented to alledge therein $ome Inconveniences, and $ome Repug- nances to the Doctrine and Opinion of <I>Ari$totle.</I> In an$wer to which Objections, I $ay, fir$t, That the being of <I>Archimedes</I> Doctrine, $im- <marg>The Authors an- $wer to the fir$t Objection.</marg> ply different from the Doctrine of <I>Ari$totle,</I> ought not to move any to $u$pect it, there being no cau$e, why the Authority of this $hould be preferred to the Authority of the other: but, becau$e, where the decrees of Nature are indifferently expo$ed to the intellectuall eyes of each, the Authority of the one and the other, lo$eth all anthentical- ne$s of Per$wa$ion, the ab$olute power re$iding in Rea$on; therefore I pa$s to that which he alledgeth in the $econd place, as an ab$urd con- <marg>The Authors an- $wer to the $e- cond Objection.</marg> $equent of the Doctrine of <I>Archimedes,</I> namely, That water $hould be more grave than Earth. But I really find not, that ever <I>Archi- medes</I> $aid $uch a thing, or that it can be rationally deduced from his Conclu$ions: and if that were manife$t unto me, I verily believe, I $hould renounce his Doctrine, as mo$t erroneous. Perhapsthis Dedu- ction of <I>Buonamico,</I> is founded upon that which he citeth of the Ve- $$el, which $wims as long as its voyd of water, but once full it $inks to the Bottom, and under$tanding it of a Ve$$el of Earth, he infers again$t <I>Archimedes</I> thus: Thou $ay$t that the Solids which $wim, are le$s grave than water: this Ve$$ell $wimmeth: therefore, this Ve$$ell is le$$e grave than water. If this be the Illation. I ea$ily an$wer, granting that this Ve$$ell is le$$e grave than water, and denying the other con$equence, <foot>namely,</foot> <p n=>421</p> namely, that Earth is le$s Grave than Water. The Ve$$el that $wims occupieth in the water, not only a place equall to the Ma$s of the Earth, of which it is formed; but equall to the Earth and to the Air together, contained in its concavity. And, if $uch a Ma$s compoun- ded of Earth and Air, $hall be le$s grave than $uch another quantity of water, it $hall $wim, and $hall accord with the Doctrine of <I>Archi- medes</I>; but if, again, removing the Air, the Ve$$ell $hall be filled with water, $o that the Solid put in the water, be nothing but Earth, nor occupieth other place, than that which is only po$$e$t by Earth, it $hall then go to the Bottom, by rea$on that the Earth is heavier than the water: and this corre$ponds well with the meaning of <I>Archimedes.</I> See the $ame effect illu$trated, with $uch another Experiment, In pre$$ing a Viall Gla$s to the Bottom of the water, when it is full of Air, it will meet with great re$i$tance, becau$e it is not the Gla$s alone, that is pre$$ed under water, but together with the Gla$s a great Ma$s of Air, and $uch, that if you $hould take as much water, as the Ma$s of the Gla$s, and of the Air contained in it, you would have a weight much greater than that of the Viall, and of its Air: and, therefore, it will not $ubmerge without great violence: but if we demit only the Gla$s into the water, which $hall be when you $hall fill the Gla$s with water, then $hall the Gla$s de$cend to the Bottom; as $uperiour in Gravity to the water.</P> <P>Returning, therefore, to our fir$t purpo$e; I $ay, that Earth is more grave than water, and that therefore, a Solid of Earth goeth to the bottom of it; but one may po$$ibly make a compo$ition of Earth and Air, which $hall be le$s grave than a like Ma$s of Water; and this $hall $wim: and yet both this and the other experiment $hall very well accord with the Doctrine of <I>Archimedes.</I> But becau$e that in my judgment it hath nothing of difficulty in it, I will not po$itive- ly affirme that <I>Signor Buonamico,</I> would by $uch a di$cour$e object unto <I>Archimedes</I> the ab$urdity of inferring by his doctrine, that Earth was le$s grave than Water, though I know not how to conceive what other accident he could have induced thence.</P> <P>Perhaps $uch a Probleme (in my judgement fal$e) was read by <I>Signor Buonamico</I> in $ome other Author, by whom peradventure it was attributed as a $ingular propertie, of $ome particular Water, and $o comes now to be u$ed with a double errour in confutation of <I>Ar- chimedes,</I> $ince he $aith no $uch thing, nor by him that did $ay it was it meant of the common Element of Water.</P> <P>The third difficulty in the doctrine of <I>Archimedes</I> was, that he <marg><I>T</I>he Authors an- $wer to the third Objection.</marg> could not render a rea$on whence it aro$e, that a piece of Wood, and a Ve$$ell of Wood, which otherwi$e floats, goeth to the bottom, if filled with Water. <I>Signor Buonamico</I> hath $uppo$ed that a Ver$$ell of Wood, and of Wood that by nature $wims, as before is $aid, <foot>goes</foot> <p n=>422</p> goes to the bottom, if it be filled with water; of which he in the fol- lowing Chapter, which is the 30 of the fifth Book copiou$ly di$cour$- eth: but I ($peaking alwayes without diminution of his $ingular Learning) dare in defence of <I>Archimedes</I> deny this experiment, being certain that a piece of Wood which by its nature $inks not in Water, $hall not $inke though it be turned and converted into the forme of a- ny Ve$$ell what$oever, and then filled with Water: and he that would readily $ee the Experiment in $ome other tractable Matter, and that is ea$ily reduced into $everal Figures, may take pure Wax, and ma- king it fir$t into a Ball or other $olid Figure, let him adde to it $o much Lead as $hall ju$t carry it to the bottome, $o that being a graine le$s it could not be able to $inke it, and making it afterwards into the forme of a Di$h, and filling it with Water, he $hall finde that with- out the $aid Lead it $hall not $inke, and that with the Lead it $hall de- $cend with much $lowne$s: & in $hort he $hall $atisfie him$elf, that the Water included makes no alteration. I $ay not all this while, but that its po$$ible of Wood to make Barkes, which being filled with water, $inke; but that proceeds not through its Gravity, encrea$ed by the Water, but rather from the Nailes and other Iron Workes, $o that it no longer hath a Body le$s grave than Water, but one mixt of Iron and Wood, more grave than a like Ma$$e of Water. Therefore let <I>Signor Buonamico</I> de$i$t from de$iring a rea$on of an effect, that is not in nature: yea if the $inking of the Woodden Ve$$ell when its full of Water, may call in que$tion the Doctrine of <I>Archimedes,</I> which he would not have you to follow, is on the contrary con$onant and a- greeable to the Doctrine of the Peripateticks, $ince it aptly a$$ignes a rea$on why $uch a Ve$$ell mu$t, when its full of Water, de$cend to the bottom; converting the Argument the other way, we may with $afety $ay that the Doctrine of <I>Archimedes</I> is true, $ince it aptly agre- eth with true experiments, and que$tion the other, who$e Deducti- ons are fa$tened upon etroneou$s Conclu$ions. As for the other point hinted in this $ame In$tance, where it $eemes that <I>Benonamico</I> under- $tands the $ame not only of a piece of wood, $haped in the forme of a Ve$$ell, but al$o of ma$$ie Wood, which filled, <I>$cilicet,</I> as I believe, he would $ay, $oaked and $teeped in Water, goes finally to the bottom that happens in $ome poro$e Woods, which, while their Poro$ity is re- pleni$hed with Air, or other Matter le$s grave than Water, are Ma$- $es $pecificially le$s grave than the $aid Water, like as is that Viall of Gla$s while$t it is full of Air: but when, $uch light Matter depart- ing, there $ucceedeth Water into the $ame Poro$ities and Cavities, there re$ults a compound of Water and Gla$s more grave than a like Ma$s of Water: but the exce$s of its Gravity con$i$ts in the Matter of the Gla$s, and not in the Water, which cannot be graver than it $elf: $o that which remaines of the Wood, the Air of its Cavi- <foot>tyes</foot> <p n=>423</p> ties departing, if it $hall be more grave <I>in $pecie</I> than Water, fil but its Poro$ities with Water, and you $hal have a Compo$t of Water and of Wood more grave than Water, but not by vertue of the Water re- ceived into and imbibed by the Poro$ities, but of that Matter of the Wood which remains when the Air is departed: and being $uch it $hall, according to the Doctrine of <I>Archimedes,</I> goe to the bottom, like as before, according to the $ame Doctrine it did $wim.</P> <P>As to that finally which pre$ents it $elf in the fourth place, namely, <marg>The Authors an$wer to the fourth Object- ion.</marg> that the <I>Ancients</I> have been heretofore confuted by <I>Ari$totle,</I> who denying Po$itive and Ab$olute Levity, and truely e$teeming all Bo- dies to be grave, $aid, that that which moved upward was driven by the circumambient Air, and therefore that al$o the Doctrine of <I>Archimedes,</I> as an adherent to $uch an Opinion was con- victed and confuted: I an$wer fir$t, that <I>Signor Buonamico</I> in my judgement hath impo$ed upon <I>Archimedes,</I> and deduced from his words more than ever he intended by them, or may from his Propo- $itions be collected, in regard that <I>Archimedes</I> neither denies, nor ad- mitteth Po$itive Levity, nor doth he $o much as mention it: $o that much le$s ought <I>Buonamico</I> to inferre, that he hath denyed that it might be the Cau$e and Principle of the A$cen$ion of Fire, and other Light Bodies: having but only demon$trated, that Solid Bodies <marg>Of Natation, Lib. 1. Prop. 7.</marg> more grave than Water de$cend in it, according to the exce$s of their Gravity above the Gravity of that, he demon$trates likewi$e, how the <marg>Of Natation, Lib. 1. Prop. 4.</marg> le$s grave a$cend in the $ame Water, accordng to its exce$s of Gra- ty, above the Gravity of them. So that the mo$t that can be gather- ed from the Dem on$tration of <I>Archimedes</I> is, that like as the exce$s of the Gravity of the Moveable above the Gravity of the Water, is the Cau$e that it de$cends therein, $o the exce$s of the Gravity of the water above that of the Moveable, is a $ufficient Cau$e why it de$- cends not, but rather betakes it $elf to $wim: not enquiring whe- ther of moving upwards there is, or is not any other Cau$e contrary to Gravity: nor doth <I>Archimedes</I> di$cour$e le$s properly than if one $hould $ay: If the South Winde $hall a$$ault the Barke with greater <I>Impetus</I> than is the violence with which the Streame of the River car- ries it towards the South, the motion of it $hall be towards the North: but if the <I>Impetus</I> of the Water $hall overcome that of the Winde, its motion $hall be towards the South. The di$cour$e is excellent and would be unworthily contradicted by $uch as $hould oppo$e it, $aying: Thou mi$-alledge$t as Cau$e of the motion of the Bark towards the South, the <I>Impetus</I> of the Stream of the Water above that of the South Winde; mi$-alledge$t I $ay, for it is the Force of the North Winde oppo$ite to the South, that is able to drive the Bark towards the South. Such an Objection would be $uperfluous, becau$e he which alledgeth for Cau$e of the Motion the $tream of the Water, denies not <foot>but</foot> <p n=>424</p> but that the Winde oppo$ite to the South may do the $ame, but only affirmeth that the force of the Water prevailing over the Sout<*> Wind, the Bark $hall move towards the South: and $aith no more than is true. And ju$t thus when <I>Archimedes</I> $aith, that the Gravity of the Water prevailing over that by which the moveable de$cends to the Bottom, $uch moveable $hall be rai$ed from the Bottom to the Sur- face alledgeth a very true Cau$e of $uch an Accident, nor doth he af- firm or deny that there is, or is not, a vertue contrary to Gravity, called by $ome Levity, that hath al$o a power of moving $ome Matters up wards. Let therefore the Weapons of <I>Signor Buonamico</I> be directed a<*> <marg><I>Plato</I> denyeth Po$itive Levi- ty.</marg> gain$t <I>Plato,</I> and other <I>Ancients,</I> who totally denying <I>Levity,</I> and taking all Bodies to be grave, $ay that the Motion upwards is made, not from an intrin$ecal Principle of the Moveable, but only by the Im- pul$e of the <I>Medium</I>; and let <I>Archimedes</I> and his Doctrine e$cape him, $ince he hath given him no Cau$e of quarelling with him But if this Apologie, produced in defence of <I>Archimedes,</I> $hould $een to $ome in$ufficient to free him from the Objections and Arguments produced by <I>Ari$totle</I> again$t <I>Plato,</I> and the other <I>Ancients,</I> as if they did al$o fight again$t <I>Archimedes,</I> alledging the Impul$e of the Water <marg>The Authors defence of the doctrine of <I>Plato</I> and the <I>Ancients,</I> who ab$olutely deny Levity:</marg> as the Cau$e of the $wimming of $ome Bodies le$s grave than it, I would not que$tion, but that I $hould be able to maintaine the Doctrine of <I>Plato</I> and tho$e others to be mo$t true, who ab$olutely deny Levity, and affirm no other Intrin$ecal Principle of Motion to be in Elemen- tary Bodies $ave only that towards the Centre of the Earth, nor no <marg>According to <I>Plato</I> there is no Principle of the Motion of de- $cent in Naturall Bodies, $ave that to the Centre.</marg> other Cau$e of moving upwards, $peaking of that which hath the re- $emblance of natural Motion, but only the repul$e of the <I>Medium,</I> $luid, and exceeding the Gravity of the Moveable: and as to the Rea$ons of <I>Ari$totle</I> on the contrary, I believe that I could be able fully to <marg>No cau$e of the motion of <I>A</I> cent, $ave the Impul$e of the <I>Medium,</I> exceed- ing the Move- able in Gravi- tie.</marg> an$wer them, and I would a$$ay to do it, if it were ab$olutely nece$$a- ry to the pre$ent Matter, or were it not too long a Digre$$ion for this $hort Treati$e. I will only $ay, that if there were in $ome of our Elle- mentary Bodies an Intrin$ecall Principle and Naturall Inclination to $hun the Centre of the Earth, and to move towards the Concave of the Moon, $uch Bodies, without doubt, would more $wiftly a$cend through tho$e <I>Mediums</I> that lea$t oppo$e the Velocity of the Moveable, and the$e are the more tenuous and $ubtle; as is, for example, the Air in compari$on of the Water, we daily proving that we can with <marg>Bodies a$cend much $wifter in the Water, than in the Air.</marg> farre more expeditious Velocity move a Hand or a Board to and a- gain in one than in the other: neverthele$s, we never could finde any Body, that did not a$cend much more $wiftly in the water than in the <marg>All Bodies a$- cending through Water, lo$e their Motion, comming to the confines of the Air.</marg> Air. Yea of Bodies which we $ee continually to a$cend in the Water, there is none that having arrived to the confines of the Air, do not whol- ly lo$e their Motion; even the Air it $elf, which ri$ing with great Ce- lerity through the Water, being once come to its Region it lo$eth all</P> <foot><I>Im-</I></foot> <p n=>425</p> <P>And, howbeit, Experience $hewes, that the Bodies, $ucce$$ively <marg>The lighter Bodies alcend more $wiftly through Water.</marg> le$s grave, do mo$t expeditiou$ly a$cend in water, it cannot be doubt- ed, but that the Ignean Exhalations do a$cend more $wiftly <marg>Fiery Exhalati- ons atcend tho<*> row the Water more $wiftly than doth the Air; & the Air a$cends more $wiftly thorow the Water, than <I>F</I>ire thorow the Air.</marg> through the water, than doth the Air: which Air is $een by Experi- ence to a$cend more $wiftly through the Water, than the Fiery Exha- lations through the Air: Therefore, we mu$t of nece$$ity conclude, that the $aid Exhalations do much more expeditiou$ly a$cend through the Water, than through the Air; and that, con$equently, they are moved by the Impul$e of the Ambient <I>Medium,</I> and not by an intrin- $ick Principle that is in them, of avoiding the Centre of the Earth; to which other grave Bodies tend.</P> <P>To that which for a finall conclu$ion, <I>Signor Buonamico</I> produceth <marg><I>T</I>he Authors confutation of the Peripateticks Cau$es of Nata- tion & Submer$i- on.</marg> of going about to reduce the de$cending or not de$cending, to the ea$ie and unea$ie Divi$ion of the <I>Medium,</I> and to the predominancy of the Elements: I an$wer, as to the fir$t part, that that cannot in any manner be admitted as a Cau$e, being that in none of the Fluid <I>Mediums,</I> as the Air, the Water, and other Liquids, there is any <marg>Water & other fluids void of Re$i$tance a- gain$t Divi$ion.</marg> Re$i$tance again$t Divi$ion, but all by every the lea$t Force, are di- vided and penetrated, as I will anon demon$trate: $o, that of $uch Re$i$tance of Divi$ion there can be no Act, $ince it $elf is not in be- ing. As to the other part, I $ay, that the predominancy of the Ele- <marg><I>T</I>he predomi- nancy of Ele- ments in Move- ables to be con- $idered only in relation to their excefs or defect of Gravity in reference to the <I>Medium.</I></marg> ments in Moveables, is to be con$idered, as far as to the exce$$e or defect of Gravity, in relation to the <I>Medium</I>: for in that Action, the Elements operate not, but only, $o far as they are grave or light: therefore, to $ay that the Wood of the Firre $inks not, becau$e Air predominateth in it, is no more than to $ay, becau$e it is le$s grave than the Water. Yea, even the immediate Cau$e, is its being le$s grave than the Water: and it being under the predominancy of the <marg><I>T</I>he immedi- ate Cau$e of Na- tation is that the Moveable is le$s grave than the Water.</marg> Air, is the Cau$e of its le$s Gravity: Therefore, he that alledgeth the predominancy of the Element for a Cau$e, brings the Cau$e of the Cau$e, and not the neere$t and immediate Cau$e. Now, who knows not that the true Cau$e is the immediate, and not the mediate? <marg><I>T</I>he <I>P</I>eripate- ticks alledge for the rea$on of Natation the Cau$e of the Cau$e.</marg> Moreover, he that alledgeth Gravity, brings a Cau$e mo$t per$picuous to Sence: The cau$e we may very ea$ily a$$ertain our $elves; whether Ebony, for example, and Firre, be more or le$s grave than water: but whether Earth or Air predominates in them, who $hall <marg>Gravity a Cau$e mo$t per- $picuous to $ence:</marg> make that manife$t? Certainly, no Experiment can better do it than to ob$erve whether they $wim or $ink. So, that he who knows, not whether $uch a Solid $wims, unle$s when he knows that Air pre- dominates in it, knows not whether it $wim, unle$s he $ees it $wim, for then he knows that it $wims, when he knows that it is Air that predominates, but knows not that Air hath the predominance, unle$s he $ees it $wim: therefore, he knows not if it $wims, till $uch time as he hath $een it $wim.</P> <foot>Iii Let</foot> <p n=>426</p> <P>Let us not then de$pi$e tho$e Hints, though very dark, which Rea$on, after $ome contemplation, offereth to our Intelligence, ar<*> lets be content to be taught by <I>Archimecles,</I> that then any Body $hal<*> <marg>Lib 1. of Na- tation Prop. 7.</marg> $ubmergein water, when it $hall be $pecifically more grave than it and that if it $hall be le$s grave, it $hall of nece$$ity $wim, and <marg>Id. Lib. 1. Prop. 4.</marg> that it will re$t indifferently in any place under water, if its Gravity<*> be perfectly like to that of the water.</P> <marg>Id. Lib. 1: Prop. 3.</marg> <P>The$e things explained and proved, I come to con$ider that which offers it $elf, touching what the Diver$ity of figure given unto the $aid Moveable hath to do with the$e Motions and Re$ts; and pro- ceed to affirme, that,</P> <head>THE OREME V.</head> <P><I>The diver$ity of Figures given to this or that Solid</I> <marg>Diver$ity of Figure no Cau$e of its ab$olute Natation or Sub- mer$ion.</marg> <I>cannot any way be a Cau$e of its ab$olute Sinking o<*> Swimming.</I></P> <P>So that if a Solid being formed, for example, into a Spherical Figure, doth $ink or $wim in the water, I $ay, that being formed into any other Fi<*>ure, the $ame figure in the $ame water, $hal<*> $ink or $wim: nor can $uch its Motion by the Expan$ion or by o<*> ther mutation of Figure, be impeded or taken away.</P> <marg>The Expan$i- on of <I>F</I>igure, re- tards the Veloci- ty of the a$cent or de$cent of the Moveable in the water; but doth not deprive it of all Motion.</marg> <P>The Expan$ion of the Figure may indeed retard its Velocity, a$<*> well of a$cent as de$cent, and more and more according as the $aid Fi- gure is reduced to a greater breadth and thinne$s: but that it may bere duced to $uch a form as that that $ame matter be wholly hindred from moving in the $ame water, that I hold to be impo$$ible. In this I have met with great contradictors, who producing $ome Experiments, and in perticular a thin Board of Ebony, and a Ball of the $ame Wood and $hewing how the Ball in Water de$cended to the bottom, and the Board being put lightly upon the Water $ubmerged not, but re$t<*> ed; have held, and with the Authority of <I>Ari$totle,</I> confirmed them $elves in their Opinions, that the Cau$e of that Re$t was the breadt<*> of the Figure, u able by its $mall weight to pierce and penetrate the Re$i$tance of the Waters Cra$$itude, which Re$i$tance is readily o<*> vercome by the other Sphericall Figure.</P> <P>This is the Principal point in the pre$ent Que$tion, in which I per- $wade my $elf to be on the right $ide.</P> <P>Therefore, beginning to inve$tigate with the examination of ex- qui$ite Experiments that really the Figure doth not a jot alter the de$- cent or A$cent of the $ame Solids, and having already demon$tra- ted that the greater or le$s Gravity of the Solid in relation to the Gra- vity of the <I>Medium</I> is the cau$e of De$cent or A$cent: when ever we <foot>would</foot> <p n=>427</p> would make proof of that, which about this Effect the diver$ity of Fi- gure worketh, its nece$$ary to make the Experiment with Matter wherein variety of Gravities hath no place. For making u$e of Mat- ters which may be different in their Specifical Gravities, and meeting with varieties of effects of A$cending and De$cending, we $hall al- wayes be left un$atisfied whether that diver$ity derive it $elf really from the $ole Figure, or el$e from the divers Gravity al$o. We may remedy this by takeing one only Matter, that is tractable and ea$ily reduceable into every $ort of Figure. Moreover, it wil be an excellent expedient to take a kinde of Matter, exactly alike in Gravity unto the Water: for that Matter, as far as pertaines to the Gravity, is in- different either to A$cend or De$cend; $o that we may pre$ently ob- $erve any the lea$t difference that derives it $elf from the diver$ity of Figure.</P> <P>Now to do this, Wax is mo$t apt, which, be$ides its incapacity of <marg>An Experi- ment in Wax, that proveth Fi- gute to have no Operation in Natation & Sub- mer$ion.</marg> receiveing any $en$ible alteration from its imbibing of Water, is duct- ile or pliant, and the $ame piece is ea$ily reduceable into all Figures: and being <I>in $pecie</I> a very incon$iderable matter inferiour in Gravity to the Water, by mixing therewith a little of the fileings of Lead it is reduced to a Gravity exactly equall to that of the Water.</P> <P>This Matter prepared, and, for example, a Ball being made there- of as bigge as an Orange or biger, and that made $o grave as to $ink to the bottom, but $o lightly, that takeing thence one only Grain of Lead, it returnes to the top, and being added, it $ubmergeth to the bottom, let the $ame Wax afterwards be made into a very broad and thin Flake or Cake; and then, returning to make the $ame Ex- periment, you $hall $ee that it being put to the bottom, it $hall, with the Grain of Lead re$t below, and that Grain deducted, it $hall a$cend to the very Surface, and added again it $hall dive to the bottom. And this $ame effect $hall happen alwaies in all $ort of Figures, as wel re- gular as irregular: nor $hall you ever finde any that will $wim with- out the removall of the Grain of Lead, or $inke to the bottom unle$s it be added: and, in $hort, about the going or not going to the Bot- tom, you $hall di$cover no diver$ity, although, indeed, you $hall about the quick and $low de$cent: for the more expatiated and di$tended Figures move more $lowly a$wel in the diveing to the bottom as in the ri$ing to the top; and the other more contracted and compact Fi- gures, more $peedily. Now I know not what may be expected from the diver$ity of Figures, if the mo$t contrary to one another operate not $o much as doth a very $mall Grain of Lead, added or removed.</P> <P>Me thinkes I hear $ome of the Adver$aries to rai$e a doubt upon <marg>An objection a- gain$t the Expe- rim<*>t in Wa<*>s</marg> my produced Experiment. And fir$t, that they offer to my con$idera- tion, that the Figure, as a Figure $imply, and disjunct from the Matter workes not any effect, but requires to be conjoyned with the Matter- <foot>Iii 2 and,</foot> <p n=>428</p> and, furthermore, not with every Matter, but with tho$e on<*> wherewith it may be able ro execute the de$ired operation. L<*> as we $ee it verified by Experience, that the Acute and $harp Angl<*> more apt to cut, than the Obtu$e; yet alwaies provided, that b<*> the one and the other, be joyned with a Matter apt to cut, as example, with Steel. Therefore, a Knife with a fine and $h<*> edge, cuts Bread or Wood with much ea$e, which it will not do the edge be blunt and thick: but he that will in$tead of Steel, t<*> Wax, and mould it into a Knife, undoubtedly $hall never know <*> effects of $harp and blunt edges: becau$e neither of them will c<*> the Wax being unable by rea$on of its flexibility, to overcome <*> hardne$s of the Wood and Bread. And, therefore, applying <*> like di$cour$e to our purpo$e, they $ay, that the difference of Fig will $hew different effects, touching Natation and Submer$ion, <*> not conjoyned with any kind of Matter, but only with tho$e Mat<*> which, by their Gravity, are apt to re$i$t the Velocity of the wat<*> whence he that would elect for the Matter, Cork or other light wo<*> unable, through its Levity, to $uperate the Cra$$itude of the wa<*> and of that Matter $hould forme Solids of divers Figures, woul<*> vain $eek to find out what operation Figure hath in Natation or S<*> mer$ion; becau$e all would $wim, and that not through any prope of this or that Figure, but through the debility of the Matter, wa<*> ing $o much Gravity, as is requi$ite to $uperate and overcome Den$ity and Cra$$itude of the water.</P> <P>Its needfull, therefore, if wee would $ee the effect wrought by Diver$ity of Figure, fir$t to make choice of a Matter of its na<*> apt to penetrate the Cra$$itude of the water. And, for this e$$<*> <marg>An Experi- ment in Ebany, brought to di$- prove the Expe- timent in Wax.</marg> they have made choice of $uch a Matter, as fit, that being readily duced into Sphericall Figure, goes to the Bottom; and it is Ebo of which they afterwards making a $mall Board or Splinter, as thi<*> a Lath, have illu$trated how that this, put upon the Surface of water, re$ts there without de$cending to the Bottom: and making the other$ide, of the $ame wood a Ball, no le$s than a hazell N<*> they $hew, that this $wims not, but de$cendes. From which Exp<*> ment, they think they may frankly conclude, that the Breadth of Figure in the flat Lath or Board, is the cau$e of its not de$cending the Bottom, fora$much as a Ball of the $ame Matter, not diffe<*> from the Board in any thing but in Figure, $ubmergeth in the $a<*> water to the Bottom. The di$cour$e and the Experiment hath rea<*> $o much of probability and likely hood of truth in it, that it would no wonder, if many per$waded by a certain cur$ory ob$ervati<*> $hould yield credit to it; neverthele$s, I think I am able to di$cov<*> how that it is not free from falacy.</P> <P>Beginning, therefore, to examine one by one, all the particulars t<*> <foot>have</foot> <p n=>429</p> have been produced, I $ay, that Figures, as $imple Figures, not only <marg>Figure is un- $eperable from Corporeall Sub- $tance.</marg> operate not in naturall things, but neither are they ever $eperated from the Corporeall $ub$tance: nor have I ever alledged them $tript of $en$ible Matter, like as al$o I freely admit, that in our endeavour- ing to examine the Diver$ity of Accidents, dependant upon the va- riety of Figures, it is nece$$ary to apply them to Matters, which ob- $truct not the various operations of tho$e various Figures: and I ad- mit and grant, that I $hould do very ill, if I would experiment the in- fluence of Acutene$$e of edge with a Knife of Wax, applying it to cut an Oak, becau$e there is no Acutene$s in Wax able to cut that very hard wood. But yet $uch an Experiment of this Knife, would not be be$ides the purpo$e, to cut curded Milk, or other very yielding Matter: yea, in $uch like Matters, the Wax is more commodious than Steel; for finding the diver$ity depending upon Angles, more or le$s Acute, for that Milk is indifferently cut with a Rai$or, and with a Knife, that hath a blunt edge. It needs, therefore, that regard be had, not only to the hardne$s, $olidity or Gravity of Bodies, which under divers figures, are to divide and penetrate $ome Matters, but it forceth al$o, that regard be had, on the other $ide, to the Re$i$tance of the Matters, to be divided and penetrated. But $ince I have in making the Experiment concerning our Conte$t, cho$en a Matter which penetrates the Re$i$tance of the water; and in all figures de$- cendes to the Bottome, the Adver$aries can charge me with no defect; yea, I have propounded $o much a more excellent Method than they, in as much as I have removed all other Cau$es, of de$cending or not de$cending to the Bottom, and retained the only $ole and pure variety of Figures, demon$trating that the $ame Figures all de$cende with the only alteration of a Grain in weight: which Grain being removed, they return to float and $wim; it is not true, therefore, (re$uming the Example by them introduced) that I have gon about to experiment the efficacy of Acutene$s, in cutting with Matters un- able to cut, but with Matters proportioned to our occa$ion; $ince they are $ubjected to no other variety, then that alone which depends on the Figure more or le$s a cute.</P> <marg>The an$wer to the Objection a- gain$t the Expe- riment of the Wax.