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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:08:12 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Archimedes</author> <title>Natation of bodies</title> <date>1662</date> <place>London</place> <translator>Thomas Salusbury</translator> <lang>en</lang> <cvs_file>archi_natat_073_en_1662.xml</cvs_file> <cvs_version></cvs_version> <locator>073.xml</locator> </info> <text> <front> <pb xlink:href="073/01/001.jpg"></pb> <section><p type="head"> <s>ARCHIMEDES <lb></lb>HIS TRACT <lb></lb>De Incidentibus Humido, <lb></lb>OR OF THE <lb></lb>NATATION OF BODIES VPON, <lb></lb>OR SVBMERSION IN, <lb></lb>THE <lb></lb>WATER <lb></lb>OR OTHER LIQUIDS.</s></p><p type="head"> <s>IN TWO BOOKS.</s></p><p type="head"> <s>Tranſlated from the Original Greek,</s></p><p type="head"> <s>Firſt into Latine, and afterwards into Italian, by <emph type="italics"></emph>NICOLO <lb></lb>TARTAGLIA,<emph.end type="italics"></emph.end> and by him familiarly demon<lb></lb>ſtrated by way of Dialogue, with <emph type="italics"></emph>Richard Wentworth,<emph.end type="italics"></emph.end><lb></lb>a Noble Engliſh Gentleman, and his Friend.</s></p><p type="head"> <s>Together with the Learned Commentaries of <emph type="italics"></emph>Federico <lb></lb>Commandino,<emph.end type="italics"></emph.end> who hath Reſtored ſuch of the Demonſtrations <lb></lb>as, thorow the Injury of Time, were obliterated.</s></p><p type="head"> <s>Now compared with the ORIGINAL, and Engliſhed <lb></lb>By <emph type="italics"></emph>THOMAS SALVSBVRY,<emph.end type="italics"></emph.end> <expan abbr="Eſq.">Eſque</expan></s></p><p type="head"> <s><emph type="italics"></emph>LONDON,<emph.end type="italics"></emph.end> Printed by <emph type="italics"></emph>W. Leybourn,<emph.end type="italics"></emph.end> 1662.</s></p></section><section><pb xlink:href="073/01/002.jpg"></pb> <pb xlink:href="073/01/003.jpg" pagenum="335[333]"></pb><p type="head"> <s>ARCHIMEDES <lb></lb>HIS TRACT <lb></lb><emph type="italics"></emph>De <lb></lb>INCIDENTIBUS HUMIDO,<emph.end type="italics"></emph.end><lb></lb>OR OF <lb></lb>The Natation of Bodies upon, or Submerſion in, <lb></lb>the Water, or other Liquids.</s></p></section> </front> <body> <chap> <pb xlink:href="073/01/004.jpg"></pb><p type="head"> <s>BOOK I.</s></p><p type="head"> <s>RICARDO.</s></p><p type="main"> <s><emph type="italics"></emph>Dear Companion,<emph.end type="italics"></emph.end> I have peruſed your <emph type="italics"></emph>Induſtrious Invention,<emph.end type="italics"></emph.end><lb></lb>in which I find not any thing that will not certainly hold <lb></lb>true; but, truth is, there are many of your Concluſions <lb></lb>of which I underſtand uot the Cauſe, and therefore, if it <lb></lb>be not a trouble to you, I would deſire you to declare them <lb></lb>to me, for, indeed, nothing pleaſeth me, if the Cauſe <lb></lb>thereof be hid from me.</s></p><p type="main"> <s>NICOLO. </s> <s>My obligations unto you are ſo many and <lb></lb>great, <emph type="italics"></emph>Honoured Campanion,<emph.end type="italics"></emph.end> that no requeſt of yours ought <lb></lb>to be troubleſome to me, and therefore tell me what thoſe Perticulars are of which <lb></lb>you know not the Cauſe, for I ſhall endeavour with the utmoſt of my power and <lb></lb>underſtanding to ſatisfie you in all your demands.</s></p><p type="main"> <s>RIC. </s> <s>In the firſt <emph type="italics"></emph>Direction<emph.end type="italics"></emph.end> of the firſt Book of that your <emph type="italics"></emph>Induſtrious Invention<emph.end type="italics"></emph.end><lb></lb>you conclude, That it is impoſſible that the Water ſhould wholly receive into it <lb></lb>any material Solid Body that is lighter than it ſeif (as to <emph type="italics"></emph>ſpeciæ<emph.end type="italics"></emph.end>) nay, you ſay, That <lb></lb>there will alwaies a part of the Body ſtay or remain above the Waters Surface <lb></lb>(that is uncovered by it;) and, That as the whole Solid Body put into the Water <lb></lb>is in proportion to that part of it that ſhall be immerged, or received, into the Wa<lb></lb>ter, ſo ſhall the Gravity of the Water be to the Gravity <emph type="italics"></emph>(in ſpeciæ)<emph.end type="italics"></emph.end> of that ſame <lb></lb>material Body: And that thoſe Solid Bodies, that are by nature more Grave than the <lb></lb>Water, being put into the Water, ſhall preſently make the ſaid Water give place; <lb></lb>and, That they do not only wholly enter or ſubmerge in the ſame, but go continu<lb></lb>ally deſcending untill they arrive at <emph type="italics"></emph>t<emph.end type="italics"></emph.end>he Bottom; and, That they ſink to the Bot<lb></lb>tom ſo much faſter, by how much they are more Grave than the Water. </s> <s>And, <lb></lb>again, That thoſe which are preciſely of the ſame Gravity with the Water, being <lb></lb>put into the ſame, are of neceſſity wholly received into, or immerged by it, but <lb></lb>yet retained in the Surface of the ſaid Water, and much leſs will the Water con<lb></lb>ſent that it do deſcend to the Bottom: and, now, albeit that all theſe things are <lb></lb>manifeſt to Senſe and Experience, yet nevertheleſs would I be very glad, if it be <lb></lb>poſſible, that you would demonſtrate to me the moſt apt and proper Cauſe of <lb></lb>theſe Effects.</s></p> <pb xlink:href="073/01/005.jpg" pagenum="334"></pb><p type="main"> <s>NIC. </s> <s>The Cauſe of all theſe Effects is aſſigned by <emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end> the <emph type="italics"></emph>Siracuſan,<emph.end type="italics"></emph.end> in <lb></lb><arrow.to.target n="marg1126"></arrow.to.target><lb></lb>that Book <emph type="italics"></emph>De Incidentibus (^{*}) Aquæ,<emph.end type="italics"></emph.end> by me publiſhed in Latine, and dedicated to <lb></lb>your ſelf, as I alſo ſaid in the beginning of that my <emph type="italics"></emph>Induſtrions Invention.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1126"></margin.target>* <emph type="italics"></emph>Aquæ,<emph.end type="italics"></emph.end> tanſlated <lb></lb>by me <emph type="italics"></emph>Humido,<emph.end type="italics"></emph.end> as <lb></lb>the more Compre<lb></lb>henſive word, for <lb></lb>his Doctrine holds <lb></lb>true in all Liquids <lb></lb>as well as in Wa<lb></lb>ter, <emph type="italics"></emph>ſoil.<emph.end type="italics"></emph.end> in Wine, <lb></lb>Oyl, Milk, <emph type="italics"></emph>&c.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>RIC. </s> <s>I have ſeen that ſame <emph type="italics"></emph>Archimedes,<emph.end type="italics"></emph.end> and have very well underſtood thoſe <lb></lb>two Books in which he treateth <emph type="italics"></emph>De Centro Gravitatis æquerepentibus,<emph.end type="italics"></emph.end> or of the <lb></lb>Center of Gravity in Figures plain, or parallel to the Horizon; and likewiſe thoſe <lb></lb><emph type="italics"></emph>De Quadratura Parabolæ,<emph.end type="italics"></emph.end> or, of Squaring the Parabola; but ^{*}<emph type="italics"></emph>that<emph.end type="italics"></emph.end> in which he treat<lb></lb>eth of Solids that Swim upon, or ſink in Liquids, is ſo obſcure, that, to ſpeak the <lb></lb>truth, there are many things in <emph type="italics"></emph>it<emph.end type="italics"></emph.end> which I do not underſtand, and therefore before <lb></lb><arrow.to.target n="marg1127"></arrow.to.target><lb></lb>we proceed any farther, I ſhould take it for a favour if you would declare it to me <lb></lb>in your Vulgar Tongue, beginning with his firſt <emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> which ſpeaketh in this <lb></lb>manner.</s></p><p type="margin"> <s><margin.target id="marg1127"></margin.target>* He ſpeaks of but <lb></lb>one Book, <emph type="italics"></emph>Tartag<lb></lb>lia<emph.end type="italics"></emph.end> having tranſla<lb></lb>ted no more.</s></p><p type="head"> <s>SVPPOSITION I.</s></p><p type="main"> <s><emph type="italics"></emph>It is ſuppoſed that the Liquid is of ſuch a nature, that <lb></lb>its parts being equi-jacent and contiguous, the leſs <lb></lb>preſſed are repulſed by the more preſſed. </s> <s>And <lb></lb>that each of its parts is preſſed or repulſed by the <lb></lb>Liquor that lyeth over it, perpendicularly, if the <lb></lb>Liquid be deſcending into any place, or preſſed any <lb></lb>whither by another.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>Every Science, Art, or Doctrine (as you know, <emph type="italics"></emph>Honoured Companion,<emph.end type="italics"></emph.end>) <lb></lb>hath its firſt undemonſtrable Principles, by which (they being <lb></lb>granted or ſuppoſed) the ſaid Science is proved, maintained, or de<lb></lb>monſtrated. </s> <s>And of theſe Principles, ſome are called <emph type="italics"></emph>Petitions,<emph.end type="italics"></emph.end><lb></lb>and others <emph type="italics"></emph>Demands,<emph.end type="italics"></emph.end> or <emph type="italics"></emph>Suppoſitions.<emph.end type="italics"></emph.end> I ſay, therefore, that the Science or Doctrine <lb></lb>of thoſe Material Solids that Swim or Sink in Liquids, hath only two undemon<lb></lb>ſtrable <emph type="italics"></emph>Suppoſitions,<emph.end type="italics"></emph.end> one of which is that above alledged, the which in compliance <lb></lb>with your deſire I have ſet down in our Vulgar Tongue.</s></p><p type="main"> <s>RIC. </s> <s>Before you proceed any farther tell me, how we are to underſtand the <lb></lb>parts of a Liquid to be <emph type="italics"></emph>Equijacent.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>When they are equidiſtant from the Center of the World, or of the <lb></lb>Earth (which is the ſame, although ^{*} ſome hold that the Centers of the Earth <lb></lb>and Worldare different.)</s></p><p type="main"> <s>RIC. </s> <s>I underſtand you not unleſs you give me ſome Example thereof in <lb></lb>Figure.<lb></lb><arrow.to.target n="marg1128"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1128"></margin.target>* The Coperni<lb></lb>cans.</s></p><p type="main"> <s>NIC. </s> <s>To exemplifie this particular, Let us ſuppoſe a quantity of Liquor (as <lb></lb>for inſtance of Water) to be upon the Earth; then let us with the Imagination <lb></lb>cut the whole Earth together with that Water into two equal parts, in ſuch a <lb></lb>manner as that the ſaid Section may paſs ^{*} by the Center of the Earth: And let <lb></lb>us ſuppoſe that one part of the Superficies of that Section, as well of the Water <lb></lb>as of the Earth, be the Superficies A B, and that the Center of the Earth be the <lb></lb>point K. </s> <s>This being done, let us in our Imagination deſcribe a Circle upon the </s></p><p type="main"> <s><arrow.to.target n="marg1129"></arrow.to.target><lb></lb>ſaid Center K, of ſuch a bigneſs as that the Circumference may paſs by the Super<lb></lb>ficies of the Section of the Water: Now let this Circumference be E F G: and <lb></lb>let many Lines be drawn from the point K to the ſaid Circumference, cutting the <lb></lb>ſame, as KE, KHO, KFQ KLP, KM. </s> <s>Now I ſay, that all theſe parts of <lb></lb>the ſaid Water, terminated in that Circumference, are Equijacent, as being all <pb xlink:href="073/01/006.jpg" pagenum="335"></pb>equidiſtant from the point K, the Center of the World, which parts are G M, <lb></lb>M L, L F, F H, H E.</s></p><p type="margin"> <s><margin.target id="marg1129"></margin.target>* Or through.</s></p><p type="main"> <s>RIC. </s> <s>I underſtand you very well, as to this particular: But tell me a little; he <lb></lb>ſaith that each of the parts of the Liquid is preſſed or repulſed by the Liquid that <lb></lb>is above it, according to the Perpendicular: I know not what that Liquid is that <lb></lb>lieth upon a part of another Perpendicularly.</s></p><p type="main"> <s>NIC. </s> <s>Imagining a Line that cometh from the Center of the Earth penetrating <lb></lb>thorow ſome Water, each part of the Water that is in that Line he ſuppoſeth to <lb></lb>be preſſed or repulſed by the Water that lieth above it in that ſame Line, and that <lb></lb>that repulſe is made according to the ſame Line, (that is, directly towards the <lb></lb>Center of the World) which Line is called a Perpendicular; becauſe every <lb></lb>Right-Line that departeth from any point, and goeth directly towards the Worlds <lb></lb>Center is called a Perpendicular. </s> <s>And that you may the better underſtand me, let <lb></lb><figure id="id.073.01.006.1.jpg" xlink:href="073/01/006/1.jpg"></figure><lb></lb>us imagine <lb></lb>the Line KHO, <lb></lb>and in that <lb></lb>let us imagine <lb></lb>ſeveral parts, <lb></lb>as ſuppoſe RS, <lb></lb>S T, T V, V H, <lb></lb>H O. </s> <s>I ſay, <lb></lb>that he ſup<lb></lb>poſeth that <lb></lb>the part V H <lb></lb>is preſſed by <lb></lb>that placed a<lb></lb>bove it, H O, <lb></lb>according to <lb></lb>the Line OK; <lb></lb>the which <lb></lb>O K, as hath been ſaid above, is called the Perpendicular paſſing thorow thoſe two <lb></lb>parts. </s> <s>In like manner, I ſay that the part T V is expulſed by the part V H, ac<lb></lb>cording to the ſaid Line O K: and ſo the part S T to be preſſed by T V, according <lb></lb>to the ſaid Perpendicular O K, and R S by S T. </s> <s>And this you are to underſtand <lb></lb>in all the other Lines that were protracted from the ſaid Point K, penetrating the <lb></lb>ſaid Water, As for Example, in <emph type="italics"></emph>K<emph.end type="italics"></emph.end> G, <emph type="italics"></emph>K<emph.end type="italics"></emph.end> M, <emph type="italics"></emph>K<emph.end type="italics"></emph.end> L, <emph type="italics"></emph>K<emph.end type="italics"></emph.end> F, <emph type="italics"></emph>K<emph.end type="italics"></emph.end> E, and infinite others of the <lb></lb>like kind.</s></p><p type="main"> <s>RIC. Indeed, <emph type="italics"></emph>Dear Companion,<emph.end type="italics"></emph.end> this your Explanation hath given megreat ſa<lb></lb>tisfaction; for, in my Judgment, it ſeemeth that all the difficulty of this Suppoſition <lb></lb>conſiſts in theſe two particulars which you have declared to me.</s></p><p type="main"> <s>NIC. </s> <s>It doth ſo; for having underſtood that the parts E H, H F, F L, L M, and <lb></lb>MG, determining in the Circumference of the ſaid Circle are equijacent, it is an <lb></lb>eaſie matter to underſtand the foreſaid <emph type="italics"></emph>Suppoſition<emph.end type="italics"></emph.end> in Order, which ſaith, <emph type="italics"></emph>That it is <lb></lb>ſuppoſed that the Liquid is of ſuch a nature, that the part thereof leſs preſſed or thrust is re<lb></lb>pulſed by the more thruſt or preſſed.<emph.end type="italics"></emph.end> As for example, if the part E H were by chance <lb></lb>more thruſt, crowded, or preſſed from above downwards by the Liquid, or ſome <lb></lb>other matter that was over it, than the part H F, contiguous to it, it is ſuppoſed <lb></lb>that the ſaid part H F, leſs preſſed, would be repulſed by the ſaid part E H. </s> <s>And <lb></lb>thus we ought to underſtand of the other parts equijacent, in caſe that they be <lb></lb>contiguous, and not ſevered. </s> <s>That each of the parts thereof is preſſed and repul. <lb></lb></s> <s>ſed by the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid that lieth over it Perpendicularly, is manifeſt by that which was <lb></lb>ſaid above, to wit, that it ſhould be repulſed, in caſe the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid be deſcending into <lb></lb>any place, and thruſt, or driven any whither by another.</s></p><p type="main"> <s>RIC. </s> <s>I underſtand this Suppoſition very well, but yet me thinks that before <lb></lb>the Suppoſition, the Author ought to have defined thoſe two particulars, which <lb></lb>you firſt declared to me, that is, how we are to underſtand the parts of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid <lb></lb>equijacent, and likewiſe the Perpendicular.</s></p> <pb xlink:href="073/01/007.jpg" pagenum="336"></pb><p type="main"> <s>NIC. </s> <s>You ſay truth.</s></p><p type="main"> <s>RIC. </s> <s>I have another queſtion to aske you, which is this, Why the Author <lb></lb>uſeth the word <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, or Humid, inſtead of Water.</s></p><p type="main"> <s>NIC. </s> <s>It may be for two of theſe two Cauſes; the one is, that Water being the <lb></lb>principal of all <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquids, therefore ſaying <emph type="italics"></emph>Humidum<emph.end type="italics"></emph.end> he is to be underſtood to mean <lb></lb>the chief Liquid, that is Water: The other, becauſe that all the Propoſitions of <lb></lb>this Book of his, do not only hold true in Water, but alſo in every other <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, <lb></lb>as in Wine, Oyl, and the like: and therefore the Author might have uſed the word <lb></lb><emph type="italics"></emph>Humidum,<emph.end type="italics"></emph.end> as being a word more general than <emph type="italics"></emph>Aqua.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>RIC. </s> <s>This I underſtand, therefore let us come to the firſt <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> which, as <lb></lb>you know, in the Original ſpeaks in this manner.</s></p><p type="head"> <s>PROP. I. THEOR. I.</s></p><p type="main"> <s><emph type="italics"></emph>If any Superficies ſhall be cut by a Plane thorough any <lb></lb>Point, and the Section be alwaies the Circumference <lb></lb>of a Circle, whoſe Center is the ſaid Point: that Su<lb></lb>perficies ſhall be Spherical.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let any Superficies be cut at pleaſure by a Plane thorow the <lb></lb>Point K; and let the Section alwaies deſcribe the Circumfe<lb></lb>rence of a Circle that hath for its Center the Point K: I ſay, <lb></lb>that that ſame Superficies is Sphærical. </s> <s>For were it poſſible that the <lb></lb>ſaid Superficies were not Sphærical, then all the Lines drawn <lb></lb>through the ſaid Point K unto that Superficies would not be equal, <lb></lb>Let therefore A and B be two <lb></lb>Points in the ſaid Superficies, ſo that <lb></lb><figure id="id.073.01.007.1.jpg" xlink:href="073/01/007/1.jpg"></figure><lb></lb>drawing the two Lines K A and <lb></lb>K B, let them, if poſſible, be une<lb></lb>qual: Then by theſe two Lines let <lb></lb>a Plane be drawn cutting the ſaid <lb></lb>Superficies, and let the Section in <lb></lb>the Superficies make the Line <lb></lb>D A B G: Now this Line D A B G <lb></lb>is, by our pre-ſuppoſal, a Circle, and <lb></lb>the Center thereof is the Point K, for ſuch the ſaid Superficies was <lb></lb>ſuppoſed to be. </s> <s>Therefore the two Lines K A and K B are equal: <lb></lb>But they were alſo ſuppoſed to be unequal; which is impoſſible: <lb></lb>It followeth therefore, of neceſſity, that the ſaid Superficies be <lb></lb>Sphærical, that is, the Superficies of a Sphære.</s></p><p type="main"> <s>RIC. </s> <s>I underſtand you very well; now let us proceed to the ſecond <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end><lb></lb>which, you know, runs thus.</s></p> <pb xlink:href="073/01/008.jpg" pagenum="337"></pb><p type="head"> <s>PROP. II. THEOR. II.</s></p><p type="main"> <s><emph type="italics"></emph>The Superficies of every Liquid that is conſiſtant and <lb></lb>ſetled ſhall be of a Sphærical Figure, which Figure <lb></lb>ſhall have the ſame Center with the Earth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let us ſuppoſe a Liquid that is of ſuch a conſiſtance as that it <lb></lb>is not moved, and that its Superficies be cut by a Plane along <lb></lb>by the Center of the Earth, and let the Center of the Earth <lb></lb>be the Point K: and let the Section of the Superficies be the Line <lb></lb>A B G D. </s> <s>I ſay that the Line A B G D is the Circumference of a <lb></lb><figure id="id.073.01.008.1.jpg" xlink:href="073/01/008/1.jpg"></figure><lb></lb>Circle, and that the Center <lb></lb>thereof is the Point K And <lb></lb>if it be poſſible that it may <lb></lb>not be the Circumference <lb></lb>of a Circle, the Right<lb></lb><arrow.to.target n="marg1130"></arrow.to.target><lb></lb>Lines drawn ^{*} by the Point <lb></lb>K to the ſaid Line A B G D <lb></lb>ſhall not be equal. </s> <s>There<lb></lb>fore let a Right-Line be <lb></lb>taken greater than ſome of thoſe produced from the Point K unto <lb></lb>the ſaid Line A B G D, and leſſer than ſome other; and upon the <lb></lb>Point K let a Circle be deſcribed at the length of that Line, <lb></lb>Now the Circumference of this Circle ſhall fall part without the <lb></lb>ſaid Line A B G D, and part within: it having been preſuppoſed <lb></lb>that its Semidiameter is greater than ſome of thoſe Lines that may <lb></lb>be drawn from the ſaid Point K unto the ſaid Line A B G D, and <lb></lb>leſſer than ſome other. </s> <s>Let the Circumference of the deſcribed <lb></lb>Circle be R B G H, and from B to K draw the Right-Line B K: and <lb></lb>drawn alſo the two Lines K R, and K E L which make a Right<lb></lb>Angle in the Point K: and upon the Center K deſcribe the Circum<lb></lb>ference X O P in the Plane and in the Liquid. </s> <s>The parts, there<lb></lb>fore, of the Liquid that are ^{*} according to the Circumference <lb></lb><arrow.to.target n="marg1131"></arrow.to.target><lb></lb>X O P, for the reaſons alledged upon the firſt <emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> are equi<lb></lb>jacent, or equipoſited, and contiguous to each other; and both <lb></lb>theſe parts are preſt or thruſt, according to the ſecond part of the <lb></lb><emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> by the Liquor which is above them. </s> <s>And becauſe the <lb></lb>two Angles E K B and B K R are ſuppoſed equal [<emph type="italics"></emph>by the<emph.end type="italics"></emph.end> 26. <emph type="italics"></emph>of<emph.end type="italics"></emph.end> 3. <lb></lb><emph type="italics"></emph>of Euclid,<emph.end type="italics"></emph.end>] the two Circumferences or Arches B E and B R ſhall <lb></lb>be equal (foraſmuch as R B G H was a Circle deſcribed for ſatis<lb></lb>faction of the Oponent, and K its Center:) And in like manner <lb></lb>the whole Triangle B E K ſhall be equal to the whole Triangle <lb></lb>B R K. </s> <s>And becauſe alſo the Triangle O P K for the ſame reaſon <pb xlink:href="073/01/009.jpg" pagenum="338"></pb>ſhall be equal to the Triangle O X K; Therefore (by common <lb></lb>Notion) ſubſtracting thoſe two ſmall Triangles O P K and O X K <lb></lb>from the two others B E K and B R K, the two Remainders ſhall <lb></lb>be equal: one of which Remainders ſhall be the Quadrangle <lb></lb>B E O P, and the other B R X O. </s> <s>And becauſe the whole Quadran<lb></lb>gle B E O P is full of Liquor, and of the Quadrangle B R X O, <lb></lb>the part B A X O only is full, and the reſidue B R A is wholly void <lb></lb>of Water: It followeth, therefore, that the Quadrangle B E O P <lb></lb>is more ponderous than the Quadrangle B R X O. </s> <s>And if the ſaid <lb></lb>Quadrangle B E O P be more Grave than the Quadrangle <lb></lb>B R X O, much more ſhall the Quadrangle B L O P exceed in Gra<lb></lb>vity the ſaid Quadrangle B R X O: whence it followeth, that the <lb></lb>part O P is more preſſed than the part O X. But, by the firſt part <lb></lb>of the Suppoſition, the part leſs preſſed ſhould be repulſed by the <lb></lb>part more preſſed: Therefore the part O X muſt be repulſed by <lb></lb>the part O P: But it was preſuppoſed that the Liquid did not <lb></lb>move: Wherefore it would follow that the leſs preſſed would not <lb></lb>be repulſed by the more preſſed: And therefore it followeth of <lb></lb>neceſſity that the Line A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G D is the Circumference of a Circle, <lb></lb>and that the Center of it is the point K. </s> <s>And in like manner ſhall <lb></lb>it be demonſtrated, if the Surface of the Liquid be cut by a Plane <lb></lb>thorow the Center of the Earth, that the Section ſhall be the Cir<lb></lb>cumference of a Circle, and that the Center of the ſame ſhall be <lb></lb>that very Point which is Center of the Earth. </s> <s>It is therefore mani<lb></lb>feſt that the Superficies of a Liquid that is conſiſtant and ſetled <lb></lb>ſhall have the Figure of a Sphære, the Center of which ſhall be <lb></lb>the ſame with that of the Earth, by the firſt <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end>; for it is <lb></lb>ſuch that being ever cut thorow the ſame Point, the Section or Di<lb></lb>viſion deſcribes the Circumference of a Circle which hath for Cen<lb></lb>ter the ſelf-ſame Point that is Center of the Earth: Which was to <lb></lb>be demonſtrated.</s></p><p type="margin"> <s><margin.target id="marg1130"></margin.target>* O: through.</s></p><p type="margin"> <s><margin.target id="marg1131"></margin.target>* <emph type="italics"></emph>i.e.<emph.end type="italics"></emph.end> Parallel.</s></p><p type="main"> <s>RIC. </s> <s>I do thorowly underſtand theſe your Reaſons, and ſince there is in them <lb></lb>no umbrage of Doubting, let us proceed to his third <emph type="italics"></emph>Propoſition.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROP. III. THEOR. III.</s></p><p type="main"> <s><emph type="italics"></emph>Solid Magnitudes that being of equal Maſs with the <lb></lb>Liquid are alſo equal to it in Gravity, being demit-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1132"></arrow.to.target><lb></lb><emph type="italics"></emph>ted into the [^{*} ſetled] Liquid do ſo ſubmerge in the <lb></lb>ſame as that they lie or appear not at all above the <lb></lb>Surface of the Liquid, nor yet do they ſink to the <lb></lb>Bottom.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/010.jpg" pagenum="339"></pb><p type="margin"> <s><margin.target id="marg1132"></margin.target>* I add the word <lb></lb>ſetled, as neceſſary <lb></lb>in making the Ex<lb></lb>periment.</s></p><p type="main"> <s>NIC. </s> <s>In this <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> it is affirmed that thoſe Solid Magnitules that hap<lb></lb>pen to be equal in ſpecifical Gravity with the Liquid being lefeat liber<lb></lb>ty in the ſaid Liquid do ſo ſubmerge in the ſame, as that they lie or ap<lb></lb>pear not at all above the Surface of the Liquid, nor yet do they go or ſink to the <lb></lb>Bottom.</s></p><p type="main"> <s>For ſuppoſing, on the contrary, that it were poſſible for one of <lb></lb>thoſe Solids being placed in the Liquid to lie in part without the <lb></lb>Liquid, that is above its Surface, (alwaies provided that the ſaid <lb></lb>Liquid be ſetled and undiſturbed,) let us imagine any Plane pro<lb></lb>duced thorow the Center of the Earth, thorow the Liquid, and <lb></lb>thorow that Solid Body: and let us imagine that the Section of the <lb></lb>Liquid is the Superficies A B G D, and the Section of the Solid <lb></lb>Body that is within it the Superſicies E Z H T, and let us ſuppoſe <lb></lb>the Center of the Earth to be the Point K: and let the part of the <lb></lb>ſaid Solid ſubmerged in the Liquid be B G H T, and let that above <lb></lb>be B E Z G: and let the Solid Body be ſuppoſed to be comprized in <lb></lb>a Pyramid that hath its Parallelogram Baſe in the upper Surface of <lb></lb>the Liquid, and its Summity or Vertex in the Center of the Earth: <lb></lb>which Pyramid let us alſo ſuppoſe to be cut or divided by the ſame <lb></lb>Plane in which is the Circumference A B G D, and let the Sections <lb></lb><figure id="id.073.01.010.1.jpg" xlink:href="073/01/010/1.jpg"></figure><lb></lb>of the Planes of the ſaid <lb></lb>Pyramid be K L and <lb></lb>K M: and in the Liquid <lb></lb>about the Center K let <lb></lb>there be deſcribed a Su<lb></lb>perficies of another <lb></lb>Sphære below E Z H T, <lb></lb>which let be X O P; <lb></lb>and let this be cut by <lb></lb>the Superficies of the Plane: And let there be another Pyramid ta<lb></lb>ken or ſuppoſed equal and like to that which compriſeth the ſaid <lb></lb>Solid Body, and contiguous and conjunct with the ſame; and let <lb></lb>the Sections of its Superficies be K M and K N: and let us ſuppoſe <lb></lb>another Solid to be taken or imagined, of Liquor, contained in that <lb></lb>ſame Pyramid, which let be R S C Y, equal and like to the partial <lb></lb>Solid B H G T, which is immerged in the ſaid Liquid: But the <lb></lb>part of the Liquid which in the firſt Pyramid is under the Super<lb></lb>ficies X O, and that, which in the other Pyramid is under the Su<lb></lb>perficies O P, are equijacent or equipoſited and contiguous, but <lb></lb>are not preſſed equally; for that which is under the Superficies <lb></lb>X O is preſſed by the Solid T H E Z, and by the Liquor that is <lb></lb>contained between the two Spherical Superficies X O and L M <lb></lb>and the Planes of the Pyramid, but that which proceeds accord<lb></lb>ing to F O is preſſed by the Solid R S C Y, and by the Liquid <pb xlink:href="073/01/011.jpg" pagenum="340"></pb>contained between the Sphærical Superficies that proceed accord<lb></lb>ing to P O and M N and the Planes of the Pyramid; and the Gra<lb></lb>vity of the Liquid, which is according to M N O P, ſhall be leſſer <lb></lb>than that which is according to L M X O; becauſe that Solid of <lb></lb>Liquor which proceeds according to R S C Y is leſs than the Solid <lb></lb>E Z H T (having been ſuppoſed to be equal in quantity to only <lb></lb>the part H B G T of that:) And the ſaid Solid E Z H T hath been <lb></lb>ſuppoſed to be equally grave with the Liquid: Therefore the Gra<lb></lb>vity of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid compriſed betwixt the two Sphærical Superfi<lb></lb>cies L M and <emph type="italics"></emph>X<emph.end type="italics"></emph.end> O, and betwixt the ſides L <emph type="italics"></emph>X<emph.end type="italics"></emph.end> and M O of the <lb></lb><figure id="id.073.01.011.1.jpg" xlink:href="073/01/011/1.jpg"></figure><lb></lb>Pyramid, together with <lb></lb>the whole Solid EZHT, <lb></lb>ſhall exceed the Gravity <lb></lb>of the Liquid compri<lb></lb>ſed betwixt the other <lb></lb>two Sphærical Superfi<lb></lb>cies M N and O P, and <lb></lb>the Sides M O and N P <lb></lb>of the Pyramid, toge<lb></lb>ther with the Solid of Liquor R S C Y by the quantity of the Gra<lb></lb>vity of the part E B Z G, ſuppoſed to remain above the Surface of <lb></lb>the Liquid: And therefore it is manifeſt that the part which pro<lb></lb>ceedeth according to the Circumference O P is preſſed, driven, and <lb></lb>repulſed, according to the <emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> by that which proceeds ac<lb></lb>cording to the Circumference X O, by which means the Liquid <lb></lb>would not be ſetled and ſtill: But we did preſuppoſe that it was <lb></lb>ſetled, namely ſo, as to be without motion: It followeth, therefore, <lb></lb>that the ſaid Solid cannot in any part of it exceed or lie above the <lb></lb>Superficies of the Liquid: And alſo that being dimerged in the Li<lb></lb>quid it cannot deſcend to the Bottom, for that all the parts of the <lb></lb>Liquid equijacent, or diſpoſed equally, are equally preſſed, becauſe <lb></lb>the Solid is equally grave with the Liquid, by what we preſuppoſed.</s></p><p type="main"> <s>RIC. </s> <s>I do underſtand your Argumentation, but I underſtand not that Phraſe <lb></lb><emph type="italics"></emph>Solid Magnitudes.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>I will declare this Term unto you. <emph type="italics"></emph>Magnitude<emph.end type="italics"></emph.end> is a general Word that <lb></lb>reſpecteth all the Species of Continual Quantity; and the Species of Continual <lb></lb>Quantity are three, that is, the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine, the Superficies, and the Body; which Body <lb></lb>is alſo called a Solid, as having in it ſelf <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ength, Breadth, and Thickneſs, or Depth: <lb></lb>and therefore that none might equivocate or take that Term <emph type="italics"></emph>Magnitudes<emph.end type="italics"></emph.end> to be <lb></lb>meant of <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines, or Superficies, but only of Solid <emph type="italics"></emph>Magnitudes,<emph.end type="italics"></emph.end> that is, Bodies, he <lb></lb>did ſpecifie it by that manner of expreſſion, as was ſaid. </s> <s>The truth is, that he <lb></lb>might have expreſt that <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> in this manner: <emph type="italics"></emph>Solids (or Bodies) which being <lb></lb>of equal Gravity with an equal Maſs of the Liquid,<emph.end type="italics"></emph.end> &c. </s> <s>And this <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> would have <lb></lb>been more cleer and intelligible, for it is as ſignificant to ſay, a <emph type="italics"></emph>Solid,<emph.end type="italics"></emph.end> or, a <emph type="italics"></emph>Body,<emph.end type="italics"></emph.end> as <lb></lb>to ſay, a <emph type="italics"></emph>Solid Magnitude:<emph.end type="italics"></emph.end> therefore wonder not if for the future I uſe theſe three <lb></lb>kinds of words indifferently.</s></p><p type="main"> <s>RIC. </s> <s>You have ſufficiently ſatisfied me, wherefore that we may loſe no time <lb></lb>let us go forwards to the fourth <emph type="italics"></emph>Propoſition.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/012.jpg" pagenum="341"></pb><p type="head"> <s>PROP. IV. THEOR. IV.</s></p><p type="main"> <s><emph type="italics"></emph>Solid Magnitudes that are lighter than the Liquid, <lb></lb>being demitted into the ſetled Liquid, will not total<lb></lb>ly ſubmerge in the ſame, but ſome part thereof will <lb></lb>lie or ſtay above the Surface of the Liquid.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>In this fourth <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> it is concluded, that every Body or Solid that is <lb></lb>lighter (as to Specifical Gravity) than the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, being put into the <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, will not totally ſubmerge in the ſame, but that ſome part of it <lb></lb>will ſtay and appear without the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, that is above its Surface.</s></p><p type="main"> <s>For ſuppoſing, on the contrary, that it were poſſible for a Solid <lb></lb>more light than the Liquid, being demitted in the Liquid to ſub<lb></lb>merge totally in the ſame, that is, ſo as that no part thereof re<lb></lb>maineth above, or without the ſaid Liquid, (evermore ſuppoſing <lb></lb>that the Liquid be ſo conſtituted as that it be not moved,) let us <lb></lb>imagine any Plane produced thorow the Center of the Earth, tho<lb></lb>row the Liquid, and thorow that Solid Body: and that the Surface <lb></lb>of the Liquid is cut by this Plane according to the Circumference <lb></lb>A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G, and the Solid <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ody according to the Figure R; and let the <lb></lb>Center of the Earth be K. </s> <s>And let there be imagined a Pyramid <lb></lb><figure id="id.073.01.012.1.jpg" xlink:href="073/01/012/1.jpg"></figure><lb></lb>that compriſeth the Figure <lb></lb>R, as was done in the pre. <lb></lb></s> <s>cedent, that hath its Ver<lb></lb>tex in the Point K, and let <lb></lb>the Superficies of that <lb></lb>Pyramid be cut by the <lb></lb>Superficies of the Plane <lb></lb>A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G, according to A K <lb></lb>and K <emph type="italics"></emph>B<emph.end type="italics"></emph.end>. </s> <s>And let us ima<lb></lb>gine another Pyramid equal and like to this, and let its Superficies <lb></lb>be cut by the Superficies A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> G according to K <emph type="italics"></emph>B<emph.end type="italics"></emph.end> and K <emph type="italics"></emph>G<emph.end type="italics"></emph.end>; and let <lb></lb>the Superficies of another Sphære be deſcribed in the Liquid, upon <lb></lb>the Center K, and beneath the Solid R; and let that be cut by the <lb></lb>ſame Plane according to <emph type="italics"></emph>X<emph.end type="italics"></emph.end> O P. And, laſtly, let us ſuppoſe ano<lb></lb>ther Solid taken ^{*} from the Liquid, in this ſecond Pyramid, which <lb></lb><arrow.to.target n="marg1133"></arrow.to.target><lb></lb>let be H, equal to the Solid R. </s> <s>Now the parts of the Liquid, name<lb></lb>ly, that which is under the Spherical Superficies that proceeds ac<lb></lb>cording to the Superficies or Circumference <emph type="italics"></emph>X<emph.end type="italics"></emph.end> O, in the firſt Py<lb></lb>ramid, and that which is under the Spherical Superficies that pro<lb></lb>ceeds according to the Circumference O P, in the ſecond Pyramid, <lb></lb>are equijacent, and contiguous, but are not preſſed equally; for <pb xlink:href="073/01/013.jpg" pagenum="342"></pb>that of the firſt Pyramid is preſſed by the Solid R, and by the Liquid <lb></lb>which that containeth, that is, that which is in the place of the Py<lb></lb>ramid according to A B O X: but that part which, in the other Py<lb></lb>ramid, is preſſed by the Solid H, ſuppoſed to be of the ſame Li<lb></lb>quid, and by the Liquid which that containeth, that is, that which <lb></lb>is in the place of the ſaid Pyramid according to P O B G: and the <lb></lb>Gravity of the Solid R is leſs than the Gravity of the Liquid <lb></lb>H, for that theſe two Magnitudes were ſuppoſed to be equal in <lb></lb>Maſs, and the Solid R was ſuppoſed to be lighter than the Liquid: <lb></lb>and the Maſſes of the two Pyramids of Liquor that containeth theſe <lb></lb><arrow.to.target n="marg1134"></arrow.to.target><lb></lb>two Solids R and H are equal ^{*} by what was preſuppoſed: There<lb></lb>fore the part of the Liquid that is under the Superficies that pro<lb></lb>ceeds according to the Circumference O P is more preſſed; and, <lb></lb>therefore, by the <emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> it ſhall repulſe that part which is leſs <lb></lb>preſſed, whereby the ſaid Liquid will not be ſetled: But it was be<lb></lb>fore ſuppoſed that it was ſetled: Therefore that Solid R ſhall not <lb></lb>totally ſubmerge, but ſome part thereof will remain without the <lb></lb>Liquid, that is, above its Surface, Which was the <emph type="italics"></emph>Propoſition.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1133"></margin.target>* That is a Maſs of <lb></lb>the Liquid.</s></p><p type="margin"> <s><margin.target id="marg1134"></margin.target>* For that the Py<lb></lb>ramids were ſuppo<lb></lb>ſed equal.</s></p><p type="main"> <s>RIC. </s> <s>I have very well underſtood you, therefore let us come to the fifth <emph type="italics"></emph>Pro<lb></lb>poſition,<emph.end type="italics"></emph.end> which, as you know, doth thus ſpeak.</s></p><p type="head"> <s>PROP. V. THEOR. V.</s></p><p type="main"> <s><emph type="italics"></emph>Solid Magnitudes that are lighter than the Liquid, <lb></lb>being demitted in the (ſetled) Liquid, will ſo far <lb></lb>ſubmerge, till that a Maſs of Liquor, equal to the <lb></lb>Part ſubmerged, doth in Gravity equalize the <lb></lb>whole Magnitude.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>It having, in the precedent, been demonſtrared that Solids lighter than <lb></lb>the Liquid, being demitted in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, alwaies a part of them remains <lb></lb>without the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, that is above its Surface; In this fifth <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> it is <lb></lb>aſſerted, that ſo much of ſuch a Solid ſhall ſubmerge, as that a Maſs of the <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid equal to the part ſubmerged, ſhall have equal Gravity with the whole <lb></lb>Solid.</s></p><p type="main"> <s>And to demonſtrate this, let us aſſume all the ſame Schemes <lb></lb>as before, in <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> 3. and likewiſe let the Liquid be ſet<lb></lb>led, and let the Solid E Z H T be lighter than the Liquid. <lb></lb></s> <s>Now if the ſaid Liquid be ſetled, the parts of it that are equija<lb></lb>cent are equally preſſed: Therefore the Liquid that is beneath <pb xlink:href="073/01/014.jpg" pagenum="343"></pb>the Superficies that proceed according to the Circumferences X O <lb></lb>and P O are equally preſſed; whereby the Gravity preſſed is equal. <lb></lb><figure id="id.073.01.014.1.jpg" xlink:href="073/01/014/1.jpg"></figure><lb></lb>But the Gravity of the <lb></lb>Liquid which is in the <lb></lb><arrow.to.target n="marg1135"></arrow.to.target><lb></lb>firſt Pyramid ^{*} without <lb></lb>the Solid B H T G, is <lb></lb>equal to the Gravity of <lb></lb>the Liquid which is in <lb></lb>the other Pyramid with<lb></lb>out the Liquid R S C Y: <lb></lb>It is manifeſt, therefore, <lb></lb>that the Gravity of the Solid E Z H T, is equal to the Gravity of <lb></lb>the Liquid R S C Y: Therefore it is manifeſt that a Maſs of Liquor <lb></lb>equal in Maſs to the part of the Solid ſubmerged is equal in Gra<lb></lb>vity to the whole Solid.</s></p><p type="margin"> <s><margin.target id="marg1135"></margin.target>* <emph type="italics"></emph>Without, i.e.<emph.end type="italics"></emph.end> that <lb></lb>being deducted.</s></p><p type="main"> <s>RIC. </s> <s>This was a pretty Demonſtration, and becauſe I very well underſtand <lb></lb>it, let us loſe no time, but proceed to the ſixth <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> ſpeaking thus.</s></p><p type="head"> <s>PROP. VI. THEOR. VI.</s></p><p type="main"> <s><emph type="italics"></emph>Solid Magnitudes lighter than the Liquid being thruſt <lb></lb>into the Liquid, are repulſed upwards with a Force <lb></lb>as great as is the exceſs of the Gravity of a Maſs <lb></lb>of Liquor equal to the Magnitude above the Gra<lb></lb>vity of the ſaid Magnitude.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>This ſixth <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> ſaith, that the Solids lighter than the Liquid <lb></lb>demitted, thruſt, or trodden by Force underneath the Liquids Sur<lb></lb>face, are returned or driven upwards with ſo much Force, by <lb></lb>how much a quantity of the Liquid equal to the. </s> <s>Solid ſhall <lb></lb>exceed the ſaid Solid in Gravity.</s></p><p type="main"> <s>And to delucidate this <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> let the Solid A be lighter <lb></lb>than the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, and let us ſuppoſe that the Gravity of the ſaid <lb></lb>Solid A is B: and let the Gravity of a <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, equal in Maſs to A, <lb></lb>be B G. </s> <s>I ſay, that the Solid A depreſſed or demitted with Force <lb></lb>into the ſaid <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, ſhall be returned and repulſed upwards with <lb></lb>a Force equal to the Gravity G. </s> <s>And to demonſtrate this <emph type="italics"></emph>Propo<lb></lb>ſition,<emph.end type="italics"></emph.end> take the Solid D, equal in Gravity to the ſaid G. </s> <s>Now <lb></lb>the Solid compounded of the two Solids A and D will be lighter <lb></lb>than the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid: for the Gravity of the Solid compounded of <lb></lb>them both is BG, and the Gravity of as much Liquor as equal<lb></lb>leth in greatneſs the Solid A, is greater than the ſaid Gravity BG, <pb xlink:href="073/01/015.jpg" pagenum="344"></pb>for that B G is the Gravity of the Liquid equal in Maſs unto it: <lb></lb>Therefore the Solid compounded of thoſe two Solids A and D <lb></lb>being dimerged, it ſhall, by the precedent, ſo much of it ſubmerge, <lb></lb>as that a quantity of the Liquid equal to the ſaid ſubmerged part <lb></lb>ſhall have equal Gravity with the ſaid compounded Solid. </s> <s>And <lb></lb><figure id="id.073.01.015.1.jpg" xlink:href="073/01/015/1.jpg"></figure><lb></lb>for an example of that <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> let the Su<lb></lb>perficies of any Liquid be that which pro<lb></lb>ceedeth according to the Circumference <lb></lb>A B G D: Becauſe now a Maſs or quantity <lb></lb>of Liquor as big as the Maſs A hath equal <lb></lb>Gravity with the whole compounded Solid <lb></lb>A D: It is manifeſt that the ſubmerged part <lb></lb>thereof ſhall be the Maſs A: and the remain<lb></lb>der, namely, the part D, ſhall be wholly a<lb></lb>top, that is, above the Surface of the Liquid. <lb></lb></s> <s>It is therefore evident, that the part A hath ſo much virtue or <lb></lb>Force to return upwards, that is, to riſe from below above the Li<lb></lb>quid, as that which is upon it, to wit, the part D, hath to preſs it <lb></lb>downwards, for that neither part is repulſed by the other: But D <lb></lb>preſſeth downwards with a Gravity equal to G, it having been ſup<lb></lb>poſed that the Gravity of that part D was equal to G: Therefore <lb></lb>that is manifeſt which was to be demonſtrated.</s></p><p type="main"> <s>RIC. </s> <s>This was a fine Demonſtration, and from this I perceive that you colle<lb></lb>cted your <emph type="italics"></emph>Induſtrious Invention<emph.end type="italics"></emph.end>; and eſpecially that part of it which you inſert in <lb></lb>the firſt Book for the recovering of a Ship ſunk: and, indeed, I have many Que<lb></lb>ſtions to ask you about that, but I will not now interrupt the Diſcourſe in hand, but <lb></lb>deſire that we may go on to the ſeventh <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> the purport whereof is this.</s></p><p type="head"> <s>PROP. VII. THEOR. VII.</s></p><p type="main"> <s><emph type="italics"></emph>Solid Magnitudes beavier than the Liquid, being de<lb></lb>mitted into the [ſetled] Liquid, are boren down<lb></lb>wards as far as they can deſcend: and ſhall be lighter <lb></lb>in the Liquid by the Gravity of a Liquid Maſs of <lb></lb>the ſame bigneſs with the Solid Magnitude.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>This ſeventh <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> hath two parts to be demonſtrated.</s></p><p type="main"> <s>The firſt is, That all Solids heavier than the Liquid, being demit<lb></lb>ted into the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, are boren by their Gravities downwards as far <lb></lb>as they can deſcend, that is untill they arrive at the Bottom. </s> <s>Which <lb></lb>firſt part is manifeſt, becauſe the Parts of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, which ſtill lie <lb></lb>under that Solid, are more preſſed than the others equijacent, <lb></lb>becauſe that that Solid is ſuppoſed more grave than the Liquid. <pb xlink:href="073/01/016.jpg" pagenum="345"></pb>But now that that Solid is lighter in the Liquid than out of it, as <lb></lb>is affirmed in the ſecond part, ſhall be demonſtrated in this man<lb></lb>ner. </s> <s>Take a Solid, as ſuppoſe A, that is more grave than the Li<lb></lb>quid, and ſuppoſe the Gravity of that ſame Solid A to be BG. <lb></lb></s> <s>And of a Maſs of <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquor of the ſame bigneſs with the Solid A, ſup<lb></lb>poſe the Gravity to be B: It is to be demonſtrated that the Solid <lb></lb>A, immerged in the Liquid, ſhall have a Gravity equal to G. </s> <s>And <lb></lb>to demonſtrate this, let us imagine another Solid, as ſuppoſe D, <lb></lb>more light than the Liquid, but of ſuch a quality as that its Gravi<lb></lb>ty is equal to B: and let this D be of ſuch a Magnitude, that a <lb></lb>Maſs of <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquor equal to it hath its Gravity equal to the Gravity <lb></lb>B G. </s> <s>Now theſe two Solids D and A being compounded toge<lb></lb>ther, all that Solid compounded of theſe two ſhall be equally <lb></lb>Grave with the Water: becauſe the Gravity of theſe two Solids <lb></lb>together ſhall be equal to theſe two Gravities, that is, to B G, and <lb></lb><figure id="id.073.01.016.1.jpg" xlink:href="073/01/016/1.jpg"></figure><lb></lb>to B; and the Gravity of a Liquid that hath its <lb></lb>Maſs equal to theſe two Solids A and D, ſhall be <lb></lb>equal to theſe two Gravities B G and B. <emph type="italics"></emph>L<emph.end type="italics"></emph.end>et <lb></lb>theſe two Solids, therefore, be put in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, <lb></lb><arrow.to.target n="marg1136"></arrow.to.target><lb></lb>and they ſhall ^{*} remain in the Surface of that <emph type="italics"></emph>L<emph.end type="italics"></emph.end>i<lb></lb>quid, (that is, they ſhall not be drawn or driven <lb></lb>upwards, nor yet downwards:) For if the Solid <lb></lb>A be more grave than the Liquid, it ſhall be <lb></lb>drawn or born by its Gravity downwards to<lb></lb>wards the Bottom, with as much Force as by the Solid D it is thruſt <lb></lb>upwards: And becauſe the Solid D is lighter than the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, it <lb></lb>ſhall raiſe it upward with a Force as great as the Gravity G: Be<lb></lb>cauſe it hath been demonſtrated, in the ſixth <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> That So<lb></lb>lid Magnitudes that are lighter than the Water, being demitted in <lb></lb>the ſame, are repulſed or driven upwards with a Force ſo much the <lb></lb>greater by how much a <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid of equal Maſs with the Solid is more <lb></lb>Grave than the ſaid Solid: But the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid which is equal in Maſs <lb></lb>with the Solid D, is more grave than the ſaid Solid D, by the Gra<lb></lb>vity G: Therefore it is manifeſt, that the Solid A is preſſed or <lb></lb>born downwards towards the Centre of the World, with a Force <lb></lb>as great as the Gravity G: Which was to be demonſtrated.