</marg> <P>But let us proceed a little farther, and ob$erve, how that indeed the Con$ideration, which, they $ay, ought to be had about the Election of the Matter, to the end, that it may be proportionate for the ma- king of our experiment, is needle$ly introduced, declaring by the ex- ample of Cutting, that like as Acutene$s is in$ufficient to cut, unle$s when it is in a Matter hard and apt to $uperate the Re$i$tance of the wood or other Matter, which we intend to cut; $o the aptitude of de$cending or notde$cending in water, ought and can only be known in tho$e Matters, that are able to overcome the Renitence, and $upe- rate the Cra$$itude of the water. Unto which, I $ay, that to make di$tinction and election, more of this than of that Matter, on which to <foot>impre$s</foot> <p n=>430</p> impre$s the Figures for cutting or penetrating this or that Body, as the $olidity or obduratene$s of the $aid Bodies $hall be greater or le$s, is very nece$$ary: but withall I $ubjoyn, that $uch di$tinct- ion, election and caution would be $uperfluous and unprofitable, if the Body to be cut or penetrated, $hould have no Re$i$tance, or $hould not at all with$tand the Cutting or Penitration: and if the Knife were to be u$ed in cutting a Mi$t or Smoak, one of Paper would be equally $erviceable with one of <I>Dama$cus</I> Steel: and $o by rea$on the water hath not any Re$i$tance again$t the Penitration of any Solid Body, all choice of Matter is $uperfluous and needle$s, and the Election which I $aid above to have been well made of a Matter reciprocall in Gravity to water, was not becau$e it was ne- ce$$ary, for the overcoming of the cra$$iitude of the water, but its Gravity, with which only it re$i$ts the $inking of Solid Bodies: and for what concerneth the Re$i$tance of the cra$$itude, if we narrowly con$ider it, we $hall find that all Solid Bodies, as well tho$e that $ink, as tho$e that $wim, are indifferently accomodated and apt to bring us to the knowledge of the truth in que$tion. Nor will I be frighted out of the belief of the$e Conclu$ions, by the Experi- ments which may be produced again$t me, of many $everall Woods, Corks, Galls, and, moreover, of $ubtle $lates and plates of all $orts of Stone and Mettall, apt by means of their Naturall Gravity, to move towards the Centre of the Earth, the which, neverthele$s, be- ing impotent, either through the Figure (as the Adver$aries thinke) or through Levity, to break and penetrate the Continuity of the parts of the water, and to di$tract its union, do continue to $wimm without $ubmerging in the lea$t: nor on the other $ide, $hall the Authority of <I>Ari$totle</I> move me, who in more than one place, a$$ir- meth the contrary to this, which Experience $hews me.</P> <marg>No Solid of $uch Levity, nor of $uch Figure, but that it doth penetrate the Cra$$itude of the Water.</marg> <P>I return, therefore, to a$$ert, that there is not any Solid of $uch Levity, nor of $uch Figure, that being put upon the water, doth not divide and penetrate its Cra$$itude: yea if any with a more per- $picatious eye, $hall return to ob$erve more exactly the thin Boards of Wood, he $hall $ee them to be with part of their thickne$s under <marg>Bodies of all Figures, laid up- on the water, do penetrate its Cra$$itude, and in what propor- tion.</marg> water, and not only with their inferiour Superficies, to ki$$e the Superiour of the water, as they of nece$$ity mu$t have believed, who have $aid, that $uch Boards $ubmerge not, as not being able to di- vide the Tenacity of the parts of the water: and, moreover, he $hall $ee, that $ubtle $hivers of Ebony, Stone or Metall, when they float, have not only broak the Continuity of the water, but are with all their thickne$s, under the Surface of it; and more and more, according as the Matters are more grave: $o that a thin Plate of Lead, $hall be lower than the Surface of the circumfu$ed water, by at lea$t twelve times the thickne$s of the Plate, and Gold $hall dive <foot>below</foot> <p n=>431</p> below the Levell of the water, almo$t twenty times the thickne$s of the Plate, as I $hall anon declare.</P> <P>But let us proceed to evince, that the water yields and $ufters it $elf to be penetrated by every the lighte$t Body; and therewithall demon$trate, how, even by Matters that $ubmerge not, we may come to know that Figure operates nothing about the going or not going to the Bottom, $eeing that the water $uffers it $elf to be penetrated equally by every Figure.</P> <P>Make a Cone, or a Piramis of Cypre$s, of Firre, or of other <marg>The Experi- ment of a Cone, demitted with its Ba$e, and af- ter with its Point down- wards.</marg> Wood of like Gravity, or of pure Wax, and let its height be $ome- what great, namely a handfull, or more, and put it into the water with the Ba$e downwards: fir$t, you $hall $ee that it will penetrate the water, nor $hall it be at all impeded by the largene$s of the Ba$e, nor yet $hall it $ink all under water, but the part towards the point $hall lye above it: by which $hall be manife$t, fir$t, that that Solid forbeares not to $ink out of an inabillity to divide the Continuity of the water, having already divided it with its broad part, that in the opinion of the Adver$aries is the le$s apt to make the divi$ion. The Piramid being thus fixed, note what part of it $hall be $ub- merged, and revert it afterwards with the point downwards, and you $hall $ee that it $hall not dive into the water more than before, but if you ob$erve how far it $hall $ink, every per$on expert in Geometry, may mea$ure, that tho$e parts that remain out of the water, both in the one and in the other Experiment are equall to an hair: whence he may manife$tly conclude, that the acute Figure which $eemed mo$t apt to part and penetrate the water, doth not part or penetrate it more than the large and $pacious.</P> <P>And he that would have a more ea$ie Experiment, let him take two Cylinders of the $ame Matter, one long and $mall, and the o- ther $hert, but very broad, and let him put them in the water, not di$tended, but erect and endways: he $hall $ee, if he diligently mea$ure the parts of the one and of the other, that in each of them the part $ubmerged, retains exactly the $ame proportion to that out of the water, and that no greater part is $ubmerged of that long and $mall one, than of the other more $pacious and broad: howbeit, this re$ts upon a very large, and that upon a very little Superficies of water: therefore the diver$ity of Figure, occa$ioneth neither facility, nor difficulty, in parting and penetrating the Con- tinuity of the water; and, con$equently, cannot be the Cau$e of the Natation or Submer$ion. He may likewi$e di$cover the non- operating of variety of Figures, in ari$ing from the Bottom of the water, towards the Surface, by taking Wax, and tempering it with a competent quantity of the filings of Lead, $o that it may become a con$iderable matter graver than the water: then let him make <foot>it</foot> <p n=>432</p> it into a Ball, and thru$t it unto the Bottom of the water; and fa$ten to it as much Cork, or other light matter, as ju$t $erveth to rai$e it, and draw it towards the Surface: for afterwards changing the $ame Wax into a thin Cake, or into any other Figure, that $ame Cork $hall rai$e it in the $ame manner to a hair.</P> <P>This $ilenceth not my Antagoni$ts, but they $ay, that all the di$cour$e hitherto made by me little importeth to them, and that it $erves their turn, that they have demon$trated in one only parti- cular, and in what matter, and under what Figure plea$eth them, namely, in a Board and in a Ball of Ebony, that this put in the water, de$cends to the Bottom, and that $tays atop to $wim: and the Matter being the $ame, and the two Bodies differing in no- thing but in Figure, they affirm, that they have with all per$picuity demon$trated and $en$ibly manife$ted what they undertook; and la$tly, that they have obtained their intent. Neverthele$s, I believe, and thinke, I can demon$trate, that that $ame Experiment proveth nothing again$t my Conclu$ion.</P> <P>And fir$t, it is fal$e, that the Ball de$cends, and the Board not: <marg>In Experi- ments of Nata- tion, the Solid is to be put into, not upon the water.</marg> for the Board $hall al$o de$cend, if you do to both the Figures, as the words of our Que$tion requireth; that is, if you put them both into the water.</P> <marg>The Que$tion of Natation $ta- ted.</marg> <P><I>The words were the$e. That the Antagoni$ts having an opinion, that the Figure would alter the Solid Bodies, in relation to the de$cending or not de$cending, a$cending or not a$cending in the $ame</I> Medium, <I>as</I> v. gr. <I>in the $ame water, in $uch $ort, that, for Example, a Solid that being of a Sphericall Figure, $hall de$cend to the Bottom, being reduced into $ome other Figure, $hall not de$cend: I holding the contrary, do affirm, that a Corporeall Solid Body, which reduced into a Sphericall Fi- gure, or any other, $hall go to the Bottom, $hall do the like under what$oever other Figure, &c.</I></P> <P>But to be in the water, implies to be placed in the water, and by <marg>Place defined according to <I>Ari$totle.</I></marg> <I>Ari$totles</I> own Definition of place, to be placed, importeth to be in- vironed by the Superficies of the Ambient Body, therefore, then $hall the two Figures be in the water, when the Superficies of the water, $hall imbrace and inviron them: but when the Adver$aries $hew the Board of Ebony not de$cending to the Bottom, they put it not into the water, but upon the water, where being by a certain im- pediment (as by and by we will $hew) retained, it is invironed, part by water, and part by air, which thing is contrary to our agreement, that was, that the Bodies $hould be in the water, and not part in water, and part in air.</P> <foot><I>The</I></foot> <p n=>433</p> <P><I>The which is again made manifest, by the que$tions being put as well about the things which go to the Bottom, as tho$e which ari$e from the Bottom to $wimme, and who $ees not that things placed in the Bottom, mu$t have water about them.</I></P> <P>It is now to be noted, that the Board of Ebany and the Ball, put <marg>The con$utati- on of the Expe- riment in the Ebany.</marg> <I>into</I> the water, both $ink, but the Ball more $wiftly, and the Board more $lowly; and $lower and $lower, according as it $hall be more broad and thin, and of this Tardity the breadth of the Figure is the true Cau$e: But the$e broad Boards that $lowly de$cend, are the $ame, that being put lightly upon the water, do $wimm: Therefore, if that were true which the Adver$aries affirm, the $ame numerical Figure, would in the $ame numericall water, cau$e one while Re$t, and another while Tardity of Motion, which is impo$$ible: for every per- <marg>Every perticular Figure hath its own peculiat Tardity.</marg> ticular Figure which de$cends to the Bottom, hath of nece$$ity its own determinate Tardity and $lowne$s, proper and naturall unto it, accor- ding to which it moveth, $o that every other Tardity, greater or le$$er is improper to its nature: if, therefore, a Board, as $uppo$e of a foot $quare, de$cendeth naturally with $ix degrees of Tardity, it is impo$$i- ble, that it $hould de$cend with ten or twenty, unle$s $ome new impe- diment do arre$t it. Much le$s can it, by rea$on of the $ame Figure re$t, and wholly cea$e to move; but it is nece$$ary, that when ever it re$teth, there do $ome greater impediment intervene than the breadth of the Figure. Therefore, it mu$t be $omewhat el$e, and not the Fi- gure, that $tayeth the Board of Ebany above water, of which Eigure the only Effect is the retardment of the Motion, according to which it de$cendeth more $lowly than the Ball. Let it be confe$$ed, there- fore, rationally di$cour$ing, that the true and $ole Cau$e of the Ebanys going to the Bottom, is the exce$s of its Gravity above the Gravity of the water: and the Cau$e of the greater or le$s Tardity, the breadth of this Figure, or the contractedne$s of that: but of its Re$t, it can by no means be allowed, that the quallity of the Figure, is the Cau$e thereof: a$well, becau$e, making the Tardity greater, according as the Figure more dilateth, there cannot be $o immen$e a Dilatation, to which there may not be found a corre$pondent immence Tardity. without redu$ing it to Nullity of Motion; as, becau$e the Figures produced by the Antagoni$ts for effecters of Re$t, are the $elf $ame that do al$o go to the Bottom.</P> <marg>* The Figure & Re$i$tance of the Medium a- gain$t Divi$ion, have nothing to do with the Ef- fect of Natation or Submer$ion, by an Experi- ment in Wall- nut tree,</marg> <P>I will not omit another rea$on, founded al$o upon Experience, and if I deceive not my $elf, manife$tly concluding, how that the Intro- ducton of the breadth or amplitude of Figure, and the Re$i$tance of the water again$t penetration, have nothing to do in the Effect of de- $cending, or a$cending, or re$ting in the water. ^{*}Take a piece of wood or other Matter, of which a Ball a$cends from the Bottom of the water <foot>Kkk to</foot> <p n=>434</p> to the Surface, more $lowly than a Ball of Ebony of the $ame bigne$$e, $o that it is manife$t, that the Ball of Ebony more readily divideth the water in de$cending, than the other in a$cending; as for Example, le<*> the Wood be Walnut-tree. Then take a Board of Walnut-tree, like and equall to that of Ebony of the Antagoni$ts, which $wims; and i<*> it be true, that this floats above water, by rea$on of the Figure, unabl<*> through its breadth, to pierce the Cra$$itude of the $ame, the other o<*> Wallnut-tree, without all que$tion, being thru$t unto the Bottom, wil<*> $tay there, as le$s apt, through the $ame impediment of Figure, to di<*> vide the $aid Re$i$tance of the water. But if we $hall find, and b<*> experience $ee, that not only the thin Board, but every other Figur<*> of the $ame Wallnut-tree will return to float, as undoubtedly we $hall then I mu$t de$ier my oppo$ers to forbear to attribute the floating o<*> the Ebony, unto the Figure of the Board, in regard that the Re$i$tane of the water is the $ame, as well to the a$cent, as to the de$cent, and th<*> force of the Wallnut-trees a$cen$ion, is le$$e than the Ebonys force i<*> going to the Bottom.</P> <P>Nay, I will $ay more, that if we $hall con$ider Gold in compari$o <marg>An Experi- ment in Gold, to prove the non- operating of Fi- gure in Natation and Submer$ion.</marg> of water, we $hall find, that it exceeds it in Gravity almo$t twenty times $o that the Force and Impetus, wherewith a Ball of Gold goes to th<*> Bottom, is very great. On the contrary, there want not matters, a<*> Virgins Wax, and $ome Woods, which are not above a fiftieth part le<*> grave than water, whereupon their A$cen$ion therein is very $low, an<*> a thou$and times weaker than the <I>Impetus</I> of the Golds de$cent: y<*> notwith$tanding, a plate of Gold $wims without de$cending to th<*> Bottom, and, on the contrary, we cannot make a Cake of Wax, or thi<*> Board of Wood, which put in the Bottom of the Water, $hall re$t the<*> without a$cending. Now if the Figure can ob$truct the Penetratio<*> and impede the de$cent of Gold, that hath $o great an <I>Impetus,</I> ho<*> can it choo$e but $uffice to re$i$t the $ame Penetration of the other ma<*> ter in a$cending, when as it hath $carce a thou$andth part of the <I>Impet<*></I> that the Gold hath in de$cending? Its therefore, nece$$ary, that th<*> which $u$pends the thin Plate of Gold, or Board of Ebony, upon t<*> water, be $ome thing that is wanting to the other Cakes and Boards <*> Matters le$s grave than the water; $ince that being put to the Botton and left at liberty, they ri$e up to the Surface, without any ob$tructio<*> But they want not for flatne$s and breadth of Figure: Therefore, t<*> $paciou$ne$$e of the Figure, is not that which makes the Gold and Ebon to $wim.</P> <P>And, becau$e, that the exce$s of their Gravity above the Gravity <*> the water, is que$tionle$s the Cau$e of the $inking of the flat piece <*> Ebony, and the thin Plate of Gold, when they go to the Bottom, ther<*> fore, of nece$$ity, when they float, the Cau$e of their $taying abo<*> water, proceeds from Levity, which in that ca$e, by $ome Acciden<*> <foot>peradventure</foot> <p n=>435</p> peradventure not hitherto ob$erved, cometh to meet with the $aid Board, rendering it no longer as it was before, whil$t it did fink more ponderous than the water, but le$s.</P> <P>Now, let us return to take the thin Plate of Gold, or of Silver, or the thin Board of Ebony, and let us lay it lightly upon the water, $o that it $tay there without $inking, and diligently ob$erve its effect. An<*> fir$t, $ee how fal$e the a$$ertion of <I>Aristotle,</I> and our oponents is, to wit, that it $tayeth above water, through its unability to pierce and pene- trate the Re$i$tance of the waters Cra$$itude: for it will manife$tly appear, not only that the $aid Plates have penetrated the water, but al$o that they are a con$iderable matter lower than the Surface of the $ame, the which continueth eminent, and maketh as it were a Rampert on all $ides, round about the $aid Plates, the profundity of which they $tay $wimming: and, according as the $aid Plates $hall be more grave than the water, two, four, ten or twenty times, it is nece$$ary, that their Superficies do $tay below the univer$all Surface of the water, $o much more, than the thickne$s of tho$e Plates, as we $hal more di$tinctly $hew anon. In the mean $pace, for the more ea$ie under$tanding of what I $ay, ob$erve with me a little the pre$ent <fig> Scheme: in which let us $uppo$e the Surface of the water to be di$tended, according to the Lines F L D B, upon which if one $hall put a board of matter $pecifically more grave than water, but $o lightly that it $ubmetge not, it $hall not re$t any thing above, but $hall enter with its whole thickne$s into the water: and, moreover, $hall $ink al$o, as we $ee by the Board A I, O I, who$e breadth is wholly $unk into the water, the little Ram- perts of water L A and D O incompa$$ing it, who$e Superficies is no- tably higher than the Superficies of the Board. See now whether it be true, that the $aid Board goes not to the Bottom, as being of Figure unapt to penetrate the Cra$$itude of the water.</P> <P>But, if it hath already penetrated, and overcome the Continuity of <marg>Why $olids having penitra- ted the Water, do not proceed to a totail Sub- mer$ion.</marg> the water, & is of its own nature more grave than the $aid water, why doth it not proceed in its $inking, but $top and $u$pend its $elf within that little dimple or cavitie, which with its pondero$ity it hath made in the water? I an$wer; becau$e that in $ubmerging it $elf, $o far as till its Superficies come to the Levell with that of the water, it lo$eth a part of its Gravity, and lo$eth the re$t of it as it $ubmergeth & de$cends be- neath the Surface of the water, which maketh Ramperts and Banks round about it, and it $u$taines this lo$s by means of its drawing after it, and carrying along with it, the Air that is above it, and by Contact ad- herent to it, which Air $ucceeds to fill the Cavity that is invironed by the Ramperts of water: $o that that which in this ca$e de$cends and is placed in the water, is not only the Board of Ebony or Plate of Iron, <foot>Kkk 2 but</foot> <p n=>436</p> but a compo$ition of Ebony and Air, from which re$ulteth a Solid no longer $uperiour in Gravity to the water, as was the $imple Ebony, or the $imple Gold. And, if we exactly con$ider, what, and how great the Solid is, that in this Experiment enters into the water, and contra$ts with the Gravity of the $ame, it will be found to be all that which we find to be beneath the Surface of the water, the which is an aggregate and Compound of a Board of Ebony, and of almo$t the like quantity of Air, or a Ma$s compounded of a Plate of Lead, and ten or twelve times as much Air. But, Genrlemen, you that are my Antagoni$ts in our Que$tion, we require the Identity of Matter, and the alteration only of the Figure; therefore, you mu$t remove that Air, which being conjoyned with the Board, makes it become another Body le$s grave than the Water, and put only the Ebony into the Water, and you $hall certainly $ee the Board de$cend to the Bottom; and, if that do not happen, you have got the day. <marg>How to $epe- rate the Air from Solids in demit- ting them into the water.</marg> And to $eperate the Air from the Ebony, there needs no more but only to bath the Superficies of the $aid Board with the $ame Water: for the Water being thus interpo$ed between the Board and the Air, the other circumfu$ed Water $hall run together without any impedi- ment, and $hall receive into it the $ole and bare Ebony, as it was to do.</P> <P>But, me thinks I hear $ome of the Adver$aries cunningly oppo$ing this, and telling me, that they will not yield, by any means, that their Board be wetted, becau$e the weight added thereto by the Water, by making it heavier than it was before, draws it to the Bottom, and that the addition of new weight is contrary to our a- greement, which was, that the Matter be the $ame.</P> <P>To this, I an$wer, fir$t; that treating of the operation of Figure in Bodies put into the Water, none can $uppo$e them to be put into the Water without being wet; nor do I de$ire more to be done to the Board, then I will give you leave to do to the Ball. Moreover, it is untrue, that the Board $inks by vertue of the new Weight added to it by the Water, in the $ingle and $light bathing of it: for I will put ten or twenty drops of Water upon the $ame Board, whil$t it is $u$tained upon the water, which drops, becau$e not conjoyned with the other Water circumfu$ed, $hall not $o encrea$e the weight of it, as to make it $ink: but if the Board being taken out, and all the water wiped off that was added thereto, I $hould bath all its Superficies with one only very $mall drop, and put it again upon the water, with- out doubt it $hall $ink, the other Water running to cover it, not be- ing retained by the $uperiour Air; which Air by the interpo$ition of the thin vail of water, that takes away its Contiguity unto the Ebony, $hall without Renitence be $eperated, nor doth it in the lea$t oppo$e the $ucce$$ion of the other Water: but rather, to $peak better, it $hall de$cend freely; becau$e it $hall be all invironed and covered <foot>with</foot> <p n=>437</p> with water, as $oon as its $uperiour Superficies, before vailed with water, doth arrive to the Levell of the univer$all Surface of the $aid water. To $ay, in the next place, that water can encrea$e the weight <marg>Water hath no Gravity in Water.</marg> of things that are demitted into it, is mo$t fal$e <*> for water hath no Gravity in water, $ince it de$cends not: yea, if we would well con$i- der what any immen$e Ma$s of water doth put upon a grave Body; <marg>Water de- mini$heth the Gravity of So- lids immerged therein.</marg> that is placed in it, we $hall find experimentally, that it, on the con- trary, will rather in a great part demini$h the weight of it, and that we may be able to lift an huge Stone from the Bottom of the water, which the water being removed, we are not able to $tir. Nor let them tell me by way of reply, that although the $uperpo$ed water augment not the Gravity of things that are in it, yet it increa$eth the pondero$ity of tho$e that $wim, and are part in the water and part <marg>The Experi- ment of a Bra$s Ketle $wiming when empty, & $inking when full, alledged to prove that water gravitates in water, an$wered.</marg> in the Air, as is $een, for Example, in a Bra$s Ketle, which whil$t it is empty of water, and repleni$hed only with Air $hall $wim, but pouring of Water therein, it $hall become $o grave, that it $hall $ink to the Bottom, and that by rea$on of the new weight added thereto. To this I will return an$wer, as above, that the Gravity of the Water, contained in the Ve$$el is not that which $inks it to the Bot- tom, but the proper Gravity of the Bra$s, $uperiour to the Specificall <marg>An Ocean $uf- ficeth not to $ink a Ve$$el $pe- cifically le$s grave than wa- ter.</marg> Gravity of the Water: for if the Ve$$el were le$s grave than water, the Ocean would not $uffice to $ubmerge it. And, give me leave to repeat it again, as the fundamentall and principall point in this Ca$e, that the Air contained in this Ve$$el before the infu$ion of the Water, was that which kept it a-float, $ince that there was made <marg>Air, the Cau$e of the Natation of empty Ve$$els of Matters gra- ver <I>in $pecie</I> than the water.</marg> of it, and of the Bra$s, a Compo$ition le$s grave than an equall quanti- ty of Water: and the place that the Ve$$el occupyeth in the Water whil$t it floats, is not equall to the Bra$s alone, but to the Bra$s and to the Air together, which filleth that part of the Ve$$el that is below the Levell of the water: Moreover, when the Water is infu$ed, the Air is removed, and there is a compo$ition made of Bra$s and of water, more grave <I>in $pecie</I> than the $imple water, but not by vertue of the water infu$ed, as having greater Specifick Gravity than the other water, but through the proper Gravity of the Bra$s, and through the alienation of the Air. Now, as he that $hould $ay that Bra$s, that by its nature goes to the Bottom, being <marg>Neither Figure, nor the breadth of Figure, is the Cau$e of Nata- tion.</marg> formed into the Figure of a Ketle, acquireth from that Figure a vertue of lying in the Water without $inking, would $ay that which is fal$e; becau$e that Bra$s fa$hioned into any whatever Figure, goeth always to the Bottom, provided, that that which is put into the water be $imple Bra$s; and it is not the Figure of the Ve$$el that makes the Bra$s to float, but it is becau$e that that is not purely Bra$s which is put into the water, but an aggregate of Bra$s and of Air: $o is it neither more nor le$s fal$e, that a thin Plate of Bra$s <foot>or</foot> <p n=>438</p> or of Ebony, $wims by vertue of its dilated & broad Figure: for <*> truth is, that it bares up without $ubmerging, becau$e that that whi<*> is put in the water, is not pure Bra$s or $imple Ebony, but an a<*> gregate of Bra$s and Air, or of Ebony and Air. And, this is n<*> contrary unto my Conclu$ion, the which, (having many a time $e<*> Ve$$els of Mettall, and thin pieces of diver$e grave Matters float, <*> vertue of the Air conjoyned with them) did affirm, That Figu<*> was not the Cau$e of the Natation or Submer$ion of $uch Solids were placed in the water. Nay more, I cannot omit, but mu$t to my Antagoni$ts, that this new conceit of denying that the Superfi- cies of the Board $hould be bathed, may beget in a third per$on a opinion of a poverty of Arguments of defence on their part, $in<*> that $uch bathing was never in$i$ted upon by them in the beginnir of our Di$pute, and was not que$tioned in the lea$t, being that the Originall of the di$cour$e aro$e upon the $wiming of Flakes of lo<*> wherein it would be $implicity to require that their Superficies mig<*> bedry: be$ides, that whether the$e pieces of Ice be wet or dry the alwayes $wim, and as the Adver$aries $ay, by rea$on of the Figu<*></P> <P>Some peradventure, by way of defence, may $ay, that wetting th<*> Board of Ebony, and that in the $uperiour Superficies, it would though of it $elf unable to pierce and penetrate the water, be bor<*> downwards, if not by the weight of the additionall water, at lea<*> by that de$ire and propen$ion that the $uperiour parts of the wate have to re-unite and rejoyn them$elves: by the Motion of whi<*> parts, the $aid Board cometh in a certain manner, to be depre$$e downwards.</P> <marg>The Bathed Solid de$cends not out of any affectation of u- nion in the upper parts of the wa- ter.</marg> <P>This weak Refuge will be removed, if we do but con$ider, th<*> the repugnancy of the inferiour parts of the water, is as great again<*> Di$-union, as the Inclination of its $uperiour parts is to union: nor ca<*> the uper unite them$elves without depre$$ing the board, nor can i<*> de$cend without di$uniting the parts of the nether Water: $o th<*> it doth follow, by nece$$ary con$equence, that for tho$e re$pects, it $hal not de$cend. Moreover, the $ame that may be $aid of the uppe<*> parts of the water, may with equall rea$on be $aid of the nether namely, that de$iring to unite, they $hall force the $aid Boar<*> upwards.</P> <P>Happily, $ome of the$e Gentlemen that di$$ent from me, will won<*> der, that I affirm, that the contiguous $uperiour Air is able to $u$tai<*> that Plate of <I>B</I>ra$s or of Silver, that $tayeth above water; as if <marg><I>A</I> Magneti$me in the <I>A</I>ir, by which it bears up tho$e Solids in the wa- ter, that are con- tiguous with it.</marg> would in a certain $ence allow the Air, a kind of Magnetick vertu<*> of $u$taining the grave <I>B</I>odies, with which it is contiguous. To $<*> tis$ie all I may, to all doubts, I have been con$idering how by $om<*> other $en$ible Experiment I might demon$trate, how truly that litt<*> contiguous and $uperiour Air $u$taines tho$e Solids, which being b<*> <foot>natur<*></foot> <p n=>439</p> <*>ture apt to de$cend to the Bottom, being placed lightly on the water $ubmerge not, unle$s they be fir$t thorowly bathed; and have found, that one of the$e Bodies having de$cended to the Bottom, by conveigh- ing to it (without touching it in the lea$t) a little Air, which conjoyneth with the top of the $ame; it becometh $ufficient, not only, as before to <*>$tain it, but al$o to rai$e it, and to carry it back to the top, where it $tays and abideth in the $ame manner, till $uch time, as the a$$i$tance of the conjoyned Air is taken away. And to this effect, I have taken a Ball of Wax, and made it with a little Lead, $o grave, that it lea$urely <*>$cends to the Bottom, making with all its Superficies very $mooth and pollite: and this being put gently into the water, almo$t wholly $ub- <marg>The Effect of the Airs Conti- guity in the Na- tation of Solids.</marg> <*>ergeth, there remaining vi$$ible only a little of the very top, the which <*>long as it is conjoyned with the Air, $hall retain the Ball a-top, but the Contiguity of the Air taken away by wetting it, it $hall de$cend to the Bottom and there remain. Now to make it by vertue of the Air, that <*>fore $u$tained it to return again to the top, and $tay there, thru$t into the water a Gla$s rever$ed with the mouth downwards, the which $hall <*>arry with it the Air it contains, and move this towards the Ball, aba$ing <*>till $uch time that you $ee, by the tran$parency of the Gla$s, that the <marg>The force of Contact.</marg> <*>ontained Air do arrive to the $ummity of the <I>B</I>all: then gently with- <*>raw the Gla$s upwards, and you $hall $ee the <I>B</I>all to ri$e, and afterwards <marg><I>A</I>n affectati- on of Conjunct- ion betwixt So- lids and the Air contiguous to them.</marg> <*>ay on the top of the water, if you carefully part the Gla$s and the water without overmuch commoving and di$turbing it. There is, therefore, a <*>ertain affinity between the Air and other <I>B</I>odies, which holds them uni- <*>d, $o, that they $eperate not without a kind of violence. The $ame <marg>The like affect- ation of Con- junction be- twixt Solids & the water.</marg> likewi$e is $een in the water; for if we $hall wholly $ubmerge $ome <I>B</I>ody <*>nit, $o that it be thorowly bathed, in the drawing of it afterwards gent- <*>y out again, we $hall $ee the water follow it, and ri$e notably above its <*>urface, before it $eperates from it. Solid <I>B</I>odies, al$o, if they be equall <marg>Al$o the like affectation and Conjunction be- twixt Solids them$eives.</marg> <*>nd alike in Superficies, $o, that they make an exact Contact without <*>he interpo$ition of the lea$t Air, that may part them in the $eperation <*>nd yield untill that the ambient <I>Medium</I> $ucceeds to repleni$h the place, <*>o hold very firmly conjoyned, and are not to be $eperated without great <*>orce but, becau$e, the Air, Water, and other Liquids, very expedi- <*>ou$ly $hape them$elves to contact with any Solid <I>B</I>odies, $o that their uperficies do exqui$itely adopt them$elves to that of the Solids, without <*>ny thing remaining between them, therefore, the effect of this Con- unction and Adherence is more manife$tly and frequently ob$erved in them, than in hard and inflexible <I>B</I>odies, who$e Superficies do very rate- y conjoyn with exactne$s of Contact. This is therefore that Magne- <marg>Contact may be the Cau$e of the Continuity of Naturall Bo- dies.</marg> <*>ck vertue, which with firm Connection conjoyneth all Bodies, that do touch without the interpo$ition of flexible fluids; and, who knows, but that that a Contact, when it is very exact, may be a $ufficient Cau$e of the Union and Continuity of the parts of a naturall <I>B</I>ody?