</s></p><p type="margin"> <s><margin.target id="marg1136"></margin.target>* Or, according to <lb></lb><emph type="italics"></emph>Commandine,<emph.end type="italics"></emph.end> ſhall <lb></lb>be equall in Gravi<lb></lb>ty to the Liquid, <lb></lb>neither moving up<lb></lb>wards or down<lb></lb>wards.</s></p><p type="main"> <s>RIC. </s> <s>This hath been an ingenuous Demonſtration; and in regard I do ſuffici<lb></lb>ently underſtand it, that we may loſe no time, we will proceed to the ſecond <emph type="italics"></emph>Suppo<lb></lb>ſition,<emph.end type="italics"></emph.end> which, as I need not tell you, ſpeaks thus.</s></p> <pb xlink:href="073/01/017.jpg" pagenum="346"></pb><p type="head"> <s>SVPPOSITION II.</s></p><p type="main"> <s><emph type="italics"></emph>It is ſuppoſed that thoſe Solids which are moved up<lb></lb>wards, do all aſcend according to the Perpendicular <lb></lb>which is produced thorow their Centre of Gravity.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"></emph>And thoſe which are moved downwards, deſcend, likewiſe, according to the Perpendicular <lb></lb>that is produced thorow their Centre of Gravity, which he pretermitted either as known, <lb></lb>or as to be collected from what went before.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>For underſtanding of this ſecond <emph type="italics"></emph>Suppoſition,<emph.end type="italics"></emph.end> it is requiſite to take notice <lb></lb>that every Solid that is lighter than the Liquid being by violence, or by ſome other <lb></lb>occaſion, ſubmerged in the Liquid, and then left at liberty, it ſhall, by that which <lb></lb>hath been proved in the ſixth <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> be thruſt or born up wards by the Liquid, <lb></lb>and that impulſe or thruſting is ſuppoſed to be directly according to the Perpendi<lb></lb>cular that is produced thorow the Centre of Gravity of that Solid; which Per<lb></lb>pendicular, if you well remember, is that which is drawn in the Imagination <lb></lb>from the Centre of the World, or of the Earth, unto the Centre of Gravity of <lb></lb>that Body, or Solid.</s></p><p type="main"> <s>RIC. </s> <s>How may one find the Centre of Gravity of a Solid?</s></p><p type="main"> <s>NIC. </s> <s>This he ſheweth in that Book, intituled <emph type="italics"></emph>De Centris Gravium, vel de Æqui<lb></lb>ponderantibus<emph.end type="italics"></emph.end>; and therefore repair thither and you ſhall be ſatisfied, for to declare <lb></lb>it to you in this place would cauſe very great confuſion.</s></p><p type="main"> <s>RIC. </s> <s>I underſtand you: ſome other time we will talk of this, becauſe I have <lb></lb>a mind at preſent to proceed to the laſt <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> the Expoſition of which ſeemeth <lb></lb>to me very confuſed, and, as I conceive, the Author hath not therein ſhewn all <lb></lb>the Subject of that <emph type="italics"></emph>Propoſition<emph.end type="italics"></emph.end> in general, but only a part: which Propoſition <lb></lb>ſpeaketh, as you know, in this form.</s></p><p type="head"> <s>PROP. VIII. THEOR. VIII.<lb></lb><arrow.to.target n="marg1137"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1137"></margin.target>A</s></p><p type="main"> <s><emph type="italics"></emph>If any Solid Magnitude, lighter than the Liquid, that <lb></lb>hath the Figure of a Portion of a Sphære, ſhall be<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1138"></arrow.to.target><lb></lb><emph type="italics"></emph>demitted into the Liquid in ſuch a manner as that <lb></lb>the Baſe of the Portion touch not the Liquid, the <lb></lb>Figure ſhall ſtand erectly, ſo, as that the Axis of <lb></lb>the ſaid Portion ſhall be according to the Perpen<lb></lb>dicular. </s> <s>And if the Figure ſhall be inclined to any <lb></lb>ſide, ſo, as that the Baſe of the Portion touch the <lb></lb>Liquid, it ſhall not continue ſo inclined as it was de<lb></lb>mitted, but ſhall return to its uprightneſs.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/018.jpg" pagenum="347"></pb><p type="margin"> <s><margin.target id="marg1138"></margin.target>B</s></p><p type="main"> <s>For the declaration of this <emph type="italics"></emph>Propoſition,<emph.end type="italics"></emph.end> let a Solid Magnitude <lb></lb>that hath the Figure of a portion of a Sphære, as hath been ſaid, <lb></lb>be imagined to be de<lb></lb><figure id="id.073.01.018.1.jpg" xlink:href="073/01/018/1.jpg"></figure><lb></lb>mitted into the Liquid; and <lb></lb>alſo, let a Plain be ſuppoſed <lb></lb>to be produced thorow the <lb></lb>Axis of that portion, and <lb></lb>thorow the Center of the <lb></lb>Earth: and let the Section <lb></lb>of the Surface of the Liquid <lb></lb>be the Circumference A B <lb></lb>C D, and of the Figure, the <lb></lb>Circumference E F H, & let <lb></lb>E H be a right line, and F T <lb></lb>the Axis of the Portion. </s> <s>If now <lb></lb>it were poſſible, for ſatisfact<lb></lb>ion of the Adverſary, Let <lb></lb>it be ſuppoſed that the ſaid Axis were not according to the <emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> Per<lb></lb><arrow.to.target n="marg1139"></arrow.to.target><lb></lb>pendicular; we are then to demonſtrate, that the Figure will not <lb></lb>continue as it was conſtituted by the Adverſary, but that it will re<lb></lb>turn, as hath been ſaid, unto its former poſition, that is, that the <lb></lb>Axis F T ſhall be according to the Perpendicular. </s> <s>It is manifeſt, by <lb></lb>the <emph type="italics"></emph>Corollary<emph.end type="italics"></emph.end> of the 1. of 3. <emph type="italics"></emph>Euclide,<emph.end type="italics"></emph.end> that the Center of the Sphære <lb></lb>is in the Line F T, foraſmuch as that is the Axis of that Figure. <lb></lb></s> <s>And in regard that the Por<lb></lb><figure id="id.073.01.018.2.jpg" xlink:href="073/01/018/2.jpg"></figure><lb></lb>tion of a Sphære, may be <lb></lb>greater or leſſer than an He<lb></lb>miſphære, and may alſo be <lb></lb>an Hemiſphære, let the Cen<lb></lb>tre of the Sphære, in the He<lb></lb>miſphære, be the Point T, <lb></lb>and in the leſſer Portion the <lb></lb>Point P, and in the greater, <lb></lb>the Point K, and let the Cen<lb></lb>tre of the Earth be the Point <lb></lb>L. </s> <s>And ſpeaking, firſt, of <lb></lb>that greater Portion which <lb></lb>hath its Baſe out of, or a<lb></lb>bove, the Liquid, thorew the Points K and L, draw the Line KL <lb></lb>cutting the Circumference E F H in the Point N, Now, becauſe <lb></lb><arrow.to.target n="marg1140"></arrow.to.target><lb></lb>every Portion of a Sphære, hath its Axis in the Line, that from the <lb></lb>Centre of the Sphære is drawn perpendicular unto its Baſe, and hath <lb></lb>its Centre of Gravity in the Axis; therefore that Portion of the Fi<lb></lb>gure which is within the Liquid, which is compounded of two Por <pb xlink:href="073/01/019.jpg" pagenum="348"></pb>tions of a Sphære, ſhall have its Axis in the Perpendicular, that is <lb></lb>drawn through the point K; and its Centre of Gravity, for the ſame <lb></lb>reaſon, ſhall be in the Line N K: let us ſuppoſe it to be the Point R: <lb></lb><arrow.to.target n="marg1141"></arrow.to.target><lb></lb>But the Centre of Gravity of the whole Portion is in the Line F T, <lb></lb>betwixt the Point R and <lb></lb><figure id="id.073.01.019.1.jpg" xlink:href="073/01/019/1.jpg"></figure><lb></lb>the Point F; let us ſuppoſe <lb></lb>it to be the Point <emph type="italics"></emph>X<emph.end type="italics"></emph.end>: The re<lb></lb>mainder, therefore, of that <lb></lb><arrow.to.target n="marg1142"></arrow.to.target><lb></lb>Figure elivated above the <lb></lb>Surface of the Liquid, hath <lb></lb>its Centre of Gravity in <lb></lb>the Line R X produced or <lb></lb>continued right out in the <lb></lb>Part towards X, taken ſo, <lb></lb>that the part prolonged may <lb></lb>have the ſame proportion to <lb></lb>X R, that the Gravity of <lb></lb>that Portion that is demer<lb></lb>ged in the Liquid hath to <lb></lb>the Gravity of that Figure which is above the Liquid; let us ſuppoſe <lb></lb><arrow.to.target n="marg1143"></arrow.to.target><lb></lb>that ^{*} that Centre of the ſaid Figure be the Point S: and thorow that <lb></lb><arrow.to.target n="marg1144"></arrow.to.target><lb></lb>ſame Centre S draw the Perpendicular L S. </s> <s>Now the Gravity of the Fi<lb></lb>gure that is above the Liquid ſhall preſſe from above downwards ac<lb></lb>cording to the Perpendicular S L; & the Gravity of the Portion that <lb></lb>is ſubmerged in the Liquid, ſhall preſſe from below upwards, accor<lb></lb>ding to the Perpendicular R L. </s> <s>Therefore that Figure will not conti<lb></lb>nue according to our Adverſaries Propoſall, but thoſe parts of the <lb></lb>ſaid Figure which are towards E, ſhall be born or drawn downwards, <lb></lb>& thoſe which are towards H ſhall be born or driven upwards, and <lb></lb>this ſhall be ſo long untill that the Axis F T comes to be according <lb></lb>to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1139"></margin.target>(a) <emph type="italics"></emph>Perpendicular <lb></lb>is taken kere, as <lb></lb>in all other places, <lb></lb>by this Author for <lb></lb>the Line K L <lb></lb>drawn thorow the <lb></lb>Centre and Cir<lb></lb>cumference of the <lb></lb>Earth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1140"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1141"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1142"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1143"></margin.target>* <emph type="italics"></emph>i. </s> <s>e,<emph.end type="italics"></emph.end> The Center <lb></lb>of Gravity.</s></p><p type="margin"> <s><margin.target id="marg1144"></margin.target>F</s></p><p type="main"> <s>And this ſame Demonſtration is in the ſame manner verified in <lb></lb>the other <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions. </s> <s>As, firſt, in the Hæmiſphere that lieth with its <lb></lb>whole Baſe above or without the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, the Centre of the Sphære <lb></lb>hath been ſuppoſed to be the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint T; and therefore, imagining T <lb></lb>to be in the place, in which, in the other above mentioned, the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint R was, arguing in all things elſe as you did in that, you ſhall <lb></lb>find that the Figure which is above the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid ſhall preſs from <lb></lb>above downwards according to the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular S <emph type="italics"></emph>L<emph.end type="italics"></emph.end>; and the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion that is ſubmerged in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid ſhall preſs from below up<lb></lb>wards according to the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular R <emph type="italics"></emph>L.<emph.end type="italics"></emph.end> And therefore it ſhall <lb></lb>follow, as in the other, namely, that the parts of the whole Figure <lb></lb>which are towards E, ſhall be born or preſſed downwards, and thoſe <lb></lb><arrow.to.target n="marg1145"></arrow.to.target><lb></lb>that are towards H, ſhall be born or driven upwards: and this ſhall <lb></lb>be ſo long untill that the Axis F T come to ſtand ^{*} <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular <pb xlink:href="073/01/020.jpg" pagenum="349"></pb>ly. </s> <s>The like ſhall alſo hold true in the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion of the Sphære <lb></lb>leſs than an Hemiſphere that lieth with its whole Baſe above the <lb></lb>Liquid.</s></p><p type="margin"> <s><margin.target id="marg1145"></margin.target>* Or according <lb></lb>to the Perpendi<lb></lb>cular.</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"></emph>The Demonſtration of this Propoſition is defaced by the Injury of Time, which we have re<lb></lb>ſtored, ſo far as by the Figures that remain, one may collect the Meaning of<emph.end type="italics"></emph.end> Archimedes, <lb></lb><emph type="italics"></emph>for we thought it not good to alter them: and what was wanting to their declaration and ex<lb></lb>planation we have ſupplyed in our Commentaries, as we have alſo determined to do in the ſe<lb></lb>cond Propoſition of the ſecond Book.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>If any Solid Magnitude lighter than the Liquid.] <emph type="italics"></emph>Theſe words, light-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1146"></arrow.to.target><lb></lb><emph type="italics"></emph>er than the Liquid, are added by us, and are not to be found in the Tranſiation; for of theſe <lb></lb>kind of Magnitudes doth<emph.end type="italics"></emph.end> Archimedes <emph type="italics"></emph>ſpeak in this Propoſition.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1146"></margin.target>A</s></p><p type="main"> <s>Shall be demitted into the Liquid in ſuch a manner as that the <lb></lb><arrow.to.target n="marg1147"></arrow.to.target><lb></lb>Baſe of the Portion touch not the Liquid.] <emph type="italics"></emph>That is, ſhall be ſo demitted into <lb></lb>the Liquid as that the Baſe ſhall be upwards, and the<emph.end type="italics"></emph.end> Vertex <emph type="italics"></emph>downwards, which he oppoſeth <lb></lb>to that which he ſaith in the Propoſition following<emph.end type="italics"></emph.end>; Be demitted into the Liquid, ſo, as <lb></lb>that its Baſe be wholly within the Liquid; <emph type="italics"></emph>For theſe words ſignifie the Portion demit<lb></lb>ted the contrary way, as namely, with the<emph.end type="italics"></emph.end> Vertex <emph type="italics"></emph>upwards and the Baſe downwards. </s> <s>The <lb></lb>ſame manner of ſpeech is frequently uſed in the ſecond Book; which treateth of the Portions <lb></lb>of Rectangle Conoids.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1147"></margin.target>B</s></p><p type="main"> <s>Now becauſe every Portion of a Sphære hath its Axis in the Line <lb></lb><arrow.to.target n="marg1148"></arrow.to.target><lb></lb>that from the Center of the Sphære is drawn perpendicular to its <lb></lb>Baſe.] <emph type="italics"></emph>For draw a Line from B to C, and let K L cut the Circumference A B C D in the <lb></lb>Point G, and the Right Line B C in M<emph.end type="italics"></emph.end>: <lb></lb><figure id="id.073.01.020.1.jpg" xlink:href="073/01/020/1.jpg"></figure><lb></lb><emph type="italics"></emph>and becauſe the two Circles A B C D, and <lb></lb>E F H do cut one another in the Points <lb></lb>B and C, the Right Line that conjoyneth <lb></lb>their Centers, namely, K L, doth cut the <lb></lb>Line B C in two equall parts, and at <lb></lb>Right Angles; as in our Commentaries <lb></lb>upon<emph.end type="italics"></emph.end> Prolomeys <emph type="italics"></emph>Planiſphære we do <lb></lb>prove: But of the Portion of the Circle <lb></lb>B N C the Diameter is M N; and of the <lb></lb>Portion B G C the Diameter is M G;<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1149"></arrow.to.target><lb></lb><emph type="italics"></emph>for the<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>Right Lines which are drawn <lb></lb>on both ſides parallel to B C do make<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1150"></arrow.to.target><lb></lb><emph type="italics"></emph>Right Angles with N G; and<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>for <lb></lb>that cauſe are thereby cut in two equall <lb></lb>parts: Therefore the Axis of the Portion <lb></lb>of the Sphære B N C is N M; and the <lb></lb>Axis of the Portion B G C is M G: <lb></lb>from whence it followeth that the Axis of <lb></lb>the Portion demerged in the Liquid is <lb></lb>in the Line K L, namely N G. </s> <s>And ſince the Center of Gravity of any Portion of a Sphære is <lb></lb>in the Axis, as we have demonstrated in our Book<emph.end type="italics"></emph.end> De Centro Gravitatis Solidorum, <emph type="italics"></emph>the <lb></lb>Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is, <lb></lb>of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra<lb></lb>vity of thoſe Portions of Sphæres. </s> <s>For ſuppoſe, if poſſible, that it be out of the Line N G, as <lb></lb>in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V <expan abbr="q.">que</expan> Becauſe <lb></lb>therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha<lb></lb>ving the ſame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the <lb></lb>Portion B G C ſhall, by the 8 of the firſt Book of<emph.end type="italics"></emph.end> Archimedes, De Centro Gravitatis <pb xlink:href="073/01/021.jpg" pagenum="350"></pb>Planotum, <emph type="italics"></emph>be in the Line V Q prolonged: But that is impoſſible; for it is in the Axis <lb></lb>G: It followeth, therefore, that the Center of Gravity of the Portion demerged in <lb></lb>Liquid be in the Line N K: which we propounded to be proved.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1148"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1149"></margin.target><emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> By 29. of the <lb></lb>firſt of <emph type="italics"></emph>Encl.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1150"></margin.target><emph type="italics"></emph>(b)<emph.end type="italics"></emph.end> By 3. of the <lb></lb>third.</s></p><p type="main"> <s>But the Centre of Gravity of the whole Portion is in the Line <lb></lb><arrow.to.target n="marg1151"></arrow.to.target><lb></lb>T, betwixt the Point R and the Point F; let us ſuppoſe it to be<lb></lb>the Point X.] <emph type="italics"></emph>Let the Sphære becompleated, ſo as that there be added of that Portion<lb></lb>the Axis T Y, and the Center of Gravity Z. </s> <s>And becauſe that from the whole Sphære,<lb></lb>whoſe Centre of Gravity is K, as we have alſo demonſtrated in the (c) Book before named, the <lb></lb>is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1152"></arrow.to.target><lb></lb><emph type="italics"></emph>of the Portion E F H ſhall be in the Line Z K prolonged: And therefore it muſt of neceſſity<lb></lb>fall betwixt K and F.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1153"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1151"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1152"></margin.target>(c) <emph type="italics"></emph>By 8 of the <lb></lb>firſt<emph.end type="italics"></emph.end> of Archimedes.</s></p><p type="margin"> <s><margin.target id="marg1153"></margin.target>E</s></p><p type="main"> <s>The remainder, therefore, of the Figure, elevated above the Sur<lb></lb>face of the Liquid, hath its Center of Gravity in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine R X<lb></lb>prolonged.] <emph type="italics"></emph>By the ſame 8 of the firſt Book of<emph.end type="italics"></emph.end> Archimedes, de Centro Gravita<lb></lb>tis Planorum.</s></p><p type="main"> <s>Now the Gravity of the Figure that is above the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid ſhall<arrow.to.target n="marg1154"></arrow.to.target><lb></lb>preſs from above downwards according to S L; and the Gravit <lb></lb>of the Portion that is ſubmerged in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid ſhall preſs from be <lb></lb>low upwards, according to the Perpendicular R L.] <emph type="italics"></emph>By the ſecond Sup<lb></lb>poſition of this. </s> <s>For the Magnitude that is demerged in the Liquid is moved upwards with as<lb></lb>much Force along R L, as that which is above the Liquid is moved downwards along S L; as<lb></lb>may be ſhewn by Propoſition 6. of this. </s> <s>And becauſe they are moved along ſeverall other Lines,<lb></lb>neither cauſeth the others being leſs moved; the which it continually doth when the Portion<lb></lb>is ſet according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes<lb></lb>do concur in one and the ſame Perpendicular, namely, in the Axis of the Portion: and look<lb></lb>with what force or<emph.end type="italics"></emph.end> Impetus <emph type="italics"></emph>that which is in the Lipuid tendeth upwards, and with the like<lb></lb>doth that which is above or without the Liquid tend downwards along the ſame Line: And<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1155"></arrow.to.target><lb></lb><emph type="italics"></emph>therefore, in regard that the one doth not ^{*} exceed the other, the Portion ſhall no longer move <lb></lb>but ſhall ſtay and reſt allwayes in one and the ſame Poſition, unleſs ſome extrinſick Cauſe<lb></lb>chance to intervene.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1154"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1155"></margin.target>* <emph type="italics"></emph>Or overcome.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROP. IX. THEOR. IX.<lb></lb><arrow.to.target n="marg1156"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1156"></margin.target>* In ſome Greek <lb></lb>Coppies this is no <lb></lb>diſtinct Propoſi<lb></lb>tion, but all <lb></lb>Commentators, <lb></lb>do divide it <lb></lb>from the Prece<lb></lb>dent, as having a <lb></lb>diſtinct demon<lb></lb>ſtration in the <lb></lb>Originall.</s></p><p type="main"> <s>^{*} <emph type="italics"></emph>But if the Figure, lighter than the Liquid, be demit<lb></lb>ted into the Liquid, ſo, as that its Baſe be wholly<lb></lb>within the ſaid Liquid, it ſhall continue in ſuch <lb></lb>manner erect, as that its Axis ſhall ſtand according <lb></lb>to the Perpendicular.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For ſuppoſe, ſuch a Magnitude as that aforenamed to be de <lb></lb>mitted into the Liquid; and imagine a Plane to be produced<lb></lb>thorow the Axis of the Portion, and thorow the Center of the <lb></lb>Earth: And let the <emph type="italics"></emph>S<emph.end type="italics"></emph.end>ection of the Surface of the Liquid, be the Cir<lb></lb>cumference A B C D, and of the Figure the Circumference E F <emph type="italics"></emph>H<emph.end type="italics"></emph.end><lb></lb>And let E H be a Right Line, and F T the Axis of the Portion. </s> <s>If<lb></lb>now it were poſſible, for ſatisfaction of the Adverſary, let it be <lb></lb>ſuppoſed that the ſaid Axis were not according to the Perpendicu<lb></lb>lar: we are now to demonſtrate that the Figure will not ſo conti <pb xlink:href="073/01/022.jpg" pagenum="351"></pb>nue, but will return to be according to the <lb></lb><figure id="id.073.01.022.1.jpg" xlink:href="073/01/022/1.jpg"></figure><lb></lb>Perpendieular. </s> <s>It is manifeſt that the Gen<lb></lb>tre of the Sphære is in the Line F T. </s> <s>And <lb></lb>again, foraſmuch as the Portion of a Sphære <lb></lb>may be greater or leſſer than an Hemiſ<lb></lb>phære, and may alſo be an Hemiſphære, let <lb></lb>the Centre of the Sphære in the Hemiſ<lb></lb>phære be the Point T, & in the leſſer Por<lb></lb>tion the Point P, and in the Greater the </s></p><p type="main"> <s><arrow.to.target n="marg1157"></arrow.to.target><lb></lb>Point R. </s> <s>And ſpeaking firſt of that greater <lb></lb>Portion which hath its Baſe within the <lb></lb>Liquid, thorow R and L, the Earths Cen<lb></lb><figure id="id.073.01.022.2.jpg" xlink:href="073/01/022/2.jpg"></figure><lb></lb>tre, draw the line RL. </s> <s>The Portion that is <lb></lb>above the Liquid, hath its Axis in the Per<lb></lb>pendicular paſſing thorow R; and by <lb></lb>what hath been ſaid before, its Centre of <lb></lb>Gravity ſhall be in the Line N R; let it <lb></lb>be the Point R: But the Centre of Gra<lb></lb>vity of the whole Portion is in the line F <lb></lb>T, betwixt R and F; let it be X: The re<lb></lb>mainder therefore of that Figure, which is <lb></lb>within the Liquid ſhall have its Centre in <lb></lb>the Right Line R <emph type="italics"></emph>X<emph.end type="italics"></emph.end> prolonged in the part <lb></lb><figure id="id.073.01.022.3.jpg" xlink:href="073/01/022/3.jpg"></figure><lb></lb>towards <emph type="italics"></emph>X,<emph.end type="italics"></emph.end> taken ſo, that the part pro<lb></lb>longed may have the ſame Proportion to <lb></lb>X R, that the Gravity of the Portion that <lb></lb>is above the Liquid hath to the Gravity <lb></lb>of the Figure that is within the Liquid. <lb></lb></s> <s>Let O be the Centre of that ſame Figure: <lb></lb>and thorow O draw the Perpendicular L <lb></lb>O. </s> <s>Now the Gravity of the Portion that <lb></lb>is above the Liquid ſhall preſs according <lb></lb>to the Right Line R L downwards; and <lb></lb>the Gravity of the Figure that is in the <lb></lb>Liquid according to the Right Line O L upwards: There the Figure <lb></lb>ſhall not continue; but the parts of it towards H ſhall move down<lb></lb>wards, and thoſe towards E upwards: & <lb></lb><figure id="id.073.01.022.4.jpg" xlink:href="073/01/022/4.jpg"></figure><lb></lb>this ſhall ever be, ſo long as F T is accord<lb></lb>ing to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1157"></margin.target>A</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s>The Portion that is above the Liquid <lb></lb><arrow.to.target n="marg1158"></arrow.to.target><lb></lb>hath its Axis in the Perpendicular paſſing <lb></lb>thorow K.] <emph type="italics"></emph>For draw B C cutting the Line N K in <lb></lb>M; and let N K out the Circumference<emph.end type="italics"></emph.end> A B <emph type="italics"></emph>C D in G. </s> <s>In <lb></lb>the ſame manner as before me will demonſtrate, that the Axis<emph.end type="italics"></emph.end> <pb xlink:href="073/01/023.jpg" pagenum="352"></pb><emph type="italics"></emph>of the Portion of the Sphære is N M; and of the Portion B G C the Axis is G M: Wherefore <lb></lb>the Centre of Gravity of them both ſhall be in the Line N M: And becauſe that from the Por<lb></lb>tion B N C the Portion B G C, not having the ſame Centre of Gravity, is cut off, the Centre <lb></lb>of Gravity of the remainder of the Magnitude that is above the Surface of the Liquid ſhall be <lb></lb>in the Line N K; namely, in the Line which conjoyneth the Centres of Gravity of the ſaid <lb></lb>Portions by the foreſaid 8 of<emph.end type="italics"></emph.end> Archimedis de Centro Gravitatis Planorum.</s></p><p type="margin"> <s><margin.target id="marg1158"></margin.target>A</s></p><p type="main"> <s>NIC. </s> <s>Truth is, that in ſome of theſe Figures C is put for X, and ſo it was in <lb></lb>the Greek Copy that I followed.</s></p><p type="main"> <s>RIC. </s> <s>This Demoſtration is very difficult, to my thinking; but I believe that <lb></lb>it is becauſe I have not in memory the Propoſitions of that Book entituled <emph type="italics"></emph>De Cen<lb></lb>tris Gravium.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>NIC. </s> <s>It is ſo.</s></p><p type="main"> <s>RIC. </s> <s>We will take a more convenient time to diſcourſe of that, and now return <lb></lb><arrow.to.target n="marg1159"></arrow.to.target><lb></lb>to ſpeak of the two laſt Propoſitions. </s> <s>And I ſay that the Figures incerted in the <lb></lb>demonſtration would in my opinion, have been better and more intelligble unto <lb></lb>me, drawing the Axis according to its proper Poſition; that is in the half Arch of <lb></lb>theſe Figures, and then, to ſecond the Objection of the Adverſary, to ſuppoſe <lb></lb>that the ſaid Figures ſtood ſomewhat Obliquely, to the end that the ſaid Axis, if it <lb></lb>were poſſible, did not ſtand according to the Perpendicular ſo often mentioned, <lb></lb>which doing, the Propoſition would be proved in the ſame manner as before: <lb></lb>and this way would be more naturall and clear.<lb></lb><arrow.to.target n="marg1160"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1159"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1160"></margin.target>B</s></p><p type="main"> <s>NIC. </s> <s>You are in the right, but becauſe thus they were in the Greek Copy, <lb></lb>I thought not fit to alter them, although unto the better.</s></p><p type="main"> <s>RIC. Companion, you have thorowly ſatisfied me in all that in the beginning <lb></lb>of our Diſcourſe I asked of you, to morrow, God permitting, we will treat of <lb></lb>ſome other ingenious Novelties.</s></p><p type="head"> <s>THE TRANSLATOR.</s></p><p type="main"> <s>I ſay that the Figures, &c. </s> <s>would have been more intelligible to </s></p><p type="main"> <s><arrow.to.target n="marg1161"></arrow.to.target><lb></lb>me, drawing the Axis Z T according to its proper Poſition, that <lb></lb>is in the half Arch of theſe Figures.] <emph type="italics"></emph>And in this conſideration I have followed <lb></lb>the Schemes of<emph.end type="italics"></emph.end> Commandine, <emph type="italics"></emph>who being the Reſtorer of the Demonſtrations of theſe two laſt <lb></lb>Propoſitions, hath well conſidered what<emph.end type="italics"></emph.end> Ricardo <emph type="italics"></emph>here propoſeth, and therefore hath drawn the <lb></lb>ſaid Axis (which in the Manuſcripts that he had by him is lettered F T, and not as in that of<emph.end type="italics"></emph.end><lb></lb>Tartaylia <emph type="italics"></emph>Z T,) according to that its proper Poſition.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1161"></margin.target>A</s></p><p type="main"> <s>But becauſe thus they were in the Greek Copy, I thought not <lb></lb><arrow.to.target n="marg1162"></arrow.to.target><lb></lb>fit to alter them although unto the better.] <emph type="italics"></emph>The Schemes of thoſe Manu-<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.023.1.jpg" xlink:href="073/01/023/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſcripts that<emph.end type="italics"></emph.end> Tartaylia <emph type="italics"></emph>had ſeen were more imperfect then thoſe <lb></lb>in Commandines Copies; but for variety ſake, take here one <lb></lb>of<emph.end type="italics"></emph.end> Tartaylia, <emph type="italics"></emph>it being that of the Portion of a Sphære, equall <lb></lb>to an Hemiſphære, with its Axis oblique, and its Baſe dimitted <lb></lb>into the Liquid, and Lettered as in this Edition.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1162"></margin.target>B</s></p><p type="main"> <s><emph type="italics"></emph>Now Courteous Readers, I hope that you may, amidſt the <lb></lb>great Obſcurity of the Originall in the Demonſtrations of theſe <lb></lb>two laſt Propoſitions, be able from the joynt light of theſe two Famous Commentators of our <lb></lb>more famous Author, to diſcern the truth of the Doctrine affirmed, namely, That Solids of the <lb></lb>Figure of Portions of Sphæres demitted into the Liquid with their Baſes upwards ſhall ſtand <lb></lb>erectly, that is, with their Axis according to the Perpendicular drawn from the Centre of the <lb></lb>Earth unto its Circumference: And that if the ſaid Portions be demitted with their Baſes <lb></lb>oblique and touching the Liquid in one Point, they ſhall not rest in that Obliquity, but ſhall <lb></lb>return to Rectitude: And that laſtly, if theſe Portions be demitted with their Baſes downwards, <lb></lb>they ſhall continue erect with their Axis according to the Perpendicular aforeſaid: ſo that no <lb></lb>more remains to be done, but that weſet before you the 2 Books of this our Admirable Author.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/024.jpg" pagenum="353"></pb><p type="head"> <s>ARCHIMEDES, <lb></lb>HIS TRACT <lb></lb><emph type="italics"></emph>DE <lb></lb>INSIDENTIBUS HUMIDO,<emph.end type="italics"></emph.end><lb></lb>OR, <lb></lb>Of the NATATION of BODIES Upon, or <lb></lb>Submerſion In the WATER, or other LIQUIDS.</s></p><p type="head"> <s><emph type="italics"></emph>BOOK<emph.end type="italics"></emph.end> II.</s></p><p type="head"> <s>PROP. I. THEOR. I.</s></p><p type="main"> <s><emph type="italics"></emph>If any Magnitude lighter than the Liquid be demitted <lb></lb>into the ſaid Liquid, it ſhall have the ſame proporti<lb></lb>on in Gravity to a Liquid of equal Maſſe, that the <lb></lb>part of the Magnitude demerged hath unto the <lb></lb>whole Magnitude.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For let any Solid Magnitude, as for in<lb></lb>ſtance F A, lighter than the Liquid, be de<lb></lb>merged in the Liquid, which let be F A: <lb></lb>And let the part thereof immerged be A, <lb></lb>and the part above the Liquid F, It is to <lb></lb>be demonſtrated that the Magnitude F A <lb></lb>hath the ſame proportion in Gravity to a <lb></lb>Liquid of Equall Maſſe that A hath to F <lb></lb>A. </s> <s>Take any Liquid Magnitude, as ſup<lb></lb>poſe N I, of equall Maſſe with F A; and let F be equall to N, and <lb></lb>A to I: and let the Gravity of the whole Magnitude F A be B, and <lb></lb>let that of the Magnitude N I be O, <lb></lb>and let that of I be R. </s> <s>Now the <lb></lb><figure id="id.073.01.024.1.jpg" xlink:href="073/01/024/1.jpg"></figure><lb></lb>Magnitude F A hath the ſame pro<lb></lb>portion unto N I that the Gravity B <lb></lb>hath to the Gravity O R: But for <lb></lb>aſmuch as the Magnitude F A demit<lb></lb>ted into the Liquid is lighter than <lb></lb>the ſaid Liquid, it is manifeſt that a Maſſe of the Liquid, I, equall <lb></lb>to the part of the Magnitude demerged, A, hath equall Gravity <lb></lb><arrow.to.target n="marg1163"></arrow.to.target><lb></lb>with the whole Magnitnde, F A: For this was <emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> above demon<lb></lb>ſtrated: But B is the Gravity of the Magnitude F A, and R of I: <pb xlink:href="073/01/025.jpg" pagenum="354"></pb>Therefore B and R are equall. </s> <s>And becauſe that of the Magni<lb></lb>tude FA the <emph type="italics"></emph>G<emph.end type="italics"></emph.end>ravity is B: Therefore of the Liquid Body <emph type="italics"></emph>N<emph.end type="italics"></emph.end> I the <lb></lb>Gravity is O R. </s> <s>As F A is to N I, ſo is B to O R, or, ſo is R to <lb></lb>O R: But as R is to O R, ſo is I to N I, and A to F A: Therefore <lb></lb><arrow.to.target n="marg1164"></arrow.to.target><lb></lb>I is to N I, as F A to N I: And as I to N I ſo is <emph type="italics"></emph>(b)<emph.end type="italics"></emph.end> A to F A. <lb></lb></s> <s>Therefore F A is to N I, as A is to F A: Which was to be demon<lb></lb>ſtrated.</s></p><p type="margin"> <s><margin.target id="marg1163"></margin.target>(a) <emph type="italics"></emph>By 5. of the <lb></lb>firſt of this.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1164"></margin.target>(b) <emph type="italics"></emph>By 11. of the <lb></lb>fifth of<emph.end type="italics"></emph.end> Eucl.</s></p><p type="head"> <s>PROP. II. THEOR. II.<lb></lb><arrow.to.target n="marg1165"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1165"></margin.target>A</s></p><p type="main"> <s>^{*} <emph type="italics"></emph>The Right Portion of a Right angled Conoide, when it <lb></lb>ſhall have its Axis leſſe than<emph.end type="italics"></emph.end> ſeſquialter ejus quæ ad <lb></lb>Axem (<emph type="italics"></emph>or of its<emph.end type="italics"></emph.end> Semi-parameter) <emph type="italics"></emph>having any what <lb></lb>ever proportion to the Liquid in Gravity, being de<lb></lb>mitted into the Liquid ſo as that its Baſe touch not <lb></lb>the ſaid Liquid, and being ſet ſtooping, it ſhall not <lb></lb>remain ſtooping, but ſhall be restored to uprightneſſe. <lb></lb></s> <s>I ſay that the ſaid Portion ſhall ſtand upright when <lb></lb>the Plane that cuts it ſhall be parallel unto the Sur<lb></lb>face of the Liquid.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be a Portion of a Rightangled Conoid, as hath been <lb></lb>ſaid; and let it lye ſtooping or inclining: It is to be demon<lb></lb>ſtrated that it will not ſo continue but ſhall be reſtored to re<lb></lb>ctitude. </s> <s>For let it be cut through the Axis by a plane erect upon <lb></lb>the Surface of the Liquid, and let the Section of the Portion be <lb></lb>A PO L, the Section of a Rightangled Cone, and let the Axis <lb></lb><figure id="id.073.01.025.1.jpg" xlink:href="073/01/025/1.jpg"></figure><lb></lb>of the Portion and Diameter of the <lb></lb>Section be N O: And let the Sect<lb></lb>ion of the Surface of the Liquid be <lb></lb>I S. </s> <s>If now the Portion be not <lb></lb>erect, then neither ſhall A L be Pa<lb></lb>rallel to I S: Wherefore N O will <lb></lb>not be at Right Angles with I S. </s></p><p type="main"> <s><arrow.to.target n="marg1166"></arrow.to.target><lb></lb>Draw therefore K <foreign lang="grc">ω,</foreign> touching the Section of the Cone I, in the <lb></lb>Point P [that is parallel to I S: and from the Point P unto I S <lb></lb><arrow.to.target n="marg1167"></arrow.to.target><lb></lb>draw P F parallel unto O N, ^{*} which ſhall be the Diameter of the <lb></lb>Section I P O S, and the Axis of the Portion demerged in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>i<lb></lb><arrow.to.target n="marg1168"></arrow.to.target><lb></lb>quid. </s> <s>In the next place take the Centres of Gravity: ^{*} and of <lb></lb>the Solid Magnitude A P O L, let the Centre of Gravity be R; and <lb></lb><arrow.to.target n="marg1169"></arrow.to.target><lb></lb>of I P O S let the Centre be B: ^{*} and draw a Line from B to R <lb></lb>prolonged unto G; which let be the Centre of Gravity of the <pb xlink:href="073/01/026.jpg" pagenum="355"></pb>remaining Figure I S L A. </s> <s>Becauſe now that N O is <emph type="italics"></emph>Seſquialter<emph.end type="italics"></emph.end><lb></lb>of R O, but leſs than <emph type="italics"></emph>Seſquialter ejus quæ uſque ad Axem<emph.end type="italics"></emph.end> (or of its <lb></lb><emph type="italics"></emph>Semi-parameter<emph.end type="italics"></emph.end>;) ^{*} R O ſhall be leſſe than <emph type="italics"></emph>quæ uſque ad Axem<emph.end type="italics"></emph.end> (or <lb></lb><arrow.to.target n="marg1170"></arrow.to.target><lb></lb>than the <emph type="italics"></emph>Semi-parameter<emph.end type="italics"></emph.end>;) ^{*} whereupon the Angle R P <foreign lang="grc">ω</foreign> ſhall be <lb></lb><arrow.to.target n="marg1171"></arrow.to.target><lb></lb>acute. </s> <s>For ſince the Line <emph type="italics"></emph>quæ uſque ad Axem<emph.end type="italics"></emph.end> (or <emph type="italics"></emph>Semi-parameter<emph.end type="italics"></emph.end>) <lb></lb>is greater than R O, that Line which is drawn from the Point R, <lb></lb>and perpendicular to K <foreign lang="grc">ω,</foreign> namely RT, meeteth with the line F P <lb></lb>without the Section, and for that cauſe muſt of neceſſity fall be<lb></lb>tween the Points <emph type="italics"></emph>P<emph.end type="italics"></emph.end> and <foreign lang="grc">ω;</foreign> Therefore if <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines be drawn through <lb></lb>B and G, parallel unto R T, they ſhall contain Right Angles with <lb></lb>the Surface of the Liquid: ^{*} and the part that is within the Li<lb></lb><arrow.to.target n="marg1172"></arrow.to.target><lb></lb>quid ſhall move upwards according to the Perpendicular that is <lb></lb>drawn thorow B, parallel to R T, and the part that is above the Li<lb></lb>quid ſhall move downwards according to that which is drawn tho<lb></lb>row G; and the Solid A P O L ſhall not abide in this Poſition; for <lb></lb>that the parts towards A will move upwards, and thoſe towards <lb></lb>B downwards; Wherefore N O ſhall be conſtituted according to <lb></lb>the Perpendicular.]</s></p><p type="margin"> <s><margin.target id="marg1166"></margin.target>* <emph type="italics"></emph>Supplied by<emph.end type="italics"></emph.end> Fe<lb></lb>derico Comman<lb></lb>dino.</s></p><p type="margin"> <s><margin.target id="marg1167"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1168"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1169"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1170"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1171"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1172"></margin.target>G</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"></emph>The Demonſtration of this propoſition hath been much deſired; which we have (in like man<lb></lb>ner as the 8 Prop. </s> <s>of the firſt Book) reſtored according to<emph.end type="italics"></emph.end> Archimedes <emph type="italics"></emph>his own Schemes, and <lb></lb>illustrated it with Commentaries.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The Right Portion of a Rightangled Conoid, when it ſhall <lb></lb><arrow.to.target n="marg1173"></arrow.to.target><lb></lb>have its Axis leſſe than <emph type="italics"></emph>Seſquialter ejus quæ uſque ad Axem<emph.end type="italics"></emph.end> (or of <lb></lb>its <emph type="italics"></emph>Semi-parameter] In the Tranſlation of<emph.end type="italics"></emph.end> Nicolo Tartaglia <emph type="italics"></emph>it is falſlyread<emph.end type="italics"></emph.end> great<lb></lb>er then Seſquialter, <emph type="italics"></emph>and ſo its rendered in the following Propoſition; but it is the Right <lb></lb>Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay <lb></lb>that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the <lb></lb>Baſe, ſhall be parallel to the Surface of the Liquid.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1173"></margin.target>A</s></p><p type="main"> <s>Which ſhall be the Diameter of the Section I P O S, and the <lb></lb><arrow.to.target n="marg1174"></arrow.to.target><lb></lb>Axis of the Portion demerged in the Liquid.] <emph type="italics"></emph>By the 46 of the firſt of <lb></lb>the Conicks of<emph.end type="italics"></emph.end> Apollonious, <emph type="italics"></emph>or by the Corol<lb></lb>lary of the 51 of the ſame.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1174"></margin.target>B</s></p><figure id="id.073.01.026.1.jpg" xlink:href="073/01/026/1.jpg"></figure><p type="main"> <s>And of the Solid Magnitude A P <lb></lb><arrow.to.target n="marg1175"></arrow.to.target><lb></lb>O L, let the Centre of Gravity be R; <lb></lb>and of I P O S let the Centre be B.] <lb></lb><emph type="italics"></emph>For the Centre of Gravity of the Portion of a Right<lb></lb>angled Conoid is in its Axis, which it ſo divideth <lb></lb>as that the part thereof terminating in the vertex, <lb></lb>be double to the other part terminating in the Baſe; as <lb></lb>in our Book<emph.end type="italics"></emph.end> De Centro Gravitatis Solidorum Propo. </s> <s>29. <emph type="italics"></emph>we have demonſtrated. </s> <s>And <lb></lb>ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there<lb></lb>fore N O ſhall be Seſquialter of O R. </s> <s>And for the ſame reaſon, B the Centre of Gravity of the Por<lb></lb>tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1175"></margin.target>C</s></p><p type="main"> <s>And draw a Line from B to R prolonged unto G; which let <lb></lb><arrow.to.target n="marg1176"></arrow.to.target><lb></lb>be the Centre of Gravity of the remaining Eigure I S L A.] <pb xlink:href="073/01/027.jpg" pagenum="356"></pb><emph type="italics"></emph>For if, the Line B R being prolonged unto G, G R hath the ſame proportion to R B as the Por<lb></lb>tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the <lb></lb>Liquid, the Toine G ſhall be its Centre of Gravity; by the 8 of the ſecond of<emph.end type="italics"></emph.end> Archimedes <lb></lb>de Centro Gravitatis Planorum, vel de <emph type="italics"></emph>Æ<emph.end type="italics"></emph.end>quiponderantibus.<lb></lb><arrow.to.target n="marg1177"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1176"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1177"></margin.target>E</s></p><p type="main"> <s>R O ſhall be leſs than <emph type="italics"></emph>quæ uſque ad Axem<emph.end type="italics"></emph.end> (or than the Semi<lb></lb>parameter.] <emph type="italics"></emph>By the 10 Propofit. </s> <s>of<emph.end type="italics"></emph.end> Euclids <emph type="italics"></emph>fifth Book of Elements. </s> <s>The Line<emph.end type="italics"></emph.end> quæ <lb></lb>uſque ad Axem, <emph type="italics"></emph>(or the Semi-parameter) according to<emph.end type="italics"></emph.end> Archimedes, <emph type="italics"></emph>is the half of that<emph.end type="italics"></emph.end><lb></lb>juxta quam poſſunt, quæ á Sectione ducuntur, (<emph type="italics"></emph>or of the Parameter;) as appeareth <lb></lb>by the 4 Propoſit of his Book<emph.end type="italics"></emph.end> De Conoidibus & Shpæroidibus: <emph type="italics"></emph>and for what reaſon it is <lb></lb>ſo called, we have declared in the Commentaries upon him by us publiſhed.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1178"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1178"></margin.target>F</s></p><p type="main"> <s>Whereupon the Angle R P <foreign lang="grc">ω</foreign> ſhall be acute.] <emph type="italics"></emph>Let the Line N O be <lb></lb>continued out to H, that ſo RH may be equall to <lb></lb>the Semi-parameter. </s> <s>If now from the Point H<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.027.1.jpg" xlink:href="073/01/027/1.jpg"></figure><lb></lb><emph type="italics"></emph>a Line be drawn at Right Angles to N H, it ſhall <lb></lb>meet with FP without the Section; for being <lb></lb>drawn thorow O parallel to A L, it ſhall fall <lb></lb>without the Section, by the 17 of our ſirst Book of<emph.end type="italics"></emph.end><lb></lb>Conicks; <emph type="italics"></emph>Therefore let it meet in V: and <lb></lb>becauſe F P is parallel to the Diameter, and H <lb></lb>V perpendicular to the ſame Diameter, and R H <lb></lb>equall to the Semi-parameter, the Line drawn <lb></lb>from the Point R to V ſhall make Right Angles <lb></lb>with that Line which the Section toucheth in the Point P: that is with K<emph.end type="italics"></emph.end> <foreign lang="grc">ω,</foreign> <emph type="italics"></emph>as ſhall anon be <lb></lb>demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and<emph.end type="italics"></emph.end> <foreign lang="grc">ω;</foreign> <emph type="italics"></emph>and the Argle R<emph.end type="italics"></emph.end><lb></lb>P <foreign lang="grc">ω</foreign> <emph type="italics"></emph>ſhall be an Acute Angle.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let A B C be the Section of a Rightangled Cone, or a Parabola, <lb></lb>and its Diameter B D; and let the Line E F touch the <lb></lb>ſame in the Point G: and in the Diameter B D take the Line <lb></lb>H K equall to the Semi-parameter: and thorow G, G L be<lb></lb>ing drawn parallel to the Diameter, draw KM from the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint K at Right Angles to B D cutting G L in M: I ſay <lb></lb>that the Line prolonged thorow Hand Mis perpendicular to <lb></lb>E F, which it cutteth in N.</s></p><p type="main"> <s><emph type="italics"></emph>For from the Point G draw the Line G O at Right Angles to E F cutting the Diameter in <lb></lb>O: and again from the ſame Point draw G P perpendicular to the Diameter: and let the <lb></lb>ſaid Diameter prolonged cut the Line E F in <expan abbr="q.">que</expan> P B ſhall be equall to B Q, by the 35 of<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1179"></arrow.to.target><lb></lb><emph type="italics"></emph>our firſt Book of<emph.end type="italics"></emph.end> Conick <emph type="italics"></emph>Sections,<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>and G<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.027.2.jpg" xlink:href="073/01/027/2.jpg"></figure><lb></lb><emph type="italics"></emph>P a Mean-proportion all betmixt Q P and PO<emph.end type="italics"></emph.end>; <lb></lb><arrow.to.target n="marg1180"></arrow.to.target><lb></lb>(b) <emph type="italics"></emph>and therefore the Square of G P ſhall be e<lb></lb>quall to the Rectangle of O P Q: But it is alſo <lb></lb>equall to the Rectangle comprehended under P B <lb></lb>and the Line<emph.end type="italics"></emph.end> juxta quam poſſunt, <emph type="italics"></emph>or the Par<lb></lb>ameter, by the 11 of our firſt Book of<emph.end type="italics"></emph.end> Conicks: <lb></lb><arrow.to.target n="marg1181"></arrow.to.target><lb></lb>(c) <emph type="italics"></emph>Therefore, look what proportion Q P hath to <lb></lb>P B, and the ſame hath the Parameter unto P O: <lb></lb>But Q P is double unto<emph.end type="italics"></emph.end> P B, <emph type="italics"></emph>for that<emph.end type="italics"></emph.end> P B <emph type="italics"></emph>and B <lb></lb>Q are equall, as hath been ſaid: And therefore <lb></lb>the Parameter ſhall be double to the ſaid P O: <lb></lb>and by the ſame Reaſon P O is equall to that which we call the Semi-parameter, that is, to K H<emph.end type="italics"></emph.end>: <lb></lb><arrow.to.target n="marg1182"></arrow.to.target><lb></lb><emph type="italics"></emph>But<emph.end type="italics"></emph.end> (d) <emph type="italics"></emph>P G is equall to K M, and<emph.end type="italics"></emph.end> (e) <emph type="italics"></emph>the Angle O P G to the Angle H K M; for they are both<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1183"></arrow.to.target><lb></lb><emph type="italics"></emph>Right Angles: And therefore O G alſo is equall to H M, and the Angle P O G unto the<emph.end type="italics"></emph.end> <pb xlink:href="073/01/028.jpg" pagenum="357"></pb><figure id="id.073.01.028.1.jpg" xlink:href="073/01/028/1.jpg"></figure><lb></lb><emph type="italics"></emph>Angle K H M: Therefore<emph.end type="italics"></emph.end> (f) O G <emph type="italics"></emph>and H N are parallel,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1184"></arrow.to.target><lb></lb><emph type="italics"></emph>and the<emph.end type="italics"></emph.end> (g) <emph type="italics"></emph>Angle H N F equall to the Angle O G F; for <lb></lb>that G O being Perpendicular to E F, H N ſhall alſo be per-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1185"></arrow.to.target><lb></lb><emph type="italics"></emph>pandicnlar to the ſame: Which was to be demon ſtrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1179"></margin.target>(a) <emph type="italics"></emph>By Cor. </s> <s>of 8. of <lb></lb>6. of<emph.end type="italics"></emph.end> Euclide.</s></p><p type="margin"> <s><margin.target id="marg1180"></margin.target>(b) <emph type="italics"></emph>By 17. of the<emph.end type="italics"></emph.end><lb></lb>6.