</P> <foot>Now,</foot> <p n=>440</p> <P>Now, pur$uing my purpo$e, I $ay; that it needs not, that w<*> recour$e to the Tenacity, that the parts of the water have among$t $elves, by which they re$i$t and oppo$e Divi$ion, Di$traction, and ration, becau$e there is no $uch Coherence and Re$i$tance of D<*> for if there were, it would be no le$s in the internall parts than in nearer the $uperiour or externall Surface, $o that the $ame Board, <*> ing alwayes the $ame Re$i$tance and Renitence, would no le$s <*> the middle of the water than about the Surface, which is fal$e. over, what Re$i$tance can we place in the Continuity of the <*> if we $ee that it is impo$$ible to $ind any Body of what$oever M Figure or Magnitude, which being put into the water, $hall be ob$t<*> and impeded by the Tenacity of the parts of the water to one an $o, but that it is moved upwards or downwards, according as the <*> of their Motion tran$ports it? And, what greater proof of it can w<*> $ier, than that which we daily $ee in Muddy waters, which being p<*> Ve$$els to be drunk, and being, after $ome hours $etling, $till, as <*> <marg><I>T</I>he $ettlement of <I>M</I>uddy Wa- ter, proveth that that Element hath no aver$i- on to Divi$ion.</marg> thick in the end, after four or $ix dayes they are wholly $etled, an come pure and clear? Nor can their Re$i$tance of Penetration $tay impalpable and in$en$ible Atomes of Sand, which by rea$on o<*> exceeding $mall force, $pend $ix dayes in de$cending the $pace o<*> a yard.</P> <P><I>Nor let them $ay, that the $eeing of $uch $mall Bodies, con$ume $ix d<*> de$cending $o little a way, is a $ufficient Argument of the Waters Re$<*> of Divi$ion; becau$e that is no re$i$ting of Divi$ion, but a retard</I> <marg>Water cannot oppo$e divi$ion, and at the $ame time permit it $elf to be divi- ded.</marg> <I>Motion; and it would be $implicity to $ay, that a thing oppo$eth Di<*> and that in the $ame in$tant, it permits it $elf to be divided: nor do<*> Retardation of Motion at all favour the Adver$aries cau$e, for that th<*> to in$tance in a thing that wholly prohibiteth Motion, and procureth<*> it is nece$$ary, therefore, to find out Bodies that $tay in the water, if one $hew its repugnancy to Divi$ion, and not $uch as move in it, ho<*>be $lowly.</I></P> <P>What then is this Cra$$itude of the water, with which it re$i$ter vi$ion? What, I be$eech you, $hould it be, if we (as we have $aid a<*> with all diligence attempting the reduction of a Matter into $o Gravity with the water, that forming it into a dilated Plate it re$t pended as we have $aid, between the two waters, it be impo$$i<*> effect it, though we bring them to $uch an Equiponderance, <*> much Lead as the fourth part of a Grain of Mu$terd-$eed, added <*> $ame expanded Plate, that in Air [<I>i. e. out of the water</I>] $hall weig<*> or fix pounds, $inketh it to the Bottom, and being $ub$tracted, it a<*> to the Surface of the water? I cannot $ee, (if what I $ay be true, mo$t certain) what minute vertue and force we can po$$ibly find o<*> gine, to which the Re$i$tance of the water again$t Divi$ion and Pe<*> <p n=>441</p> tion is not inferiour; whereupon, we mu$t of nece$$ity conclude that it is nothing: becan$e, if it were of any $en$ible power, $ome large Plate might be found or compounded of a Matter alike in Gra- vity to the water, which not only would $tay between the two wa- ters; but, moreover, $hould not be able to de$cend or a$cend with- out notable force. We may likewi$e collect the $ame from an o- <marg>An hair will draw a great Ma$s thorow the Water; which proveth, that it hath no Re$i$t- ance again$t tran$ver$all Di- vi$ion.</marg> ther Experiment, $hewing that the Water gives way al$o in the $ame manner to tran$ver$all Divi$ion; for if in a $etled and $tanding water we $hould place any great Ma$s that goeth not to the bottom, draw- ing it with a $ingle (Womans) Hair, we might carry it from place to place without any oppo$ition, and this whatever Figure it hath, though that it po$$e$s a great $pace of water, as for in$tance, a great Beam would do moved $ide-ways. Perhaps $ome might oppo$e me and $ay, that if the Re$i$tance of water again$t Divi$ion, as I affirm, were nothing; Ships $hould not need $uch a force of Oars and Sayles for the moving of them from place to place in a tranquile Sea, or $tanding Lake. To him that $hould make $uch an objection, I would <marg>How $hips are moved in the water.</marg> reply, that the water contra$teth not again$t, nor $imply re$i$teth Divi$ion, but a $udden Divi$ion, and with $o much greater Reni- tence, by how much greater the Velocity is: and the Cau$e of this Re$i$tance depends not on Cra$$itude, or any other thing that ab$o- lutely oppo$eth Divi$ion, but becau$e that the parts of the water divided, in giving way to that Solid that is moved in it, are them- $elves al$o nece$$itated locally to move, $ome to the one $ide, and $ome to the other, and $ome downwards: and this mu$t no le$s be done by the waves before the Ship, or other Body $wimming through the water, than by the po$teriour and $ub$equent; becau$e, the Ship proceeding forwards, to make it $elf a way to receive its Bulk, it is requi$ite, that with the Prow it repul$e the adjacent parts of the water, as well on one hand as on the other, and that it move them as much tran$ver$ly, as is the half of the breadth of the Hull: and the like removall mu$t tho$e waves make, that $ucceeding the Poump do run from the remoter parts of the Ship towards tho$e of the middle, $ucce$$ively to repleni$h the places, which the Ship in ad- vancing forwards, goeth, leaving vacant. Now, becau$e, all Moti- <marg>Bodies moved a certain $pace in a certain Time, by a certain power, cannot be moved the $ame $pace, and in a $horter time, but by a greater power.</marg> tions are made in Time, and the longer in greater time: and it being moreover true, that tho$e Bodies that in a certain time are moved by a certain power $uch a certain $pace, $hall not be moved the $ame $pace, and in a $horter Time, unle$s by a greater Power: therefore, the broader Ships move $lower than the narrower, being put on by an equall Force: and the $ame Ve$$el requires $o much greater force of Wind, or Oars, the fa$ter it is to move.</P> <foot>Lll <I>But</I></foot> <p n=>442</p> <P><I>But yet for all this, any great Ma$s $wimming in a $tanding Lake, <*> be moved by any petit force; only it is true, that a le$$er force n<*> $lowly moves it: but if the waters Re$i$tance of Divi$ion, were in <*> manner $en$ible, it would follow, that the $aid Ma$s, $hould, not<*> $tanding the percu$$ion of $ome $en$ible force, continue immoveable, whi</I><*> <marg>The parts of Liquids, $o farte from re$i$ting Divi$ion, that they contain not any thing that may be divided.</marg> <I>not $o. Yea, I will $ay farther, that $hould we retire our $elves into<*> more internall contemplation of the Nature of water and other Flui<*> perhaps we $hould di$cover the Con$titution of their parts to be $uch, <*> they not only do not oppo$e Divi$ion, but that they have not any thing <*> them to be divided: $o that the Re$i$tance that is ob$erved in mov<*></I> <marg>The Re$i$t- ance a Solid findeth in mo- ving through the water, like to that we meet with in pa$$ing through a throng of peo- ple;</marg> <I>through the water, is like to that which we meet with in pa$$ing thr<*> a great Throng of People, wherein we find impediment, and not by <*> difficulty in the Divi$ion, for that none of tho$e per$ons are divi<*> whereof the Croud is compo$ed, but only in moving of tho$e per$ons<*> ways which were before divided and disjoyned: and thus we <*> Re$i$tance in thru$ting a Stick into an heap of Sand, not becau$e any p<*> of the Sand is to be cut in pieces, but only to be moved and rai$ed. <*></I> <marg>Or in thru$t- ing a Stick into an heap of Sand.</marg> <I>manners of Penetration, therefore, offer them$elves to us, one in Bod<*> who$e parts were continuall, and here Divi$ion $eemeth nece$$ary; <*></I> <marg>Two kinds of Penetration, one in Bodies conti- nuall, the other in Bodies only contiguous.</marg> <I>other in the aggregates of parts not continuall, but contiguous only, <*> here there is no nece$$ity of dividing but of moving only. Now, I <*> not well re$olved, whether water and other Fluids may be e$teeme <*> be of parts continuall or contiguous only; yet I find my $elf indeed in <*></I> <marg>Water con$i$ts not of continu- all, but only of contiguous parts.</marg> <I>ned to think that they are rather contiguous (if there be in Natur<*> other manner of aggregating, than by the union, or by the touching of <*> extreams:) and I am induced thereto by the great difference that I <*> between the Conjunction of the parts of an hard or Solid Body, and <*></I> <marg><I>Set what $atis- faction he hath given, as to this point, in Lib. de Motu. Dial.</I> 2.</marg> <I>Conjunction of the $ame parts when the $ame Body $hall be made Liq<*> and Fluid: for if, for example, I take a Ma$s of Silver or other S<*> and hard Mettall, I $hall in dividing it into two parts, find not only <*></I> <marg>Great differ- ence betwixt the Conjunction of the parts of a Bo- dy when Solid, and when fluid.</marg> <I>re$i$tance that is found in the moving of it only, but an other incompar<*> greater, dependent on that vertue, whatever it be, which holds the p<*> united: and $o if we would divide again tho$e two parts into other <*> and $ucce$$ively into others and others, we $hould $till find a like Re<*> ance, but ever le$s by how much $maller the parts to be divided $hall <*> but if, la$tly, employing mo$t $ubtile and acute In$truments, $uch as <*> the mo$t tenuous parts of the Fire, we $hall re$olve it (perhaps) int<*> la$t and lea$t Particles, there $hall not be left in them any longer ci<*> Re$i$tance of Divi$ion, or $o much as a capacity of being farther d<*> ded, e$pecially by In$truments more gro$$e than the acuities of Fire: <*> what Knife or Ra$or put into well melted Silver can we finde, that <*> divide a thing which $urpa$$eth the $eparating power of Fire? Certa<*> none: becau$e either the whole $hall be reduced to the mo$t minuie <*> ultimate Divi$ions, or if there remain parts capable $till of other Su<*></I> <foot><I>vi$i<*></I></foot> <p n=>443</p> <I>divi$ions, they cannot receive them, but only from acuter Divi$ors than Fire; but a Stick or Rod of Iron, moved in the melted Met all, is not $uch a one. Of a like Con$titution and Con$i$tence, I account the parts</I> <marg>Water con$i$ts of parts that ad- mit of no fat- ther divi$ion.</marg> <I>of Water, and other Liquids to be, namely, incapable of Divi$ion by rea$on of their Temtity; or if not ab$olutely indivi$ible, yet at lea$t not to be divided by a Board, or other Solid Body, palpable unto the band, the Sector being alwayes required to be more $harp than the Solid to be cut. Solid Bodies, therefore, do only move, and not divide the</I> <marg>Solids dimit- ted into the wa- ter, do onely move, and not divide it.</marg> <I>Water, when put into it; who$e parts being before divided to the ex- treame$t minuity, and therefore capable of being moved, either many of them at once, or few, or very few, they $oon give place to every $mall Cor- pu$cle, that de$cends in the $ame: for that, it being little and light, de- $cending in the Air, and arriving to the Surface of the Water, it meets with Particles of Water more $mall, and of le$s Re$i$tance again$t Motion and Extru$ion, than is its own prement and extru$ive force, whereupon it $ubmergeth, and moveth $uch a portion of them, as is pro- portionate to its Power. There is not, therefore, any Re$i$tance in Water again$t Divi$ion, nay, there is not in it any divi$ible parts. I adde, moreover, that in ca$e yet there $bould be any $mall Re$i$tance</I> <marg>If there were any Re$i$tance of Divi$ion in water, it mu$t needs be $mall, in that it is over- come by an Hair, a Grain of Lead, or a $light bathing of the Solid.</marg> <I>found (which is ab$olutely fal$e) haply in attempting with an Hair to move a very great natant Machine, or in e$$aying by the addition of one $mall Grain of Lead to $ink, or by removall of it to rai$e a very broad Plate of Matter, equall in Gravity with Water, (which likewi$e will not happen, in ca$e we proceed with dexterity) we may ob$erve that that Re$i$tance is a very different thing from that which the Adver$aries pro- duce for the Cau$e of the Natation of the Plate of Lead or Board of Ebo- ny, for that one may make a Board of Ebony, which being put upon the Water $wimmeth, and cannot be $ubmerged, no not by the addition of an bundred Grains of Lead put upon the $ame, and afterwards being ba- thed, not only $inks, though the $aid Lead be taken away, but though moreover a quantity of Cork, or of $ome other light Body fa$tened to it, $ufficeth not to hinder it from $inking unto the bottome: $o that you $ee, that although it were granted that there is a certain $mall Re$i$t- ance of Divi$ion found in the $ubstance of the Water, yet this hath no- thing to do with that Cau$e which $upports the Board above the Water, with a Re$i$tance an hundred times greater than that which men can find in the parts of the Water: nor let them tell me, that only the Sur-</I> <marg>The uper parts of the Water, do no more re$i$t Divi$ion, than the middle or lowe$t parts.</marg> <I>face of the Water hath $uch Re$i$tance, and not the internall parts, or that $uch Re$i$tance is found greate$t in the beginning of the Submer$ion, as it al$o $eems that in the beginning, Motion meets with greater oppo$iti- on, than in the continuance of it; becau$e, fir$t, I will permit, that the</I> <marg>Waters Re- $i$tance of divi- $ion, not greater in the begin- ning of the Sub- mer$ion.</marg> <I>Water be $tirred, and that the $uperiour parts be mingled with the mid- dle, and inferiour parts, or that tho$e above be wholly removed, and tho$e in the middle only made u$e off, and yet you $hall $ee the effect for</I> <foot><I>Ell</I> 2 <I>all</I></foot> <p n=>444</p> <I>all that, to be still the $ame: Moreover, that Hair which dren <*> Beam through the Water, is likewi$e to divide the upperparts, <*> al$o to begin the Motion, and yet it begins it, and yet it divides it: <*> finally, let the Board of Ebony be put in the midway, betwixt the botto<*> and the top of the Water, and let it there for a while be $u$pended <*> $etled, and afterwards let it be left at liberty, and it will instantly b<*> its Motion, and will continue it unto the bottome. Nay, more, the Bo<*> $o $oon as it is dimitted upon the Water, hath not only begun to m<*> and divide it, but is for a good $pace dimerged into it.</I></P> <P>Let us receive it, therefore, for a true and undoubted Conclu<*> on, That the Water hath not any Renitence again$t $imple Div<*> on, and that it is not po$$ible to find any Solid Body, be it of wh<*> Figure it will, which being put into the Water, its Motion upwa<*> or downwards, according as it exceedeth, or $hall be exceeded <*> the Water in Gravity (although $uch exce$$e and difference be <*> $en$ible) $hall be prohibited, and taken away, by the Cra$$itude <*> the $aid Water. When, therefore, we $ee the Board of Ebony, <*> of other Matter, more grave than the Water, to $tay in the Co<*> fines of the Water and Air, without $ubmerging, we mu$t have <*> cour$e to $ome other Originall, for the inve$ting the Cau$e of th<*> Effect, than to the breadth of the Figure, unable to overcome <*> Renitence with which the Water oppo$eth Divi$ion, $ince there <*> no Re$i$tance; and from that which is not in being, we can exp<*> no Action. It remains mo$t true, therefore, as we have $aid before, <*> this $o $ucceds, for that that which in $uch manner put upon the v<*> ter, not the $ame Body with that which is put <I>into</I> the Water: beca<*> this which is put <I>into</I> the Water, is the pure Board of Ebony, wh<*> for that it is more grave than the Water, $inketh, and that which put <I>upon</I> the Water, is a Compo$ition of Ebony, and of $o m<*> Air, that both together are $pecifically le$s grave than the Wa<*> and therefore they do not de$cend.</P> <P>I will farther confirm this which I $ay. Gentlemen, my Anta<*> ni$ts, we are agreed, that the exce$s or defect of the Gravity of <*> Solid, unto the Gravity of the Water, is the true and proper Ca<*> of Natation or Submer$ion.</P> <marg>Great Caution to be had in ex- perimenting the operation of Fi- gure in Natati- on.</marg> <P>Now, if you will $hew that be$ides the former Cau$e, there is a<*> ther which is $o powerfull, that it can hinder and remove the S<*> mer$ion of tho$e very Solids, that by their Gravity $ink, and if y<*> will $ay, that this is the breadth or amplene$s of Figure, you are o<*> lieged, when ever you would $hew $uch an Experiment, fir$t to ma<*> the circum$tances certain, that that Solid which you put into <*> Water, be not le$s grave <I>in $pecie</I> than it, for if you $hould not do<*> any one might with rea$on $ay, that not the Figure, but the Levi<*> was the cau$e of that Natation. But I $ay, that when you $hall <*> <foot>n</foot> <p n=>445</p> mit a Board of Ebony into the Water, you do not put therein a Solid more grave <I>in $pecie</I> than the Water, but one lighter, for be $ides the Ebony, there is in the Water a Ma$s of Air, united with the Ebony, and $uch, and $o light, that of both there re$ults a Compo$ition le$s grave than the Water: See, therefore, that you remove the Air, and put the Ebony alone into the Water, for $o you $hall immerge a So- lid more grave then the Water, and if this $hall not go to the Bottom, you have well Philo$ophized, and I ill.</P> <P>Now, $ince we have found the true Cau$e of the Natation of tho$e Bodies, which otherwi$e as being graver than the Water, would de- $cend to the bottom, I think, that for the perfect and di$tinct know- ledge of this bu$ine$s, it would be good to proceed in a way of di$- covering demon$tratively tho$e particular Accidents that do attend the$e effects, and,</P> <head>PROBL. I.</head> <P><I>To finde what proportion $everall Figures of different</I> <marg>To finde the proportion Fi- gures ought to have to the wa- ters Gravity, that by help of the contiguous Air, they may $wim.</marg> <I>Matters ought to have, unto the Gravity of the Water, that $o they may be able by vertue of the Contigucus Air to $tay afloat.</I></P> <P>Let, therefore, for better illu$tration, D F N E be a Ve$$ell, wherein the water is contained, and $uppo$e a Plate or Board, who$e thickne$s is comprehended between the Lines I C and O S, and let it be of Matter exceeding the water in Gravity, $o that being put upon the water, it dimergeth and aba$eth below the Levell of the $aid water, leaving the little Banks A I and B C, which are at the greate$t height they can be, $o that if the Plate I S $hould but de$cend any little $pace farther, the little Banks or Ramparts would no longer con$i$t, but expul$ing the Air A I C B, they would dif- fu$e them$elves over the Superficies I C, and would $ubmerge the Plate. The height AIBC is therefore the greate$t profundity that the <fig> little <I>B</I>anks of water admit of. Now I $ay, that from this, and from the proportion in Gra- vity, that the Matter of the Plate hath to the water, we may ea$ily $inde of what thickne$s, at mo$t, we may make the $aid Plates, to the end, they may be able to bear up above water: for if the Matter of the Plate or <I>B</I>oard I S were, for Example, as heavy again as the water, a <I>B</I>oard of that Matter $hall be, at the mo$t of a thickne$s equall to the greate$t height of the <I>B</I>anks, that is, as thick as A I is high: which we will thus demon$trate. Lot the So- lid I S be donble in Gravity to the water, and let it be a regular <foot>Pri$me</foot> <p n=>446</p> Pri$me, or Cylinder, to wit, that hath its two flat Superficies, $uper<*> our and inferiour, alike and equall, and at Right Angles with the <*> ther laterall Superficies, and let its thickne$s I O be equall to t<*> greate$t Altitude of the Banks of water: I $ay, that if it be put upo<*> the water, it will not $ubmerge: for the Altitud <*> A I being equall to the Altitude I O, the Ma<*> of the Air A B C I $hall be equall to the Ma$s <*> <fig> the Solid C I O S: and the whole Ma$s A O S <*> double to the Ma$s I S; And $ince the Ma<*> of the Air A C, neither encrea$eth nor dim<*> ni$heth the Gravity of the Ma$s I S, and the Solid I S was $uppo$e<*> double in Gravity to the water; Therefore as much water as th<*> Ma$s $ubmerged A O S B, compounded of the Air A I C B, and <*> the Solid I O S C, weighs ju$t as much as the $ame $ubmerged Ma<*> A O S B: but when $uch a Ma$s of water, as is the $ubmerged part o<*> the Solid, weighs as much as the $aid Solid, it de$cends not farther <*> <marg>Of Natation Lib. 1. Prop. 3.</marg> but re$teth, as by <I>(a) Archimedes,</I> and above by us, hath been d<*> mon$trated: Therefore, I S $hall de$cend no farther, but $hall re$t <*> And if the Solid I S $hall be Se$quialter in Gravity to the water, i<*> $hall float, as long as its thickne$s be not above twice as much as th<*> greate$t Altitude of the Ramparts of water, that is, of A I. For I<*> being Se$quialter in Gravity to the water, and the Altitude O <*> being double to I A, the Solid $ubmerged A O S B, $hall be al$<*> Se$quialter in Ma$s to the Solid I S. And becau$e the Air A <*> neither increa$eth nor dimini$heth the pondero$ity of the Solid I <*> Therefore, as much water in quantity as the $ubmerged Ma$s AOSB <*> weighs as much as the $aid Ma$s $ubmerged: And, therefore, tha<*> Ma$s $hall re$t. And briefly in generall.</P> <head>THEOREME. VI.</head> <P><I>When ever the exce$s of the Gravity of the Solid above</I> <marg>The proporti- on of the great- e$t thickne$s of Solids, beyond which encrea- $ed they $ink.</marg> <I>the Gravity of the Water, $hall have the $ame pro- portion to the Gravity of the Water, that the Alti- tude of the Rampart, hath to the thickne$s of the Solid, that Solid $hall not $ink, but being never $o lit- tle thicker it $hall.</I></P> <P>Let the Solid I S be $uperior in Gravity to the water, and of $uch thickne$s, that the Altitude of the Rampart A I, be in proporti- on to the thickne$s of the Solid I O, as the exce$s of the Gravi- ty of the $aid Solid I S, above the Gravity of a Ma$s of water equall to the Ma$s I S, is to the Gravity of the Ma$s of water equall to the <foot>Ma$s</foot> <p n=>447</p> Ma$s I S. I $ay, that the Solid I S $hall not $inke, but being never $o little thicker it $hall go to the bottom: For being that as A I is <fig> to I O, $o is the Exce$s of the Gravity of the Solid I S, above the Gravity of a Ma$s of water equall to the Ma$s I S, to the Gravity of the $aid Ma$s of water: Therefore, compounding, as A O is to O I, $o $hall the Gravity of the Solid I S, be to the Gravity of a Ma$s of water equall to the Ma$s I S: And, converting, as I O is to O A, $o $hall the Gravity of a Ma$s of water equall to the Ma$s I S, be to the Gravity of the Solid I S: But as I O is to O A, $o is a Ma$s of water I S, to a Ma$s of water equall to the Ma$s A B S O: and $o is the Gravity of a Ma$s of water I S, to the Gravity of a Ma$s of water A S: Therefore as the Gravity of a Ma$s of water, equall to the Ma$s I S, is to the Gravity of the Solid I S, $o is the $ame Gravity of a Ma$s of water I S, to the Gravity of a Ma$s of Water A S: Therefore the Gra- vity of the Solid I S, is equall to the Gravity of a Ma$s of water e- quall to the Ma$s A S: But the Gravity of the Solid I S, is the $ame with the Gravity of the Solid A S, compounded of the Solid I S, and of the Air A B C I. Therefore the whole compounded Solid A O S B, weighs as much as the water that would be compri$ed in the place of the $aid Compound A O S B: And, therefore, it $hall make an <I>Equilibrium</I> and re$t, and that $ame Solid I O S C $hall $inke no farther. But if its thickne$s I O $hould be increa$ed, it would be ne- ce$$ary al$o to encrea$e the Altitude of the Rampart A I, to main- tain the due proportion: But by what hath been $uppo$ed, the Alti- tude of the Rampart A I, is the greate$t that the Nature of the Water and Air do admit, without the waters repul$ing the Air ad- herent to the Superficies of the Solid I C, and po$$e$$ing the $pace A I C B: Therefore, a Solid of greater thickne$s than I O, and of the $ame Matter with the Solid I S, $hall not re$t without $ubmerging, but $hall de$cend to the bottome: which was to be demon$trated. In con$equence of this that hath been demon$trated, $undry and va- rious Conclu$ions may be gathered, by which the truth of my prin- cipall Propo$ition comes to be more and more confirmed, and the imperfection of all former Argumentations touching the pre$ent Que$tion cometh to be di$covered.</P> <P><I>And fir$t we gather from the things demonstrated, that,</I></P> <foot>COROL-</foot> <p n=>448</p> <head>THEOREME VII.</head> <marg>The heavie$t Bodies may $wimme.</marg> <P><I>All Matters, how heavy $oever, even to Gold it $elf, t<*> heavie$t of all Bodies, known by us, may float up <*> the Water.</I></P> <P>Becau$e its Gravity being con$idered to be almo$t twenty t<*> greater than that of the water, and, moreover, the greate$t A <*> tude that the Rampart of water can be extended to, without bre<*> ing the Contiguity of the Air, adherent to the Surface of the So<*> that is put upon the water being predetermined, if we $hould m<*> a Plate of Gold $o thin, that it exceeds not the nineteenth part of<*> Altitude of the $aid Rampart, this put lightly upon the water $<*> re$t, without going to the bottom: and if Ebony $hall chance to<*> in $e$qui$eptimall proportion more grave than the water, the great <*> thickne$s that can be allowed to a Board of Ebony, $o that it may <*> able to $tay above water without $inking, would be $eaven ti<*> more than the height of the Rampart Tinn, <I>v. gr.</I> eight times m<*> grave than water, $hall $wimm as oft as the thickne$s of its Pla<*> <marg><I>He el$ewhere cites this as a Propo$ition, there- fore I make it of that number.</I></marg> exceeds not the 7th part of the Altitude of the Rampart.</P> <P>And here I will not omit to note, as a $econd Corrollary depend<*> upon the things demon$trated, that,</P> <head>THEOREME VIII.</head> <marg>Natation and Submer$ion, col- lected from the thickne$s, exclu- ding the length and breadth of Plates.</marg> <P><I>The Expan$ion of Figure not only is not the Cau$e of t<*> Natation of tho$e grave Bodies, which otherwi<*> do $ubmerge, but al$o the determining what be tho<*> Boards of Ebony, or Plates of Iron or Gold that w<*> $wimme, depends not on it, rather that $ame determin<*> tion is to be collected from the only thickne$s of tho<*> Figures of Ebony or Gold, wholly excluding the co<*> $ideration of length and breadth, as having no way any $hare in this Effect.</I></P> <P>It hath already been manife$ted, that the only cau$e of the Na<*> tion of the $aid Plates, is the reduction of them to be le$s gra<*> than the water, by means of the connexion of that Air, which d<*> $cendeth together with them, and po$$e$$eth place in the wate<*> which place $o occupyed, if before the circumfu$ed water diffu$e<*> it $elf to fill it, it be capable of as much water, as $hall weigh equ<*> with the Plate, the Plate $hall remain $u$pended, and $inke <*> farther.</P> <foot>N<*></foot> <p n=>449</p> <P>Now let us $ee on which of the$e three dimen$ions of the Solid depends the terminating, what and how much the Ma$s of that ought to be, that $o the a$$i$tance of the Air contiguous unto it, may $uffice to render it $pecifically le$s grave than the water, whereupon it may re$t without Submer$ion. It $hall undoubtedly be found, that the length and breadth have not any thing to do in the $aid determina- tion, but only the height, or if you will the thickne$s: for, if we take a Plate or Board, as for Example, of Ebony, who$e Altitude hath unto the greate$t po$$ible Altitude of the Rampart, the proportion above declared, for which cau$e it $wims indeed, but yet not if we never $o little increa$e its thickne$s; I $ay, that retaining its thick- ne$s, and encrea$ing its Superficies to twice, four times, or ten times its bigne$s, or dmini$ning it by dividing it into four, or $ix, or twenty, or a hundred parts, it $hall $till in the $ame manner continue to float: but encrea$ing its thickne$s only a Hairs breadth, it will alwaies $ubmerge, although we $hould multiply the Superficies a hundred and a hundred times. Now fora$much as that this is a Cau$e, which being added, we adde al$o the Effect, and being remo- ved, it is removed; and by augmenting or le$$ening the length or breadth in any manner, the effect of going, or not going to the bot- tom, is not added or removed: I conclude, that the greatne$s and $malne$s of the Superficies hath no influence upon the Natation or Submer$ion. And that the proportion of the Altitude of the Ram- parts of Water, to the Altitude of the Solid, being con$tituted in the manner afore$aid, the greatne$s or $malne$s of the Superficies, makes not any variation, is manife$t from that which hath been above demon$trated, and from this, that, <I>The Pri$ms and Cylinders which</I> <marg>Pri$mes and Cylinders ha- ving the $ame Ba$e, are to one another as their heights.</marg> <I>have the $ame Ba$e, are in proportion to one another as their heights:</I> Whence Cylinders or Prifmes, namely, the Board, be they great or little, $o that they be all of equall thickne$s, have the $ame proportion to their Conterminall Air, which hath for Ba$e the $aid Superficies of the Board, and for height the Ramparts of water; $o that alwayes of that Air, and of the Board, Solids are compounded, that in Gravity equall a Ma$s of water equall to the Ma$s of the Solids, compounded of Air, and of the Board: whereupon all the $aid Solids do in the $ame manner continue afloat. We will conclude in the third place, that,</P> <foot>Mmm THEO-</foot> <p n=>450</p> <head>THEOREME. IX.</head> <marg>All Figures of all Matters, float by hep of the Rampart re- pleni$hed with Air, and $ome but only touch the water.</marg> <P><I>All $orts of Figures of what$oever Matter, albeit mo grave than the Water, do by Benefit of the $aid R<*> part, not only float, but $ome Figures, though of t<*> grave$t Matter, do $tay wholly above Water, wetti<*> only the inferiour Surface that toucheth the Water.</I></P> <P>And the$e $hall be all Figures, which from the inferiour Ba$e<*> wards, grow le$$er and le$$er; the which we $hall exemplifie<*> this time in Piramides or Cones, of which Figures the pa$$ions<*> common. We will demon$trate therefore, that,</P> <P><I>It is po$$ible to form a Piramide, of any what$oever Matter propo$ which being put with its Ba$e upon the Water, re$ts not only with<*> $ubmerging, but without wetting it more then its Ba$e.</I></P> <P>For the explication of which it is requi$ite, that we fir$t demon$tr<*> the $ub$equent Lemma, namely, that,</P> <head>LEMMA II.</head> <P><I>Solids who$e Ma$$es an$wer in proportion contrarily</I> <marg>Solids who$e Ma$$es are in contrary pro- portion to their Specifick Gra- vities, are equall in ab$olute Gra vity.</marg> <I>their Specificall Gravities, are equall in Ab$olu Gravities.