</s></p><p type="margin"> <s><margin.target id="marg1181"></margin.target>(c) <emph type="italics"></emph>By 14. of the<emph.end type="italics"></emph.end><lb></lb>6.</s></p><p type="margin"> <s><margin.target id="marg1182"></margin.target>(d) <emph type="italics"></emph>By 33. of the<emph.end type="italics"></emph.end><lb></lb>1.</s></p><p type="margin"> <s><margin.target id="marg1183"></margin.target>(e) <emph type="italics"></emph>By 4. of the<emph.end type="italics"></emph.end> 1.</s></p><p type="margin"> <s><margin.target id="marg1184"></margin.target>(f) <emph type="italics"></emph>By 28. of the<emph.end type="italics"></emph.end><lb></lb>1.</s></p><p type="margin"> <s><margin.target id="marg1185"></margin.target>(g) <emph type="italics"></emph>By 29. of th<emph.end type="italics"></emph.end><lb></lb>1</s></p><p type="main"> <s>And the part which is within the Liquid <lb></lb><arrow.to.target n="marg1186"></arrow.to.target><lb></lb>doth move upwards according to the Per<lb></lb>pendicular that is drawn thorow B parallel <lb></lb>to R T.] <emph type="italics"></emph>The reaſon why this moveth upwards, and that <lb></lb>other downwards, along the Perpendicular Line, hath been ſhewn above in the 8 of the firſt <lb></lb>Book of this; ſo that we have judged it needleſſe to repeat it either in this, or in the reſt <lb></lb>that follow.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1186"></margin.target>G</s></p><p type="head"> <s>THE TRANSLATOR.</s></p><p type="main"> <s><emph type="italics"></emph>In the<emph.end type="italics"></emph.end> Antient <emph type="italics"></emph>Parabola (namely that aſſumed in a Rightangled <lb></lb>Cone) the Line<emph.end type="italics"></emph.end> juxta quam Poſſunt quæ in Sectione ordinatim du<lb></lb>cuntur <emph type="italics"></emph>(which I, following<emph.end type="italics"></emph.end> Mydorgius, <emph type="italics"></emph>do call the<emph.end type="italics"></emph.end> Parameter<emph type="italics"></emph>) is<emph.end type="italics"></emph.end> (a) <lb></lb><arrow.to.target n="marg1187"></arrow.to.target><lb></lb><emph type="italics"></emph>double to that<emph.end type="italics"></emph.end> quæ ducta eſt à Vertice Sectionis uſque ad Axem, <emph type="italics"></emph>or in<emph.end type="italics"></emph.end><lb></lb>Archimedes <emph type="italics"></emph>phraſe,<emph.end type="italics"></emph.end> <foreign lang="grc">τᾱς υσ́χρι τοῡ ἄξον<34>;</foreign> <emph type="italics"></emph>which I for that cauſe, and <lb></lb>for want of a better word, name the<emph.end type="italics"></emph.end> Semiparameter: <emph type="italics"></emph>but in<emph.end type="italics"></emph.end> Modern <lb></lb><emph type="italics"></emph>Parabola's it is greater or leſſer then double. </s> <s>Now that throughout this <lb></lb>Book<emph.end type="italics"></emph.end> Archimedes <emph type="italics"></emph>ſpeaketh of the Parabola in a Rectangled Cone, is mani<lb></lb>feſt both by the firſt words of each Propoſition, & by this that no Parabola <lb></lb>hath its Parameter double to the Line<emph.end type="italics"></emph.end> quæ eſt a Sectione ad Axem, <emph type="italics"></emph>ſave <lb></lb>that which is taken in a Rightangled Cone. </s> <s>And in any other Parabola, for <lb></lb>the Line<emph.end type="italics"></emph.end> <foreign lang="grc">τᾱς μσ́χριτοῡ ἄεον<34></foreign> <emph type="italics"></emph>or<emph.end type="italics"></emph.end> quæ uſque ad Axem <emph type="italics"></emph>to uſurpe the Word<emph.end type="italics"></emph.end> Se<lb></lb>miparameter <emph type="italics"></emph>would be neither proper nor true: but in this caſe it may paſs<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1187"></margin.target>(a) Rîvalt. <emph type="italics"></emph>in<emph.end type="italics"></emph.end> Ar<lb></lb>chimed. <emph type="italics"></emph>de Cunoid <lb></lb>& Sphæroid.<emph.end type="italics"></emph.end> Prop. <lb></lb></s> <s>3. Lem. </s> <s>1.</s></p><p type="head"> <s>PROP. III. THEOR. III.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid, when it <lb></lb>ſhall have its Axis leſſe than ſeſquialter of the Se<lb></lb>mi-parameter, the Axis having any what ever pro<lb></lb>portion to the Liquid in Gravity, being demitted into <lb></lb>the Liquid ſo as that its Baſe be wholly within the <lb></lb>ſaid Liquid, and being ſet inclining, it ſhall not re<lb></lb>main inclined, but ſhall be ſo reſtored, as that its Ax<lb></lb>is do ſtand upright, or according to the Perpendicular.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let any Portion be demitted into the Liquid, as was ſaid; and <lb></lb>let its Baſe be in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid; <lb></lb><figure id="id.073.01.028.2.jpg" xlink:href="073/01/028/2.jpg"></figure><lb></lb>and let it be cut thorow the <lb></lb>Axis, by a Plain erect upon the Sur<lb></lb>face of the Liquid, and let the Se<lb></lb>ction be A P O <emph type="italics"></emph>L,<emph.end type="italics"></emph.end> the Section of a <lb></lb>Right angled Cone: and let the Axis <lb></lb>of the Portion and Diameter of the <pb xlink:href="073/01/029.jpg" pagenum="356"></pb>Section of the Portion be A P O L, the Section of a Rightangled <lb></lb>Cone; and let the Axis of the Portion and Diameter of the Section <lb></lb>be N O, and the Section of the Surface of the Liquid I S. </s> <s>If now <lb></lb>the Portion be not erect, then N O ſhall not be at equall Angles with <lb></lb>I S. </s> <s>Draw R <foreign lang="grc">ω</foreign> touching the Section of the Rightangled Conoid <lb></lb>in P, and parallel to I S: and from the Point P and parall to O N <lb></lb>draw <emph type="italics"></emph>P<emph.end type="italics"></emph.end> F: and take the Centers of Gravity; and of the Solid A <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end> O L let the Centre be R; and of that which lyeth within the <lb></lb>Liquid let the Centre be B; and draw a Line from B to R pro<lb></lb>longing it to G, that G may be the Centre of Gravity of the Solid <lb></lb>that is above the Liquid. </s> <s>And becauſe N O is ſeſquialter of R <lb></lb>O, and is greater than ſeſquialter of the Semi-Parameter; it is ma<lb></lb><arrow.to.target n="marg1188"></arrow.to.target><lb></lb>nifeſt that <emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> R O is greater than the <lb></lb><figure id="id.073.01.029.1.jpg" xlink:href="073/01/029/1.jpg"></figure><lb></lb>Semi-parameter. ^{*}Let therefore R <lb></lb><arrow.to.target n="marg1189"></arrow.to.target><lb></lb>H be equall to the Semi-Parameter, <lb></lb><arrow.to.target n="marg1190"></arrow.to.target><lb></lb>^{*} and O <emph type="italics"></emph>H<emph.end type="italics"></emph.end> double to H M. </s> <s>Foraſ<lb></lb>much therefore as N O is ſeſquialter <lb></lb><arrow.to.target n="marg1191"></arrow.to.target><lb></lb>of R O, and M O of O H, <emph type="italics"></emph>(b)<emph.end type="italics"></emph.end> the <lb></lb>Remainder N M ſhall be ſeſquialter <lb></lb>of the Remainder R H: Therefore <lb></lb>the Axis is greater than ſeſquialter <lb></lb>of the Semi parameter by the quan<lb></lb>tity of the Line M O. </s> <s>And let it be <lb></lb>ſuppoſed that the Portion hath not leſſe proportion in Gravity unto <lb></lb>the Liquid of equall Maſſe, than the Square that is made of the <lb></lb>Exceſſe by which the Axis is greater than ſeſquialter of the Semi<lb></lb>parameter hath to the Square made of the Axis: It is therefore ma<lb></lb>nifeſt that the Portion hath not leſſe proportion in Gravity to the <lb></lb>Liquid than the Square of the Line M O hath to the Square of N <lb></lb>O: But look what proportion the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion hath to the Liquid in <lb></lb>Gravity, the ſame hath the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion ſubmerged to the whole Solid: <lb></lb>for this hath been demonſtrated <emph type="italics"></emph>(c)<emph.end type="italics"></emph.end> above: ^{*}And look what pro<lb></lb><arrow.to.target n="marg1192"></arrow.to.target><lb></lb>portion the ſubmerged Portion hath to the whole <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion, the <lb></lb><arrow.to.target n="marg1193"></arrow.to.target><lb></lb>ſame hath the Square of <emph type="italics"></emph>P<emph.end type="italics"></emph.end> F unto the Square of N O: For it hath <lb></lb>been demonſtrated in <emph type="italics"></emph>(d) Lib. de Conoidibus,<emph.end type="italics"></emph.end> that if from a Right<lb></lb><arrow.to.target n="marg1194"></arrow.to.target><lb></lb>angled Conoid two <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions be cut by Planes in any faſhion pro<lb></lb>duced, theſe <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions ſhall have the ſame Proportion to each <lb></lb>other as the Squares of their Axes: The Square of P F, therefore, <lb></lb>hath not leſſe proportion to the Square of N O than the Square of <lb></lb>M O hath to the Square of N O: ^{*}Wherefore P F is not leſſe than <lb></lb><arrow.to.target n="marg1195"></arrow.to.target><lb></lb>M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn <lb></lb><arrow.to.target n="marg1196"></arrow.to.target><lb></lb>from H at Right Angles unto N O, it ſhall meet with B <emph type="italics"></emph>P,<emph.end type="italics"></emph.end> and ſhall <lb></lb><arrow.to.target n="marg1197"></arrow.to.target><lb></lb>fall betwixt B and P; let it fall in T: <emph type="italics"></emph>(e)<emph.end type="italics"></emph.end> And becauſe <emph type="italics"></emph>P<emph.end type="italics"></emph.end> F is <lb></lb><arrow.to.target n="marg1198"></arrow.to.target><lb></lb>parallel to the Diameter, and H T is perpendicular unto the ſame <lb></lb>Diameter, and R H equall to the Semi-parameter; a Line drawn <lb></lb>from R to T and prolonged, maketh Right Angles with the Line <pb xlink:href="073/01/030.jpg" pagenum="360"></pb>contingent unto the Section in the Point P: Wherefore it alſo <lb></lb>maketh Right Angles with the Surface of the Liquid: and that <lb></lb>part of the Conoidall Solid which is within the Liquid ſhall move <lb></lb>upwards according to the Perpendicular drawn thorow B parallel <lb></lb>to R T; and that part which is above the Liquid ſhall move down<lb></lb>wards according to that drawn thorow G, parallel to the ſaid R T: <lb></lb>And thus it ſhall continue to do ſo long untill that the Conoid be <lb></lb>reſtored to uprightneſſe, or to ſtand according to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1188"></margin.target>(a) <emph type="italics"></emph>By 10. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1189"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1190"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1191"></margin.target>(b) <emph type="italics"></emph>By 19. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1192"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1193"></margin.target>(c) <emph type="italics"></emph>By 1. of this <lb></lb>ſecond Book.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1194"></margin.target>(d) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> 6. De Co<lb></lb>noilibus & <emph type="italics"></emph>S<emph.end type="italics"></emph.end>phæ<lb></lb>roidibus <emph type="italics"></emph>of<emph.end type="italics"></emph.end> Archi<lb></lb>medes.</s></p><p type="margin"> <s><margin.target id="marg1195"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1196"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1197"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1198"></margin.target>(e) <emph type="italics"></emph>By 2. of this <lb></lb>ſecond Book.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>COMMANDINE.<lb></lb><arrow.to.target n="marg1199"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1199"></margin.target>A</s></p><p type="main"> <s>Let therefore R H be equall to the Semi-parameter.] <emph type="italics"></emph>So it is to be <lb></lb>read, and not R M, as<emph.end type="italics"></emph.end> Tartaglia's <emph type="italics"></emph>Tranſlation hath is; which may be made appear from <lb></lb>that which followeth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1200"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1200"></margin.target>B</s></p><p type="main"> <s>And O H double to H M.] <emph type="italics"></emph>In the Tranſlation aforenamed it is falſly render<lb></lb>ed,<emph.end type="italics"></emph.end> O N <emph type="italics"></emph>double to<emph.end type="italics"></emph.end> R M.</s></p><p type="main"> <s><arrow.to.target n="marg1201"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1201"></margin.target>C</s></p><p type="main"> <s>And look what proportion the Submerged Portion hath to the whole <lb></lb>Portion, the ſame hath the Square of P F unto the Square of N O.] <lb></lb><emph type="italics"></emph>This place we have reſtored in our Tranſlation, at the requeſt of ſome friends: But it is demon<lb></lb>ſtrated by<emph.end type="italics"></emph.end> Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. </s> <s>26.</s></p><p type="main"> <s><arrow.to.target n="marg1202"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1202"></margin.target>D</s></p><p type="main"> <s>Wherefore P F is not leſſe than M O.] <emph type="italics"></emph>For by 10 of the fifth it followeth <lb></lb>that the Square of P F is not leſſe than the Square of M O: and therefore neither ſhall the <lb></lb>Line P F be leße than the Line M O, by 22 of the<emph.end type="italics"></emph.end></s></p><figure id="id.073.01.030.1.jpg" xlink:href="073/01/030/1.jpg"></figure><p type="main"> <s><arrow.to.target n="marg1203"></arrow.to.target><lb></lb><emph type="italics"></emph>ſixth.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1204"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1203"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1204"></margin.target>(a) <emph type="italics"></emph>By 14. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Nor B P than H O,] <emph type="italics"></emph>For as P F is to <lb></lb>P B, ſo is M O to H O: and, by Permutation, as<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1205"></arrow.to.target><lb></lb><emph type="italics"></emph>P F is to M O, ſo is B P to H O; But P F is not <lb></lb>leſſe than M O as hath bin proved; (a) Therefore <lb></lb>neither ſhall B P be leſſe than H O.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1205"></margin.target>F</s></p><p type="main"> <s>If therefore a Right Line be drawn <lb></lb>from H at Right Angles unto N O, it <lb></lb>ſhall meet with B P, and ſhall fall be<lb></lb>twixt B and P.] <emph type="italics"></emph>This Place was corrupt in the <lb></lb>Tranſlation of<emph.end type="italics"></emph.end> Tartaglia<emph type="italics"></emph>: But it is thus demonstra<lb></lb>ted. </s> <s>In regard that P F is not leſſe than O M, nor P B than O H, if we ſuppoſe P F equall to <lb></lb>O M, P B ſhall be likewiſe equall to O H: Wherefore the Line drawn thorow O, parallel to A L <lb></lb>ſhall fall without the Section, by 17 of the firſt of our Treatiſe of Conicks; And in regard that <lb></lb>B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth <lb></lb>alſo meet with the ſame beneath B, and it doth of neceſſity fall betwixt B and P: But the <lb></lb>ſame is much more to follow, if we ſuppoſe P F to be greater than O M.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/031.jpg" pagenum="361"></pb><p type="head"> <s>PROP. V. THEOR. V.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Right-Angled Conoid lighter <lb></lb>than the Liquid, when it ſhall have its Axis great<lb></lb>er than<emph.end type="italics"></emph.end> Seſquialter <emph type="italics"></emph>of the Semi-parameter, if it have <lb></lb>not greater proportion in Gravity to the Liquid [of <lb></lb>equal Maſs] than the Exceſſe by which the Square <lb></lb>made of the Axis is greater than the Square made <lb></lb>of the Exceſſe by which the Axis is greater than<emph.end type="italics"></emph.end><lb></lb>ſeſquialter <emph type="italics"></emph>of the Semi-Parameter hath to the <lb></lb>Square made of the Axis being demitted into the Li<lb></lb>quid, ſo as that its Baſe be wholly within the Liquid, <lb></lb>and being ſet inclining, it ſhall not remain ſo inclined, <lb></lb>but ſhall turn about till that its Axis ſhall be accor<lb></lb>ding to the Perpendicular.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>For let any Portion be demitted into the Liquid, as hath been <lb></lb>ſaid; and let its Baſe be wholly within the Liquid, And being <lb></lb>cut thorow its Axis by a Plain erect upon the Surface of the <lb></lb>Liquid; its Section ſhall be the Section <lb></lb><figure id="id.073.01.031.1.jpg" xlink:href="073/01/031/1.jpg"></figure><lb></lb>of a Rightangled Cone: Let it be <lb></lb>A P O L, and let the Axis of the Por<lb></lb>tion and Diameter of the Section be <lb></lb>N O; and the Section of the Surface of <lb></lb>the Liquid I S. </s> <s>And becauſe the Axis <lb></lb>is not according to the Perpendicu<lb></lb>lar, N O will not be at equall angles <lb></lb>with I S. </s> <s>Draw K <foreign lang="grc">ω</foreign> touching the Se<lb></lb>ction A P O L in P, and parallel unto <lb></lb>I S: and thorow P, draw P F parallel unto N O: and take the <lb></lb>Centres of Gravity; and of the Solid A P O L let the Centre be <lb></lb>R; and of that which lyeth above the Liquid let the Centre be B; <lb></lb>and draw a Line from B to R, prolonging it to G; which let be the <lb></lb>Centre of Gravity of the Solid demerged within the Liquid: and <lb></lb>moreover, take R H equall to the Semi-parameter, and let O H be <lb></lb>double to H M; and do in the reſt as hath been ſaid <emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> above. <lb></lb><arrow.to.target n="marg1206"></arrow.to.target><lb></lb>Now foraſmuch as it was ſuppoſed that the Portion hath not greater <lb></lb>proportion in Gravity to the Liquid, than the Exceſſe by which <lb></lb>the Square N O is greater than the Square M O, hath to the ſaid <lb></lb>Square N O: And in regard that whatever proportion in Gravity <pb xlink:href="073/01/032.jpg" pagenum="362"></pb>the Portion hath to the Liquid of equall Maſſe, the ſame hath the <lb></lb>Magnitude of the Portion ſubmerged unto the whole Portion; as <lb></lb>hath been demonſtrated in the firſt Propoſition; The Magnitude <lb></lb>ſubmerged, therefore, ſhall not have greater proportion to the <lb></lb><arrow.to.target n="marg1207"></arrow.to.target><lb></lb>whole <emph type="italics"></emph>(b)<emph.end type="italics"></emph.end> Portion, than that which hath been mentioned: ^{*}And <lb></lb>therefore the whole Portion hath not greater proportion unto that <lb></lb><arrow.to.target n="marg1208"></arrow.to.target><lb></lb>which is above the Liquid, than the Square N O hath to the Square <lb></lb><arrow.to.target n="marg1209"></arrow.to.target><lb></lb>M O: But the <emph type="italics"></emph>(c)<emph.end type="italics"></emph.end> whole Portion hath the ſame proportion unto <lb></lb>that which is above the Liquid that the Square N O hath to the <lb></lb>Square P F: Therefore the Square N O hath not greater propor<lb></lb><arrow.to.target n="marg1210"></arrow.to.target><lb></lb>tion unto the Square P F, than it hath unto the Square M O: ^{*}And <lb></lb>hence it followeth that P F is not leſſe than O M, nor P B than O <lb></lb><arrow.to.target n="marg1211"></arrow.to.target><lb></lb>H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O <lb></lb>ſhall meet with B P betwixt P and B: Let it be in T: And be<lb></lb>cauſe that in the Section of the Rectangled Cone P F is parallel unto <lb></lb>the Diameter N O; and H T perpendicular unto the ſaid Diame<lb></lb>ter; and R H equall to the Semi-parameter: It is manifeſt that <lb></lb>R T prolonged doth make Right Angles with K P <foreign lang="grc">ω</foreign>: And there<lb></lb>fore doth alſo make Right Angles with I S: Therefore R T is per<lb></lb>pendicular unto the Surface of the Liquid; And if thorow the <lb></lb>Points B and G Lines be drawn parallel unto R T, they ſhall be <lb></lb>perpendicular unto the Liquids Surface. </s> <s>The Portion, therefore, <lb></lb>which is above the Liquid ſhall move downwards in the Liquid ac<lb></lb>cording to the Perpendicular drawn thorow B; and that part <lb></lb>which is within the Liquid ſhall move upwards according to the <lb></lb>Perpendicular drawn thorow G; and the Solid Portion A P O L <lb></lb>ſhall not continue ſo inclined, [<emph type="italics"></emph>as it was at its demerſion<emph.end type="italics"></emph.end>], but ſhall <lb></lb>move within the Liquid untill ſuch time that N O do ſtand accor<lb></lb>ding to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1206"></margin.target>(a) <emph type="italics"></emph>In 4. Prop. of <lb></lb>this.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1207"></margin.target>(a) <emph type="italics"></emph>By 11. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1208"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1209"></margin.target>(b) <emph type="italics"></emph>By 26. of the <lb></lb>Book<emph.end type="italics"></emph.end> De Conoid. <lb></lb></s> <s>& Sphæroid.</s></p><p type="margin"> <s><margin.target id="marg1210"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1211"></margin.target>C</s></p><p type="head"> <s>COMMANDINE.<lb></lb><arrow.to.target n="marg1212"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1212"></margin.target>A</s></p><p type="main"> <s>And therefore the whole Portion hath not greater proportion <lb></lb>unto that which is above the Liquid, than the Square N O hath to <lb></lb>the Square M O.] <emph type="italics"></emph>For in regard that the Magnitude of the Portion demerged <lb></lb>within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which <lb></lb>the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of <lb></lb>the Proportion, by the 26. of the fifth of<emph.end type="italics"></emph.end> Euclid, <emph type="italics"></emph>of<emph.end type="italics"></emph.end> Campanus <emph type="italics"></emph>his Tranſlation, the whole <lb></lb>Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O <lb></lb>hath unto the Exceſſe by which N O is greater than the Square M O. </s> <s>Let a Portion be taken; <lb></lb>and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which <lb></lb>is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe <lb></lb>by which the Square N O is greater than the Square M O be the fourth. </s> <s>Now of theſe Mag<lb></lb>nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third & <lb></lb>fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square <lb></lb>N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi<lb></lb>on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the <lb></lb>firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame<emph.end type="italics"></emph.end> <pb xlink:href="073/01/033.jpg" pagenum="363"></pb><emph type="italics"></emph>the proportion of the whole Portion unto that part thereof which is above the Liquid ſhall not be <lb></lb>greater than that of the Square N O unto the Square M O: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And hence it followeth that P F is not leſſe than O M, nor P B </s></p><p type="main"> <s><arrow.to.target n="marg1213"></arrow.to.target><lb></lb>than O H.] <emph type="italics"></emph>This followeth by the 10 and 14 of the fifth, and by the 22 of the ſixth of<emph.end type="italics"></emph.end><lb></lb>Euclid, <emph type="italics"></emph>as hath been ſaid above.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1213"></margin.target>B</s></p><p type="main"> <s>A <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine, therefore, drawn from Hat Right Angles unto N O ſhall <lb></lb><arrow.to.target n="marg1214"></arrow.to.target><lb></lb>meet with P B betwixt P and B.] <emph type="italics"></emph>Why this ſo falleth out, we will ſhew in the <lb></lb>next.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1214"></margin.target>C</s></p><p type="head"> <s>PROP. VI. THEOR. VI.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid lighter <lb></lb>than the Liquid, when it ſhall have its Axis greater <lb></lb>than ſeſquialter of the Semi-parameter, but leſſe than <lb></lb>to be unto the Semi-parameter in proportion as fifteen <lb></lb>to fower, being demitted into the Liquid ſo as that <lb></lb>its Baſe do touch the Liquid, it ſhall never stand ſo <lb></lb>enclined as that its Baſe toucheth the Liquid in one <lb></lb>Point only.