</I></P> <P>Let A C and B be two Solids, and let the Ma$s A C be to <*> Ma$s B, as the Specificall Gravity of the Solid B, is to the Spe ficall Gravity of the Solid A C: I $ay, the Solids A C and B <*> equall in ab$olute weight, that is, equally grave, F <fig> if the Ma$s A C be equall to the Ma$s B, then, by <*> A$$umption, the Specificall Gravity of B, $hall be quall to the Specificall Gravity of A C, and being quall in Ma$s, and of the $ame Specificall Gravity th<*> $hall ab$olutely weigh one as much as another. B if their Ma$$es $hall be unequall, let the Ma$s A C be greater, and in take the part C, equall to the Ma$s B. And, becau$e the Ma$$es and C are equall; the Ab$olute weight of B, $hall have the $ame pr<*> portion to the Ab$olute weight of C, that the Specificall Gravity B, hath to the Specificall Gravity of C; or of C A, which is <*> $ame <I>in $pecie</I>: But look what proportion the Specificall Gravity B, hath to the Specificall Gravity of C A, the like proportion, by t<*> A$$umption, hath the Ma$s C A, to the Ma$s B; that is, to the Ma$s <*> <foot>Therefore</foot> <p n=>451</p> Therefore, the ab$olute weight of B, to the ab$olute weight of C, is as the Ma$s A C to the Ma$s <I>C</I>: But as the Ma$s AC, is to the Ma$s C, $o is the ab$olute weight of A C, to the ab$olute weight of C: There- fore the ab$olute weight of B, hath the $ame proportion to the ab$o- lute weight of C, that the ab$olute weight of A C, hath to the ab- $olute weight of C: Therefore, the two Solids A C and B are equall in ab$olute Gravity: which was to be demon$trated. Having de- mon$trated this, I $ay,</P> <head>THEOREME X.</head> <P><I>That it is po$$ible of any a$$igned Matter, to form a Pi-</I> <marg>There may be Cones and Pira- mides of any <I>M</I>atter, which demittedinto the water, re$t only their Ba$es.</marg> <I>ramide or Cone upon any Ba$e, which being put upon the Water $hall not $ubmerge, nor wet any more than its Ba$e.</I></P> <P>Let the greate$t po$$ible Altitude of the Rampart be the Line D B, and the Diameter of the Ba$e of the Cone to be made of any Mat- ter a$$igned B C, at right angles to D B: And as the Specificall Gravity of the Matter of the Piramide or Cone to be made, is to the Specificall Gravity of the water, $o let the Altitude of the <fig> Rampart D B, be to the third part of the Piramide or Cone A B C, de$cribed upon the Ba$e, who$e Diameter is B C: I $ay, that the $aid Cone A B C, and any other Cone, lower then the $ame, $hall re$t upon the Surface of the water B C without $inking. Draw D F parallel to B C, and $uppo$e the Pri$me or Cylinder E C, which $hall be tripple to the Cone A B C. And, becau$e the Cylinder D C hath the $ame proportion to the Cylinder C E, that the Altitude D B, hath to the Altitude B E: But the Cylinder C E, is to the Cone A B C, as the Altitude E B is to the third part of the Altitude of the Cone: Therefore, by Equality of proportion, the Cylinder D C is to the Cone A B C, as D B is to the third part of the Altitude B E: But as D B is to the third part of B E, $o is the Specificall Gravity of the Cone A B C, to the Specificall Gra- vity of the water: Therefore, as the Ma$s of the Solid D C, is to the Ma$s of the Cone A <I>B</I> C, $o is the Specificall Gravity of the $aid Cone, to the Specificall Gravity of the water: Therefore, by the precedent Lemma, the Cone A B C weighs in ab$olute Gravity as much as a Ma$s of Water equall to the Ma$s D C: But the water which by the impo$ition of the Cone A B C, is driven out of its place, is as much as would preci$ely lie in the place D C, and is equall in weight to the Cone that di$placeth it: Therefore, there $hall be an <I>Equilibrium,</I> and the Cone $hall re$t without farther $ubmerging. And its ma- nife$t,</P> <foot>Mmm 2 COROL.</foot> <p n=>452</p> <head>COROLARY I.</head> <marg>Among$t Cones of the $ame Ba$e, tho$e of lea$t Al- titude $hall $ink the lea$t.</marg> <P><I>That making upon the $ame Ba$is, a Cone of a le$s Altitude, it $hall <*> al$o le$s grave, and $hall $o much the more re$t without Submer$ion.</I></P> <head>COROLARY II.</head> <P><I>It is manife$t, al$o, that one may make Cones and Piramids of any Matt<*></I> <marg>There may be Cones and Pira- mides of any Matter, which demitted with the Point down- wards do float a- top.</marg> <I>what$oever, more grave than the water, which being put into t<*> water, with the Apix or Point downwards, re$t without Submer$io<*></I></P> <P>Becau$e if we rea$$ume what hath been above demon$trated, <*> Pri$ms and Cylinders, and that on Ba$es equall to tho$e of th<*> $aid Cylinders, we make Cones of the $ame Matter, and thr<*> times as high as the Cylinders, they $hall re$t afloat, for that in Ma<*> and Gravity they $hall be equall to tho$e Cylinders, and by havin<*> their Ba$es equall to tho$e of the Cylinders, they $hall leave equa Ma$$es of Air included within the Ramparts. This, which for Exan<*> ple $ake hath been demon$trated, in Pri$ms, Cylinders, Cones an<*> Piramids, might be proved in all other Solid Figures, but it woul<*> require a whole Volume ($uch is the multitude and variety of the Symptoms and Accidents) to comprehend the particuler demon$tratio of them all, and of their $everall Segments: but I will to avoid prolixit in the pre$ent Di$cour$e, content my $elf, that by what I have declare every one of ordinary Capacity may comprehend, that there is no any Matter $o grave, no not Gold it $elf, of which one may not for<*> all $orts of Figures, which by vertue of the $uperiour Air adherent t<*> them, and not by the Waters Re$i$tance of Penetration, do remai<*> afloat, $o that they $ink not. Nay, farther, I will $hew, for removin that Error, that,</P> <head>THEOREME XI.</head> <marg>A Piramide or Cone, demitted with the Point downwards $hal $wim, with its Ba$e downward $hall $ink.</marg> <P><I>A Piramide or Cone put into the Water, with the Poin<*> downward $hall $wimme, and the $ame put with th<*> Ba$e downwards $hall $inke, and it $hall be impo$$ibl<*> to make it float.</I></P> <P>Now the quite contrary would happen, if the difficulty of Pene<*> trating the water, were that which had hindred the de$cent, fo that the $aid Cone is far apter to pierce and penetrate with its $har<*> Point, than with its broad and $pacious Ba$e.</P> <P>And, to demon$trate this, let the Cone be <I>A B C,</I> twice as grav<*> as the water, and let its height be tripple to the height of the Rampar<*> <I>D A E C</I>: I $ay, fir$t, that being put lightly into the water with the <foot>Point</foot> <p n=>453</p> Point downwards, it $hall not de$cend to the bot- tom: for the Aeriall Cylinder contained betwixt <fig> the Ramparts <I>D A C E,</I> is equall in Ma$s to the Cone <I>A B C</I>; $o that the whole Ma$s of the Solid compounded of the Air <I>D A C E,</I> and of the Cone <I>A B C,</I> $hall be double to the Cone <I>A C B:</I> And, becau$e the Cone <I>A B C</I> is $uppo$ed to be of Matter double in Gra- vity to the water, therefore as much water as the whole Ma$$e <I>D A B C E,</I> placed beneath the Levell of the water, weighs as much as the Cone <I>A B C</I>: and, therefore, there $hall be an <I>Equilibrium,</I> and the Cone <I>A B C</I> $hall de$cend no lower. Now, I $ay farther, that the $ame Cone placed with the Ba$e downwards, $hall $ink to the bottom, without any po$$ibility of returning again, by any means to $wimme.</P> <P>Let, therefore, the Cone be <I>A B D,</I> double in Gravity to the water, and let its height be tripple the height <fig> of the Rampart of water L B: It is already manife$t, that it $hall not $tay wholly out of the water, becau$e the Cylinder being com- prehended betwixt the Ramparts <I>L B D P,</I> equall to the Cone <I>A B D,</I> and the Matter of the Cone, beig double in Gravity to the water, it is evident that the weight of the $aid Cone $hall be double to the weight of the Ma$s of water equall to the Cylinder <I>L B D P</I>: Therefore it $hall not re$t in this $tate, but $hall de$cend.</P> <head>COROLARY I.</head> <P><I>I $ay farther; that much le$$e $hall the $aid Cone stay afloat, if one</I> <marg>Much le$s $hall the $aid Cone $wim, if one im- merge a part thereof.</marg> <I>immerge a part thereof.</I></P> <P>Which you may $ee, comparing with the water as well the part that $hall immerge as the other above water. Let us therefore of the Cone A B D, $ubmergeth part N T O S, and advance the Point N S F above water. The Altitude of the Cone F N S, $hall either be more than half the whole Altitude of the Cone F T O, or it $hall not be more: if it $hall be more than half, the Cone F N S $hall be more than half of the Cylinder E N S C: for the Altitude of the Cone F N S, $hall be more than Se$quialter of the Altitude of the Cylinder E N S C: And, becau$e the Matter of the Cone is $uppo$ed to be double in Specificall Gravity to the water, the water which would be contained within the Rampart E N S C, would be le$s grave ab$olutely than the Cone F N S; $o that the whole Cone F N S cannot be $u$tained by the Rampart: But the part immerged N T O S, by being double in Specificall Gravity to the water, $hall <foot>tend</foot> <p n=>454</p> tend to the bottom: Therefore, the whole <I>C</I>one F T O, as well in re$pect of the part $ubmerged, as the part above water $hall de- $cend to the bottom. But if the Altitude of the Point F N S, $hall be half the Altitude of the whole Cone F T O, the $ame Altitude of the $aid <I>C</I>one F N S $hall be Se$quialter to the Altitude E N: and, therefore, E N S C $hall be double to the Cone F N S; and as much water in Ma$s as the <I>C</I>ylinder E N S C, would weigh as much as the part of the <I>C</I>one F N S. But, becau$e the other immerged part N T O S, is double in Gravity to the water, a Ma$s of water equall to that compounded of the <I>C</I>ylinder E N S C, and of the Solid N T O S, $hall weigh le$s than the <I>C</I>one F T O, by as much as the weight of a Ma$s of water equall to the Solid N T O S: Therefore, the <I>C</I>one $ha l al$o de$cend. Again, becau$e the Solid N T O S, is $eptuple to the <I>C</I>one F N S, to which the <I>C</I>ylinder E S is double, the propor- tion of the Solid N T O S, $hall be to the <I>C</I>ylinder E N S C, as $eaven to two: Therefore, the whole Solid compounded of the <I>C</I>ylinder E N S C, and of the Solid N T O S, is much le$s than double the Solid N T O S: Therefore, the $ingle Solid N T O S, is much graver than a Ma$s of water equall to the Ma$s, compounded of the <I>C</I>y- linder E N S C, and of N T O S.</P> <head>COROLARY II.</head> <marg>Part of the Cones towards the Cu$pis remo- ved, it $hall $till $ink.</marg> <P><I>From whence it followeth, that though one $hould remove and take a- way the part of the Cone F N S, the $ole remainder N T O S would go to the bottom.</I></P> <head>COROLARY III.</head> <P><I>And if we $hould more depre$s the Cone F T O, it would be $o much the</I> <marg>The more the Cone is immer- ged, the more impo$$ible is its floating.</marg> <I>more impo$$ible that it $hould $u$tain it $elf afloat, the part $ubmerged N T O S $till encrea$ing, and the Ma$s of Air contained in the Rampart dimini$hing, which ever grows le$s, the more the Cone $ubmergeth.</I></P> <P>That Cone, therefore, that with its Ba$e upwards, and its <I>Cu$pis</I> downwards doth $wimme, being dimitted with its Ba$e downward mu$t of nece$$ity $inke. They have argued farre from the truth, therefore, who have a$cribed the cau$e of Natation to waters re$i$tance of Divi$ion, as to a pa$$ive principle, and to the breadth of the Figure, with which the divi$ion is to be made, as the Efficient.</P> <P>I come in the fourth place, to collect and conclude the rea$on of that which I have propo$ed to the Adver$aries, namely,</P> <foot>THE|O-</foot> <p n=>455</p> <head>THE OREME XII.</head> <P><I>That it is po$$ible to fo m Solid Bodies, of what Figure</I> <marg>Solids of any Figure & great- ne$$e, that natu- rally $ink, may by help of the Air in the Ram- part $wimme.</marg> <I>and greatne$s $oever, that of their own Nature goe to the Bottome; But by the help of the Air con- tained in the Rampart, re$t without $ubmerging.</I></P> <P>The truth of this Propo$ition is $ufficiently manife$t in all tho$e Solid Figures, that determine in their uppermo$t part in a plane Superficies: for making $uch Figures of $ome Matter $pecifi- cally as grave as the water, putting them into the water, $o that the whole Ma$s be covered, it is manife$t, that they $hall re$t in all places, provided, that $uch a Matter equall in weight to the water, may be exactly adju$ted: and they $hall by con$equence, re$t or lie even with the Levell of the water, without making any Rampart. If, therefore, in re$pect of the Matter, $uch Figures are apt to re$t without $ubmerging, though deprived of the help of the Rampart, it is manife$t, that they may admit $o much encrea$e of Gravity, (without encrea$ing their Ma$$es) as is the weight of as much water as would be contained within the Rampart, that is made about their upper plane Surface: by the help of which being $u$tained, they $hall re$t afloat, but being bathed, they $hall de$cend, having been made graver than the water. In Figures, therefore, that determine above in a plane, we may cleerly comprehend, that the Rampart added or removed, may prohibit or permit the de$cent: but in tho$e Figures that go le$$ening upwards towards the top, $ome Per$ons may, and that not without much $eeming Rea$on, doubt whether the $ame may be done, and e$pecially by tho$e which terminate in a very acute Point, $uch as are your Cones and $mall Piramids. Touch- ing the$e, therefore, as more dubious than the re$t, I will endeavour to demon$trate, that they al$o lie under the $ame Accident of going, or not going to the Bottom, be they of any whatever bigne$s. Let therefore the Cone be A B D, made of a matter $pecifically as grave as the water; it is manife$t <fig> that being put all under water, it $hall re$t in all places (alwayes provided, that it $hall weigh exactly as much as the water, which is almo$t impo$$ible to effect) and that any $mall weight being added to it, it $hall $ink to the bottom: but if it $hall de$cend downwards gently, I $ay, that it $hall make the Rampart E S T O, and that there $hall $tay out of the water the point A S T, tripple in height to the Rampart E S: which is manife$t, for the Matter of the <foot>Cone</foot> <p n=>456</p> Cone weighing equally with the water, the part $ubmerged S B D T, becomes indifferent to move downwards or upwards; and the Cone <I>A S T,</I> being equall in Ma$s to the water that would be contained in the concave of the Rampart <I>E S T O,</I> $hall be al$o equall unto it in Gravity: and, therefore, there $hall be a perfect <I>Equilibrium,</I> and, con$equently, a Re$t. Now here ari$eth a doubt, whether the Cone <I>A B D</I> may be made heavier, in $uch $ort, that when it is put wholly under water, it goes to the bottom, but yet not in $uch $ort, as to take from the Rampart the vertue of $u$taining it that it $ink not, and, the rea$on of the doubt is this: that although at $uch time as the Cone <I>A B D</I> is $pecifically as grave as the water, the Rampart <I>E S T O</I> $u$taines it, not only when the point <I>A S T</I> is tripple in height to the Altitude of the Rampart <I>E S,</I> but al$o when a le$$er part is above water; [for although in the De$cent of the Cone the Point <I>A S T</I> by little and little dimini$heth, and $o likewi$e the Rampart <I>E S T O,</I> yet the Point dimini$heth in <fig> greater proportion than the Rampart, in that it dimini$heth according to all the three Di- men$ions, but the Rampart according to two only, the Altitude $till remaining the $ame; or, if you will, becau$e the Cone <I>S T</I> goes di- mini$hing, according to the proportion of the cubes of the Lines that do $ucce$$ively become the Diameters of the Ba$es of emergent Cones, and the Ramparts dimini$h according to the proportion of the Squares of the $ame Lines; whereupon the proportions of the Points are alwayes Se$quialter of the proportions of the Cylinders, con- tained within the Rampart; $o that if, for Example, the height of the emergent Point were double, or equall to the height of the Rampart, in the$e ca$es, the Cylinder contained within the Ram- part, would be much greater than the $aid Point, becau$e it would be either $e$quialter or tripple, by rea$on of which it would perhaps $erve over and above to fu$tain the whole Cone, $ince the part $ub- merged would no longer weigh any thing;] yet, neverthele$s, when any Gravity is added to the whole Ma$s of the Cone, $o that al$o the part $ubmerged is not without $ome exce$$e of Gravity above the Gravity of the water, it is not manife$t, whether the Cylinder con- tained within the Rampart, in the de$cent that the Cone $hall make, can be reduced to $uch a proportion unto the emergent Point, and to $uch an exce$$e of Ma$s above the Ma$s of it, as to compen$ate the exce$$e of the Cones Specificall Gravity above the Gravity of the wa- ter: and the Scruple ari$eth, becau$e that howbeit in the de$cent made by the Cone, the emergent Point <I>A S T</I> dimini$heth, whereby there is al$o a diminution of the exce$s of the Cones Gravity above <foot>the</foot> <p n=>459</p> the Gravity of the water, yet the ca$e $tands $o, that the Rampart doth al$o contract it $elf, and the Cylinder contained in it doth de- mini$h. Neverthele$s it $hall be demon$trated, how that the Cone <I>A B D</I> being of any $uppo$ed bigne$$e, and made at the fir$t of a Matter exactly equall in Gravity to the Water, if there may be affixed to it $ome Weight, by means of which it may de$cend to the bottom, when $ubmerged under water, it may al$o by vertue of the Rampart $tay above without $inking.</P> <P>Let, therefore, the Cone <I>A B D</I> be of any $uppo$ed greatne$$e, and alike in $pecificall Gravity to the water. It is manife$t, that being put lightly into the water, it $hall re$t without de$cending; and it $hall advance above water, the Point <fig> <I>AS T,</I> tripple in height to the height of the Rampart <I>E S</I>: Now, $uppo$e the Cone <I>A B D</I> more depre$$ed, $o that it advance above wa- ter, only the Point <I>A I R,</I> higher by half than the Point <I>A S T,</I> with the Rampart about it <I>C I R N.</I> And, becau$e, the Cone <I>A B D</I> is to the Cone <I>A I R,</I> as the cube of the Line <I>S T</I> is to the cube of the Line <I>I R,</I> but the Cylin- der <I>E S T O,</I> is to the Cylinder <I>C I R N,</I> as the Square of <I>S T</I> to the Square of <I>I R,</I> the Cone <I>A S T</I> $hall be Octuple to the Cone <I>A I R,</I> and the Cylinder <I>E S T O,</I> quadruple to the Cylinder <I>C I R N</I>: But the Cone <I>A S T,</I> is equall to the Cylinder E <I>S T O</I>: Therefore, the Cylinder <I>C I R N,</I> $hall be double to the Cone <I>A I R:</I> and the water which might be contained in the Rampart <I>C I R N,</I> would be double in Ma$s and in Weight to the Cone <I>A I R,</I> and, therefore, would be able to $u$tain the double of the Weight of the Cone <I>AIR</I>: Therefore, if to the whole Cone <I>A B D,</I> there be added as much Weight as the Gravity of the Cone <I>A I R,</I> that is to $ay, the eighth part of the weight of the Cone <I>A S T,</I> it al$o $hall be $u$tained by the Rampart <I>C I R N,</I> but without that it $hall go to the bottome: the Cone <I>A B D,</I> being, by the addition of the eighth part of the weight of the Cone <I>A S T,</I> made $pecifically more grave than the water. But if the Altitude of the Cone <I>A I R,</I> were two thirds of the Altitude of the Cone <I>A S T,</I> the Cone <I>A S T</I> would be to the Cone <I>A I R,</I> as twenty $even to eight; and the Cylinder <I>E S T O,</I> to the Cylinder <I>C I R N,</I> as nine to four, that is, as twenty $even to twelve; and, therefore, the Cylinder <I>C I R N,</I> to the Cone <I>A I R,</I> as twelve to eight; and the exce$s of the Cylinder <I>C I R N,</I> above the Cone <I>A I R,</I> to the Cone <I>A S T,</I> as four to twenty $even: there- fore if to the Cone <I>A B D</I> be added $o much weight as is the four twenty $evenths of the weight of the Cone <I>A S T,</I> which is a little more then its $eventh part, it al$o $hall continue to $wimme, and <foot>Nnn the</foot> <p n=>460</p> the height of the emergent Point $hall be double to the height of the Rampart. This that hath been demon$trated in Cones, exactly holds in Piramides, although the one or the other $hould be very $harp in <marg>Natatiou ea$i- e$t effected in Figures broad toward the top.</marg> their Point or Cu$pis: From whence we conclude, that the $ame Accident $hall $o much the more ea$ily happen in all other Figures, by how much the le$s $harp the Tops $hall be, in which they deter- mine, being a$$i$ted by more $pacious Ramparts.</P> <head>THEOREME XIII.</head> <marg>All Figures $ink or $wim, upon bathing or not bathing of their tops.</marg> <P><I>All Figures, therefore, of whatever greatne$$e, may go, and not go, to the Bottom, according as their Sumi- ties or Tops $hall be bathed or not bathed.</I></P> <P>And this Accident being common to all $orts of Figures, without exception of $o much as one. Figure hath, therefore, no part in the production of this Effect, of $ometimes $inking, and $ome- times again not $inking, but only the being $ometimes conjoyned to, and $ometimes $eperated from, the $upereminent Air: which cau$e, in fine, who $o $hall rightly, and, as we $ay, with both his Eyes, con$ider this bu$ine$s, will find that it is reduced to, yea, that it really is the $ame with, the true, Naturall and primary cau$e of Natation or Submer$ion; to wit, the exce$s or deficiency of the Gravity of the water, in relation to the Gravity of that Solid Mag- nitude, that is demitted into the water. For like as a Plate of Lead, as thick as the back of a Knife, which being put into the water by it $elf alone goes to the bottom, if upon it you fa$ten a piece of Cork four fingers thick, doth continue afloat, for that now the Solid that is demitted in the water, is not, as before, more grave than the water, but le$s, $o the Board of Ebony, of its own nature more grave than water; and, therefore, de$cending to the bottom, when it is demit- ted by it $elf alone into the water, if it $hall be put upon the water, conjoyned with an Expanded vail of Air, that together with the Ebony doth de$cend, and that it be $uch, as that it doth make with it a compound le$s grave than $o much water in Ma$s, as equalleth the Ma$s already $ubmerged and depre$$ed beneath the Levell of the waters Surface, it $hall not de$cend any farther, but $hall re$t, for no other than the univer$all and mo$t common cau$e, which is that Solid Magnitudes, le$s grave <I>in$pecie</I> than the water, go not to the bottom.</P> <P>So that if one $hould take a Plate of Lead, as for Example, a finger thick, and an handfull broad every way, and $hould attempt to make it $wimme, with putting it lightly on the water, he would lo$e his Labour, becau$e that if it $hould be depre$$ed an Hairs breadth be- <foot>yond</foot> <p n=>461</p> yond the po$$ible Altitude of the Ramparts of water, it would dive and $ink; but if whil$t it is going downwards, one $hould make certain Banks or Ramparts about it, that $hould hinder the do fu$ion of the water upon the $aid Plate, the which Banks $hould ri$e $o high, as that they might be able to contain as much water, as $hould weigh equally with the $aid Plate, it would, without all Que$tion, de$cend no lower, but would re$t, as being $u$tained by vertue of the Air contained within the afore$aid Ramparts: and, in $hort, there would be a Ve$$ell by this means formed with the bottom of Lead. But if the thinne$s of the Lead $hall be $uch, that a very $mall height of Rampart would $uffice to contain $o much Air, as might keep it afloat, it $hall al$o re$t without the Artificiall Banks or Ram- parts, but yet not without the Air, becau$e the Air by it $elf makes Banks $ufficient for a $mall height, to re$i$t the Superfu$ion of the water: $o that that which in this ca$e $wimmes, is as it were a Ve$$ell filled with Air, by vertue of which it continueth afloat.</P> <P>I will, in the la$t place, with an other Experimeut, attempt to remove all difficulties, if $o be there $hould yet be any doubt le$t in any one, touching the opperation of this ^{*}Continuity of the Air, with <marg>*Orrather Cor- tiguity,</marg> the thi<*> Plate which $wims, and afterwards put an end to this part of my di$cour$e.</P> <P>I $uppo$e my $elf to be que$tioning with $ome of my Oponents.</P> <P>Whether Figure have any influence upon the encrea$e or diminu- <marg>An Experi- ment of the op- peration of Fi- gures, in en- crea$ing or le$- $ening of the Airs Re$i$tance of Divi$ion.</marg> tion of the Re$i$tance in any Weight again$t its being rai$ed in the Air, and I $uppo$e, that I am to maintain the Affirmative, a$$ert- ing that a Ma$s of Lead, reduced to the Figure of a Ball, $hall be rai$ed with le$s force, then if the $ame had been made into a thinne and broad Plate, becau$e that it in this $pacious Figure, hath a great quantity of Air to penetrate, and in that other, more compacted and contracted very little: and to demon$trate the truth of $uch my O- pinion, I will hang in a $mall thred fi<*>$t the Ball or Bullet, and put that into the water, tying the thred that upholds it to one end of the Ballance that I hold in the Air, and to the other end I by degrees adde $o much Weight, till that at la$t it brings up the Ball of Lead out of the water: to do which, $uppo$e a Gravity of thirty Ounces $ufficeth; I afcerwards reduce the $aid Lead into a flat and thinne Plate, the which I likewi$e put into the water, $u$pended by three threds, which hold it parallel to the Surface of the water, and put- ting in the $ame manner, Weights to the other end, till $uch time as the Place comes to be rai$ed and drawn out of the water: I finde that thirty $ix ounces will not $uffice to $eperate it from the water, and rai$e it tho<*>ow the Air: and arguing from this Experiment, I af- firm, that I have fully demon$trated the truth of my Propo$ition. He re my Oponents de$ires me to look down, $hewing me a thing <foot>Nnn 2 which</foot> <p n=>462</p> which I had not before ob$erved, to wit, that in the A$cent of the Plate out of the water, it draws after it another Plate <I>(if I may $o call it)</I> of water, which before it divides and parts from the inferiour Surface of the Plate of Lead, is rai$ed above the Levell of the other water, more than the thickne$s of the back of a Knife: Then he goeth to repeat the Experiment with the Ball, and makes me $ee, that it is but a very $mall quantity of water, which cleaves to its compacted and contracted Figure: and then he $ubjoynes, that its no wonder, if in $eperating the thinne and broad Plate from the water, we meet with much greater Re$i$tance, than in $eperating the Ball, $ince together with the Plate, we are to rai$e a great quantity of water, which occurreth not in the Ball: He telleth me moreover, how that our Que$tion is, whether the Re$i$tance of Elevation be greater in a dilated Plate of Lead, than in a Ball, and not whether more re$i$teth a Plate of Lead with a great quantity of water, or a Ball with a very little water: He $heweth me in the clo$e, that the putting the Plate and the Ball fir$t into the water, to make proofe thereby of their Re$i$tance in the Air, is be$ides our ca$e, which treats of Elivating in the Air, and of things placed in the Air, and not of the Re$i$tance that is made in the Confines of the Air and water, and by things which are part in Air and part in water: and la$tly, they make me feel with my hand, that when the thinne Plate is in the Air, and free from the weight of the water, it is rai$ed with the very $ame Force that rai$eth the Ball. Seeing, and under$tand- ing the$e things, I know not what to do, unle$s to grant my $elf con- vinced, and to thank $uch a Friend, for having made me to $ee that which I never till then ob$erved: and, being adverti$ed by this $ame Accident, to tell my Adver$aries, that our Que$tion is, whether a Board and a Ball of Ebony, equally go to the bottom in water, and not a Ball of Ebony and a Board of Ebony, joyned with another flat Body of Air: and, farthermore, that we $peak of $inking, and not $inking to the bottom, in water, and not of that which happeneth in the Confines of the water and Air to Bodies that be part in the Air, and part in the water; nor much le$s do we treat of the greater or le$$er Force requi$ite in $eperating this or that Body from the Air; not omitting to tell them, in the la$t place, that the Air doth re$i$t, and gravitate downwards in the water, ju$t $o much as the water (if I may $o $peak) gravitates and re$i$ts upwards in the Air, and that the $ame force is required to $inke a Bladder under water, that is full of Air, as to rai$e it in the Air, being full of water, removing the con- $ideration of the weight of that Filme or Skinne, and confidering the water and the Air only. And it is likewi$e true, that the $ame Force is required to $ink a Cup or $uch like Ve$$ell under water, whil$t it is full of Air, as to rai$e it above the Superficies of the water, keeping <foot>it</foot> <p n=>463</p> it with the mouth downwards; whil$t it is full of water, which is con$trained in the $ame manner to follow the Cup which contains it, and to ri$e above the other water into the Region of the Air, as the Air is forced to follow the $ame Ve$$ell under the Surface of the wa- ter, till that in this ca$e the water, $urmounting the brimme of the Cup, breaks in, driving thence the Air, and in that ca$e, the $aid brimme coming out of the water, and arriving to the Confines of the Air, the water falls down, and the Air $ub-enters to fill the cavity of the Cup: upon which en$ues, that he no le$s tran$gre$$es the Arti- cles of the <I>Convention,</I> who produceth a Plate conjoyned with much Air, to $ee if it de $eend to the bottom in water, then he that makes proof of the Re$i$tance again$t Elevation in Air with a Plate of Lead, joyned with a like quantity of water.</P> <P>I have $aid all that I could at pre$ent think of, to maintain the <marg><I>Ari$totles</I> opi- nion touching the Operation of Figure ex- amined.</marg> A$$ertion I have undertook. It remains, that I examine that which <I>Ari$totle</I> hath writ of this matter towards the end of his Book <I>De Cælo</I>; wherein I $hall note two things: the one that it being true as hath <marg><I>Ari$tot de Cælo,</I> Lib. 4. Cap. 66.</marg> been demon$trated, that Figure hath nothing to do about the moving or not moving it $elf upwards or downwards, it $eemes that <I>Aristotle</I> at his fir$t falling upon this Sp. culation, was of the $ame opinion, as in my opinion may be collected from the examination of his words. Tis true, indeed, that in e$$aying afterwards to render a rea$on of $uch effect, as not having in my conceit hit upon the right, (which in the $econd place I will examine) it $eems that he is brought to admit the largene$$e of Figure, to be intere$$ed in this operation. As to the fir$t particuler, hear the preci$e words of <I>Aristotle.</I></P> <P><I>Figures are not the Cau$es of moving $imply upwards or downwards,</I> <marg><I>Ari$totle</I> makes not Figure the cau$e of Motion ab$olutely, but of $wi$t or $low motion,</marg> <I>but of moving more $lowly or $wiftly, and by what means this comes to pa$s, it is not difficult to $ee.</I></P> <P>Here fir$t I note, that the terms being four, which fall under the pre$ent con$ideration, namely, Motion, Re$t, Slowly and Swiftly: <marg>Lib. 4. Cap. 61 Text. 42.