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be a Portion, as was ſaid; and demit it into the Li<lb></lb>quid in ſuch faſhion as that its Baſe do touch the Liquid in <lb></lb>one only Point: It is to be demonſtrated that the ſaid Portion <lb></lb><arrow.to.target n="marg1215"></arrow.to.target><lb></lb>ſhall not continue ſo, but ſhall turn about in ſuch manner as that <lb></lb>its Baſe do in no wiſe touch the Surface of the Liquid. </s> <s>For let it be <lb></lb>cut thorow its Axis by a Plane erect <lb></lb><figure id="id.073.01.033.1.jpg" xlink:href="073/01/033/1.jpg"></figure><lb></lb>upon the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquids Surface: and let <lb></lb>the Section of the Superficies of the <lb></lb>Portion be A P O L, the Section of <lb></lb>a Rightangled Cone; and the Sect<lb></lb>ion of the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid be <lb></lb>A S; and the Axis of the Portion <lb></lb>and Diameter of the Section N O: <lb></lb>and let it be cut in F, ſo as that O <lb></lb>F be double to F N; and in <foreign lang="grc">ω</foreign> ſo, as that N O may be to F <foreign lang="grc">ω</foreign> in the <lb></lb>ſame proportion as fifteen to four; and at Right Angles to N O <lb></lb>draw <foreign lang="grc">ω</foreign> <emph type="italics"></emph>N<emph.end type="italics"></emph.end>ow becauſe N O hath greater proportion unto F <foreign lang="grc">ω</foreign> than <lb></lb>unto the Semi-parameter, let the Semi-parameter be equall to F B: <lb></lb><arrow.to.target n="marg1216"></arrow.to.target><lb></lb>and draw P C parallel unto A S, and touching the Section A P O L <lb></lb>in P; and P I parallel unto <emph type="italics"></emph>N O<emph.end type="italics"></emph.end>; and firſt let P I cut K<foreign lang="grc">ω</foreign> in H. For<lb></lb><arrow.to.target n="marg1217"></arrow.to.target><lb></lb>aſmuch, therefore, as in the Portion A P O L, contained betwixt <lb></lb>the Right <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine and the Section of the Rightangled Cone, K <foreign lang="grc">ω</foreign> is <lb></lb>parallel to A L, and P I parallel unto the Diameter, and cut by the <pb xlink:href="073/01/034.jpg" pagenum="364"></pb>ſaid K <foreign lang="grc">ω</foreign> in H, and A S is parallel unto the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine that toucheth in <lb></lb>P; It is neceſſary that P I hath unto P H either the ſame proportion <lb></lb>that <emph type="italics"></emph>N<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> hath to <foreign lang="grc">ω</foreign> O, or greater; for this hath already been de<lb></lb>monſtrated: But <emph type="italics"></emph>N<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> is ſeſquialter of <foreign lang="grc">ω</foreign> O; and P I, therefore, is <lb></lb>either Seſquialter of H P, or more than ſeſquialter: Wherefore <lb></lb><arrow.to.target n="marg1218"></arrow.to.target><lb></lb>P H is to H I either double, or leſſe than double. <emph type="italics"></emph>L<emph.end type="italics"></emph.end>et P T be <lb></lb>double to T I: the Centre of Gravity of the part which is within <lb></lb>the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid ſhall be the Point T. </s> <s>Therefore draw a <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine from T <lb></lb>to F prolonging it; and let the Centre of <lb></lb><figure id="id.073.01.034.1.jpg" xlink:href="073/01/034/1.jpg"></figure><lb></lb>Gravity of the part which is above the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid <lb></lb>be G: and from the Point B at Right Angles <lb></lb>unto <emph type="italics"></emph>N O<emph.end type="italics"></emph.end> draw B R. </s> <s>And ſeeing that P I is <lb></lb>parallel unto the Diameter <emph type="italics"></emph>N O,<emph.end type="italics"></emph.end> and B R <lb></lb>perpendicular unto the ſaid Diameter, and F <lb></lb>B equall to the Semi-parameter; It is mani<lb></lb>feſt that the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine drawn thorow the Points <lb></lb>F and R being prolonged, maketh equall <lb></lb>Angles with that which toucheth the Section <lb></lb>A P O L in the Point P: and therefore doth alſo make Right An<lb></lb>gles with A S, and with the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid: and the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines <lb></lb>drawn thorow T and G parallel unto F R ſhall be alſo perpendicu<lb></lb>lar to the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid: and of the Solid Magnitude A P <lb></lb>O L, the part which is within the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid moveth upwards according <lb></lb>to the Perpendicular drawn thorow T; and the part which is above <lb></lb>the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid moveth downwards according to that drawn thorow G: <lb></lb><arrow.to.target n="marg1219"></arrow.to.target><lb></lb>The Solid A <emph type="italics"></emph>P<emph.end type="italics"></emph.end> O L, therefore, ſhall turn about, and its Baſe ſhall <lb></lb>not in the leaſt touch the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid, And if <emph type="italics"></emph>P<emph.end type="italics"></emph.end> I do not <lb></lb>cut the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine K <foreign lang="grc">ω,</foreign> as in the ſecond Figure, it is manifeſt that the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint T, which is the Centre of Gravity of the ſubmerged <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion, <lb></lb>falleth betwixt <emph type="italics"></emph>P<emph.end type="italics"></emph.end> and I: And for the other particulars remaining, <lb></lb>they are demonſtrated like as before.</s></p><p type="margin"> <s><margin.target id="marg1215"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1216"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1217"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1218"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1219"></margin.target>E</s></p><p type="head"> <s>COMMANDINE.<lb></lb><arrow.to.target n="marg1220"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1220"></margin.target>A</s></p><p type="main"> <s>It is to be demonſtrated that the ſaid <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion ſhall not continue <lb></lb>ſo, but ſhall turn about in ſuch manner as that its Baſe do in no wiſe <lb></lb>touch the Surface of the Liquid.] <emph type="italics"></emph>Theſe words are added by us, as having been <lb></lb>omitted by<emph.end type="italics"></emph.end> Tartaglia.</s></p><p type="main"> <s><emph type="italics"></emph>N<emph.end type="italics"></emph.end>ow becauſe N O hath greater proportion to F <foreign lang="grc">ω</foreign> than unto </s></p><p type="main"> <s><arrow.to.target n="marg1221"></arrow.to.target><lb></lb>the Semi parameter.] <emph type="italics"></emph>For the Diameter of the Portion N O hath unto F<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>the <lb></lb>ſame proportion as fifteen to fower: But it was ſuppoſed to have leſſe proportion unto the <lb></lb>Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign><lb></lb><emph type="italics"></emph>than unto the Semi-parameter: And therefore<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>the Semi-parameter ſhall be greater<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1222"></arrow.to.target><lb></lb><emph type="italics"></emph>than the ſaid F<emph.end type="italics"></emph.end> <foreign lang="grc">ω.</foreign></s></p><p type="margin"> <s><margin.target id="marg1221"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1222"></margin.target>(a) <emph type="italics"></emph>By 10. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Foraſmuch, therefore, as in the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion <emph type="italics"></emph>A P O L,<emph.end type="italics"></emph.end> contained, be<lb></lb><arrow.to.target n="marg1223"></arrow.to.target><lb></lb>twixt the Right <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine and the Section of the Rightangled Cone K <lb></lb><foreign lang="grc">ω</foreign> is parallel to A L, and <emph type="italics"></emph>P I<emph.end type="italics"></emph.end> parallel unto the Diameter, and cut by <pb xlink:href="073/01/035.jpg" pagenum="365"></pb>the ſaid K <foreign lang="grc">ω</foreign> in H, and A S is parallel unto the Line that toucheth <lb></lb>in P; It is neceſſary that P I hath unto P H either the ſame propor<lb></lb>tion that N <foreign lang="grc">ω</foreign> hath to <foreign lang="grc">ω</foreign> O, or greater; for this hath already been <lb></lb>demonſtrated.] <emph type="italics"></emph>Where this is demonſtrated either by<emph.end type="italics"></emph.end> Archimedes <emph type="italics"></emph>himſelf, or by <lb></lb>any other, doth not appear; touching which we will here inſert a Demonſtration, after that <lb></lb>we have explained ſome things that pertaine thereto.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1223"></margin.target>C</s></p><p type="head"> <s>LEMMA I.</s></p><p type="main"> <s>Let the Lines A B and A C contain the Angle B A C; and from <lb></lb>the point D, taken in the Line A C, draw D E and D F at <lb></lb>pleaſure unto A B: and in the ſame Line any Points G and L <lb></lb>being taken, draw G H & L M parallel to D E, & G K and <lb></lb>L N parallel unto F D: Then from the Points D & G as farre <lb></lb>as to the Line M L draw D O P, cutting G H in O, and G Q <lb></lb>parallel unto B A. </s> <s>I ſay that the Lines that lye betwixt the Pa<lb></lb>rallels unto F D have unto thoſe that lye betwixt the Par<lb></lb>allels unto D E (namely K N to G Q or to O P; F K to D O; <lb></lb>and F N to D P) the ſame mutuall proportion: that is to ſay, <lb></lb>the ſame that A F hath to A E.</s></p><p type="main"> <s><emph type="italics"></emph>For in regard that the Triangles A F D, A K G, and A N L<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.035.1.jpg" xlink:href="073/01/035/1.jpg"></figure><lb></lb><emph type="italics"></emph>are alike, and E F D, H K G, and M N L are alſo alike: There-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1224"></arrow.to.target><lb></lb><emph type="italics"></emph>fore,<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>as A F is to F D, ſo ſhall A K be to K G; and as F D is to <lb></lb>F E, ſo ſhall K G be to K H: Wherefore,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>as A F is to F <lb></lb>E, ſo ſhall A K be to K H: And, by Converſion of proportion, as <lb></lb>A F is to A E, ſo ſhall A K be to K H. </s> <s>It is in the ſame manner <lb></lb>proved that, as A F is to A E, ſo ſhall A N be to A M. </s> <s>Now A<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1225"></arrow.to.target><lb></lb><emph type="italics"></emph>N being to A M, as A K is to A H; The<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>Remainder K N ſhall <lb></lb>be unto the Remainder H M, that is unto G Q, or unto O P, as <lb></lb>A N is to A M; that is, as A F is to A E: Again, A K is to <lb></lb>A H, as A F is to A E; Therefore the Remainder F K ſhall be to <lb></lb>the Remainder E H, namely to D O, as A F is to A E. </s> <s>We might in <lb></lb>like manner demonstrate that ſo is F N to D P: Which is that that <lb></lb>was required to be demonstrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1224"></margin.target>(a) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1225"></margin.target>(b) <emph type="italics"></emph>By 5. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s>In the ſame Line A B let there be two Points R and S, ſo diſpo<lb></lb>ſed, that A S may have the ſame Proportion to A R that <lb></lb>A F hath to A E; and thorow R draw R T parallel to E D, <lb></lb>and thorow S draw S T parallel to F D, ſo, as that it may <lb></lb>meet with R T in the Point T. </s> <s>I ſay that the Point T fall<lb></lb>eth in the Line A C.</s></p> <pb xlink:href="073/01/036.jpg" pagenum="366"></pb><figure id="id.073.01.036.1.jpg" xlink:href="073/01/036/1.jpg"></figure><p type="main"> <s><emph type="italics"></emph>For if it be poſſible, let it fall ſhort of it: and let R T be pro<lb></lb>longed as farre as to A C in V: and then thorow V draw V X pa<lb></lb>rallel to F D. Now, by the thing we have last demonſtrated, A X <lb></lb>ſhall have the ſame proportion unto A R, as A F hath to A E. <lb></lb></s> <s>But A S hath alſo the ſame proportion to A R: Wherefore<emph.end type="italics"></emph.end> (a) <lb></lb><arrow.to.target n="marg1226"></arrow.to.target><lb></lb>A S <emph type="italics"></emph>is equall to A X, the part to the whole, which is impoſſi<lb></lb>ble. </s> <s>The ſame abſurdity will follow if we ſuppoſe the Toint <lb></lb>T to fall beyond the Line A C: It is therefore neceſſary that <lb></lb>it do fall in the ſaid A C. </s> <s>Which we propounded to be demonstrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1226"></margin.target>(a) <emph type="italics"></emph>By 9. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA III.</s></p><p type="main"> <s>Let there be a Parabola, whoſe Diameter <lb></lb><arrow.to.target n="marg1227"></arrow.to.target><lb></lb>let be A B; and let the Right Lines A C and B D be ^{*} con<lb></lb>tingent to it, A C in the Point C, and B D in B: And two <lb></lb>Lines being drawn thorow C, the one C E, parallel unto <lb></lb>the Diameter; the other C F, parallel to B D; take any <lb></lb>Point in the Diameter, as G; and as F B is to B G, ſo let B <lb></lb>G be to B H: and thorow G and H draw G K L, and H E <lb></lb>M, parallel unto B D; and thorow M draw M N O parallel <lb></lb>to <emph type="italics"></emph>A C,<emph.end type="italics"></emph.end> and cutting the Diameter in O: and the Line <emph type="italics"></emph>N P<emph.end type="italics"></emph.end><lb></lb>being drawn thorow <emph type="italics"></emph>N<emph.end type="italics"></emph.end> unto the Diameter let it be parallel <lb></lb>to B D. </s> <s>I ſay that H O is double to G B.</s></p><p type="margin"> <s><margin.target id="marg1227"></margin.target>* Or touch it.</s></p><p type="main"> <s><emph type="italics"></emph>For the Line M N O cutteth the Diameter either in G, or in other Points: and if it do <lb></lb>cut it in G, one and the ſame Point ſhall be noted by the two letters G and O. </s> <s>Therfore F C, <lb></lb>P N, and H E M being Parallels, and A C being Parallels to M N O, they ſhall make the<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.036.2.jpg" xlink:href="073/01/036/2.jpg"></figure><lb></lb><emph type="italics"></emph>Triangles A F C, O P N and O H M like to<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1228"></arrow.to.target><lb></lb><emph type="italics"></emph>each other: Wherefore<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>O H ſhall be to <lb></lb>H M, as A F to FC: and<emph.end type="italics"></emph.end> ^{*} Permutando, <lb></lb><arrow.to.target n="marg1229"></arrow.to.target><lb></lb><emph type="italics"></emph>O H ſhall be to A F, as H M to F C: But <lb></lb>the Square H M is to the Square G L as the Line <lb></lb>H B is to the Line B G, by 20. of our firſt Book <lb></lb>of<emph.end type="italics"></emph.end> Conicks; <emph type="italics"></emph>and the Square G L is unto the <lb></lb>Square F C, as the Line G B is to the Line B F: <lb></lb>and the Lines H B, B G and B F are thereupon<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1230"></arrow.to.target><lb></lb><emph type="italics"></emph>Proportionals: Therefore the<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>Squares <lb></lb>H M, G L and F C and there Sides, ſhall alſo be <lb></lb>Proportionals: And, therefore, as the (c) <lb></lb>Square H M is to the Square G L, ſo is the Line<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1231"></arrow.to.target><lb></lb><emph type="italics"></emph>H M to the Line F C: But as H M is to F C, ſo <lb></lb>is O H to A F; and as the Square H M is to <lb></lb>the Square G L, ſo is the Line H B to B G; that <lb></lb>is, B G to B F: From whence it followeth that <lb></lb>O H is to A F, as B G to B F: And<emph.end type="italics"></emph.end> Permu<lb></lb>tando, <emph type="italics"></emph>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 <lb></lb>are equall, by 35. of our firſt Book of<emph.end type="italics"></emph.end> Conicks: <emph type="italics"></emph>And therefore N O is double to G B: <lb></lb>Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/037.jpg" pagenum="367"></pb><p type="margin"> <s><margin.target id="marg1228"></margin.target>(a) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1229"></margin.target>* Or permitting.</s></p><p type="margin"> <s><margin.target id="marg1230"></margin.target>(b) <emph type="italics"></emph>By 22. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1231"></margin.target>(c) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> Cor. <emph type="italics"></emph>of 20. <lb></lb>of the ſixth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA IV.</s></p><p type="main"> <s>The ſame things aſſumed again, and M Q being drawn from the <lb></lb>Point M unto the Diameter, let it touch the Section in the <lb></lb>Point M. </s> <s>I ſay that H Q hath to Q O, the ſame proportion <lb></lb>that G H hath to C N.</s></p><p type="main"> <s><emph type="italics"></emph>For make H R equall to G F; and ſeeing that<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.037.1.jpg" xlink:href="073/01/037/1.jpg"></figure><lb></lb><emph type="italics"></emph>the Triangles A F C and O P N are alike, and <lb></lb>P N equall to F C, we might in like manner de<lb></lb>monſtrate P O and F A to be equall to each other: <lb></lb>Wherefore P O ſhall be double to F B: But H O <lb></lb>is double to G B: Therefore the Remainder P H <lb></lb>is alſo double to the Remainder F G; that is, to <lb></lb>R H: And therefore is followeth that P R, R H <lb></lb>and F G are equall to one another; as alſo that <lb></lb>R G and P F are equall: For P G is common to <lb></lb>both R P and G F. </s> <s>Since therefore, that H B is <lb></lb>to B G, as G B is to B F, by Converſion of Pro<lb></lb>portion, B H ſhall be to H G, as B G is to G F: <lb></lb>But Q H is to H B, as H O to B G. </s> <s>For by 35 <lb></lb>of our firſt Book of<emph.end type="italics"></emph.end> Conicks, <emph type="italics"></emph>in regard that Q <lb></lb>M toucheth the Section in the Point M, H B and <lb></lb>B Q ſhall be equall, and Q H double to H B: <lb></lb>Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>Q H ſhall be to H G, as <lb></lb>H O to G F; that is, to H R: and,<emph.end type="italics"></emph.end> Permu<lb></lb>tando, <emph type="italics"></emph>Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q <lb></lb>O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de<lb></lb>monſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Theſe things therefore being explained, we come now to that <lb></lb>which was propounded. </s> <s>I ſay, therefore, firſt that <emph type="italics"></emph>N C<emph.end type="italics"></emph.end> hath <lb></lb>to C K the ſame proportion that H G hath to G B.</s></p><p type="main"> <s><emph type="italics"></emph>For ſince that H Q is to Q O, as H G to C N<emph.end type="italics"></emph.end>; <lb></lb><figure id="id.073.01.037.2.jpg" xlink:href="073/01/037/2.jpg"></figure><lb></lb><emph type="italics"></emph>that is, to A O, equall to the ſaid C N: The Re<lb></lb>mainder G Q ſhall be to the Remainder Q A, as <lb></lb>H Q to Q O: and, for the ſame cauſe, the Lines <lb></lb>A C and G L prolonged, by the things that wee <lb></lb>have above demonstrated, ſhall interſect or meet <lb></lb>in the Line Q M. Again, G Q is to Q A, <lb></lb>as H Q to Q O: that is, as H G to F P; as<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1232"></arrow.to.target><lb></lb>(a) <emph type="italics"></emph>was bnt now demonstrated, But unto<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>G<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1233"></arrow.to.target><lb></lb><emph type="italics"></emph>Q two Lines taken together, Q B that is H B, and <lb></lb>B G are equall: and to Q A H F is equall; for <lb></lb>if from the equall Magnitudes H B and B Q there <lb></lb>be taken the equall Magnitudes F B and B A, the <lb></lb>Re mainder ſhall be equall; Therefore taking H <lb></lb>G from the two Lines H B and B G, there ſhall re<lb></lb>main a Magnitude double to B G; that is, O H: <lb></lb>and P F taken from F H, the Remainder is H P: <lb></lb>Wherefore<emph.end type="italics"></emph.end> (c) <emph type="italics"></emph>O H is to H P, as G Q to Q A:<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1234"></arrow.to.target><lb></lb><emph type="italics"></emph>But as G Q is to Q A, ſo is H Q to Q O;<emph.end type="italics"></emph.end> <pb xlink:href="073/01/038.jpg" pagenum="368"></pb><arrow.to.target n="marg1235"></arrow.to.target><lb></lb><emph type="italics"></emph>that is, H G to N C: and as<emph.end type="italics"></emph.end> (d) <emph type="italics"></emph>O H is to H P, ſo is G B to C K; For O H is double <lb></lb>to G B, and H P alſo double to G F; that is, to C K; Therefore H G hath the ſame propor<lb></lb>tion to N C, that G B hath to C K: And<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>N C hath to C K the ſame proportion <lb></lb>that H G hath to G B.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1232"></margin.target>(a) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> 2. Lemma.</s></p><p type="margin"> <s><margin.target id="marg1233"></margin.target>(b) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> 4. Lemma.</s></p><p type="margin"> <s><margin.target id="marg1234"></margin.target>(b) <emph type="italics"></emph>By 19. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1235"></margin.target>(d) <emph type="italics"></emph>By 15. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Then take ſome other Point at pleaſure in the Section, which <lb></lb>let be S: and thorow S draw two Lines, the one S T paral<lb></lb>lel to D B, and cutting the Diameter in the Point T; the <lb></lb>other S V parallel to A C, and cutting C E in V. </s> <s>I ſay <lb></lb>that V C hath greater proportion to C K, than T G hath <lb></lb>to G B.</s></p><p type="main"> <s><emph type="italics"></emph>For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the <lb></lb>Diameter parallel to B D: G T ſhall be leſſe than G Y, in regard that V S is leße than V X: <lb></lb>And, by the firſt Lemma, Y G ſhall be to V C, as H G to N C; that is, as G B to C K, which <lb></lb>was demonſtrated but now: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>Y G ſhall be to G B, as V C to C K: But <lb></lb>T G, for that it is leſſe than Y G, hath leſſe proportion to G B, than Y G hath to the ſame; <lb></lb>Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de<lb></lb>monſtrated. </s> <s>Therefore a Poſition given G K, there ſhall be in the Section one only Point, to <lb></lb>wit M, from which two Lines M E H and M N O being drawn, N C ſhall have the ſame pro<lb></lb>portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall<lb></lb>eth betwixt A C, and the Line parallel unto it ſhall alwayes have greater proportion to C K, <lb></lb>than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there<lb></lb>fore, is manifeſt which was affirmed by<emph.end type="italics"></emph.end> Archimedes, <emph type="italics"></emph>to wit, that the Line P I hath unto P H, <lb></lb>either the ſame proportion that N<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>hath to<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>O, or greater.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1236"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1236"></margin.target>D</s></p><p type="main"> <s>Wherefore P H is to H I either double, or leſſe than double.] <lb></lb><emph type="italics"></emph>If leſſe than double, let P T be double to T I: The Centre of Gravity of that part of the <lb></lb>Portion that is within the Liquid ſhall be the<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.038.1.jpg" xlink:href="073/01/038/1.jpg"></figure><lb></lb><emph type="italics"></emph>Point T: But if P H be double to H I, H ſhall <lb></lb>be the Centre of Gravity; And draw H F, and <lb></lb>prolong it unto the Centre of that part of the Por<lb></lb>tion which is above the Liquid, namely, unto G, <lb></lb>and the reſt is demonſtrated as before. </s> <s>And the <lb></lb>ſame is to be underſtood in the Propoſition that <lb></lb>followeth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>The Solid A P O L, therefore, <lb></lb>ſhall turn about, and its Baſe ſhall <lb></lb>not in the leaſt touch the Surface <lb></lb>of the Liquid.] <emph type="italics"></emph>In<emph.end type="italics"></emph.end> Tartaglia's <emph type="italics"></emph>Tranſlation it is rendered<emph.end type="italics"></emph.end> ut Baſis ipſius non tangent <lb></lb>ſuperficiem humidi ſecundum unum ſignum; <emph type="italics"></emph>but we have choſen to read<emph.end type="italics"></emph.end> ut Baſis ipſius <lb></lb>nullo modo humidi ſuperficiem contingent, <emph type="italics"></emph>both here, and in the following Propoſitions, <lb></lb>becauſe the Greekes frequently uſe<emph.end type="italics"></emph.end> <foreign lang="grc">ὡδὲεἶς, ὡδὲ<gap></gap></foreign> <emph type="italics"></emph>pro<emph.end type="italics"></emph.end> <foreign lang="grc">ὠδεὶσ<gap></gap> & οὐδὶν</foreign>: <emph type="italics"></emph>ſo that<emph.end type="italics"></emph.end> <foreign lang="grc">οὐδἔσινουδείς,</foreign> nullus <lb></lb>eſt; <foreign lang="grc">οὐδ<gap></gap>ὑπ̓ἑρὸς</foreign> à nullo, <emph type="italics"></emph>and ſo of others of the like nature.