</marg> And <I>Ari$totle</I> naming Figures as Cau$es of Tardity and Velocity, ex- cluding them from being the Cau$e of ab$olute and $imple Motion, it $eems nece$$ary, that he exclude them on the other $ide, from being the Cau$e of Re$t, $o that his meaning is this. Figures are not the Cau$es of moving or not moving ab$olutely, but of moving quickly or $lowly: and, here, if any $hould $ay the mind of <I>Ari$totle</I> is to exclude Figures from being Cau$es of Motion, but yet not from being Cau$es of Re$t, $o that the $ence would be to remove from Figures, there being the Cau$es of moving $imply, but yet not there being Cau$es of Re$t, I would demand, whether we ought with <I>Aristotle</I> to under$tand, that all Figures univer$ally, are, in $ome manner, the cau$es of Re$t in tho$e Bodies, which otherwi$e would move, or el$e $ome particular Figures only, as for Example, broad <foot>and</foot> <p n=>464</p> and thinne Figures: If all indifferently, then every Body $hall re$t: becau$e every Body hath $ome Figure, which is fal$e: but if $ome particular Figures only may be in $ome manner a Cau$e of Re$t, as, for Example, the broad, then the others would be in $ome manner the Cau$es of Motion: for if from $eeing $ome Bodies of a contracted Figure move, which after dilated into Plates re$t, may be inferred, that the Amplitude of Figure hath a part in the Cau$e of that Re$t; $o from $eeing $uch like Figures re$t, which afterwards contracted move, it may with the $ame rea$on be affirmed, that the united and contracted Figure, hath a part in cau$ing Motion, as the remover of that which impeded it: The which again is directly oppo$ite to what <I>Ari$totle</I> $aith, namely, that Figures are not the Cau$es of Motion. Be$ides, if <I>Ari$totle</I> had admitted and not excluded Figures from be- ing Cau$es of not moving in $ome Bodies, which moulded into ano- ther Figure would move, he would have impertinently propounded in a dubitative manner, in the words immediately following, whence it is, that the large and thinne Plates of Lead or Iron, re$t upon the water, $ince the Cau$e was apparent, namely, the Amplitude of Figure. Let us conclude, therefore, that the meaning of <I>Ari$totle</I> in this place is to affirm, that Figures are not the Cau$es of ab$olutely moving or not moving, but only of moving $wif<*>ly or $lowly: which we ought the rather to believe, in regard it is indeed a me$t true con- ceipt and opinion. Now the mird of <I>Ari$totle</I> being $uch, and ap- pearing by con$equence, rather contrary at the fir$t $ight, then fa- vourable to the a$$ertion of the Oponents, it is nece$$ary, that their Interpretation be not exactly the $ame with that, but $uch, as being in part under$tood by $ome of them, and in part by others, was $et down: and it may ea$ily be indeed $o, being an Interpretation con$onent to the $ence of the more famous Interpretors, which is, that the Adverbe <I>Simply</I> or <I>Ab$olutely,</I> put in the Text, orght not to be joyned to the Verbe to <I>Move,</I> but with the Noun <I>Cau$es</I>: $o that the purport of <I>Ari$totles</I> words, is to affirm, That Figures are not the Cau$es ab$olutely of moving or not moving, but yet are Cau$es <I>Se- cundum quid, viz</I> in $ome $ort; by which means, they are called Auxiliary and Concomitant Cau$es: and this Propo$ition is received and a$$erted as true by <I>Signor Buonamico Lib.</I> 5. <I>Cap.</I> 28. where he thus writes. <I>There are other Cau$es concomitant, by which $ome things float, and others $ink, among which the Figures of Bodies hath the fir$t place,</I> &c.</P> <P>Concerning this Propo$ition, I meet with many doubts and diffi- culties, for which me thinks the words of <I>Ari$totle</I> are not capable of $uch a con$truction and $ence, and the difficulties are the$e.</P> <P>Fir$t in the order and di$po$ure of the words of <I>Ari$totle,</I> the par- ticle <I>Simpliciter,</I> or if you will <I>ab$oluté,</I> is conjoyned with the Verb <foot><I>to</I></foot> <p n=>465</p> <I>to move,</I> and $eperated from the Noun <I>Cau$es,</I> the which is a great pre$umption in my favour, $eeing that the writing and the Text $aith, Figures are not the Cau$e of moving $imply upwards or downwards, but of quicker or $lower Motion: and, $aith not, Figures are not $imply the Cau$es of moving upwards or down- wards, and when the words of a Text receive, tran$po$ed, a $ence different from that which they found, taken in the order wherein the Author di$po$eth them, it is not convenient to inverte them. And who will affirm that <I>Ari$totle</I> de$iring to write a Propo$ition, would di$po$e the words in $uch $ort, that they $hould import a different, nay, a contrary $ence? contrary, I $ay, becau$e under- $tood as they are written; they $ay, that Figures are not the Cau$es of Motion, but inverted, they $ay, that Figures are the Cau$es of Motion, &c.</P> <P>Moreover, if the intent of <I>Aristotle</I> had been to $ay, that Figures are not $imply the Cau$es of moving upwards or downwards, but only Cau$es <I>Secundum quid,</I> he would not have adjoyned tho$e words, <I>but they are Cau$es of the more $wift or $low Motion</I>; yea, the $ubjoining this would have been not only $uperfluous but fal$e, for that the whole tenour of the Propo$ition would import thus much. Figures are not the ab$olute Cau$es of moving upwards or down- wards, but are the ab$olute Cau$e of the $wift or $low Motion; which is not true: becau$e the primary Cau$es of greater or le$$er Velocity, are by <I>Ari$totle</I> in the 4th of his <I>Phy$icks, Text.</I> 71. attri- buted to the greater or le$$er Gravity of Moveables, compared a- mong them$elves, and to the greater or le$$er Re$i$tance of the <I>Medium's,</I> depending on their greater or le$s Cra$$itude: and the$e are in$erted by <I>Ari$totle</I> as the primary Cau$es; and the$e two only are in that place nominated: and Figure comes afterwards to be con$idered, <I>Text.</I> 74. rather as an In$trumentall Cau$e of the force of the Gravity, the which divides either with the Figure, or with the <I>Impetus</I>; and, indeed, Figure by it $elf without the force of Gravity or Levity, would opperate nothing.</P> <P>Iadde, that if <I>Ari$totle</I> had an opinion that Figure had been in $ome $ort the Cau$e of moving or not moving, the inqui$ition which he makes immediately in a doubtfull manner, whence it comes, that a Plate of Lead flotes, would have been impertinent; for if but ju$t before he had $aid, that Figure was in a certain $ort the Cau$e of moving or not moving, he needed not to call in Que$tion, by what Cau$e the Plate of Lead $wims, and then a$cri- bing the Cau$e to its Figure; and framing a di$cour$e in this manner. Figure is a Cau$e <I>Secundum quid</I> of not $inking: but, now, if it be doubted, for what Cau$e a thin Plate of Lead goes not to the bottom; it $hall be an$wered, that that proceeds from its Figure: a di$cour$e <foot>which</foot> <p n=>466</p> which would be indecent in a Child, much more in <I>Ari$totle</I>; For where is the occa$ion of doubting? And who $ees not, that if <I>Ari$totle</I> had held, that Figure was in $ome $ort a Cau$e of Natation, he would without the lea$t He$itation have writ; That Figure is in a certain $ort the Cau$e of Natation, and therefore the Plate of Lead in re$pect of its large and expatiated Figure $wims; but if we take the propo$ition of <I>Ari$totle</I> as I $ay, and as it is writte n, and as in- deed it is true, the en$uing words come in very oppo$itely, as well in the introduction of $wift and $low, as in the que$tion, which very pertinently offers it $elf, and would $ay thus much.</P> <P>Figures are not the Cau$e of moving or not moving $imply up- wards or downwards, but of moving more quickly or $lowly: But if it be $o, the Cau$e is doubtfull, whence it proceeds, that a Plate of Lead or of Iron broad and thin doth $wim, &c. And the occa$ion of the doubt is obvious, becau$e it $eems at the fir$t glance, that the Figure is the Cau$e of this Natation, $ince the $ame Lead, or a le$s quantity, but in another Figure, goes to the bottom, and we have already affirmed, that the Figure hath no $hare in this effect.</P> <P>La$tly, if the intent of <I>Ari$totle</I> in this place had been to $ay, that Figures, although not ab$olutely, are at lea$t in $ome mea$ure the Cau$e of moving or not moving: I would have it con$idered, that he names no le$s the Motion upwards, than the other down- wards: and becau$e in exemplifying it afterwards, he produceth no other Experiments than of a Plate of Lead, and Board of Ebony, Matters that of their own Nature go to the bottom, but by vertue (as our Adver$aries $ay) of their Figure, re$t afloat; it is $it that they $hould produce $ome other Experiment of tho$e Matters, which by their Nature $wims, but retained by their Figure re$t at the bottom. But $ince this is impo$$ible to be done, we conclude, that <I>Ari$totle</I> in this place, hath not attributed any action to the Figure of $imply moving or not moving.</P> <P>But though he hath exqui$itely Philo$ophiz'd, in inve$tigating the $olution of the doubts he propo$eth, yet will I not undertake to maintain, rather various difficulties, that pre$ent them$elves unto me, give me occa$ion of $u$pecting that he hath not entirely di$plaid unto us, the true Cau$e of the pre$ent Conclu$ion: which difficulties I will propound one by one, ready to change opinion, when ever I am $hewed, that the Truth is different from what I $ay; to the confe$$ion whereof I am much more inclinable than to contra- diction.</P> <marg><I>Ari$totle</I> erred in affirming a Needle dimitted long wayes to $ink.</marg> <P><I>Ari$totle</I> having propounded the Que$tion, whence it proceeds, that broad Plates of Iron or Lead, float or $wim; he addeth (as it were $trengthening the occa$ion of doubting) fora$much as other things, le$s, and le$s grave, be they round or long, as for in$tance a <foot>Needle</foot> <p n=>467</p> Needle go to the bottom. Now I here doubt, or rather am certain, that a Needle put lightly upon the water, re$ts afloat, no le$s than the thin Plates of Iron or Lead. I cannot believe, albeit it hath been told me, that $ome to defend <I>Ari$totle</I> $hould $ay, that he intends a Needle demitted not longwayes but endwayes, and with the Point downwards; neverthele$s, not to leave them $o much as this, though very weak refuge, and which in my judgement <I>Ari$totle</I> him$elf would refu$e, I $ay it ought to be under$tood, that the Needle mu$t be demitted, according to the Dimen$ion named by <I>Ari$totle,</I> which is the length: becau$e, if any other Dimen$ion than that which is named, might or ought to be taken, I would $ay, that even the Plates of Iron and Lead, $ink to the bottom, if they be put into the water edgewayes and not flatwayes. But becau$e <I>Ari$totle</I> $aith, broad Figures go not to the bottom, it is to be under$tood, being demitted broadwayes: and, therefore, when he $aith, long Figures as a Needle, albeit light, re$t not afloat, it ought to be under$tood of them when demitted longwayes.</P> <P><I>Morcover, to $ay that</I> Ari$totle <I>is to be under$tood of the Needle de- mitted with the Point downwards, is to father upon him a great imper- tinency; for in this place he $aith, that little Particles of Lead or Iron, if they be round or long as a Needle, do $ink to the bottome; $o that by his Opinion, a Particle or $mall Grain of Iron cannot $wim: and if he thus believed, what a great folly would it be to $ubjoyn, that neither would a Needle demitted endwayes $wim? And what other is $uch a Needle, but many $uch like Graines accumulated one upon another? It was too unworthy of $uch a man to $ay, that one $ingle Grain of Iron could not $wim, and that neither can it $wim, though you put a hundred more upon it.</I></P> <P>La$tly, either <I>Ari$totle</I> believed, that a Needle demitted long- wayes upon the water, would $wim, or he believed that it would not $wim: If he believed it would not $wim, he might well $peak as indeed he did; but if he believed and knew that it would $loat, why, together with the dubious Problem of the Natation of broad Figure, though of ponderous Matter, hath he not al$o introduced the Que$tion; whence it proceeds, that even long and $lender Fi- gures, howbeit of Iron or Lead do $wim? And the rather, for that the occa$ion of doubting $eems greater in long and narrow Figures, than in broad and thin, as from <I>Aristotles</I> not having doubted of it, is manife$ted.</P> <P>No le$$er an inconvenience would they fa$ten upon <I>Ari$totle,</I> who in his defence $hould $ay, that he means a Needle pretty thick, and not a $mall one; for take it for granted to be intended of a $mall one<*> <foot>Ooo and</foot> <p n=>468</p> and it $hall $uffice to reply, that he believed that it would $wim; and I will again charge him with having avoided a more wonderfull and intricate Probleme, and introduced the more facile and le$s wonderfull.</P> <P>We $ay freely therefore; that <I>Ari$totle</I> did hold, that only the broad Figure did $wim, but the long and $lender, $uch as a Needle, not. The which neverthele$s is fal$e, as it is al$o fal$e in round Bodies: becau$e, as from what hath been predemon$trated, may be ga- thered, little Balls of Lead and Iron, do in like manner $wim.</P> <P>He propo$eth likewi$e another Conclu$ion, which likewi$e $eems <marg><I>Ari$totle</I> af- fir meth $ome Bodies volatile for their Minu- ity, Text. 42.</marg> different from the truth, and it is, That $ome things, by rea$on of their littlene$s fly in the Air, as the $mall du$t of the Earth, and the thin leaves of beaten Gold: but in my Opinion, Experience $hews us, that that happens not only in the Air, but al$o in the water, in which do de$cend, even tho$e Particles or Atomes of Earth, that di$tur be it, who$e minuity is $uch, that they are not de$ervable, $ave only when they are many hundreds together. Therefore, the du$t of the Earth, and beaten Gold, do not any way $u$tain them$elves in the Air, but de$cend downwards, and only fly to and again in the $ame, when $trong Windes rai$e them, or other agitations of the Air commove them: and this al$o happens in the commotion of the water, which rai$eth its Sand from the bottom, and makes it muddy. But <I>Ari$totle</I> cannot mean this impediment of the commotion, of which he makes no mention, nor names other than the lightne$s of $uch Minutiæ or Atomes, and the Re$i$tance of the Cra$$itudes of the Water and Air, by which we $ee, that he $peakes of a calme, and not di$turbed and agitated Air: but in that ca$e, neither Gold nor Earth, be they never $o $mall, are $u$tained, but $peedily de$cend.</P> <marg><I>Democritus</I> pla- ced the Cau$e of Natation in certain $iery A- tomes.</marg> <P>He pa$$eth next to confute <I>Democritus,</I> which, by his Te$timony would have it, that $ome Fiery Atomes, which continually a$cend through the water, do $pring upwards, and $u$tain tho$e grave Bodies, which are very broad, and that the narrow de$cend to the bottom, <marg><I>Ari$tot. De Cœlo</I> lib. 4. cap. 6. text. 43.</marg> for that but a $mall quantity of tho$e Atomes, encounter and re$i$t them.</P> <P>I $ay, <I>Ari$totle</I> confutes this po$ition, $aying, that that $hould <marg><I>Democritus</I> con- futed by <I>Ari- $totle,</I> text 43.</marg> much more occurre in the Air, as the $ame <I>Democritus</I> in$tances a- gain$t him$elf, but after he had moved the objection, he $lightly re- $olves it, with $aying, that tho$e Corpu$cles which a$cend in the Air, make not their <I>Impetus</I> conjunctly. Here I will not $ay, that the <marg><I>Ari$totles</I> con- futation of <I>De- mocritus</I> refuted by the Author.</marg> rea$on alledged by <I>Democritus</I> is true, but I will only $ay, it $eems in my judgement, that it is not wholly confuted by <I>Ari$totle,</I> whil$t he $aith, that were it true, that the calid a$cending Atomes, $hould $u$tain Bodies grave, but very broad, it would much more be done in the Air, than in Water, for that haply in the Opinion of <I>Ari$totle,</I> <foot>the</foot> <p n=>469</p> the $aid calid Atomes a$cend with much greater Force and Velocity through the Air, than through the water. And if this be $o, as I veri- ly believe it is, the Objection of <I>Ari$totle</I> in my judgement $eems to give occa$ion of $u$pecting, that he may po$$ibly be deceived in more than one particular: Fir$t, becau$e tho$e calid Atomes, (whether they be Fiery Corpu$cles, or whether they be Exhalations, or in $hort, whatever other matter they be, that a$cends upwards through the Air) cannot be believed to mount fa$ter through Air, than through water: but rather on the contrary, they peradventure move more impetuou$ly through the water, than through the Air, as hath been in part demon$trated above. And here I cannot finde the rea- $on, why <I>Ari$totle</I> $eeing, that the de$eending Motion of the $ame Moveable, is more $wift in Air, than in water, hath not adverti$ed us, that from the contrary Motion, the contrary $hould nece$$arily follow; to wit, that it is more $wift in the water, than in the Air: for $ince that the Moveable which de$cendeth, moves $wifter through the Air, than through the water, if we $hould $uppo$e its Gravity gradually to dimini$h, it would fir$t become $uch, that de$cending $wiftly through the Air, it would de$cend but $lowly through the water: and then again, it might be $uch, that de$cending in the Air, it $hould a$cend in the water: and being made yet le$s grave, it $hall a$cend $wiftly through the water, and yet de$cend likewi$e through the Air: and in $hort, before it can begin to a$cend, though but $lowly through the Air, it $hall a$cend $wiftly through the water: how then is it true, that a$cending Moveables move $wifter through the Air, than through the water?</P> <P>That which hath made <I>Ari$totle</I> believe, the Motion of A$cent to be $wifter in Air, than in water, was fir$t, the having referred the Cau$es of $low and quick, as well in the Motion of A$cent, as of De$cent, only to the diver$ity of the Figures of the Moveable, and to the more or le$s Re$i$tance of the greater or le$$er Cra$$itude, or Ra- rity of the <I>Medium</I>; not regarding the compari$on of the Exce$$es of the Gravities of the Moveables, and of the <I>Mediums</I>: the which notwith$tanding, is the mo$t principal point in this affair: for if the augmentation and diminution of the Tardity or Velocity, $hould have only re$pect to the Den$ity or Rarity of the <I>Medium,</I> every Body that de$cends in Air, would de$cend in water: becau$e whatever difference is found between the Cra$$itude of the water, and that of the Air, may well be found between the Velocity of the $ame Move- able in the Air, and $ome other Velocity: and this $hould be its proper Velocity in the water, which is ab$olutely fal$e. The other occa$ion is, that he did believe, that like as there is a po$itive and in- trin$ecall Quality, whereby Elementary Bodies have a propen$ion of moving towards the Centre of the Earth, $o there is another like- <foot>Ooo 2 wi$e</foot> <p n=>470</p> wi$e intrin$ecall, whereby $ome of tho$e Bodies have an <I>Impetus</I> of <marg>Lib. 4. Cap. 5.</marg> flying the Centre, and moving upwards: by Vertue of which in- trin$e call Principle, called by him Levity, the Moveables which have that $ame Motion more ea$ily penetrate the more $ubtle <I>Medium,</I> than the more den$e: but $uch a Propo$ition appears likewi$e un- certain, as I have above hinted in part, and as with Rea$ons and Experiments, I could demon$trate, did not the pre$ent Argument im- portune me, or could I di$patch it in few words.</P> <P>The Objection therefore of <I>Ari$totle</I> again$t <I>Democritus,</I> whil$t he $aith, that if the Fiery a$cending Atomes $hould $u$tain Bodies grave, but of a di$tended Figure, it would be more ob$ervable in the Air than in the water, becau$e $uch Corpu$cles move $wifter in that, than in this, is not good; yea the contrary would evene, for that they a$cend more $lowly through the Air: and, be$ides their moving $lowly, they a$cend, not united together, as in the water, but di$continue, and, as we $ay, $catter: And, therefore, as <I>Democritus</I> well replyes, re$olving the in$tance they make not their pu$h or <I>Impetus</I> conjunctly.</P> <P><I>Ari$totle,</I> in the $econd place, deceives him$elf, whil$t he will have the $aid grave Bodies to be more ea$ily $u$tained by the $aid Fiery a$cending Atomes in the Air than in the Water: not ob$erv- ing, that the $aid Bodies are much more grave in that, than in this, and that $uch a Body weighs ten pounds in the Air, which will not in the water weigh 1/2 an ounce; how can it then be more ea$ily $u$tained in the Air, than in the Water?</P> <P>Let us conclude, therefore, that <I>Democritus</I> hath in this particular better Philo$ophated than <I>Ari$totle.</I> But yet will not I affirm, that <I>De-</I> <marg><I>Democritus</I> con- futed by the Authour.</marg> <I>mocritus</I> hath rea$on'd rightly, but I rather $ay, that there is a ma- nife$t Experiment that overthrows his Rea$on, and this it is, That if it were true, that calid a$cending Atomes $hould uphold a Body, that if they did not hinder, would go to the bottom, it would follow, that we may find a Matter very little $uperiour in Gravity to the water, the which being reduced into a Ball, or other contracted Figure, $hould go to the bottom, as encountring but few Fiery A- tomes; and which being di$tended afterwards into a dilated and thin Plate, $hould come to be thru$t upwards by the impul$ion of a great Multitude of tho$e Corpu$cles, and at la$t carried to the very Surface of the water: which wee $ee not to happen; Experience $hewing us, that a Body <I>v. gra.</I> of a Sphericall Figure, which very hardly, and with very great lea$ure goeth to the bottom, will re$t there, and will al$o de$cend thither, being reduced into what$oever other di$tended Figure. We mu$t needs $ay then, either that in the water, there are no $uch a$cending Fiery Atoms, or if that $uch there be, that they are not able to rai$e and lift up any Plate of a Matter, <foot>that</foot> <p n=>471</p> that without them would go to the bottom: Of which two Pofitions, I e$teem the $econd to be true, under$tanding it of water, con$tituted in its naturall Coldne$s. But if we take a Ve$$el of Gla$s, or Bra$s, or any other hard matter, full of cold water, within which is put a Solid of a flat or concave Figure, but that in Gravity exceeds the water $o little, that it goes $lowly to the bottom; I $ay, that putting $ome burning Coals under the $aid Ve$$el, as $oon as the new Fiery Atomes $hall have penetrated the $ub$tance of the Ve$$el, they $hall without doubt, a$cend through that of the water, and thru$ting a- gain$t the fore$aid Solid, they $hall drive it to the Superficies, and there detain it, as long as the incur$ions of the $aid Corpu$cles $hall la$t, which cea$ing after the removall of the Fire, the Solid being a- bandoned by its $upporters, $hall return to the bottom.</P> <P>But <I>Democritus</I> notes, that this Caufe only takes place when we treat of rai$ing and $u$taining of Plates of Matters, but very little heavier than the water, or extreamly thin: but in Matters very grave, and of $ome thickne$s, as Plates of Lead or other Mettal, that $ame Effect wholly cea$eth: In Te$timony of which, let's ob$erve that $uch Plates, being rai$ed by the Fiery Atomes, a$cend through all the depth of the water, and $top at the Confines of the Air, $till $taying under water: but the Plates of the Opponents $tay not, but only when they have their upper Superficies dry, nor is there any means to be u$ed, that when they are within the water, they may not $ink to the bottom. The cau$e, therefore, of the Supernatation of the things of which <I>Democritus</I> $peaks is one, and that of the Super- natation of the things of which we $peak is another. But, returning <marg><I>Ari$totle</I> $hews his de$ire of finding <I>Demo- critus</I> in an Er- ror, to exceed that of di$co- veting Truth.</marg> to <I>Ari$totle,</I> methinks that he hath more weakly confuted <I>Democritus,</I> than <I>Democritus</I> him$elf hath done: For <I>Ari$totle</I> having propounded the Objection which he maketh again$t him, and oppo$ed him with $aying, that if the calid a$cendent Corpu$cles were tho$e that rai$ed the thin Plate, much more then would $uch a Solid be rai$ed and born upwards through the Air, it $heweth that the de$ire in <I>Ari$totle</I> to detect <I>Democritus,</I> was predominate over the exqui$itene$s of Solid Philo$ophizing: which de$ire of his he hath di$covered in other oc- ca$ions, and that we may not digre$s too far from this place, in the Text precedent to this Chapter which we have in hand; where he <marg>Cap. 5. Text 41.</marg> attempts to confute the $ame <I>Democritus,</I> for that he, not content- ing him$elf with names only, had e$$ayed more particularly to de- clare what things Gravity and Levity were; that is, the Cau$es of de$cending and a$cending, (and had introduced Repletion and Va- cuity) a$cribing this to Fire, by which it moves upwards, and that to the Earth, by which it de$cends; afterwards attributing to the Air more of Fire, and to the water more of Earth. But <I>Ari$totle</I> de$iring a po$itive Cau$e, even of a$cending Motion, and not as <I>Plato,</I> <foot>or</foot> <p n=>472</p> or the$e others, a $imple negation, or privation, $uch as Vacuity <marg>Id. ibid.</marg> would be in reference to Repletion, argueth again$t <I>Democritus</I> and $aith: If it be true, as you $uppo$e, then there $hall be a great Ma$s of water, which $hall have more of Fire, than a $mall Ma$s of Air, and a great Ma$s of Air, which $hall have more of Earth than a lit- tle Ma$s of water, whereby it would en$ue, that a great Ma$s of Air, $hould come more $wiftly downwards, than a little quantity of water: But that is never in any ca$e $oever: Therefore <I>Democritus</I> di$cour$eth erroneou$ly.</P> <P>But in my opinion, the Doctrine of <I>Democritus,</I> is not by this alle- gation overthrown, but if I erre not, the manner of <I>Ari$totle</I> deduction either concludes not, or if it do conclude any thing, it may with e- quall force be re$tored again$t him$elf. <I>Democritus</I> will grant to <I>Ari$totle,</I> that there may be a great Ma$s of Air taken, which con- tains more Earth, than a $mall quantity of water, but yet will deny, that $uch a Ma$s of Air, $hall go fa$ter downwards than a little water, and that for many rea$ons. Fir$t, becau$e if the greater quantity of Earth, contained in the great Ma$s of Air, ought to cau$e a greater Velocity than a le$s quantity of Earth, contained in a little quantity of water, it would be nece$$ary, fir$t, that it were true, that a greater Ma$s of pure Earth, $hould move more $wiftly than a le$s: But this is fal$e, though <I>Ari$totle</I> in many places affirms it to be true: becau$e not the greater ab$olute, but the greater $pecificall Gravity, <marg>The greater Specificall, not the greater ab- $olute Gravity, is the Cau$e of Velocity.</marg> is the cau$e of greater Velocity: nor doth a Ball of Wood, weigh- ing ten pounds, de$cend more $wiftly than one weighing ten Ounces, and that is of the $ame Matter: but indeed a Bullet of Lead of four Ounces, de$cendeth more $wiftly than a Ball of Wood of twenty Pounds: becau$e the Lead is more grave <I>in $pecie</I> than the Wood. Therefore, its not nece$$ary, that a great Ma$s of Air, by rea$on of the much Earth contained in it, do de$cend more $wiftly than a little <marg>Any Ma$s of water $hal move more $wiftly, than any of Air, and why.</marg> Ma$s of water, but on the contrary, any what$oever Ma$s of water, $hall move more $wiftly than any other of Air, by rea$on the partici- pation of the terrene parts <I>in $pecie</I> is greater in the water, than in the Air. Let us note, in the $econd place, how that in multiplying the Ma$s of the Air, we not only multiply that which is therein of terrene, but its Fire al$o: whence the Cau$e of a$cending, no le$s encrea$eth, by vertue of the Fire, than that of de$cending on the account of its multiplied Earth. It was requi$ite in increa$ing the greatne$s of the Air, to multiply that which it hath of terrene only, leaving its Fire in its fir$t $tate, for then the terrene parts of the augmented Air, overcoming the terrene parts of the $mall quantity of water, it might with more probability have been pretended, that the great quanti- ty of Air, ought to de$cend with a greater <I>Impetus,</I> than the little quantity of water.</P> <foot>Therefore,</foot> <p n=>467</p> <P>Therefore, the Fallacy lyes more in the Di$cour$e of <I>Ari$totle,</I> than in that of <I>Democritus,</I> who with $everall other Rea$ons might oppo$e <I>Ari$totle,</I> and alledge; If it be true, that the extreame Elements be one $imply grave, and the other $imply light, and that the mean Elements participate of the one, and of the other Nature; but the Air more of Levity, and the water more of Gravity, then there $hall be a great Ma$s of Air, who$e Gravity $hall exceed the Gravity of a little quantity of water; and therefore $uch a Ma$s of Air $hall de- $cend more $wiftly than that little water: But that is never $een to occurr: Therefore its not true, that the mean Elements do partici- pate of the one, and the other quality. This argument is fallacious, no le$s than the other again$t <I>Democritus.</I></P> <P>La$tly, <I>Aristotle</I> having $aid, that if the Po$ition of <I>Democritus</I> were true, it would follow, that a great Ma$s of Air $hould move more $wiftly than a $mall Ma$s of water, and afterwards $ubjoyned, that that is never $een in any Ca$e: methinks others may become de- $irous to know of him in what place this $hould evene, which he de- duceth again$t <I>Democritus,</I> and what Experiment teacheth us, that it never falls out $o. To $uppo$e to $ee it in the Element of water, or in that of the Air is vain, becau$e neither doth water through water, nor Air through Air move, nor would they ever by any whatever participation others a$$ign them, of Earth or of Fire: the Earth, in that it is not a Body fluid, and yielding to the mobility of other Bodies, is a mo$t improper place and <I>Medium</I> for $uch an Ex- periment: <I>Vacuum,</I> according to the $ame <I>Ari$totle</I> him$elf, there is none, and were there, nothing would move in it: there remaine the Region of Fire, but being $o far di$tant from us, what Experi- ment can a$$ure us, or hath a$$ertained <I>Ari$totle</I> in $uch $ort, that he $hould as of a thing mo$t obvious to $ence, affirm what he produ- ceth in confutation of <I>Democritus,</I> to wit, that a great Ma$s of Air, is moved no $wifter than a little one of water? But I will dwell no longer upon this matter, whereon I have $poke $ufficiently: but leaving <I>Democritus,</I> I return to the Text of <I>Ari$totle,</I> wherein he goes about to render the true rea$on, how it comes to pa$s, that the thin Plates of Iron or Lead do $wim on the water; and, moreover, that Gold it $elf being beaten into thin Leaves, not only $wims in water, but flyeth too and again in the Air. He $uppo$eth that of <marg><I>De Cal<*></I> l. 4. c. 6. t. 44.</marg> Continualls, $ome are ea$ily divi$ible, others not: and that of the ea$ily divi$ible, $ome are more $o, and $ome le$s: and the$e he affirms we $hould e$teem the Cau$es. He addes that that is ea$ily divi$ible, which is well terminated, and the more the more divi$ible, and that the Air is more $o, than the water, and the water than the Earth. And, la$tly, he $uppo$eth that in each kind, the le$$e quan- tity is ea$lyer divided and broken than the greater.