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/039.jpg" pagenum="369"></pb><p type="head"> <s>PROP. VII. THE OR. VII.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid lighter <lb></lb>than the Liquid, when it ſhall have its Axis greater <lb></lb>than Seſquialter of the Semi-parameter, but leſſe <lb></lb>than to be unto the ſaid Semi-parameter in proportion <lb></lb>as fiſteen to fower, being demitted into the Liquid ſo <lb></lb>as that its Baſe be wholly within the Liquid, it ſhall <lb></lb>never ſtand ſo as that its Baſe do touch the Surface <lb></lb>of the Liquid, but ſo, that it be wholly within the <lb></lb>Liquid, and ſhall not in the leaſt touch its Surface.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be a Portion as hath been ſaid; and let it be de<lb></lb>mitted into the Liquid, as we have ſuppoſed, ſo as that its <lb></lb>Baſe do touch the Surface in one Point only: It is to be de<lb></lb>monſtrated that the ſame ſhall not ſo <lb></lb><figure id="id.073.01.039.1.jpg" xlink:href="073/01/039/1.jpg"></figure><lb></lb>continue, but ſhall turn about in <lb></lb>ſuch manner as that its Baſe do in no <lb></lb>wiſe touch the Surface of the Liquid. <lb></lb></s> <s>For let it be cut thorow its Axis by <lb></lb>a Plane erect upon the Liquids Sur<lb></lb>face: and let the Section be A P O L, <lb></lb>the Section of a Rightangled <lb></lb>Cone; the Section of the Liquids <lb></lb>Surface S L; and the Axis of the <lb></lb>Portion and Diameter of the Section P F: and let P F be cut in <lb></lb>R, ſo, as that R P may be double to R F, and in <foreign lang="grc">ω</foreign> ſo as that P F <lb></lb>may be to R <foreign lang="grc">ω</foreign> as fifteen to fower: and draw <foreign lang="grc">ω</foreign> K at Right Angles </s></p><p type="main"> <s><arrow.to.target n="marg1237"></arrow.to.target><lb></lb>to P F: <emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> R <foreign lang="grc">ω</foreign> ſhall be leſſe than the Semi-parameter. </s> <s>There<lb></lb>fore let R H be ſuppoſed equall to the Semi-parameter: and <lb></lb>draw C O touching the Section in O and parallel unto S L; and <lb></lb>let N O be parallel unto P F; and firſt let N O cut K <foreign lang="grc">ω</foreign> in the Point <lb></lb>I, as in the former Schemes: It ſhall be demonſtrated that N O is <lb></lb>to O I either ſeſquialter, or greater than ſeſquialter. </s> <s>Let O I be <lb></lb>leſſe than double to I N; and let O B be double to B N: and let <lb></lb>them be diſpoſed like as before. </s> <s>We might likewiſe demonſtrate <lb></lb>that if a Line be drawn thorow R and T it will make Right Angles <lb></lb>with the Line C O, and with the Surface of the Liquid: Where<lb></lb>fore Lines being drawn from the Points B and G parallels unto <lb></lb>R T, they alſo ſhall be Perpendiculars to the Surface of the Liquid: <lb></lb>The Portion therefore which is above the Liquid ſhall move down <pb xlink:href="073/01/040.jpg" pagenum="370"></pb><figure id="id.073.01.040.1.jpg" xlink:href="073/01/040/1.jpg"></figure><lb></lb>wards according to that ſame Perpendicular <lb></lb>which paſſeth thorow B; and the Portion <lb></lb>which is within the Liquid ſhall move up<lb></lb>wards acording to that paſſing thorow G: <lb></lb>From whence it is manifeſt that the Solid <lb></lb>ſhall turn about in ſuch manner, as that <lb></lb>its Baſe ſhall in no wiſe touch the Surface <lb></lb>of the Liquid; for that now when it touch<lb></lb>eth but in one Point only, it moveth down<lb></lb>wards on the part towards L. </s> <s>And though <lb></lb>N O ſhould not cut <foreign lang="grc">ω</foreign> K, yet ſhall the ſame hold true.</s></p><p type="margin"> <s><margin.target id="marg1237"></margin.target>(a) <emph type="italics"></emph>By 10 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROP. VIII. THE OR. VIII.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid, when it <lb></lb>ſhall have its Axis greater than ſeſquialter of the Se<lb></lb>mi-parameter, but leſſe than to be unto the ſaid Semi<lb></lb>parameter, in proportion as fifteen to fower, if it <lb></lb>have a leſſer proportion in Gravity to the Liquid, than <lb></lb>the Square made of the Exceſſe by which the Axis is <lb></lb>greater than Seſquialter of the Semi-parameter hath <lb></lb>to the Square made of the Axis, being demitted into <lb></lb>the Liquid, ſo as that its Baſe touch not the Liquid, <lb></lb>it ſhall neither return to Perpendicularity, nor conti<lb></lb>nue inclined, ſave only when the Axis makes an <lb></lb>Angle with the Surface of the Liquid, equall to that <lb></lb>which we ſhall preſently ſpeak of.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be a Portion as hath been ſaid; and let B D be equall <lb></lb>to the Axis: and let B K be double to K D; and R K equall <lb></lb><arrow.to.target n="marg1238"></arrow.to.target><lb></lb>to the Semi-parameter: and let C B be Seſquialter of B R: <lb></lb>C D ſhall be alſo Sefquialter of K R. </s> <s>And as the Portion is to the <lb></lb>Liquid in Gravity, ſo let the Square F Q be to the Square D B; <lb></lb>and let F be double to Q: It is manifeſt, therefore, that F Q hath <lb></lb>to D B, leſs proportion than C B hath to B D; For C B is the <lb></lb>Exceſs by which the Axis is greater than Seſquialter of the Semi<lb></lb><arrow.to.target n="marg1239"></arrow.to.target><lb></lb>parameter: And, therefore, F Q is leſs than B C; and, for the <lb></lb><arrow.to.target n="marg1240"></arrow.to.target><lb></lb>ſame reaſon, F is leſs than B R. </s> <s>Let R <foreign lang="grc">ψ</foreign> be equall to F; and draw <lb></lb><foreign lang="grc">ψ</foreign> E perpendicular to B D; which let be in power or contence the <lb></lb>half of that which the Lines K R and <foreign lang="grc">ψ</foreign> B containeth; and <lb></lb>draw a Line from B to E: It is to be demonſtrated, that the <pb xlink:href="073/01/041.jpg" pagenum="371"></pb>Portion demitted into the Liquid, like as hath been ſaid, ſhall ſtand <lb></lb>enclined ſo as that its Axis do make an Angle with the Surface of <lb></lb>the Liquid equall unto the Angle E B <foreign lang="grc">Ψ.</foreign> For demit any Portion <lb></lb>into the Liquid ſo as that its Baſe <lb></lb><figure id="id.073.01.041.1.jpg" xlink:href="073/01/041/1.jpg"></figure><lb></lb>touch not the Liquids Surface; <lb></lb>and, if it can be done, let the <lb></lb>Axis not make an Angle with the <lb></lb>Liquids Surface equall to the <lb></lb>Angle E B <foreign lang="grc">Ψ</foreign>; but firſt, let it be <lb></lb>greater: and the Portion being <lb></lb>cut thorow the Axis by a Plane e<lb></lb>rect unto [<emph type="italics"></emph>or upon<emph.end type="italics"></emph.end>] the Surface of <lb></lb>the Liquid, let the Section be A P <lb></lb>O L the Section of a Rightangled <lb></lb>Cone; the Section of the Surface of the Liquid X S; and let the <lb></lb>Axis of the Portion and Diameter of the Section be N O: and <lb></lb>draw P Y parallel to X S, and touching the Section A P O L in P; <lb></lb>and P M parallel to N O; and P I perpendicular to N O: and <lb></lb>moreover, let B R be equall to O <foreign lang="grc">ω,</foreign> and R K to T <foreign lang="grc">ω;</foreign> and let <foreign lang="grc">ω</foreign> H <lb></lb>be perpendicular to the Axis. </s> <s>Now becauſe it hath been ſuppoſed <lb></lb><arrow.to.target n="marg1241"></arrow.to.target><lb></lb>that the Axis of the Portion doth make an Angle with the Surface <lb></lb>of the Liquid greater than the Angle B, the Angle P Y I ſhall be <lb></lb>greater than the Angle B: Therefore the Square P I hath greater <lb></lb><arrow.to.target n="marg1242"></arrow.to.target><lb></lb>proportion to the Square Y I, than the Square E <foreign lang="grc">Ψ</foreign> hath to the <lb></lb>Square <foreign lang="grc">Ψ</foreign> B: But as the Square P I is to the Square Y I, ſo is the <lb></lb><arrow.to.target n="marg1243"></arrow.to.target><lb></lb>Line K R unto the Line I Y; and as the Square E <foreign lang="grc">Ψ</foreign> is to the Square <lb></lb><arrow.to.target n="marg1244"></arrow.to.target><lb></lb><foreign lang="grc">Ψ</foreign> B, ſo is half of the Line K R unto the Line <foreign lang="grc">Ψ</foreign> B: Wherefore <lb></lb><emph type="italics"></emph>(a)<emph.end type="italics"></emph.end> K R hath greater proportion to I Y, than the half of K R hath <lb></lb><arrow.to.target n="marg1245"></arrow.to.target><lb></lb>to <foreign lang="grc">Ψ</foreign> B: And, conſequently, I Y isleſſe than the double of <foreign lang="grc">Ψ</foreign> B, <lb></lb>and is the double of O I: Therefore O I is leſſe than <foreign lang="grc">Ψ</foreign> B; and I <foreign lang="grc">ω</foreign><lb></lb><arrow.to.target n="marg1246"></arrow.to.target><lb></lb>greater than <foreign lang="grc">Ψ</foreign> R: but <foreign lang="grc">Ψ</foreign> R is equall to F: Therefore I <foreign lang="grc">ω</foreign> is greater <lb></lb><arrow.to.target n="marg1247"></arrow.to.target><lb></lb>than F. </s> <s>And becauſe that the Portion is ſuppoſed to be in Gra<lb></lb>vity unto the Liquid, as the Square F Q is to the Square B D; and <lb></lb>ſince that as the Portion is to the Liquid in Gravity, ſo is the part <lb></lb>thereof ſubmerged unto the whole Portion; and in regard that as <lb></lb>the part thereof ſubmerged is to the whole, ſo is the Square P M to <lb></lb>the Square O N; It followeth, that the Square P M is to the Square <lb></lb>N O, as the Square F Q is to the Square B D: And therefore F <lb></lb><arrow.to.target n="marg1248"></arrow.to.target><lb></lb>Q is equall to P M: But it hath been demonſtrated that P H is <lb></lb><arrow.to.target n="marg1249"></arrow.to.target><lb></lb>greater than F: It is manifeſt, therefore, that P M is leſſe than <lb></lb>ſeſquialter of P H: And conſequently that P H is greater than <lb></lb>the double of H M. </s> <s>Let P Z be double to Z M: T ſhall be the Cen<lb></lb>tre of Gravity of the whole Solid; the Centre of that part of it <lb></lb>which is within the Liquid, the Point Z; and of the remaining <lb></lb><arrow.to.target n="marg1250"></arrow.to.target><lb></lb>part the Centre ſhall be in the Line Z T prolonged unto G. </s> <s>In <pb xlink:href="073/01/042.jpg" pagenum="372"></pb>the ſame manner we might demon<lb></lb><figure id="id.073.01.042.1.jpg" xlink:href="073/01/042/1.jpg"></figure><lb></lb>ſtrate the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine T H to be perpendi<lb></lb>cular unto the Surface of the Liquid: <lb></lb>and that the Portion demerged with<lb></lb>in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid moveth or aſcend<lb></lb>eth out of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid according to <lb></lb>the Perpendicular that ſhall be <lb></lb>drawn thorow Z unto the Surface <lb></lb>of the Liquid; and that the part <lb></lb>that is above the Liquid deſcendeth <lb></lb>into the Liquid according to that <lb></lb>drawn thorow G: therefore the Portion will not continue ſo inclined <lb></lb>as was ſuppoſed: But neither ſhall it return to Rectitude or Per<lb></lb>pendicularity; For that of the Perpendiculars drawn thorow Z and <lb></lb>G, that paſſing thorow Z doth fall on thoſe parts which are to<lb></lb>wards L; and that that paſſeth thorow G on thoſe towards A: <lb></lb>Wherefore it followeth that the Centre Z do move upwards, <lb></lb>and G downwards: Therefore the parts of the whole Solid which <lb></lb>are towards A ſhall move downwards, and thoſe towards L up<lb></lb>wards. </s> <s>Again let the Propoſition run in other termes; and let <lb></lb>the Axis of the Portion make an Angle with the Surface of the <lb></lb><arrow.to.target n="marg1251"></arrow.to.target><lb></lb>Liquid leſſe than that which is at B. </s> <s>Therefore the Square P I <lb></lb>hath leſſer Proportion unto the Square <lb></lb><figure id="id.073.01.042.2.jpg" xlink:href="073/01/042/2.jpg"></figure><lb></lb>I Y, than the Square E <foreign lang="grc">Ψ</foreign> hath to the <lb></lb>Square <foreign lang="grc">Ψ</foreign> B: Wherefore K R hath <lb></lb>leſſer proportion to I Y, than the half <lb></lb>of K R hath to <foreign lang="grc">Ψ</foreign> B: And, for the <lb></lb>ſame reaſon, I Y is greater than dou<lb></lb>ble of <foreign lang="grc">Ψ</foreign> B: but it is double of O I: <lb></lb>Therefore O I ſhall be greater than <lb></lb><foreign lang="grc">Ψ</foreign> B: But the Totall O <foreign lang="grc">ω</foreign> is equall <lb></lb>to R B, and the Remainder <foreign lang="grc">ω</foreign> I leſſe <lb></lb>than <foreign lang="grc">ψ</foreign> R: Wherefore P H ſhall alſo <lb></lb>be leſſe than F. And, in regard that <lb></lb>M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui<lb></lb>alter of P H; and that P H is leſſe than double of <emph type="italics"></emph>H<emph.end type="italics"></emph.end> M. <emph type="italics"></emph>L<emph.end type="italics"></emph.end>et P Z <lb></lb>be double to Z M. </s> <s>The Centre of Gravity of the whole Solid ſhall <lb></lb>again be T; that of the part which is within the Liquid Z; and <lb></lb>drawing a Line from Z to T, the Centre of Gravity of that which <lb></lb>is above the Liquid ſhall be found in that Line portracted, that is <lb></lb>in G: Therefore, Perpendiculars being drawn thorow Z and G <lb></lb><arrow.to.target n="marg1252"></arrow.to.target><lb></lb>unto the Surface of the Liquid that are parallel to T H, it followeth <lb></lb>that the ſaid Portion ſhall not ſtay, but ſhall turn about till <lb></lb>that its Axis do make an Angle with the Waters Surface greater than <lb></lb>that which it now maketh. </s> <s>And becauſe that when before we <pb xlink:href="073/01/043.jpg" pagenum="373"></pb>did ſuppoſe that it made an Angle greater than the Angle B, the <lb></lb>Poriton did not reſt then neither; It is manifeſt that it ſhall ſtay <lb></lb><arrow.to.target n="marg1253"></arrow.to.target><lb></lb>or reſt when it ſhall make an Angle eqnall to B. </s> <s>For ſo ſhall I O <lb></lb>be equall to <foreign lang="grc">Ψ</foreign> <emph type="italics"></emph>B<emph.end type="italics"></emph.end>; and <foreign lang="grc">ω</foreign> I equall to <lb></lb><figure id="id.073.01.043.1.jpg" xlink:href="073/01/043/1.jpg"></figure><lb></lb><foreign lang="grc">Ψ</foreign> R; and P H equall to F: There<lb></lb>fore <emph type="italics"></emph>M P<emph.end type="italics"></emph.end> ſhall be ſeſquialter of <emph type="italics"></emph>P H,<emph.end type="italics"></emph.end><lb></lb>and <emph type="italics"></emph>P H<emph.end type="italics"></emph.end> double of H M: And there<lb></lb>fore ſince H is the Centre of Gravity <lb></lb>of that part of it which is within the <lb></lb>Liquid, it ſhall move upwards along <lb></lb>the ſame <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular according to <lb></lb>which the whole <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion moveth; <lb></lb>and along the ſame alſo ſhall the part <lb></lb>which is above move downwards: <lb></lb>The <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion therefore ſhall reſt; for<lb></lb>aſmuch as the parts are not repulſed by each other.</s></p><p type="margin"> <s><margin.target id="marg1238"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1239"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1240"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1241"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1242"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1243"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1244"></margin.target>G</s></p><p type="margin"> <s><margin.target id="marg1245"></margin.target>(a) <emph type="italics"></emph>By 13. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1246"></margin.target>H</s></p><p type="margin"> <s><margin.target id="marg1247"></margin.target>K</s></p><p type="margin"> <s><margin.target id="marg1248"></margin.target>L</s></p><p type="margin"> <s><margin.target id="marg1249"></margin.target>M</s></p><p type="margin"> <s><margin.target id="marg1250"></margin.target>N</s></p><p type="margin"> <s><margin.target id="marg1251"></margin.target>O</s></p><p type="margin"> <s><margin.target id="marg1252"></margin.target>P</s></p><p type="margin"> <s><margin.target id="marg1253"></margin.target>Q</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s>And let <emph type="italics"></emph>C B<emph.end type="italics"></emph.end> be ſeſquialter of <emph type="italics"></emph>B R<emph.end type="italics"></emph.end>: C D ſhall alſo be ſeſquialter <lb></lb><arrow.to.target n="marg1254"></arrow.to.target><lb></lb>of K R.] <emph type="italics"></emph>In the Tranſlation it is read thus:<emph.end type="italics"></emph.end> Sit autem & CB quidem hemeolia <lb></lb>ipſius B R: C D autem ipſius K R. <emph type="italics"></emph>But we at the reading of this paſſage have thought <lb></lb>fit thus to correctit; for it is not ſuppoſed ſo to be, but from the things ſuppoſed is proved to <lb></lb>be ſo. </s> <s>For if B<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>be double of<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>D, D B ſhall be ſeſquialter of B<emph.end type="italics"></emph.end> <foreign lang="grc">ψ.</foreign> <emph type="italics"></emph>And becauſe E B is <lb></lb>ſeſquialter of B R, it followeth that the<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>Remainder C D is ſeſquialter of<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>R; that is, of<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1255"></arrow.to.target><lb></lb><emph type="italics"></emph>the Semi-parameter: Wherefore B C ſhall be the Exceſſe by which the Axis is greater than <lb></lb>ſeſquialter of the Semi-parameter.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1254"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1255"></margin.target>(a) <emph type="italics"></emph>By 19. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And therefore F Q is leſſe than <emph type="italics"></emph>B C.] For in regard that the Portion hath<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1256"></arrow.to.target><lb></lb><emph type="italics"></emph>the ſame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B; <lb></lb>and hath leſſer proportion than the Square made of the Exceſſe by which the Axis <lb></lb>is greater than Seſquialter of the Semi parameter, hath to the Square made of the Axis; that <lb></lb>is, leßer than the Square C B hath to the Square B D; for the Line B D was ſuppoſed to be <lb></lb>equall unto the Axis: Therefore the Square F Q ſhall have to the Square D B leſſer proporti<lb></lb>on than the Sqnare C B to the ſame Square B D: And therefore the Square<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>F Q ſhall be<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1257"></arrow.to.target><lb></lb><emph type="italics"></emph>leße than the Square C B: And, for that reaſon, the Line F Q ſhall be leße than B C.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1256"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1257"></margin.target>(b) <emph type="italics"></emph>By 8 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And, for the ſame reaſon, F is leſſe than <emph type="italics"></emph>B R.] For C B being ſeſqui-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1258"></arrow.to.target><lb></lb><emph type="italics"></emph>alter of B R, and F Q ſeſquialter of F<emph.end type="italics"></emph.end>: (c) F <emph type="italics"></emph>Q ſhall be likewiſe leſſe than B C; and F<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1259"></arrow.to.target><lb></lb><emph type="italics"></emph>leße than B R.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1258"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1259"></margin.target>(c) <emph type="italics"></emph>By 14 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Now becauſe it hath been ſuppoſed that the Axis of the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion <lb></lb><arrow.to.target n="marg1260"></arrow.to.target><lb></lb>doth make an Angle with the Surface of the Liquid greater than <lb></lb>the Angle <emph type="italics"></emph>B,<emph.end type="italics"></emph.end> the Angle <emph type="italics"></emph>P Y I<emph.end type="italics"></emph.end> ſhall be greater than the Angle <emph type="italics"></emph>B.] <lb></lb>For the Line P Y being parallel to the Surface of the Liquid, that is, to XS<emph.end type="italics"></emph.end>; (d) <emph type="italics"></emph>the Angle<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1261"></arrow.to.target><lb></lb><emph type="italics"></emph>P Y I ſhall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the <lb></lb>Line X S: And therefore ſhall be greater than the Angle B.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1260"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1261"></margin.target>(d) <emph type="italics"></emph>By 29 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Therefore the Square <emph type="italics"></emph>P I<emph.end type="italics"></emph.end> hath greater proportion to the Square <lb></lb><arrow.to.target n="marg1262"></arrow.to.target><lb></lb>Y I, than the Square E <foreign lang="grc">Ψ</foreign> hath to the Square <foreign lang="grc">Ψ</foreign> <emph type="italics"></emph>B] Let the Triangles P I Y <lb></lb>and E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B, be deſcribed apart: And ſeeing that the Angle P Y I is greater <lb></lb>than the Angle E B<emph.end type="italics"></emph.end> <foreign lang="grc">ψ,</foreign> <emph type="italics"></emph>unto the Line I Y, and at the Point Y aſſigned in<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.043.2.jpg" xlink:href="073/01/043/2.jpg"></figure><lb></lb><emph type="italics"></emph>the ſame, make the Angle V Y I equall to the Angle E B<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign>; <emph type="italics"></emph>But <lb></lb>the Right Angle at I, is equall unto the Right Angle at<emph.end type="italics"></emph.end> <foreign lang="grc">ψ;</foreign> <emph type="italics"></emph>therefore the<emph.end type="italics"></emph.end> <pb xlink:href="073/01/044.jpg" pagenum="374"></pb><emph type="italics"></emph>Remaining Angle Y V I is equall to the Remaining Angle B E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ.</foreign> <emph type="italics"></emph>And therefore the<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1263"></arrow.to.target><lb></lb>(e) <emph type="italics"></emph>Line V I hath to the Line I Y the ſame proportion that the Line E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>hath to<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B: But <lb></lb>the<emph.end type="italics"></emph.end> (f) <emph type="italics"></emph>Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1264"></arrow.to.target><lb></lb><emph type="italics"></emph>to the ſame: Therefore<emph.end type="italics"></emph.end> (g) <emph type="italics"></emph>T I ſhall have greater proportion unto I Y, than E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>hath to<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B: <lb></lb>And, by the ſame reaſon, the Square T I ſhall have greater proportion to the Square I Y, than<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1265"></arrow.to.target><lb></lb><emph type="italics"></emph>the Square E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>hath to the Square<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1266"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1262"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1263"></margin.target>(e) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1264"></margin.target>(f) <emph type="italics"></emph>By 8. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1265"></margin.target>(g) <emph type="italics"></emph>By 13 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1266"></margin.target>F</s></p><p type="main"> <s>But as the Square P I is to the Square Y I, ſo is the Line K R unto <lb></lb>the Line I Y] <emph type="italics"></emph>For by 11. of the firſt of our<emph.end type="italics"></emph.end> Conicks, <emph type="italics"></emph>the Square P I is equall <lb></lb>to the Rectangle contained under the Line I O, and under the Parameter; which <lb></lb>we ſuppoſed to be eqnall to the Semi-parameter; that is, the double of K R<emph.end type="italics"></emph.end>: </s></p><p type="main"> <s><arrow.to.target n="marg1267"></arrow.to.target><lb></lb><emph type="italics"></emph>But I Y is double of I O, by 33 of the ſame: And, therefore, the<emph.end type="italics"></emph.end> (h) <emph type="italics"></emph>Rectangle made of K R <lb></lb>and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1268"></arrow.to.target><lb></lb><emph type="italics"></emph>that is, to the Square P I: But as the<emph.end type="italics"></emph.end> (i) <emph type="italics"></emph>Rectangle compounded of K R and I Y is to the <lb></lb>Square I Y, ſo is the Line K R unto the Line I Y: Therefore the Line K R ſhall have unto I <lb></lb>Y, the ſame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I <lb></lb>hath to the Square I Y.