</P> <foot>Here</foot> <p n=>474</p> <P>Here I note, that the Conclu$ions of <I>Ari$totle</I> in generall are all true, but methinks, that he applyeth them to particulars, in which they have no place, as indeed they have in others, as for Example, Wax is more ea$ily divi$ible than Lead, and Lead than Silver, in- a$much as Wax receives all the terms more ea$iler than Lead, and Lead than Silver. Its true, moreover, that a little quantity of Sil- ver is ea$lier divided than a great Ma$s: and all the$e Propo$itions are true, becau$e true it is, that in Silver, Lead and Wax, there is $imply a Re$i$tance again$t Divi$ion, and where there is the ab$o- lute, there is al$o the re$pective. But if as well in water as in Air, there be no Renitence again$t $imple Divi$ion, how can we $ay, that the water is ea$lier divided than the Air? We know not how to ex- tricate our $elves from the Equivocation: whereupon I return to an$wer, that Re$i$tance of ab$olute Divi$ion is one thing, and Re- $i$tance of Divi$ion made with $uch and $uch Velocity is another. But to produce Re$t, and to abate the Motion, the Re$i$tance of ab$olute Divi$ion is nece$$ary; and the Re$i$tance of $peedy Di- vi$ion, cau$eth not Re$t, but $lowne$s of Motion. But that as well in the Air, as in water, there is no Re$i$tance of $imple Divi$ion, is manife$t, for that there is not found any Solid Body which divides not the Air, and al$o the water: and that beaten Gold, or $mall du$t, are not able to $uperate the Re$i$tance of the Air, is contrary to that which Experience $hews us, for we $ee Gold and Du$t to go waving to and again in the Air, and at la$t to de$cend down- wards, and to do the $ame in the water, if it be put therein, and $e- parated from the Air. And, becau$e, as I $ay, neither the water, nor the Air do re$i$t $imple Divi$ion, it cannot be $aid, that the water re$i$ts more than the Air. Nor let any object unto me, the Exam- ple of mo$t light Bodies, as a Feather, or a little of the pith of El- der, or water-reed that divides the Air and not the water, and from this infer, that the Ait is ea$lier divi$ible than the water; for I $ay unto them, that if they do well ob$erve, they $hall $ee the $ame <marg><I>Archimed. De In$ident, humi</I> lib. 2. prop. 1.</marg> Body likewi$e divide the Continuity of the water, and $ubmerge in part, and in $uch a part, as that $o much water in Ma$s would weigh as much as the whole Solid. And if they $hal yet per$i$t in their doubt, that $uch a Solid $inks not through inability to divide the water, I will return them this reply, that if they put it under water, and then let it go, they $hall $ee it divide the water, and pre$ently a$cend with no le$s celerity, than that with which it divided the Air in de$cending: $o that to $ay that this Solid a$cends in the Air, but that coming to the water, it cea$eth its Motion, and therefore the water is more difficult to be divided, concludes nothing: for I, on the contrary, will propo$e them a piece of Wood, or of Wax, which ri$eth from the bottom of the water, and ea$ily divides its Re$i$tance, which afterwards being arri- <foot>ved</foot> <p n=>475</p> ved at the Air, $tayeth there, and hardly toucheth it; whence I may aswell $ay, that the water is more ea$ier divided than the Air</P> <P>I will not on this occa$ion forbear to give warning of another fal- lacy of the$e per$ons, who attribute the rea$on of $inking or $wimming to the greater or le$$e Re$i$tance of the Cra$$itude of the water again$t Divi$ion, making u$e of the example of an Egg, which in $weet water goeth to the bottom, but in $alt water $wims; and alledging for the cau$e thereof, the faint Re$i$tance of fre$h water again$t Divi$ion, and the $trong Re$i$tance of $alt water But if I mi$take not, from the $ame Experiment, we may aswell deduce the quite contrary; namely, that the fre$h water is more den$e, and the $alt more tenuous and $ubtle, $ince an Egg from the bottom of $alt water $peedily a$cends to the top, and divides its Re$i$tance, which it cannot do in the fre$h, in who$e bottom it $tays, being unable to ri$e upwards. Into $uch like perplex- ities, do fal$e Principles Lead men: but he that rightly Philo$ophating, $hall acknowledge the exce$$es of the Gravities of the Moveables and of the Mediums, to be the Cau$es of tho$e effects, will $ay, that the Egg $inks to the bottom in fre$h water, for that it is more grave than it, and $wimeth in the $alt, for that its le$s grave than it: and $hall without any ab$urdity, very $olidly e$tabli$h his Conclu$ions.</P> <P>Therefore the rea$on totally cea$eth, that <I>Ari$totle</I> $ubjoyns in the <marg>Text 45.</marg> Text $aying; The things, therefore, which have great breadth remain above, becau$e they comprehend much, and that which is greater, is not ea$ily divided. Such di$cour$ing cea$eth, I $ay, becau$e its not true, that there is in water or in Air any Re$i$tance of Divi$ion; be- $ides that the Plate of Lead when it $tays, hath already divided and penetrated the Cra$$itude of the water, and profounded it $elf ten or twelve times more than its own thickne$s: be$ides that $uch Re$i$tance of Divi$ion, were it $uppo$ed to be in the water, could not rationally be affirmed to be more in its $uperiour parts than in the middle, and lower: but if there were any difference, the inferiour $hould be the more den$e, $o that the Plate would be no le$s unable to penetrate the lower, than the $uperiour parts of the water; neverthele$s we $ee that no $ooner do we wet the $uperious Superficies of the Board or thin Piece of Wood, but it precipitatly, and without any reten$ion, de$cends to the bottom.</P> <P>I believe not after all this, that any (thinking perhaps thereby to defend <I>Aristotle</I>) will $ay, that it being true, that the much water re- $i$ts more than the little, the $aid Board being put lower de$cendeth, becau$e there remaineth a le$s Ma$s of water to be divided by it: be- cau$e if after the having $een the $ame Board $wim in four Inches of water, and al$o after that in the $ame to $ink, he $hall try the $ame Experiment upon a profundity of ten or twenty fathom water, he $hall $ee the very $elf $ame effect. And here I will take occa$ion to <foot>Ppp remember</foot> <p n=>476</p> remember, for the removall of an Error that is too common; That that Ship or other what$oever Body, that on the depth of an hundred or a thou$and fathom, $wims with $ubmerging only $ix fathom of its own height, [<I>or in the Sea dialect, that draws $ix fathom water</I>] $hall $wim in the $ame manner in water, that hath but $ix fathom and half <marg>A Ship that in 100 Fathome water draweth 6 Fathome, $hall float in 6 Fa- thome and 1/2 an Inch of depth.</marg> an Inch of depth. Nor do I on the other $ide, think that it can be $aid, that the $uperiour parts of the water are the more den$e, al- though a mo$t grave Authour hath e$teemed the $uperiour water in the Sea to be $o, grounding his opinion upon its being more $alt, than that at the bottom: but I doubt the Experiment, whether hitherto in taking the water from the bottom, the Ob$ervatour did not light upon $ome $pring of fre$h water there $pouting up: but we plainly $ee on the contrary, the fre$h Waters of Rivers to dilate them$elves for $ome miles beyond their place of meeting with the $alt water of the Sea, without de$cending in it, or mixing with it, unle$s by the intervention of $ome commotion or turbulency of the Windes.</P> <P>But returning to <I>Aristotle,</I> I $ay, that the breadth of Figure hath nothing to do in this bu$ine$s more or le$s, becau$e the $aid Plate of <marg>Thickne$s not breadth of Fi- gure to be re- $pected in Na- tation.</marg> Lead, or other Matter, cut into long Slices, $wim neither more nor le$s; and the $ame $hall the Slices do, being cut anew into little pieces, becau$e its not the breadth but the thickne$s that operates in this bu$ine$s. I $ay farther, that in ca$e it were really true, that the <marg>Were Reni- tence the cau$e of Natation, breadth of Fi- gure would hinder the $wiming of Bo- dies.</marg> Renitence to Divi$ion were the proper Cau$e of $wimming, the Fi- gures more narrow and $hort, would much better $wim than the more $pacious and broad, $o that augmenting the breadth of the Figure, the facility of $upernatation will be demini$hed, and decrea$ing, that this will encrea$e.</P> <P>And for declaration of what I $ay, con$ider that when a thin Plate of Lead de$cends, dividing the water, the Divi$ion and di$continu- ation is made between the parts of the water, invironing the perime- ter or Circumference of the $aid Plate, and according to the big- ne$s greater or le$$er of that circuit, it hath to divide a greater or le$$er quantity of water, $o that if the circuit, $uppo$e of a Board, be ten Feet in $inking it flatways, it is to make the $eperation and divi$ion, and to $o $peak, an inci$$ion upon ten Feet of water; and likewi$e a le$$er Board that is four Feet in Perimeter, mu$t make an ince$$ion of four Feet. This granted, he that hath any knowledge in Geometry, will comprehend, not only that a Board $awed in many long thin pieces, will much better float than when it was entire, but that all Figures, the more $hort and narrow they be, $hall $o much the better $wim. Let the Board ABCD be, for Example, eight Palmes long, and five broad, its circuit $hall be twenty $ix Palmes; and $o many mu$t the ince$$ion be, which it $hall make in the water to de$cend therein: but if we do $aw ir, as $uppo$e into eight little <foot>pieces</foot> <p n=>469</p> pieces, according to the Lines E F, G H, <I>&c.</I> making $even Segments, we mu$t adde to the twenty $ix Palmes of the circuit of the whole Board, $eventy others; whereupon the eight little pieces $o cut and $eperated, have to cut ninty $ix Palmes of water. And, if moreover, we cur each of the $aid pieces into five parts, re- <fig> ducing them into Squares, to the circuit of ninty $ix Palmes, with four cuts of eight Palmes apiece; we $hall adde al$o $ixty four Palmes, whereupon the $aid Squares to de$cend in the water, mu$t divide one hundred and $ixty Palmes of water, but the Re$i$tance is much greater than that of twenty $ix; therefore to the le$$er Superficies, we $hall reduce them, $o much the more ea$ily will they float: and the $ame will happen in all other Figures, who$e Superficies are $imular among$t them$elves, but different in bigne$s: becau$e the $aid Superficies, being either demini- $hed or encrea$ed, always dimini$h or encrea$e their Perimeters in $ubduple proportion; to wit, the Re$i$tance that they find in pene- trating the water; therefore the little pieces gradually $wim, with more and more facility as their breadth is le$$ened.</P> <P><I>This is manife$t; for keeping $till the $ame height of the Solid, with the $ame proportion as the Ba$e encrea$eth or demini$heth, doth the $aid Solid al$o encrea$e or dimini$h; whereupon the Solid more dimini$hing than the Circuit, the Cau$e of Submer$ion more dimini$heth than the Cau$e of Natation: And on the contrary, the Solid more encrea$ing than the Circuit, the Cau$e of Submer$ion encrea$eth more, that of Natation le$s.</I></P> <P>And this may all be dedueed out of the Doctrine of <I>Ari$totle</I> a- gain$t his own Doctrine.</P> <P>La$tly, to that which we read in the latter part of the Text, that <marg>Lib. 4. c. 6. Text 45.</marg> is to $ay, that we mu$t compare the Gravity of the Moveable with the Re$i$tance of the Medium again$t Divi$ion, becau$e if the force of the Gravity exceed the Re$i$tance of the <I>Medium,</I> the Moveable will de$cend, if not it will float. I need not make any other an$wer, but that which hath been already delivered; namely, that its not the Re$i$tance of ab$olute Divi$ion, (which neither is in Water nor Air) but the Gravity of the <I>Medium</I> that mu$t be compared with the Gravity of the Moveables; and if that of the <I>Medium</I> be greater, the Moveable $hall not de$cend, nor $o much as make a totall Submer$ion, but a partiall only: becau$e in the place which it would occupy in the water, there mu$t not remain a Body that weighs le$s than a like quantity of water: but if the Moveable be more grave, it $hall de$- cend to the bottom, and po$$e$s a place where it is more conformable <foot>Ppp 2 for</foot> <pb> for it to remain, than another Body that is le$s grave. And this is the only true proper and ab$olute Cau$e of Natation and Sub- mer$ion, $o that nothing el$e hath part therein: and the Board of the Adver$aries $wimmeth, when it is conjoyned with as much Air, as, together with it, doth form a Body le$s grave than $o much water as would fill the place that the $aid Compound occupyes in the water; but when they $hall demit the $imple Ebony into the water, according to the Tenour of our Que- $tion, it $hall alwayes go to the bottom, though it were as thin as a Paper.</P> <head><I>FINIS.</I></head> <fig> <pb> <head>THE TROUBLESOME INVENTION <I>OF</I> Nicolas Tartalea:</head> <head>BEING A Generall way to recover from the bottome of the <I>Water,</I> any <I>SHIP</I> that's <I>Sunke,</I> Or any other <I>Ponderous Ma$$e,</I> though it were a <I>Solid TOWER of Metal.</I></head> <head><I>TOGETHER WITH</I> An Artificiall way of DIVING, and $taying a long time under <I>Water,</I> to $eeke any thing <I>Sunke</I> in the greate$t <I>DEPTHS.</I></head> <head><I>AS ALSO, A SVPPLEMENT,</I> Shewing a Generall and Secure Way to <I>Grapple, &c.</I> any <I>Submerged SHIP.</I></head> <head>Engli$hed, By <I>THO. SALUSBURY,</I> E$q;</head> <fig> <head><I>LONDON,</I> Printed by WILLIAM LEYBOURN, <I>Anno Dom.</I> <I>MDC LXIV.</I></head> <pb> <P>To the mo$t <I>Serene,</I> and mo$t <I>Illustrious</I> Prince, FRANCESCO DONATO Duke of VENICE.</P> <P><I>It having been told me here at</I> Bre$cia, <I>Mo$t Serene and Mo$t Illu$trious Prince, that about ten years $ince, that a Ship full-laden did $inke near to</I> Malamoccho, <I>in about</I> 5 <I>Fathome of Water, and that to endeavour the recovering and getting it from thence, there had been u$ed all tho$e Means, and boun- tifull Offers and Tenders that could be imagined, a$wel by the Illu$trious Signory, for the Pre$ervation of the Port, as by the chief Owners of the Ship and its Cargo: and that although there were many that had tried, and attempted the $ame, by $undry and divers wayes, of no $mall expence, and that it had been $ever all times well grappled and begirt, yet neverthele$s as far as I could hear, none of them were able to rai$e her from that $mall depth: And it being al$o told me, that of late there was another $unk again in le$s than four Fathome of Water, $o that all its Poope and Prow, and a greate part of its Hull, was above Water, and that yet not with $tanding this al$o was judged by the fruitle$s Experiments and Ex- pen$es made about the former, to be irrecoverable, $o</I> <foot><I>that</I></foot> <pb> <I>that for the clearing of the Port, it is pre$ently re$olved, that the $aid Ship $hould be broken up, & taken to pieces at low Water: and $o, for ought that I hear, it hath been. Now I having con$idered of how great prejudice the breaking up of $uch a Ve$$el was, be$ides the lo$s of the Cargo, I deliberated about the finding of a way or Rule, that might remedy $uch detriment all Occurrences: And having found out one thats generall and unquestionable, I thought fit, for the common benefit of this renowned City, to declare, and by Figures to dilucidate the $ame in the pre$ent Tractate, and to offer and dedicate the $ame to your Highne$s; not as a pre$ent worthy of yon (for indeed the$e Mechanicall Matters are exceeding di$proporti- onate to your Highne$s Merits) but only with an Ambi- tion to Enoble and Dignifie my Book with your Glorious Name; In confidence that like as the Sun doth not di$- dain that all $orts of Per$ons $hould make u$e of its light and heat, $oneither will Your accu$tomed Humanity be offended with this my Pre$umption; and therefore I humbly lay my $elf at your Highne$s Feet,</I></P> <P>Nicolas Tartalea.</P> <p n=>483</p> <head>THE Indu$trious or Trouble$ome INVENTION OF Nicolas Tartalea:</head> <head><I>BOOKEI.</I></head> <cap><I>The Figure of a Ship $unke according to the Relation made of that which was cau$ed to be broken up neere</I> Malamoccho, <I>as being judged irrecoverable.</I></cap> <fig> <head><I>EXPLANATION I.</I></head> <P>Before I come to declare the promi$ed way to recover any laden or empty Ship when it is $unke; I thinke it convenient (<I>Mo$t Serene and Illu$trious Prince,</I>) fir$t to de- clare the reall cau$e of its $inking.</P> <foot>Qqq I $ay,</foot> <p n=>484</p> <marg><I>Archimed.</I> of Natation, Lib. 2. Prop. 1.</marg> <P>I $ay then; That its impo$$ible that the water $hould wholly $wallow or receive into it any materiall Body lighter than it $elf (as to $pecies;) but it will leave or cau$e one part thereof to lie above the Superficies of the $aid water, that is uncovered by it. And as the whole Body demitted into the water, is to the part thereof, which $hall be received or admitted by the water, $o $hall the Spe- cificall Gravity of the water, be unto the Specificall Gravity of the $aid Solid Body.</P> <marg><I>Archimed.</I> of Natation, Lib. 1. Prop. 7.</marg> <P>But tho$e Solid Bodies which are more grave than the water; be- ing demitted into the $aid water, $uddenly make the water to give place; and not only enter wholly into the $ame, but they do go continually de$cending, till they arrive at the bottom: And they de$cend with $o much greater Velocity, by how much they exceed the water in $pecificall Gravity.</P> <marg><I>A chimed.</I> of Natation, Lib. 1. Prop. 111.</marg> <P>And tho$e again which happen to be of the $ame Gravity with the water, of nece$$ary con$equence being put into it, are admitted and received totally into the $ame, but yet they $tay in the Surface of the $aid water; that is, they $uffer not any part to lie above the Superficies of the $aid water, nor much le$s doth the water con$ent to their de$cent to the bottom.</P> <P>And all this is demon$trated by <I>Archimedes</I> of <I>Syracu$a,</I> in that his Tract <I>De in$identibus aquæ,</I> by us tran$lated. And becau$e the greate$t part of woods are lighter, or le$s grave than the water; he therefore that $hall build a Ship or other Ve$$el meerly of wood, lighter than water, its manife$t that he cannot (though he $hould fill the $ame with water, as full as it would hold) make the $ame totally to $ink, but that nece$$arily $ome one part or other of the $aid Ship or Ve$$el $hall $tand above the Surface of the water: For its a thing very clear, that all that $ame Body, compounded of wood and of water, would be much lighter than if it were all only of water without wood: Such a compound Body therefore being le$s grave than the water, its nece$$ary (for the rea$ons above produced) that a part of the $ame remain above the Surface of the water.</P> <P>And if the $aid Ship or Bark $hall be built (as it is u$ual) with Bolts, Nailes, and other Materials of Iron, and that $uch Iron- works be not of $uch quantity, as to make that Body compounded of wood and Iron, graver than the water, but that it continue $till le$s grave than the water (as I judge all Ships and Barks to be;) The $ame will follow as did before, namely, that filling the $aid Ship with water, as full as is po$ible, it cannot by any means go to the bottom If then a Ship or other Ve$$el being wholly fill'd with water, cannot be thereby $unk to the bottom; It is a thing evident, that if $uch a Ship or Ve$$el $hall be totally fill'd with a Matter lighter than the water; not only its totall $inking under that weight <foot>will</foot> <p n=>485</p> will be impo$$ible, but al$o its floating in $ome part above the Sur- face of the water will be nece$$ary: And $o much the greater part $hall be vi$ible above the water, by how much the Matter of the Lading, is lighter than the water.</P> <P>Therefore, if all the Cargo of a Ship (for in$tance) Buts of Oyl, and that no other Matters of a graver Nature than water were intro- duced, and that the $aid Ship $hould by $ome Accident be filled up with water, it is not only manife$t that the Ship cannot be there- by $unk to the bottom, but that a part thereof mu$t nece$$arily float above the Surface of the water: Becau$e all that Compo$ition of Wood, Water and Oyl, would be lighter than if it had been all $imply of water. The very $ame would follow, if the Cargo had been $oley of Wine, Wax, Camphor, Spices, or the like Matters, lighter than the water. But becau$e the Merchandizes that fraight Ships, or other Ve$$els, are $ome ($pecifically) graver, and $ome ($pecifically) lighter than the water: (The graver are all forts of Mettals, as Iron, Tinn, Lead, Bra$s, Copper, Silver, Gold, and infi- nite other Species of Commodities; likewi$e the per$ons of Men, Stones, Balla$ts, and the like:) And that al$o there are $ome $orts of Commodities that chance to differ very little in Gravity from the water: Therefore I conclude, that as oft as any Ship accidentally is fill'd with water, and $o $inks by degrees to the bottom, it is ne- ce$$ary to grant that all the Compo$ition, namely, of the Fraight, of the Ve$$el, and of the water that entered into it, is more grave, than if the compo$ition had been all $imply of water, by the rea$ons before alledg'd.</P> <P>And therefore in $uch a ca$e things graver than the water, mu$t of nece$$ity exceed in force tho$e that be lighter: and by how much things graver than the water, exceed the lighter, $o much the more Force will be required to recover $uch a Ship or other Ve$$el being $unk, and on the contrary, $o much le$s Force will be required, when the Ma$s of the Materials more grave than the water, $hall not differ much from the Ma$s of the le$s grave: provided the Re- covery be undertaken in $ome $hort time after the Ship $hall be $unk, For if the Ship lie many dayes under water, the delay will intro. duce many difficulties: One will be, that it will con$olidate with and dock or work it $elf farther into the Mudd or Sand, which will not a little hinder its Recovery; and again, the water will continu- ally carry into the $aid Ship, Ouze, Mudd, and Sand, which Mat- ter is much graver than the water, whereby the Ship is continually made graver as to the water, than it was at the beginning when it was fir$t $ubmerg'd. And moreover the corruptible Matters, which are by nature lighter than the water, will corrupt, and corrupting will change into other earthy $ub$tances much graver than the <foot>Qqq 2 water:</foot> <p n=>486</p> water: in$omuch that at the length, it ought to be pre$uppo$ed in order to the recovery of the $aid Ship, as if it were $olely laden with Mire, Dirt, and Sand: which doing, you will not be deceived in the operation, that is to $ay, preparing and working with a Force equivalent to that its Gravity. The way to know how to prepare Forces equivalent to the Gravity $hall be $hewn in the eight Expla- nation of this.</P> <head><I>EXPLANATION</I> II.</head> <P>Now to give beginning to the bu$ine$s propo$ed, I $ay, that in the Recovery of a Foundred Ship laden, or any other la- den Ve$$el that is foundered or $unk, there interveneth more e$pecially the$e three great Ob$tructions. The fir$t difficulty is, how to imbreech and grapple it with $uch, and $o many Ropes, as may $uffice to bear it up; for if this either by ill chance cannot be done (whether through its being in a place two deep, or too far dockt in the Mudd or Sand) all our other labour will be fru$trate and vain.</P> <P>The $econd difficulty, when once it is grappled, is how with dex- terity to $eperate it from the bottom of the Sea; and this d<*>ff<*>y will be much greater, the Ship being in a Miry or Sandy bottom, than if it $hall be in a Stony place; and it $hall be al$o a greater difficulty to $eperate it from a very deep bottom, than from a Shal- low; (alwayes $uppo$ing that the two bottoms be both alike, name- ly, either both Stony or both Sandy;) and al$o far greater $hall the $aid difficulty be in a Ship long $unk, than in one newly four dered; (as we have already $aid in the precedent Explanation:) But when $he is once water-born, or $eperated from the bottom, its an ca$ie matter to rai$e her up to the Surface of the water; for then $he $hall not be a little aleviated in her Gravity: But the truth is, the draw- ing of it after wards above the Superficies of the water, is no very ca- $ie matter, but is extream hard to be done; and this is the third difficulty; the principal cau$e of which two la$t difficulties $hall be a$$igned by and by.</P> <P>But becau$e the means to obviate and $uperate the fir$t difficulties <marg>* <I>The Author be- lieved (as he de- clareth in the E- pi$tle to the en$u- ing Suppliment of this his</I> Inventi- on) <I>that the Ma- riners conver$ant in the$e affairs, had many wayes to imbreech a Ve$$el uuder water; and for that rea$on he over pa$$eth it here, and is very cur$ive upon the $ame Point, in the $econd Book, but giveth a generall Rule for it in the $aid Suppliment: to which the Reader is referred for fuller Satisfaction.</I></marg> as more ^{*} common, we $hall forbear to $peak of them untill the next Book. To provide, and that briefly, to the $econd and third impediments (which are the lea$t known) that is, not only to $e- perate the Ship from the bottom, but to rai$e it al$o $omewhat above the Surface of the water.</P> <foot>And</foot> <p n=>487</p> <P>And this is the Rule that you mu$t ob$erve; If the Ship be newly $unk, you mu$t immediately, if it be po$$ible, find two other Ships, that be each of them rather of greater bulk than the foundered Ship than le$s: and when you have found the$e two Ships, you mu$t free them of all the inward and outward lading, and rigging, e$pe- cially of tho$e things which are by nature more grave than the water, as are the Guns, the Shot, and any kind of Balla$t, which is pre$up- po$ed to be in the Hold, and of other things of impediment; and when the$e Ships are thus cleared, you mu$t $top all the Loop-holes, Cat-holes, Skuppers and Hau$es, which you $hall finde between or above Decks, graving and calking them $o with Okum, and paying them with Pitch, that the water can neither get in nor out thereat. And next you mu$t join or grapple the$e two Ships together with five or more Tires or Orders of thick and $trong Beames tripplicated; that is, that each of the $aid Orders con$i$t of three Beams, joyned lengthways; and that each of the three Beams be $omewhat longer than the bredth of the Deck or Hull of each Ship; and that theybe thick and $trong, as being to $upport the Foundered Ship, as you $hall $ee it made to appear pre$ently: and couple the $aid Ships to- gether, at $uch a di$tance from each other, that you give berth, or leave room enough betwixt for the foundered Ship to play; and you mu$t make this couppling in $uch $ort, that the length or $ide of the one Ship, look towards the length or $ide of the other; and albeit this conjunction or grappling may be made with many Orders or Tires of tho$e Bcams tripplicated lengthways, as was $aid above, <cap><I>The Figurall repre$entation of the two empty Ships, conjoyned with five Orders of Beams, and towed ju$t over the place where the Foundered Ship is.</I></cap> <fig> yet that we may not cau$e confu$ion in the Figure, we would have this colligation to be made only of five Rows, as appeareth in the <foot>Scheme</foot> <p n=>488</p> Scheme: and although the $aid Rows of Beames cannot be all placed equidi$tant from the Surface of the water, for that the Wailes or Rifings of the two Ships are not flu$h, but cuved, it is not of any importance, $o that they be well fa$tened and $trength- ened in tho$e places where they re$t upon the $aid Rifings: upon which Ri$ings, you $hall conjoyn the $aid Beams, namely, the two ends of them, which two ends $hall be the $tronge$t place, able to $upport any great weight. Yet the truth is, that to fit the$e Tires of Beams, you need not have regard to make them pa$s through from $ide to $ide, in that weak part of the Ships Poop and Prow, to re$t them on the Rifings or Gun-wales of the Deck of tho$e Ships, and to go cro$s the Hull in tho$e places. And next you are to make upon the$e Beams, that is upon the mouths of both the Ships, a Plat-form of Planks for to $tand upon whil$t you are about the work; leaving diver$e Scuttles or Spaces open, whereby to de$cend, aud for other u$es: And all this being done, you are to tow or hall the$e Ve$$els to the place where the Ship is that did $ink, and to lay them Board and Board in $uch fa$hion, that the one may lie on one $ide of it, and the other upon the other, as in the Scheme is apparent.</P> <P>This being done, fill tho$e two Ships as full of water as they can hold or $wim, (the way to free them with great facility and expe- dition, $hall be $hewn in the twelfth Explanation;) and being full, wait the time of low water; that is, when the Tide returning, the Sea doth low as much as it can do; and at that in$tant of time, make the Ship very fa$t with tho$e ends of Cords or Cables (with which it was Swite or bound) to tho$e five, or more Tires of Beams, wherewith the fore$aid two Ships were imbreecht or grappled: And having well belayd or fa$tned tho$e Cables, you mu$t bale or take out a $mall part of the water out of one of the two Ships, and then let it re$t $o, till $uch time as you have baled or taken a little more than that quantity out of the other Ship; and then again take a little more out of the fir$t Ship, and leave it $o till you have taken another $uch a quantity from the other Ship, and thus proceed gra- dually, till you find the Foundered Ship, water-born or loo$ned from the bottom: but being water-born (if it be in a Showle bot- tom, as was that at <I>Malamoccho)</I> you are to take out the $aid water, equally from both the Ships, at one and the $aid time, to the end the Ship may ri$e evenly without $wagging or $haking; and thus you are to proceed till you have taken all the water from the one & the other of the two Ships: In $o doing, you $hall $ee the two Shpis lea- $urely and gently rai$e the Ship that was $unk, $o high above the Surface of the water, that you may commodiou$ly free it, and di$charge it of its lading, as appeareth in the following Figures. And if you would not keep the two Ships $o long imploy'd, you may <foot>warpe</foot> <p n=>489</p> warpe or towe the Foundered Ship at high-water to $ome place where it may lie a-ground: and by that means upon the Ebbe or Rece$$ion of the Tide, it will lie much more above water; and then you may $afely unfa$ten it from tho$e five or more Tires of Beames, to which it was at fir$t tyed, to hall it to a place of $afety, as it was our purpo$e to do; and this $hall $ucceed as well in an ouzie bot- tom, as in a Stony, This though you may take notice of, that if the Cargo of this new Foundred Ship was $uch, that the things more grave than the water, did not much exceed the le$s grave, <*> would be ea$ie to effect the recovery with two Ships, very much le$s than tho$e which we have $poken of above; yet neverthele$s it will be good prudence to take them rather bigger than le$$er, that $o they may exceed 200000 pounds in Power, rather than want one only ounce in Act; e$pecially in ca$e you would in a deep place at the fir$t motion hoi$t it by meer Force $omewhat above the Surface of the water, for in that point alone it will require incomparably much more force, than in all the other operations.</P> <P>How you are to preceed, in ca$e the Ship $hould be $unk in a place very deep, $hall be declared in the $eaventh Explanation. The Figures of this Explanation are the$e two that folllow.</P> <cap><I>The Figure of the two Ships filled with water, to rai$e the Ship that is $unk</I></cap> <fig> <foot>The</foot> <p n=>490</p> <cap><I>The Figure of the two Ships emptied as they lie, with the other Ship rai$ed up above water.</I></cap> <fig> <head><I>EXPLANATION</I> III.</head> <P>But if it $o fall out, that you cannot on $nch an in$tant, finde two Ships of the $ame Bulk with the Ship $unk, you may take four $maller; provided, that all the four together hold twice as much burden as the Ship $unk, and rather more than le$s. Which four $mall Ships being all fir$t cleer'd of their lading, and well $topt in all their Skuppers and Portholes (as was $aid in the two) you mu$t couple them with Beams and good Planks, by two and two, as you u$e to do with two Lighters, when you would make a Bridge of them: and the$e two pair of Hoys or Barkes thus coupled together, you mu$t afterwards fa$ten one pair to another, with $even of tho$e Tires or Rows of thick and $trong Beams tripplicated, as was $aid in the precedent Explanation; and place them at $uch a di$tance one pair from another, as that you may leave berth or $pace enough for the $unk or foundered Ship to ri$e between them, and $ome what more, (as was $aid of the two.) And though this conjunction of the two pair of Ships, may be made three $everall wayes, yet I will have you make the two Poops or Hin decks of the one couple, to lie op- po$ite to the two Poops of the other couple. And to make this conjunction, you are to place two Tires of tho$e great Beams along the upper parts of the $aid Poops, $o, that they may re$t in the in- $ide on tho$e le$$er Beams and Planks, where with each of tho$e two pair of Ships were coupled: and each of the$e Orders or Tires of <foot>Beames</foot> <p n=>491</p> Beames ought to be compo$ed of three Beams conjoyned length- wayes, as was $aid in the precedent Explanation; and make two of the Tires lie upon the Ships; and to tho$e Tires, let that $unk Ship be grappled: and another Tire of the $aid Beams is to be placed in the mid$t between the one and the other couple; and two other Tires of the $aid Beams ought to be fa$tened upon the one and other $ide, that is, upon the Rifings or Bends of tho$e two couples of Ships; and that being done, there will be in all $even Tires or Or- ders of Beams; which $eaven Orders of Beams ought conjunctly to be prolonged, on the one and on the other $ide. almo$t to the length of the Hull of each Ship, as in the Figure is reprere<*>d: and <cap><I>The Figur all <*>ple how to recover a Foundered Ship with four $mall Ship</I></cap> <fig> this being done, you are to proceed, as hath been $hewn in the two, that is, fill them top full of water, and at low water, imbreech the Ship $unk very well, withall tho$e ends of Ropes or Cables, that you did belay to tho$e $even Tires of Beams: and when tho$e Grapplings $hall be well made fa$t; you $hall at high water bale or free the water by little and little out of the Ships, one pair after a- nother, till you feel the foundered Ship is di$engaged from the bot- tom, and water-born, as was $aid in the two. And having $epera- ted it from the bottom (if it be in a $hallow place, as was that where the Ship was foundered neer <I>M<*>lamoccbo</I>) you are to proceed to let out the re$t of the $aid water, but take it equally and gradually from the one and the other pair, that they may de$cend evenly, and with- out heeling, as was $aid of the two; and in $o doing, the $aid Ship $hall not only be hoi$ted up to the Surface of the water, but much <foot>Rrr above</foot> <p n=>492</p> above the $ame; $o that you may in that po$ture free or drain it and di$charge it of the Cargo. But if you cannot $o long $pare tho$e four Ships from other u$es, then you may at high water tow it to $ome place, where running it on ground, you may at the ebbe of the Tide (for that then there will lie much more of it above wa- ter) $afely loo$e it from tho$e Beames, as was al$o $aid in the prece- dent Explanation of the two Ships.</P> <P>But in ca$e the Foundered Ship $hould chance to be in a very deep Sea, in the $eventh Explanation (to be the briefer in this place) $hall be $hewn how you are to proceed.</P> <head><I>EXPLANATION</I> IV.</head> <P>And if it happen that it $hould be in a place where there are no Ships to be got, either great cr little; you may take of other kind of Pinaces, Barks or Barges, but endeavour to get $uch as are floaty, and highe$t built in there Rifings, that $o they may, at $uch time as they are full of water, de$cend very far under water, (or according to the Mariners phra$e, may draw much wa- ter) and of the$e you mu$t $top all the Skuppers, Haw$es, Cat-holes and Port holes, that you finde, as in the Ships, that they may hold the more water, and con$equently draw the more water, or be de- pre$$ed deeper into the $ame; and take $o many couple of the$e Botes, that they may all together contain double the burden of the Ship to be recovered, and rather much more, than any thing le$s. And of all the$e Boats or Barks, make two Squadrons, conjoyning each Squadron with good $mall Timbers & Planks, as you u$e to do, when you would make a Bridge of Boats: And the$e $ame Ve$$els of the one and other divi$ion, $hould be placed board and board, that $o the great Beams, which are to conjoyn one Squadron to the other, may bear upon the Rifings, Bends or Wales, of the $aid Ve$$els. And this being done, you are to couple the$e two Squadrons, to each other with tho$e thick and $trong Tires of Beams, mentioned in the former Explanations, which Orders of Beams $hould be fixed between two & two of tho$e Botes, as is $aid above, to the end, that they may bear or re$t upon the Bends of tho$e Boats; and place another Tire upon the out$ides of both the Divi$ions, upon the ends of the cro$s $mall Beams which hold the $everall Ve$$els together: So that if the Squadrons con- $i$ted each of four Barks, the Tires of the $aid Beams would come to be five,; and if there $hould be five in a Squadron, the Tires of Beams would be $ix, and $o forwards; that is, the Orders of Beams, by this means, $hall be alwayes one more than the number of Botes in each Squadron. But in the Ships you mu$t ob$erve another method, becau$e of tho$e two Orders, which are placed in each Poop; by <foot>which</foot> <p n=>493</p> <cap><I>The way to recover a Foundered Ship with many Barks or Wherryes.</I></cap> <fig> which means in every two Ships to a Divi$ion (which in all make four Ships) there mu$t be $even Orders of Beams, and in three Ships to a Squadron, there mu$t be ten Orders of Beams, and in four Ships to a Squadron thirteen; and thus proceeding forwards to a greater number of Ships in a Squadron. And having under$tood the way of coupling many Barks or Wherryes in Squadrons; as al$o the manner how to joyn or fa$ten them to each other, and with how many Orders of Beams; you are to proceed in the re$t, as in the precedent Explanations hath been demon$trated in $howle bottoms, but the directions how to manage this affair in deep places, $hall be declared in the $eventh Explanation.</P> <head><I>EXPLANATION</I> V.</head> <P>To remove this inconvenience of taking Ships or other Ve$$els; and of $tanding to lighten them of their Guns & lading, and of $topping their Loop-holes; you may in$uch a misfortune cau$e to be made two great Ve$$els, almo$t in form of ^{*} Che$ts without co- <marg>* Of the$e Ve$- $els Cardinall <I>Richleis</I> made u$e at the Siege of <I>Rochell</I> to <*> up the Haven.</marg> vers, the length of each to be equal to the Hull of a middle rate Ship, and the breadth equall to that of the $ame Ship at the Main-ma$t, and the height al$o the $ame with that of the Ship in the Bow, $o that each of the$e Plat forms or Che$ts, $hall hold much more than a common Ship, and thus both will contain more than the double burden of $uch a Ship. And for the making of the$e Ve$$els, you mu$t fir$t make the Models in Carvel-manner of thick and $trong Timber, with their Eutertaces, Tran$omes and Knees, to hold their $ides and ends together: and this done, $pike down to them certain <foot>Rrr2 thick</foot> <p n=>494</p> thick and $trong Planks; and then cau$e them to be well graved and calked in the Seames or Strakes by a Calker, with Okum, and paid with Pitch, as you u$e to do Ships or Gallyes, and then apply them to your purpo$e. And when you would u$e them, you need only fa$ten them together with tho$e five or more Orders of thick and lu$ty Beams, trippled lengthwayes, that is, prolonged both wayes, $o as that they may lie athwart the Decks of the $aid two Ve$$els, and place the $aid Ships $o far di$tant from each other, as you gue$$e the bredth of the Foundered Ship to be, and $omething more: And then make upon the Deck of each of them, that is, upon tho$e Beams, a Plat-form of Planks, as was $aid in the two Ships of the $econd Explanation, and afterwards proceed as in tho$e two Ships.</P> <head><I>EXPLANATION</I> VI.</head> <P>And inca$e you think the making of a couple of $uch great Modles or Ve$$els, as we mentioned in the foregoing Ex- planation, would be too great a trouble or expence; you may make two pair of $uch Che$ts, each of them but of hal$ the bulk of one of the former: but if you judge the$e two pair too trouble$ome, you may make three, four, or more pairs; alwayes provided, that among$t them all they hold about twi$e the burden of the Ship $unk; and the$e Frames when you would u$e them, mu$t be joyned together in two Ranks, with le$$er Beams and Planks, as was $aid of the four Boats or Wherryes; and then fa$ten the$e two Ranks to each other at the requi$ite di$tance, with great and $trong tripplicated Beams, as was $aid of the Ships, Barks and Boats; and then operate as you was to do with tho$e: alwayes remembring in the freeing or emptying the $aid Ve$$els, to bale out the water by little and little fir$t from one Rank, and then from the other; and $o proceed interchangeably till you percieve that the Ship is clear of the bottom: and being di$engaged, if it be in a $hallow place, continue taking the water equally out of the one and other Divi$i- on of Ve$$els, till all the water be drained out of them, as was requi- red upon the former Explanations: but if it be $unk in a deep Sea, the next Explanation $hall $hew how you are to proceed; and that briefly.</P> <head><I>EXPLANATION</I> VII.</head> <P>And in ca$e the $aid Ship newly $unk, chance to be in a very deep bottom; It will be nece$$ary fir$t to fix upon tho$e two or four Ships, or upon tho$e two Squadrons of Barks, Fly-boats or Wherryes, at lea$t $ix or eight Cap$tains, Ship-Cranes <foot>or</foot> <p n=>495</p> or Windla$$es, with their nece$$ary Garnets or Pullies, requi$ite to $nch a weight: and you may ea$ily accomodate the$e Pullies, to tho$e Orders of great Beams, wherewith the $aid Ve$$els were conjoyned. And having prepared the$e Cap$tains, you are to proceed in all things, as hath been directed you in the precedent Explanations, excepting only in this, that whil$t you are freeing the water alter- nately by degrees out of the two or more Ships, or from the two Squadrons of Barks, Fly-boats or Wherryes, as $oon as you finde the Foundered Ship to be water-born or got clear of the bottom of the Sea, I would have you cea$e to take any more water forth of the $aid Ships, or le$$er Ve$$els before filled; and I would have you with tho$e Cap$tains, attempt to draw the $aid Ship that was funk unto the Levell or Surtace of the water, or to lie Horizontal unto it, which may be ea$ily done, for that its pondero$ity will be much di- mini$hed. And when you have drawn it to the Surface of the water, then I would have you di$charge all the other water out of the two Ships, or the two Squadrons of $mall Ve$$els. And this $econd wa- ter, I would have raken equally, and at the $ame time, from the one and the other Ship, or from each Rank of Barks or Boats, as hath been $aid of the other. And thus tho$e Ships or Squadrons of Boats $hall hoi$t the $aid Foundered Ship, $o high above the Superficies of the water, that you may free it of the water which was got into it, and unlade its Cargo, which was our purpo$e.</P> <P>You mu$t note, that all that hath been hitherto $aid of a Ship newly $unk, ought to be under$tood of all other kind of Foundered Ships, proceeding alwayes proportionately as was directed in that Ship. And again, I give you no Figure how you are to fit and fix the Cap$tains and Pullies, as being a thing common and manife$t.</P> <head><I>EXPLANATION</I> VIII.</head> <P>But if it $o fall out, that the $aid Ship or other Ve$$el hath been $unk many Months; albeit that there might have been many matters in the Cargo of a lighter nature than water, yet you mu$t $uppo$e the ca$e as if the Ship were as heavy as if it had been fil'd with Sand or Gravel; yea and much heavier, for many Rea$ons, as hath been alledg'd in the fir$t Explanation. Therefore that you may not deceive your $elves in the de$igned recovering of it, you would do well to double the Forces required to the recovery of a new $unk Ship; that is, you mu$t take four Ships, each as big as the Foundered <I>S</I>hip, and combine the$e four <I>S</I>hips, as you were re- quired to joyn the four $mall <I>S</I>hips in the third Explanation. And if you cannot procure them of that burthen, take eight le$$er, pro- vided that altogether they be quadruple in contence to the <I>S</I>hip to <foot>be</foot> <p n=>496</p> to be recovered: and divide the$e eight le$$er <I>S</I>hips or Barks into two <I>S</I>quadrons, of four in a <I>S</I>quadron, according as you was di- rected in the four Ships in the third direction. And if you cannot pro- cure <I>S</I>hips great or $mal, take $o many pair of other Ve$$els, Fly boats or Wherryes, that in all they may at lea$t contain four times the bur- then of the Foundered <I>S</I>hip: And reduce the$e Barks, Boats or Wherryes into two Divi$ions, as you are taught in the fourth Ex- planation: and in all other particulars, proceed according to the method pre$cribed in the recovery of the <I>S</I>hip newly $unk; and that as well in deep, as $hallow places; that is, placing in a deep <I>S</I>ea upon the $aid Ships, or <I>S</I>quadrons of Boats, at lea$t twelve or $ixteen Cap$tains, which it will be ea$ie to do, for that you will have a large $pace upon tho$e <I>S</I>hips or ranks of Boats, as al$o there will not want room to fa$ten their Pullies to tho$e Tires of Beams, which combine the $aid <I>S</I>hips or ranks of Boats. In all things el$e proceed preci$ely according as you have been directed in the $econd, third, fourth, fifth, $ixth and $eventh Explanations.</P> <P>This indeed mu$t be granted, that inca$e the $aid <I>S</I>hip long $unk, $hould be in a <I>S</I>tony bottom, or where $he hath a great current, the which current $uffereth not any great bed or $helves of Mudd to gather about the $aid <I>S</I>hip, it may then ea$ily be got clear of the bot- tom, with the $ame Forces as were imploy'd in that newly $unk, to recover it; and al$o may as ea$ily be drawn to the Surface of the water: But whether you can rai$e it with part of its Hull above the <I>S</I>uperficies of the water, is a thing much to be doubted; yet if it $hould prove $o upon the Experiment, namely, that you cannot elevate its Hull above the <I>S</I>urface of the water, you may in $uch a ca$e hall it at high water to $hore, or to $ome place where it may lie a ground, whereby at the retreat of the Tide, it will lye with part of its Hull above water, $o that you may commodiou$ly clear it of the imbibed water and Cargo.</P> <head><I>EXPLANATION</I> IX.</head> <P>And to the end that this invention may be of generall u$e for the re covery or rai$ing any kind of Collo$$us, that may happen to be $unk, to wit, of all <I>S</I>pecies of <I>S</I>olid Bodies, whether of <I>S</I>tone, Iron, Pewter, Bra$s, Lead, <I>S</I>ilver or Gold (as you may have many occa$ions voluntarily to $ink them in time of war, to pre$erve them) and then that you may know how to get them up again, you mu$t ob$erve this Rule: If the <I>S</I>olid long time $ubmer- ged were of Brick; $o $oon as it is imbreecht, you mu$t take $o ma- ny couple of <I>S</I>hips, Barks, Hoyes or Wherryes, that the $um of their contents put together, may exceed the <I>S</I>quare of the <I>S</I>olid Area of the $ubmerged <I>S</I>olid: and if the <I>S</I>olid $o long $unk were of Mar- <foot>ble</foot> <p n=>497</p> ble, the <I>S</I>olid Content of all the <I>Vacua</I> of tho$e <I>S</I>hips or Ve$$els ad- ded together, mu$t not be le$s than Septuple to the Solid Content of the $ubmerged Body; namely, $even times as much. And if that long $unk Solid chance to be of Iron; you mu$t make the Solid Content of all the <I>Vacuum's</I> of tho$e Ve$$els to be no le$$e in the Aggregate than 12 3/2 times as much as the Solid Content of that $ub- merged Solid: and the like mu$t be done, if the $ubmerged Solid be of Pewter, for that Iron and Pewter differ not much in Gravity. But and if the drowned <I>S</I>olid be of Copper, it is requi$ite that the <I>S</I>olid Content of all the Ve$$els Cavities in $um, be no le$s than thirteen times as much as the <I>S</I>olid Content of the $aid <I>S</I>olid $unk. And if the $ubmerged <I>S</I>olid were of Lead, the <I>S</I>olid Content of all the <I>Vacua</I> of tho$e <I>S</I>hips, wherewith you would recover it, $hould be no le$s than twenty times as much as the <I>S</I>olid content of the drowned <I>S</I>olid, and rather more than le$s; and almo$t the $ame proportion ought to be ob$erved, if the $ubmerged Solid were of fine Silver, for that Lead and pure Silver differ not much in Gravity: truth is, that Lead is $omewhat more weighty than Silver, but not much.</P> <P>But if the Solid which was $unk, $hould chance to be of pure Gold, you mu$t for its recovery take $o many couple of Barks or Boats, that the Solid Content of their <I>Vacua,</I> taken in aggregate, may be no le$s than 34 times as much as the Solid content of the $aid Golden Solid $ubmerged. And that you may the better under- $tand me, I will put an Example, that you were to recover or rai$e out of the water, a Solid Body re$embling a great Tower, which I imagine to be in length an 100 Paces, and in breadth 10, and in thickne$s al$o ten: and I $uppo$e that it is all one <I>S</I>olid, that is to $ay, not hollow within. And fir$t we put the ca$e that this Tower were made of Brick. Now becau$e the <I>S</I>olid Content of this $up- po$ed <I>S</I>olid would be 10000 cubical Paces: therefore in this ca$e, if you would recover this $ame Body, that is, not only loo$en it from the bottom of the <I>S</I>ea, but al$o rai$e it a good height above water, it will be requi$ite, as is $aid above, to take $o many pair of Ships, Barks, Boats, or other Ve$$els, (as hath been $hewn in the 5 and 6 Explanation) that the <I>S</I>olid Content of all the <I>Vacua</I> of them put together, be not le$s than four times the $aid $um of 10000 cubick Paces; that is, it mu$t not be under 40000 cubicall Paces, as was above determined. And $o ìf it happen that the $aid $ubmerged So- lid $hould be all of Marble, the <I>S</I>olid Content of all the Vacuities of the $aid <I>S</I>hips, ought not to be le$s than 70000 cubicall Paces, namely Septuple, as was before concluded. And thus if the $unk. <I>S</I>olid were all of Iron or Pewter, the aggregate of all the <I>S</I>olid Con- tent of all tho$e <I>Vacuums</I> put together, mu$t be rather more than <foot>le$s</foot> <p n=>498</p> le$s then 126666 2/3 cubical Paces. And in ca$e the Solid were all of Copper, the Solid Content of the $aid <I>Vacua</I> ought to be about 130000 cubick Paces. And likewi$e if the Solid were all of Lead or Silver, the Solid Content of all the $aid <I>Vacua</I> is to be no le$s than 200000 Paces cubical. La$tly, if $uch $ubmerged Solid be pro- pounded all of fine Gold, the $um of tho$e Cavities ought to be no le$s than 340000 cubick Paces.</P> <P>The manner how to proceed in the recovery of tho$e $everall kinds of Solids, is to be under$tood to be like to that which was pre$cribed in the recovery of the Ship: and that as well in deep, as $hallow waters. And the greater number of Ships or Boats are re- quired to opperate in the recovery of the $aid $ubmerged Solid in a deep Channell, $o much the more room mu$t yon take upon the one and the other Squadron, for to be able to pitch $uch a number of Cap$tens as $hall be needfull, and more if occa$ion be. Yet you mu$t ob$erve, that in the taking the water alternately from the one and other Squadron, when you perceive the $aid Solid to be di$- engaged from the bottom, you are to forbear taking out any more from either of them; as was appointed touching the Ship, in the $eventh Explanation. And make u$e of as many Pullies as you $hall $ee cau$e for, not only to lift it to, but al$o to draw it above the waters Surface: and that if notwholly, yet for the greater part: and when it is lifted as high as is po$ible, then take the remaining water by equall mea$ures, out of the one and other Squadron, or Rank of Ships; which being done, it $hall be hoi$ted $o high out of the water, that you may put under it as many Lighters or Flat-boats, as $hall be $ufficient to bear it up, and to carry it to any place, as occa$ion $hall require.</P> <head><I>EXPLANATION X.</I></head> <P>Albeit <I>Vitruvius, Vegetius</I> and <I>Valturius</I> do teach diver$e and $un- dry wayes to carry water up on high, many whereof may $tand us in much $tead in this our Invention, for the commo- dious filling and emptying all the $everall kinds of Ve$$els $poken of above; of which al$o, many are very well known and familiar to every one; to wit, with Bur-pumps, Chain pumps, common-pumps, and many others: yet neverthele$s to fill the $aid Ships or other Ve$$els with water, with great facility and dexterity; I judge this more expedient than any of them; namely, to make a Hole in the bottom of each of tho$e Ships or other Ve$$els, of two or three inches Diameter at lea$t, and for every Ship to appoint a Boome or long tapered Pole like a Plugg or Tapp, $o that being thru$t into the $aid Hole, it will $top it $o clo$e, that unle$s you con$ent thereto, no <foot>water</foot> <p n=>499</p> water can enter in thereat, and this Pole is to be $omewhat longer than to reach from the Keel to the upper deck of the $aid Ship; and near the other end, put another piece of a Pole cro$s wayes; that you may be able by means of that to rule it; namely, to pull it up, when you would un$top the Hole, to let in the water that $hould fill the Ship, and to thru$t it down when you would $top the Hole that no more water may enter; and this $ame Pole $hould pa$s through two Rings, fixed in the Hold of the Ship, which are to keep the $aid Pole directly over the Hole, that if you would $top it, the Plugg or Spiggot may not go be$ides the Hole, when you thru$t the Pole downwards. And that I may be the better under$tood, I have here below drawn the $ame Pole, with its Tapp or Plugg at the end. And when you go about to recover any Ship, you mu$t $top the $aid Holes, till $uch time as the $aid Ships are carried <fig> and fitted upon the place, as is $hewn above. And when you would fill them with water, it is but with- drawing the $aid Poles, and opening the Holes; and fa$ten them at that $tay, till you have a mind to $top the Holes; and then look downwards, and ob$erve when the Ships are as full as they can $wim, or when they are full enough, which will be in a very $hort time: and then let down tho$e Poles, and $top the Holes very clo$e. And when they are as full as they need, in the ebb of the Tide, combine the Ship with the Pullies, to tho$e five or more Orders of Beams often mentioned: and then draw out the water with Pumps by little and little, and one while out of one, and another while out of the other Ship, as was appointed in the $econd Explication: and in all other particulars proceed, as was al$o there directed But if the Gravity of tho$e Ve$$els, cau$eth them not to fill fa$t enough, you mu$t fill them at the top, that is by baling in water by the Deck (I mean the $aid Poles being fir$t thru$t down) to make the $aid Ve$$els to de$cend fa$ter, and to rai$e the Matter $ubmerged with more Force; many other new wayes might be $hewn, as well to empty, as to fill the$e Ve$$els; but for the pre$ent this $hall $uffice.</P> <head><I>EXPLANATION</I> XI.</head> <P>If you would attempt to recover a Ship or other Ve$$el by the wayes here pre$cribed: you mu$t go about the $ame, when the <marg><I>i.e.</I> At a $pring, tide, which is greate$t the third day after the fuil and change.</marg> Moon is in the Auge of the Excentrick, for at that time the Sea ebbeth and floweth more than at any other time in the Moneth; and this happens in her Coujunction and Oppo$ition, which is a matter of great avail in the$e operations: and herewith we conclude this our fir$t Book.</P> <foot>S$$ The</foot> <p n=>500</p> <head>THE Indu$trious or Trouble$ome INVENTION OF Nicholaus Tartalea:</head> <head><I>BOOKE</I> II.</head> <P>In which are taught, $ome artificial wayes of <I>Diving</I> and $taying long under Water: whereby one may ea$ily de$cend to the Bottom, to finde out, not on- ly a Ship $unke, but al$o, any other $mall thing of Value: And the place being darke, many wayes are $hewn how to enlighten it: And the thing $unk being found, $everall wayes and means are pre$cribed how to imbreach them, as well in a Deepe, as Shallow Channel.</P> <head><I>EXPLANATION</I> I.</head> <P>Having under$tood, <I>Mo$t Serene Prince,</I> from $un- dry Sea men, that there are many now adayes, who without any particular Artifice or help, do upon occa$ion dive and continue a long time under Water, and in places very deep; I had thought to have added nothing touching the way of Artificiall Diving, and $taying under water, to $eeke and finde out a Ship, Boare, or other thing of Value $ubmerged, and that for two Rea$ons. Fir$t, Fearing that I $hould be derided by tho$e kinde of men, it being to them a $uperfluous thing to go about to do tho$e things by Art, which they know how to execute without any arrificiall help. <foot>Se-</foot> <p n=>501</p> Secondly, doubting, by rea$on of my $mall experience in Maratine Affairs, to incurre $ome Soleci$me: but there coming into my mind an excellent expre$$ion of a famous Philo$opher of this Renowned City; who upon a time per$wading me to write $omething that was new, and I having an$wered (it being incident for humanly to erre) that I was afraid lea$t my $o great de$ire to publi$h my fund y new Conjectures, might run me into $ome fanta$tical conceits, that might make me become the $ubject of vulgar di$cour$e, this excel- lent per$on replied: That if Nature $hould forbear her operations for fear of producing $ometimes $ome mon$trous things, the worlds de- $truction would en$ue, for that they onely are free from erring who do nothing, who$e $peech hath emboldened me to $peak of a point, which I never thought to have medled with; namely, To declare $ome of my conjectural wayes of artificial diving, and continuing under water, to $eek out any thing that was $unk in the $ame, though in places very deep. And I judge the$e the mo$t expedient that can be devi$ed: and becau$e the$e and the like wayes may be varied into $everal forms, and $orts, one more ingenious, and artificial than another; the prettie$t, and mo$t ingenious is this, I would have you <marg>A Place near to <I>Venice,</I> where the famous Glahes are made.</marg> get, made at <I>Murano,</I> a hollow Globe of Tran$parent Gla$$e, the di- ameter of which I would have to be at lea$t two foot, with a round mouth, that the Diameter of the $aid mouth may be at lea$t one foot, or wrather more; that is, $o much as one may ea$ily put his head therein, and at plea$ure draw it forth; and next you mu$t make two round Boards of a Diameter $omething bigger then that of the $aid Globe, and with the$e two round Boards, and four $len- der pieces of Wood, as long as a man is high, and a little more, you mu$t make a little Modell for a man to $tand betwixt the$e four pie- <marg>Like the Frame of an Houre- gla$$e.</marg> ces of Wood; and with one of the round Boards above, and the o- ther beneath; and the$e round Boards are to be very fa$t nailed or otherwi$e fa$tened to the four pieces of the Frame, and in the top of this Machine, you mu$t fit and fix the $aid Sphere of Gla$$e with the mouth downwards, $o, that if a man $tand upright in the $aid Frame, he may hold his head in the $aid gla$$e without $tooping. And this being done, take neer upon as much Lead as all this Machine weighs, and make it into a round figure, of the compa$$e of the round Boards, and then fa$ten and nail it to the bottome of the $aid Mo- dell, namely, underneath the lowermo$t Board on which your feet $tand when you put it into the Water: And then, (or before) make an hole as big as a Shilling in the Centre of this Lead and Board, pa$$ing through them both; and this $ame Lead will be able to draw almo$t all the Machine together with him that $hall be therein under Water. Truth is, that the Experiment requireth that the $aid Lead be $o limitted that it may be able to draw the Ma- <foot>S$$ 2 chine</foot> <p n=>502</p> chine and per$on in it under Water, but $o, that the $upreme or up per part of the $ame, that is the uppermo$t round Board, may $tay at the Superficies of the Water; that is, if the Lead chance to be $o ponderous, that it cau$e the Engine to $ink lei$urely to the bottome, you mu$t take away $ome of the $aid Lead; and on the contrary, if it chance that the Lead be not able to draw it all in that manner under Water, $o as to make the $aid upper round Board to lye and $tay exactly level with the Surface of the Water, but that a part of it re$ts vi$ible above the Water, you mu$t encrea$e the $aid Lead $o, that the upper Board may lye and abide preci$ely, as was $aid be- fore, in the Surface of the Water: and when you have thus adju- $ted the $aid Lead, I would have you take a Ball or Bullet of Lead weighing two or three pounds, (that is to $ay of $uch a weight, that it may be $ufficient to make the Machine and per$on diving to de- $cend to the bottome as oft as it is interpo$ed, or added,) with an Iron Ring in the $aid Ball, to which bend or fa$ten a Rope as long as the $aid Water is deep, in which the Diver is to de$cend, and $ome- what more; and reeve or pa$$e the other end of the $aid Cord through the hole made in the Board and Lead through the bot- <fig> tom of the Model; and fa$ten that $ame end of the Cord in a place of the Machine, $o, that the Diver may take it, and draw it, or $lack it as he plea$eth: and this being done, the $aid Machine will be fini$hed. And that you may better under- $tand it, I have here in- $erted it graphically: yet I $hould have told you, that for many rea- ons you $hould in the beginning have fa$tened a Ring in the Cen- tre of the upper Board, on the out$ide, to tye a Cord to the $ame as occa$ion $erveth.</P> <foot><I>EXPLA-</I></foot> <p n=>503</p> <head><I>EXPLANATION</I> II.</head> <P>Having under$tood the manner how to make this $ame En- gine, it remains to $hew how it is to be u$ed; And for your direction therein, I $ay, That he that would dive or go under Water to $eek any thing that was $unk, $hould carry the $aid Ma- chine to the place where he re$olves to de$cend, and fir$t to let that Ball of Lead with the Line go to the bottome, and then to put in the Machine it $elf, which by means of its heavy bottome of Lead will re$t upright in the Water, with almo$t all the Globe of Gla$$e above Water, in $uch $ort, that he that would may ea$ily enter into the $ame: yet you mu$t be dexterous in going into it, that you do not much $way the Machine $idewayes, for that, if it lye too oblique the Water will enter into the Globe of Gla$$e, and drive the Aire thence that was in the $ame, or at lea$t in part, but holding it up- right when you enter the $ame, the Water $hall keep in the Aire on all $ides, whereby the water will be kept from entring. And therefore if he that $hall enter into the $aid Machine, do nimbly thru$t his head into the $aid Globe by the hole thereof, he $hall finde it quite fil- led with Ayre; in which place he may breath for verry many Re- $pirations, without the lea$t ob$truction from the Water: And be- cau$e this Machine will $tay with its upper end level with the Wa- ters $urface (the affixed Lead having been $o limited) therefore de$iring to de$cend to the bottom, the Diver $hould hale the Ball and Line upwards, which was $ent before to the Bottom, in haling of which the $aid Machine will de$cend as much under Water as he hales the Corde; and if he continue haling it, till there be none of it left, he $hall de$cend to the Bottome; and in the de$cent, and after that he $hall be got to the bottom, he mu$t look round about him through that tran$parent Globe for to finde out the thing he $eeks, and $eeing it, he may many wayes with ca$e transferre him$elf thither without ri$ing again to the top; And when he would re- turn upwards to the toppe of the Water, he needs do no more but $lacken that corde fa$tned to the Ball of Lead, for thereupon the Machine $hall begin to ri$e upwards, and letting the $aid Corde goe, it $hall not $tay till the Machines upper parte arrive at the $urface of the Water; and being a$cended thither, the Diver may come out thereof, and $wim to the top, and provide him$elf afterwards of $uch things as are nece$$ary for embreching the $aid Ship or other matter $unke: And in ca$e the Diver cannot $wim, it will be nece$$a- ry to fa$ten a Corde to the Ring placed in the Centre of the upper Board, and thereby to draw the Modell above the Surface of the <foot>S$$ 3 Water;</foot> <p n=>504</p> Water; but knowing how to $wim, he may enter, a$cend, and de$cend of him$elf, without any help.</P> <head><I>EXPLANATION</I> III.</head> <P>But if you chance to be in a place where you cannot procure the $aid Globe to be made of Gla$$e, it may be made of Wood; but then you mu$t make therein great Sights, or Eyeholes of clear Gla$$e of each $ide to look four $everall wayes; and pay it without, and al$o within if you $ee cau$e with Pitch. And if you cannot get $uch a Ball of Wood, you may make $hift with a little Cubicall Che$t or Boxe, like one of tho$e Che$ts wherein they plant Ceaders, which mu$t be well joyned graved and pitch't, with four $uch Sights of Gla$$e as before, namely one upon every lateral flat or plain, $o placed, that the Diver may $ee through them every way, and be able to look downwards, it would be good to make the Box $omewhat narrower towards the mouth, that $o the four late- rall Planes may look $omewhat $loping: and in the entrance, de- $cent, a$cent, and coming forth, you are to u$e the $ame Rules as be- fore; aud if you have a de$ire to de$cend fa$ter, you mu$t make the Ball of Lead $omewhat heavier, that was tyed to the end of the Corde, and this done the Machine $hall de$cend fa$ter to the bottom upon halling the $aid Corde and Ball; and when you vere or let loo$e the Cord, the Engine will re-a$cend but according to its former $peed: But if you would al$o make it $wifter in its a$cent you are to proceed quite contrary, that is, you mu$t $omewhat dimini$h the Lead, which is under the Ba$e of the fiame; and the more you di- mini$h the $aid Lead, the $wifter $hall it be in a$cending. But you mu$t remember withall to encrea$e the Ball of Lead, $o that it may be able to draw the $aid Machine to the bottome $peedily or lei$ure- ly according as occa$ion requires.</P> <head><I>EXPLANATION</I> IV.</head> <P>But if there be any likelihood of any obnoxious Fi$h in the place where the Diver is to de$cend, that may hurt him, being quite na- ked; though that in the former kind of Machine with four pillars you may $e u e him with a wire Grate, made in the manner of doors to the $ame, yet to the end that you may know that this Invention may be varied $undry ways; you may in this ca$e have a Globe of tran$parent gla$s made at <I>Murano,</I> of $uch a bigne$s, that a man $tanding on his feet, or el$e $itting, may be contain'd therein, having amouth or round hole of capacity $ufficient for a man, commodiou$ly to enter and goe out thereby, and $omewhat larger: & then coffin or enclo$e the $aid Globe <foot>between</foot> <p n=>505</p> between two round Boards of $omewhat a greater Diameter than the Globe, with four pillars, as in the en$uing figure doth graphically appear. But in the round Board which is put over the hole or mouth of the $aid Globe, you mu$t al$o make a round hole $omewhat nar- rower than that of the Globe, but yet big enough for a man to pa$$e in and out thereat. Afterwards under this round Board $o bored, you mu$t place and fix another round bored piece of Lead of $uch thickne$$e, as that it may be able to draw the $aid Ball or Globe of Gla$$e, together with the Diver in $uch manner under Water, that the upper round Board do re$t in the Surface of the Water, namely, that it may not be $o heavy as to $ink the Globe and Diver to the bottome, but only to retain it beneath the Surface of the Water, which by tryal may be ea$ily proportioned, namely, by adding or taking away Lead from the Ba$e, according as occa$ion $hall require. Next you are to frame a $eat for the Diver to $it commodiou$ly in the $aid Ball or Globe, and next fa$ten a Ball of Lead to the end of a Rope, as many fathom long as the water is deep into which you would de$cend, and $omewhat more, as was $aid in the preceding Explanation. And that Ball of Lead $hould be of $uch bigne$$e, that applied to the $aid Model, it may be $ufficient to make it de$cend to the bottome lei$urely, or $wiftly, as he $eeth cau$e who is to dive. And make an handle or peg in the $aid Globe whereat to fa$ten or belay the other end of the $aid Rope, and to draw it ea$ily upwards, <fig> or let it loo$e at the plea$ure of him that is within, and this may be ea$ily done by joyning and fa$tening four pieces of wood upright in the mouth or hole of that bored Board and Lead, which $hall be about the mouth of the $aid Globe; and that I may be the better under$tood, I will give it you in figure with the Diver $itting therein. If you would de$cend to the bottome of $ome deep water by help of this Machine, you are to proceed according to the directions gi- ven in the precedent Explanation.</P> <foot><I>EXPLA-</I></foot> <p n=>506</p> <head><I>EXPLANATION</I> V.</head> <P>In ca$e you $hould be in a place where you could not have $uch a Globe made of Gla$$e, you may procure one of Copper or <marg>* <I>Brenta,</I> a Ve$$el in which they in Italy carry Grapes to the Pre$s.</marg> Lead, round in fa$hion of a greater ^{*} Churne, wide in the bot- tome and narrow in the mouth, and at lea$t five foot high, and four foot broad. It may indeed be made quadrangular, that is, $o that the mouth be at lea$t three foot $quare every way, and the bottome at lea$t four foot every $ide, and not under five foot high, and this $ame ve$$el, making it of Lead, mu$t be $o contrived, or proportio- ned, that the corporeal or $olid <I>Area,</I> or Content of its interiour va- cuity, or $pace, be about <*> or uple to the $olid <I>Area</I> of the Lead, which is imployed in making the $aid Ve$$el; that is, make the Lead of $uch a thickne$$e, that the Ve$$els vacuity may be nine tenths of the $olid Area of all the whole Frame, which may be ea$ily done by any one that is not ignorant of practical Geometry: and this Ve$$el being made, you $hould place or $et therein four great Fye holes or Sights of tran$parent or cri$taline Gla$$e, $o placed as to $ee any way as you $hall need or de$ire: and furthermore, in the framing of this $ame Ve$$el, you mu$t make $ome provi$ion for the $etling or $tay- ing your feet, and to $it down, and likewi$e you mu$t make a Pulley to hall the Ball of Lead up, or let it down, which is fa$tened to the end of the long cord, as was $aid in the two precedent ca$es. And moreover, in the making of this Ve$$el, you are to fa$ten four Rings of Iron to the bottome without, namely, to the four Angles, it being Quadrangular; (and being round, let them divide the Cir- cumference into four equal parts) and betwixt the$e four Rings, you mu$t place a $quare or round Deal Board. And this Ve$$el thus modellized $hall be $o contrived, that putting it into the water with the mouth downwards, with him in it who is to Dive, it $hall but ju$t $tay in the Surface of the water with that bottome of wood; but if it chance that it $hall not $tay at the Surface of the water by helpof that bottome of Board, but that it will de$cend, you mu$t upon that bottome fa$ten another, or two, or more $quare or round Boards to the four Rings, in $uch wi$e, that by means of the $aid Boaids it may be reduced to $uch a quality, that it may re$t with the $aid round Boards in the Surface of the water, and de$cend no farther. Having with judgement and experience provided all the$e things, and the Diver being de$irous to de$cend of him$elf, and likewi$e to return to the top when he plea$eth, this may be performed with that Ball of Lead tied to the end of that long Rope, as hath been $aid in the precedent Explanations, that is, to $end the Ball fir$t to the bottom in he place where the Diver would de$cend, and then to enter into the <foot>Machine,</foot> <p n=>507</p> Machine, and to $ettle him$elf therein; and then to pull the Ball upwards, which $hould be of that Gravity, that it may be apt to make $uch a Ve$$el or Machine de$cend together with the Diver; and if the Machine chance to be ju$tly contrived, as hath been $aid a- bove, I hold that a Ball of $ive or $ix pounds may be $ufficient to make it de$cend nimbly upon the pulling of the Cord, and lifting the Ball from the bottome, and continuing to draw the $aid Cord, as long as there is any remaining, he $hall arrive at the bottome; and whenever he would return upwards, he need but only vere or $lack- en that Cord, and letting it all go he will not cea$e a$cending till the Machine attains with its top (covered with tho$e $quare or round Boards) unto the Surface of the Water, as hath been $aid of the o- thers. I will not $tand to $hew you the many particularities which might be in$erted for the tran$porting your $elves from one pl<*>ce to another, keeping at the bottome, that is, without returning to the top, for that they are almo$t infinite, but it $hall $uffice to let you know, that he may ea$ily do it, carrying with him a long Hitcher, or a Boom, or a Spike with a Hook at the end.</P> <P>Many other particulars there might be in$i$ted on, and e$pecially how many may $imply (that is, without any of the for<*>d Ma- chines) go to the bottome, and $tay for many hours under Water, which, be$ides the many profitable conclu$ions that might from thence be inferred for Diving in indifferent depths, being accompa- nied with the helps pre$cribed in the foregoing Explanations, they would be much to the purpo$e, for that the Liver being once condu- cted with the Machine near unto the thing $unk, he might come out of the $aid Machine, and go and $tay for a long time about the $ame, to fa$ten, or prepare tho$e things that are nece$$ary for the rai$ing it: And farthermore, there is $omething to be $aid, when the thing $unk is in a muddy or dark Water, how the Diver may in $undry wayes, kindle there a great and flaming light, which flaming fire, be$ides that it would make him di$cern the thing $unk, it would al$o $ecure him in his going forth of the Machine from any devouring Fi$hes, for that all $uch as $hould chance to be near that place would be affrighted at $uch an unu$ual $pectacle, and would make far from it. I might al$o $hew many wayes to embreech and grapple a Ship when it is found, as well in deep as $hallow Channels, which particulars I $hall re$erve for another time.</P> <P>I will not $tand to $hew how this kind of Diving Machine might be made of Boards, and that in $undry fa$hions, well calked and pitcht, with four Lights or Sights, fa$tening about the mouth of the $ame as much Lead as $hould be nece$$ary, ora$inuch as by what hath been $poken in the third Explanation, it is $ufficiently manife$t. <foot>Ttt A</foot> <p n=>508</p> <head>A SUPPLEMENT OF THE Indu$trious or Trouble$ome INVENTION OF Nicholaus Tartalea:</head> <P>In which is $hewn a general and $afe way to im- breech Cables, and hitch Grappling irons to any Ship that's $unk, a$well in a deep as $hallow Bot- tome, provided you know the exact place where the $aid Ship is. Together with another new way of rai$ing or recovering the $ame.</P> <P><I>Whereunto is, la$t of all, added $ome new ways to conduct a Light, or Flaming Matter, unto the Bottome of the Water, to enlighten, upon oc- ca$ion, any dark Bottome, for the di$covery, not onely, of a Ship or Bark, but al$o any $mall thing of value that is $unk, and that in the night <*> well as in the day.</I></P> <head>To the Mo$t Illu$trious and mo$t Serene</head> <head>PRINCE France$co Donato, Duke of VENICE.</head> <P><I>Having not long $ince, Most Serene, and Mo$t Illu$trious Prince, publi$hed under the Glorious Name of your Highne$$e, $undry and diver$e way storai$e a Ship $unk, with its Cargo in it (when once</I> <foot><I>it</I></foot> <p n=>509</p> <I>it is Grappled) I mu$t confe$$e I was not then $ollicitous to find a way to imbreach or grapple the $aid Ship (though it is nece$$ary to be known) and the cau$e thereof was, for that I concluded that among$t Mariners there were a thou$and means to effect it, and I was loath to enquire af- ter $uch things as are commonly known to many, although I be ignorant of them; but delight to $earch into tho$e things which none el$e can do. Now, having been $ince told and a$$ured by many, that Mariners, and all other per$ons of ingenuity find far greater difficulty in imbrea- ching and Grappling $uch a Ship, than they do, (when once they have hold of it) to rai$e the $ame: I under$tan- ding the $ame, pre$ently deliberated upon $ome way that $hould be general and $ecure, and to adde it in the end of my Treati$e, that $o it might not, for want thereof, be vain and u$ele$s. And thus; of many that I have found, that which to me hath $eemed mo$t univer$al and ea$y to be explained by writing; I have here $ubjoined, together with another new way to recover the $aid Ship: and the manner how to illuminate the bottome of a dark Water, but still under the Illustrious Name of your Serene Highne$$e, at who$e feet I once more humbly throw my $elf</I></P> <P>NICOLO TARTAGLIA.</P> <p n=>510</p> <head>A Supplement.</head> <head><I>EXPLANATION</I> I.</head> <P>To hitch therefore, and $ling, or grapple fa$t a laden Ship that is $unk, being in a $howle bottome, as was that broken up near to <I>Malamoccho,</I> you are to take a very $trong Sheat-anchor Cable, of $uch a length as is $ufficient for the U$es hereafter to be under$tood, and at one end of $uch a Cable you are to $eiz or fa$ten very well a thick and $trong Iron Ring, big enough for the other end of the Cable to pa$$e through with ea$e, and make thereof a running Parbunckle: and then, near to this Ring (that is under this Cable at the place where it $hall be bent to the Ring) you mu$t $eiz or fa$ten one of the Flooks of a thick and $trong Anchor, and about three fathoms $pace from that fir$t An- chor hitch the Flook of another $econd Anchor into the $aid Cable, $eizing or fa$tening it that it $tir not: and about two fathoms di- $tance from this $econd Anchor, $eiz, as before the Flook of a third Anchor, and $o two fathom from that a fourth Anchor; and $o pro- ceed, placing in that manner as many Anchors as $uffice to go round the Hull of the $aid Ship under its Wails, and rather le$$e than more, to the end the la$t Anchor may be no hinderance to the running of the Parbunckle at the Ring at $uch time as it is to be rou$ed or vered, that is, to be drawn or let $lip. The truth is, that in the part of the Cable marked E, in the Figure following, and in the oppo$ite part marked G (which parts you are to place $o that they may fall one at the Stem, the other at the Stern) no Anchor is to be placed, but you mu$t leave at lea$t three fathom interval betwixt tho$e An- chors at G, as was required to be done betwixt the fir$t and $econd at E. And then form the $aid Running Parbunckle, that is, reeve the other end of the Cable through the Ring of Iron; and, that being made, you are to place many per$ons upon Flat-bottome Boats fa- $tened in an Oval Figure round the place where the Ship lyeth: and then vere or $lacken the Parbunckle, but in an Oval Form, to that widene$$e, that it may at four or five foot di$tance, inviron the foun- dered Ship: and this done, you mu$t let all the Anchors, together with this Girdle or Parbunckle, (being kept at that widene$$e) gent- ly and equally fall to the bottome of the Sea, keeping the Ship in the mid$t of the Ovall: and when you perceive all the Anchors de- $cended to the bottome, you mu$t vere there $everal Cables, that they may $ink deep into the $and or Ouze; and then after this you <foot>mu$t</foot> <p n=>511</p> mu$t draw, and bring them by degrees clo$e underneath the Hull of the Ve$$el, and then hall or $train hard the end of the Sheat Anchor Cable which was reeved through the Ring; and begirt the Hull of the Ship therewith, as with a Girdle (and to $train it very taught, it would not be ami$$e to make u$e of a Cap$tan) and when this Girdle is drawn to its due exactne$$e, to the end it may not $lip (in the elevation of the Ship) fa$ten to that part which you hold above Water another Ring of Iron, and pa$$e through this Ring one of the Anchor-Cables that is on the $ame $ide as the fir$t Ring is on, and almo$t as far from the $aid Ring, as the $econd Ring is di$tant from the fir$t; whereupon making this $econd Ring to $lip along the $aid Anchor Cable, and then in the Elevation halling the $ame, it $hall make the $aid Girdle taught under the $aid Ship: and that I may be the better under$tood, I have here underneath repre$ented the $aid Girdle pul'd together in an Oval Figure as it is to lye under the Rake of the Ships Hull with fourteen Flooks of fourteen An- chors under the $ame (except in the part inked E, and in its oppo- $ite part G,) well $ea$- ed; of which Girdle, or Parbunckle, the fir$t <fig> Ring $hall be A, through which the Sheat-Anchor Cable pa$$eth, namely, the Cable A B, to which Cable was fa$tened a $econd Ring in the point B, through which $econd Ring, (to the end the Girdle might not $lio) we will reeve the Cable of the An- chor C; which Anchor C we $uppo$e to be $omewhat farther from the Ring A, than the $econd Ring B is from the fir$t Ring A, and then make the $aid Ring B to $lip along the Cable of the $aid <I>A</I>nchot C, till it come to the point C. <I>A</I>nd thus the Ship $hall be $ecurely and $trongly grappled and begirt. <I>W</I>hich done, proceeding as we directed in the fir$t Book of our <I>Indn$trious Invention,</I> you will exe- cute your purpo$e; That is, when the two or more coupled Ships $hall be full of water, at the ebbing of the Tide you are to fa$ten and belay to tho$e Tires of Beams that couple the $aid Ships, all tho$e fourteen Cables, taking a little more care in tying, and belay- <foot>Ttt 3 ing</foot> <p n=>512</p> ing that of the Anchor C, which will keep the Girdle from $lipping in the Elevation.</P> <P>But if you doubt that that $ingle Cable, to which the Anchors are fa$tened, is not $ufficient for $o great a weight, you may above that, place another with a Ring al$o, through which (as before) the end of it may pa$$e, by that means begirting the Ship with two of tho$e Girdles, and ob$erving the $ame Rules you may take three or four of tho$e $lipping $heat- anchor Cables, each with its Ring wherein to run in the manner of a Noo$e. <I>A</I>nd when the $aid new Girdle is pulled $trait and clo$e to the Ship, fa$ten to the $aid Cable, (or to each of them if you u$e more) another $econd Ring, to gird and hold the $aid Noo$e fa$t, that it $lip not with the Cable of the <I>A</I>nchor C, or with more of tho$e <I>A</I>nchor-Cables if there be occa- $ion.</P> <P><I>A</I>nd in ca$e that tho$e fourteen Cables be thought in$ufficient to bear $o great a burden, you may take twenty or thirty of them, or as many as you plea$e, tying them clo$er to one the other, under the running Cable, and make half of them to be placed on one $ide, and the other half on the other $ide of the $aid Ship.</P> <P><I>A</I>nd if again it be doubted that the $ingle Cable of the <I>A</I>nchor C s not able to hold the Noo$e fa$t, you may take two or three of them, for you may judge what the $tre$s of that anchor is by means of the height of the water. Truth is, this office might be di$tributed among$t more <I>A</I>nchors, by adding a third Ring to the main Cable, as far from the $econd, as the <I>A</I>nchor D is di$tant from the <I>A</I>nchor C, $o that the Cable of the <I>A</I>nchor D, pa$$ing through that third Ring, and $lipping the $aid Ring along till it come to D, it will fol- low that tho$e two Cables of tho$e two Anchors C and D, will keep the Parbunckle $traight; aud in this manner you may proceed by ad- ding new Rings, and imploying more Anchor-Cables, for the great- er $ecurity.</P> <head><I>EXPLANATION</I> II.</head> <P>The $ame method may al$o be ob$erved when the Ship is in a deep place, provided that the depth exceed not the length of the Hull of the Ship, becau$e then there may be alwaies found $ome one or more Cables $ufficient to reeve through the $econd and follow- ing Rings of the Main Cable to $ecure the Noo$e from $lipping, or growing $lack, as in the preceding declaration hath been $aid. But if it chance that the depth of the place be far greater than the length of the Ship, you can no longer $ecure the Noo$e with that $econd Ring, but mu$t find out $ome other way, and though there might be many found out, I $hall in$tance but in this one.</P> <foot>After</foot> <p n=>513</p> <P>After you have $trained, drawn the $aid Girdle as taught as you can, you may take the Cable thereof, and the Cable of the anchor next adjoyning on the $ame $ide that the fir$t Ring is on (namely, the Cable marked F,) and twi$t and wind them together, and then reeve the $ingle Cable of the Girdle <I>A B,</I> through the Ring of a Sheat-anchor, (without its Cable) and let the anchor $lide down- wards along the $aid Main Cable, which by rea$on of its weight will run almo$t clo$e to the Ring <I>A,</I> of the Main Cable, pre$$ing the twi$t of the two Cables clo$e at <I>A</I>; and this done, once more twine or twi$t a little the two former Cables, namely the Sheat anchor-cable B, and the le$$er Cable F, and then $ea$e tho$e two Cables $everally to the Orders of Beams, that is, one to one Order, and the other to another at $ome di$tance from the former, to the end they drive down the twi$ting near to the Ring of the anchor: which twi$ting will keep the Noo$e from $lipping or opening in elevating the Ship. <I>A</I>nd if there be any occa$ion to u$e a Cap$tain (as was $aid in the $eventh Explanation of the fir$t Book) you mu$t always take care to $train the$e two Cables equally, and much a$under, which doing, the Girdle $hall be kept $trait. Many other ways might be $hewen for to keep the $aid Grand Cable from $lipping, but e$teeming them $uperfluous, I omit them.</P> <head><I>EXPLANATION</I> III.</head> <P>He that is de$irous to recover a foundered Ship laden with Fraight, by other ways than tho$e pre$cribed in the fir$t Book, namely, without $tanding to fill tho$e two or more Ships, or other Ve$$els with water, and then to empty them, may only by force of Cap$tains or Cranes ea$ily effect the $ame in the manner following, ($till making u$e of the Parbunckle and flooks of anchors explained in the fir$t Explanation of this) namely: By taking from their an- chors Rings all their Cables, except that which is to make fa$t the Main-Cable Noo$e that begirts the Ship, and in their places make fa$t to each Ring a $trong Pulley or Block, in $uch $ort, that all the $aid Pulleys or Blocks have equal number of Shivers, or wheels, and tho$e as many as you can make them: and through the$e Shivers or wheels reeve their proper and convenient Cables or Ropes, incatena- ting each Pulley with its $uperiour; and this done, make two $qua- drons of Barks, or Lighters, or Flat-boats, according to the method laid down in the fourth Explanation of the fir$t Book, collated and bound together with tho$e Tires of thick and $trong Beams tripled, and with a great and $pacious platform of thick Planks upon each $quadron, and upon tho$e two $pacious platforms place as many Cap$ters or Ship-cranes as you $hall judge nece$$ary for $uch a <foot>weight,</foot> <p n=>514</p> weight, and rather much more, then ever $o little le$s, and then let fall the $aid Anchors lei$urely, with the Girdle opened in an Oval Figure, untill they come to the bottome of the Sea, $o that the Girdle do encircle or $urround the foundered Ship. And having once be- girt it carefully, approximate all the Anchors with the Girdle to the Hull of the Ship, and then $harpen or make taught the Girdle-cable by halling it hard and $treight to the Ships hull, and when it is drawn clo$e, belay it that it may not $lacken, with that $ingle An- chor-cable, or more, according to that $ecure way $poken of but now, or by $ome other than $hall $eem more expedient, (for many more, if one think thereon may be found:) and this being done, $eek to loo$en the Ship by degrees from its bed of Ouze, a little on one $ide, and a little on the other with the afore$aid Cap$ters, and, being once water born, then draw it upwards equally on both $ides, and proceed in this manner till $uch time as you have hoi$ted it $uf- ficiently above the Waters $urface, and then pump out the Water, and unlade its Cargo.</P> <head><I>EXPLANATION</I> IV.</head> <P>Having in the $econd Book $hewn $everal ways of Diving under Water in $earch of things $unk, in this place I have thought fit to add, in ca$e that $ome little thing of value $hould fall into a Water in $ome $hady place, and where its bottome is ob$cure and dark, a way how to conveigh a Light thither that may give light enough for the di$cerning of that little thing, provided that it be not buried in, or covered with the old Ouze. Now to perform this, and that with expedition, we may in $mall depths take one of tho$e bra$s Buckets or Pails, which are u$ed in carrying and keeping of Water for hou$ehold u$es: and tho$e of them that are $haped long and deep, with feet $hall be better then tho$e that are made round and $hallow, without feet; and the bigger and higher it is, $o much the better it $hall be. And having made choice of $uch a Bucket, you are to fa$ten to the Ears of it two $mall Ropes of about two yardes apiece, in $uch a fa$hion, as that they may one cro$s the other at the mouth of the Bucket, making upon it a perfect cro$s, and that the Knot of the Ropes may be in the mid$t of the Buckets bottom with- out, making of the ropes a Hoop over the bottome whereat to fa$ten another Rope of greater length; $o that the Bucket being held by that la$t Rope may come to hang with its mouth perpendicularly downwards. And this done, fa$ten as much Lead to the two Eares of the Bucket as may ju$t make it $ink to the Bottome, and then $et and fa$ten a little Wax candle lighted in the inter$ection that tho$e two Ropes make over the mouth of the Bucket, that is, in the centre <foot>of</foot> <p n=>515</p> of that perfect cro$s; $o that the candle with its light may be with- in, and near the bottome of the $aid Bucket. This being done, let down the Bucket, with the candle in it gently unto the bottome, which doing, you $hall $ee the burning candle clearly enlighten the bottome of the Water. And this Bucket you may remove from place to place, without drawing it upwards. The truth is, that this candle will not long continue burning, but will $erve for a little while, and when it $hall go out of it $elf, it may be drawn up, re- lighted, and let down, as occa$ion requires: but the greater that the Bucket, and the le$$er that the candle $hall be, $o much the longer time $hall it keep its light under Water: and therefore if the $aid bottome were very deep, it would be requi$ite to perform that e$$ect with $o much a greater Ve$$el, as a great Caldron, but yet of Brals, or by that means the candle $hall continue longer lighted.</P> <head><I>EXPLANATION</I> V.</head> <P>But in ca$e that a Ship or Bark were foundered in $ome $pacious and profound Gulph, and that the exact place where it $unk were unknown, and that the bottome of the $aid $pacious Gulph were very ob$cure, it is manife$t that $o little a light as that $poke of in the precedent Explanation would hardly $erve. And therefore if you would convey thither one much bigger, you may do it $eve- rall wayes, of which one is this. Take nine ounces of refined Salt- peter, $ix ounces (Greek weight) of Brim$tone that is clear and tran$parent, three ounces of Camphire refined, and one ounce of Ma$tick; and beat all the$e things $everally, not very $mall; and when you have beaten them, mix them all together in an Earthen Pan; and when they are well mingled, put thereto three pounds of common Gunpowder, and then remingle them very well together; and afterwards put therein four ounces of oyl of Stone, and mix all very well; and this done, take a Cartredge thereof, and give fire to it; and if it burn too $lowly, put a little more Gunpowder to it, but if it burn too vehemently and $uddenly, add thereto more oyl. Put this Compo$ition, after this, into a little Bag of double Canvis, of $uch a widene$$e, that when all the mixture is out, therein it may be as broad, as high, and cram the Compo$ition hard down into the Bag; and then with very good Pack thread $ew up the mouth of the Sack, cutting away the $uperfluous Canvas. Then winde a good hempen cord round about it very hard every way, reducing it to the form of a round Ball, and after it is very well bound and $wathed a- bout many $everall times, you mu$t melt Brim$tone into a great Ve$- $el, and when it is melted, roll the $aid Ball therein $o, as that it may be covered all over with a cru$t of Brim$tone. And this being done <foot>Vvv affix</foot> <p n=>516</p> affix a piece of Lead unto the Ball by an iron Wire, and make it ve- ry fa$t, and frame in the top of the Ball a Bow or Noo$e with the $aid Wire, and to that fa$ten a long Rope, and then in the oppo$ite place where the Lead is fixed, make an hole with an iron rod into the middle of the Ball, and $top that hole with a little fine Gunpow- der, holding it $u$pended by the Rope: and when you would have that Light de$cend into the bottome of the Sea or Gulph, goe to the place, and give fire to the little hole, and when it is inkindled, let down the Ball and Lead, lengthwayes, almo$t to the bottome, where he $hall be that would find the thing $unk, and you $hall find that the $aid fire will illuminate very much round about the $aid bottom, and $hall la$t a long time, and more or le$s, according to the hole made in the Ball. 'Tis to be noted, that the Ball is to be held over the head of him that diveth, for that the $moke proceeding from it will much ob$cure the Waters above it, $o as that it will give Light only downwards; and this fire will be a dreadful $ight unto the Fi$h, $o that they will fly from $o new a $pectacle.</P> <head><I>The END of the fir$t part of the Second TOME.</I></head>