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1269"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1267"></margin.target>(h) <emph type="italics"></emph>By 26. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1268"></margin.target>(i) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> Lem. </s> <s>22 <emph type="italics"></emph>of <lb></lb>the tenth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1269"></margin.target>G</s></p><p type="main"> <s>And as the Square E <foreign lang="grc">Ψ</foreign> is to the Square <foreign lang="grc">Ψ</foreign> B, ſo is half of the <lb></lb>Line K R unto the Line <foreign lang="grc">ψ</foreign> B.] <emph type="italics"></emph>For the Square E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>having been ſuppoſed equall <lb></lb>to half the Rectangle contained under the Line K R and<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B; that is, to that contained under <lb></lb>the half of K R and the Line<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B; and ſeeing that as the<emph.end type="italics"></emph.end> (k) <emph type="italics"></emph>Rectangle made of half K R<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1270"></arrow.to.target><lb></lb><emph type="italics"></emph>and of B<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>is to the Square<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B, ſo is half K R unto the Line<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B; the half of K R ſhall have <lb></lb>the ſame proportion to<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B, as the Square E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>hath to the Square<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1271"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1270"></margin.target>(k) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> Lem. </s> <s>22 <emph type="italics"></emph>of <lb></lb>the tenth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1271"></margin.target>H</s></p><p type="main"> <s>And, conſequently, I Y is leſſe than the double of <foreign lang="grc">ψ</foreign> B.] <lb></lb><emph type="italics"></emph>For, as half K R is to<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B, ſo is K R to another Line: it ſhall be<emph.end type="italics"></emph.end> (1) <emph type="italics"></emph>greater than I Y; that<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1272"></arrow.to.target><lb></lb><emph type="italics"></emph>is, than that to which K R hath leſſer proportion; and it ſhall be double of<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B: Therefore <lb></lb>I Y is leſſe than the double of<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1273"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1272"></margin.target>(l) <emph type="italics"></emph>By 10 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1273"></margin.target>K</s></p><p type="main"> <s>And I <foreign lang="grc">ω</foreign> greater than <foreign lang="grc">ψ</foreign> R.] <emph type="italics"></emph>For O having been ſuppoſed equall to B R, <lb></lb>if from B R,<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B be taken, and from O<emph.end type="italics"></emph.end> <foreign lang="grc">ω,</foreign> <emph type="italics"></emph>O I, which is leſſer than B, be taken; the <lb></lb>Remainder I<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>ſhall be greater than the Remainder<emph.end type="italics"></emph.end> <foreign lang="grc">Ψ</foreign> <emph type="italics"></emph>R.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1274"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1274"></margin.target>L</s></p><p type="main"> <s>And, therefore, F Q is equall to P M.] <emph type="italics"></emph>By the fourteenth of the fifth of<emph.end type="italics"></emph.end><lb></lb>Euclids <emph type="italics"></emph>Elements: For the Line O N is equall to B D.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1275"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1275"></margin.target>M</s></p><p type="main"> <s>But it hath been demonſtrated that P H is greater than F.] <lb></lb><emph type="italics"></emph>For it was demonſtrated that I<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>is greater than F: And P H is equall to I<emph.end type="italics"></emph.end> <foreign lang="grc">ω.</foreign></s></p><p type="main"> <s><arrow.to.target n="marg1276"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1276"></margin.target>N</s></p><p type="main"> <s>In the ſame manner we might demonſtrate the Line T H <lb></lb>to be Perpendicular unto the Surface of the Liquid.] <emph type="italics"></emph>For T<emph.end type="italics"></emph.end> <foreign lang="grc">α</foreign> <emph type="italics"></emph>is equall <lb></lb>to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated, <lb></lb>the Line T H ſhall be drawn Perpendicular unto the Liquids Surface.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1277"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1277"></margin.target>O</s></p><p type="main"> <s>Therefore, the Square P I hath leſſer proportion unto the <lb></lb>Square I Y, than the Square E <foreign lang="grc"><gap></gap></foreign> hath to the Square <foreign lang="grc">ψ</foreign> B.] <lb></lb><emph type="italics"></emph>Theſe, and other particulars of the like nature, that follow both in this and the following <lb></lb>Propoſitions, ſhall be demonſtrated by us no otherwiſe than we have done above.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1278"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1278"></margin.target>P</s></p><p type="main"> <s>Therefore Perpendiculars being drawn thorow Z and G, unto <lb></lb>the Surface of the Liquid, that are parallel to T H, it followeth <lb></lb>that the ſaid Portion ſhall not ſtay, but ſhall turn about till that its <lb></lb>Axis do make an Angle with the Waters Surface greater than that <lb></lb>which it now maketh.] <emph type="italics"></emph>For in that the Line drawn thorow G, doth fall perpendicu<lb></lb>larly towards thoſe parts which are next to L; but that thorow Z, towards thoſe next to A; <lb></lb>It is neceſſary that the Centre G do move downwards, and Z upwards: and, therefore, the <lb></lb>parts of the Solid next to L ſhall move downwards, and thoſe towards A upwards, that the <lb></lb>Axis may makea greater Angle with the Surface of the Liquid.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1279"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1279"></margin.target>Q</s></p><p type="main"> <s>For ſo ſhall I O be equall to <foreign lang="grc">ψ</foreign> B; and <foreign lang="grc">ω</foreign> I equall to I R; and <lb></lb>P H equall to F.] <emph type="italics"></emph>This plainly appeareth in the third Figure, which is added by us.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/045.jpg" pagenum="375"></pb><p type="head"> <s>PROP. IX. THE OR. IX.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid, when it <lb></lb>ſhall have its Axis greater than Seſquialter of the <lb></lb>Semi-parameter, but leſſer than to be unto the ſaid <lb></lb>Semi-parameter in proportion as fifteen to four, and <lb></lb>hath greater proportion in Gravity to the Liquid, than <lb></lb>the exceſs by which the Square made of the Axis is <lb></lb>greater than the Square made of the Exceſs, by which <lb></lb>the Axis is greater than Seſquialter of the Semi<lb></lb>parameter, hath to the Square made of the Axis, <lb></lb>being demitted into the Liquid, ſo as that its Baſe <lb></lb>be wholly within the Liquid, and being ſet inclining<lb></lb>it ſhall neither turn about, ſo as that its Axis ſtand <lb></lb>according to the Perpendicular, nor remain inclined, <lb></lb>ſave only when the Axis makes an Angle with <lb></lb>the Surface of the Liquid, equall to that aßigned <lb></lb>as before.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let there be a Portion as was ſaid; and ſuppoſe D B equall to <lb></lb>the Axis of the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion: and let B K be double to K D; and <lb></lb>K R equall to the Semi-parameter: and C B Seſquialter of <lb></lb>B R. </s> <s>And as the Portion is to the Liquid in Gravity, ſo let the Ex<lb></lb>ceſſe by which the Square B D exceeds the Square F Q be to the <lb></lb>Square B D: and let F be double to Q: It is manifeſt, therefore, <lb></lb>that the Exceſſe by which the <lb></lb><figure id="id.073.01.045.1.jpg" xlink:href="073/01/045/1.jpg"></figure><lb></lb>Square B D is greater than the <lb></lb>Square B C hath leſser proportion <lb></lb>to the Square B D, than the Exceſs <lb></lb>by which the Square B D is greater <lb></lb>than the Square F Q hath to the <lb></lb>Square B D; for B C is the Exceſs <lb></lb>by which the Axis of the Portion is <lb></lb>greater than Seſquialter of the <lb></lb>Semi-parameter: And, therefore, </s></p><p type="main"> <s><arrow.to.target n="marg1280"></arrow.to.target><lb></lb>the Square B D doth more exceed <lb></lb>the Square F Q, than doth the <lb></lb>Square B C: And, conſequently, the Line F Q is leſs than B C; <pb xlink:href="073/01/046.jpg" pagenum="376"></pb>and F leſs than B R. </s> <s>Let R <foreign lang="grc">Ψ</foreign> be equall to F; and draw <foreign lang="grc">Ψ</foreign> E <lb></lb>perpendicular to B D; which let be in power the half of that <lb></lb>which the Lines K R and <foreign lang="grc">Ψ</foreign> B containeth; and draw a Line from <lb></lb>B to E: I ſay that the Portion demitted into the Liquid, ſo as that <lb></lb>its Baſe be wholly within the Liquid, ſhall ſo ſtand, as that its Axis <lb></lb>do make an Angle with the Liquids Surface, equall to the Angle B. <lb></lb></s> <s>For let the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion be demitted into the Liquid, as hath been ſaid; <lb></lb>and let the Axis not make an Angle with the Liquids Surface, equall <lb></lb>to B, but firſt a greater: and the ſame being cut thorow the Axis <lb></lb>by a Plane erect unto the Surface of the Liquid, let the Section of <lb></lb>the Portion be A P O L, the Section of a Rightangled Cone; the <lb></lb>Section of the Surface of the Liquid <foreign lang="grc">Γ</foreign> I; and the Axis of the <lb></lb>Portion and Diameter of the Section N O; which let be cut in <lb></lb>the Points <foreign lang="grc">ω</foreign> and T, as before: and draw Y P, parallelto <foreign lang="grc">Γ</foreign> I, and <lb></lb>touching the Section in P, and MP parallel to N O, and P S perpen<lb></lb>dicular to the Axis. </s> <s>And becauſe now that the Axis of the Portion <lb></lb>maketh an <emph type="italics"></emph>A<emph.end type="italics"></emph.end>ngle with the Liquids Surface greater than the Angle <lb></lb>B, the Angle S Y P ſhall alſo be greater than the Angle B: And, <lb></lb>therefore, the Square P S hath greater proportion to the Square <lb></lb><arrow.to.target n="marg1281"></arrow.to.target><lb></lb>S Y, than the Square <foreign lang="grc">Ψ</foreign> E hath to the Square <foreign lang="grc">Ψ</foreign> B: And, for that <lb></lb>cauſe, K R hath greater proportion to S Y, than the half of K R <lb></lb>hath to <foreign lang="grc">Ψ</foreign> B: Therefore, S Y is leſs than the double of <foreign lang="grc">Ψ</foreign> B; and <lb></lb><arrow.to.target n="marg1282"></arrow.to.target><lb></lb>S O leſs than <foreign lang="grc">ψ</foreign> B: <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd, therefore, S <foreign lang="grc">ω</foreign> is greater than R <foreign lang="grc">ψ</foreign>; and <lb></lb><arrow.to.target n="marg1283"></arrow.to.target><lb></lb>P H greater than F. <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd, becauſe that the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion hath the <lb></lb>ſame proportion in Gravity unto the Liquid, that the Exceſs by <lb></lb>which the Square B D, is greater than the Square F Q, hath unto <lb></lb>the Square B D; and that as the Portion is in proportion to the <lb></lb>Liquid in Gravity, ſo is the part thereof ſubmerged unto the whole <lb></lb>Portion; It followeth that the part ſubmerged, hath the ſame <lb></lb>proportion to the whole <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion, that the Exceſs by which the <lb></lb>Square B D is greater than the Square F Q hath unto the Square <lb></lb>B D: <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd, therefore, the whole <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion ſhall have the ſame propor<lb></lb><arrow.to.target n="marg1284"></arrow.to.target><lb></lb>tion to that part which is above the <lb></lb><figure id="id.073.01.046.1.jpg" xlink:href="073/01/046/1.jpg"></figure><lb></lb>Liquid, that the Square B D hath to <lb></lb>the Square F Q: But as the whole <lb></lb>Portion is to that part which is above <lb></lb>the Liquid, ſo is the Square N O unto <lb></lb>the Square P M: Therefore, P M <lb></lb>ſhall be equall to F Q: But it <lb></lb>hath been demonſtrated, that P H is <lb></lb>greater than F. And, therefore, <lb></lb>MH ſhall be leſs than <expan abbr="q;">que</expan> and P H <lb></lb>greater than double of H M. </s> <s>Let <lb></lb>therefore, P Z be double to Z M: <pb xlink:href="073/01/047.jpg" pagenum="377"></pb>and drawing a Line from Z to T pro<lb></lb><figure id="id.073.01.047.1.jpg" xlink:href="073/01/047/1.jpg"></figure><lb></lb>long it unto G. </s> <s>The Centre of <lb></lb>Gravity of the whole Portion ſhall <lb></lb>be T; of that part which is above <lb></lb>the Liquid Z; and of the Remain<lb></lb>der which is within the Liquid, the <lb></lb>Centre ſhall be in the Line Z T pro<lb></lb>longed; let it be in G: It ſhall be <lb></lb>demonſtrated, as before, that T H <lb></lb>is perpendicular to the Surface of <lb></lb>the Liquid, and that the Lines <lb></lb>drawn thorow Z and G parallel to the ſaid T H, are alſo perpen<lb></lb>diculars unto the ſame: Therefore, the Part which is above the <lb></lb>Liquid ſhall move downwards, along that which paſseth thorow Z; <lb></lb>and that which is within it, ſhall move upwards, along that which <lb></lb>paſseth thorow G: And, therefore, the Portion ſhall not remain <lb></lb>ſo inclined, nor ſhall ſo turn about, as that its Axis be perpendicular <lb></lb><arrow.to.target n="marg1285"></arrow.to.target><lb></lb>unto the Surface of the Liquid; for the parts towards L ſhall move <lb></lb>downwards, and thoſe towards <emph type="italics"></emph>A<emph.end type="italics"></emph.end> upwards; as may appear by <lb></lb>the things already demonſtrated. </s> <s>And, if the Axis ſhould make <lb></lb>an Angle with the Surface of the Liquid, leſs than the Angle B; <lb></lb>it ſhall in like manner be demonſtrated, that the Portion will not <lb></lb><arrow.to.target n="marg1286"></arrow.to.target><lb></lb>reſt, but incline untill that its Axis do make an Angle with the <lb></lb>Surface of the Liquid, equall to the Angle B.</s></p><p type="margin"> <s><margin.target id="marg1280"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1281"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1282"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1283"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1284"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1285"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1286"></margin.target>G</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s>And, therefore, the Square B D doth more exceed the Square <lb></lb><arrow.to.target n="marg1287"></arrow.to.target><lb></lb>F Q, than doth the Square B C: And, conſequently, the Line <lb></lb>F Q, is leſs than B C; and F leſs than B R.] <emph type="italics"></emph>Becauſe the Exceſs by <lb></lb>which the Square B D exceedeth the Square B C; having leſs proportion unto the Square B D, <lb></lb>than the Exceſs by which the Square B D exceedeth the Square F Q, hath to the ſaid Square<emph.end type="italics"></emph.end>; <lb></lb>(a) <emph type="italics"></emph>the Exceſs by which the Square B D exceedeth the Square B C ſhall be leſs than the Exceſs<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1288"></arrow.to.target><lb></lb><emph type="italics"></emph>by which it exceedeth the Square F Q: Therefore, the Square F Q is leſs than the Square B C: <lb></lb>and, conſquently, the Line F Q leſs than the Line BC: But F Q hath the ſameproportion <lb></lb>to F, that B C hath to B R; for the Antecedents are each Seſquialter of their conſequents: <lb></lb>And<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>F Q being leſs than B C, F ſhall alſo be leſs than B R.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1289"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1287"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1288"></margin.target>(a) <emph type="italics"></emph>By 8. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1289"></margin.target>(b) <emph type="italics"></emph>By 14. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And, for that cauſe, K R hath greater proportion to S Y, than <lb></lb>the half of K R hath to <foreign lang="grc">ψ</foreign> B.] <emph type="italics"></emph>For K R is to S Y, as the Square P S is to the Square<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1290"></arrow.to.target><lb></lb><emph type="italics"></emph>S Y: and the half of the Line K R is to the Line<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B, as the Square E<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>is to the Square<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>B.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1290"></margin.target>B</s></p><p type="main"> <s>And S O leſs than <foreign lang="grc">ψ</foreign> B.] <emph type="italics"></emph>For S Y is double of S O.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1291"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1291"></margin.target>C</s></p><p type="main"> <s>And P H greater than F.] <emph type="italics"></emph>For P H is equall to S<emph.end type="italics"></emph.end> <foreign lang="grc">ω,</foreign> <emph type="italics"></emph>and R<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>equall to F.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1292"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1292"></margin.target>D</s></p><p type="main"> <s>And, therefore, the whole Portion ſhall have the ſame propor</s></p><p type="main"> <s><arrow.to.target n="marg1293"></arrow.to.target><lb></lb>tion to that part which is above the Liquid, that the Square B D <lb></lb>hath to the Square F Q] <emph type="italics"></emph>Becauſe that the part ſubmerged, being to the whole Portion <lb></lb>as the Exceſs by which the Square B D is greater than the Square F Q, is to the Square B D; <lb></lb>the whole Portion, Converting, ſhall be to the part thereof ſubmerged, as the Square B D is to<emph.end type="italics"></emph.end> <pb xlink:href="073/01/048.jpg" pagenum="378"></pb><emph type="italics"></emph>the Exceſs by which it exceedeth the Square F Q: And, therefore, by Converſion of Proportion, <lb></lb>the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square, <lb></lb>F <expan abbr="q;">que</expan> for the Square B D is ſo much greater than the Exceſs by which it exceedeth the Squar, <lb></lb>F Q as is the ſaid Square F <expan abbr="q.">que</expan><emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1294"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1293"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1294"></margin.target>F</s></p><p type="main"> <s>For the parts towards L ſhall move downwards, and thoſe to<lb></lb>wards A upwards.] <emph type="italics"></emph>We thus carrect theſe words, for in<emph.end type="italics"></emph.end> Tartaglia's <emph type="italics"></emph>Tranſlation it <lb></lb>is falſly, as I conceive, read<emph.end type="italics"></emph.end> Quoniam quæ ex parte L ad ſuperiora ferentur, <emph type="italics"></emph>becauſe <lb></lb>the Line thàt paſſeth thorow Z falls perpendicularly on the parts towards L, and that thorow<lb></lb>G falleth perpendicularly on the parts towards A: Whereupon the Centre Z, together with thoſe <lb></lb>parts which are towards L ſhall move downwards; and the Centre G, together with the parts <lb></lb>which are towards A upwards.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1295"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1295"></margin.target>G</s></p><p type="main"> <s>It ſhall in like manner be demonſtrated that the Portion ſhall not <lb></lb>reſt, but incline untill that its Axis do make an Angle with the <lb></lb>Surface of the Liquid, equall to the Angle B.] <emph type="italics"></emph>This may be eaſily demon<lb></lb>ſtratred, as nell from what hath been ſaid in the precedent Propoſition, as alſo from the two <lb></lb>latter Figures, by us inſerted<emph.end type="italics"></emph.end></s></p><p type="head"> <s>PROP. X. THEOR. X.</s></p><p type="main"> <s><emph type="italics"></emph>The Right Portion of a Rightangled Conoid, lighter <lb></lb>than the Liquid, when it ſhall have its Axis greater <lb></lb>than to be unto the Semiparameter, in proportion as <lb></lb>fifteen to four, being demitted into the Liquid, ſo as<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1296"></arrow.to.target><lb></lb><emph type="italics"></emph>that its Baſe touch not the ſame, it ſhall ſometimes<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1297"></arrow.to.target><lb></lb><emph type="italics"></emph>ſtand perpendicular; ſometimes inclined; and ſome<lb></lb>times ſo inclined, as that its Baſe touch the Surface <lb></lb>of the Liquid in one Point only, and that in two Po-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1298"></arrow.to.target><lb></lb><emph type="italics"></emph>ſitions; ſometimes ſo that its Baſe be more ſubmer-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1299"></arrow.to.target><lb></lb><emph type="italics"></emph>ged in the Liquid; and ſometimes ſo as that it doth <lb></lb>not in the leaſt touch the Surface of the Liquid;<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1300"></arrow.to.target><lb></lb><emph type="italics"></emph>according to the proportion that it hath to the Liquid <lb></lb>in Gravity. </s> <s>Every one of which Caſes ſhall be anon <lb></lb>demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1296"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1297"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1298"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1299"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1300"></margin.target>E</s></p><p type="main"> <s>Let there be a Portion, as hath been ſaid; and it being cut <lb></lb>thorow its Axis, by a Plane erect unto the Superficies of the <lb></lb>Liquid, let the Section be A P O L, the Section of a Right <lb></lb>angled Cone; and the Axis of the Portion and Diameter of the <lb></lb>Section B D: and let B D be cut in the Point K, ſo as that B K <lb></lb>be double of K D; and in C, ſo as that B D may have the ſame <lb></lb><arrow.to.target n="marg1301"></arrow.to.target><lb></lb>proportion to K C, as fifteen to four: It is manifeſt, therefore, <lb></lb><arrow.to.target n="marg1302"></arrow.to.target><lb></lb>that K C is greater than the Semi-parameter: Let the Semi <pb xlink:href="073/01/049.jpg" pagenum="379"></pb>parameter be equall to K R: and <lb></lb><figure id="id.073.01.049.1.jpg" xlink:href="073/01/049/1.jpg"></figure><lb></lb><arrow.to.target n="marg1303"></arrow.to.target><lb></lb>let D S be Seſquialter of K R: but <lb></lb>S B is alſo Seſquialter of B R: <lb></lb>Therefore, draw a Line from A to <lb></lb>B; and thorow C draw C E Per<lb></lb>pendicular to B D, cutting the Line <lb></lb>A B in the Point E; and thorow E <lb></lb>draw E Z parallel unto B D. Again, <lb></lb>A B being divided into two equall <lb></lb>parts in T, draw T H parallel to the <lb></lb>ſame B D: and let Sections of <lb></lb>Rightangled Cones be deſcribed, A E I about the Diameter E Z; <lb></lb>and A T D about the Diameter T H; and let them be like to the <lb></lb><arrow.to.target n="marg1304"></arrow.to.target><lb></lb>Portion A B L: Now the Section of the Cone A E I, ſhall paſs <lb></lb><arrow.to.target n="marg1305"></arrow.to.target><lb></lb>thorow K; and the Line drawn from R perpendicular unto B D, <lb></lb>ſhall cut the ſaid A E I; let it cut it in the Points Y G: and <lb></lb>thorow Y and G draw P Y Q and O G N parallels unto B D, and <lb></lb>cutting A T D in the Points F and X: laſtly, draw P <foreign lang="grc">Φ</foreign> and O X <lb></lb>touching the Section A P O L in the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>oints P and O. </s> <s>In regard, <lb></lb><arrow.to.target n="marg1306"></arrow.to.target><lb></lb>therefore, that the three <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions A P O L, A E I, and A T D are <lb></lb>contained betwixt Right Lines, and the Sections of Rightangled <lb></lb>Cones, and are right alike and unequall, touching one another, upon <lb></lb>one and the ſame Baſe; and N X G O being drawn from the <lb></lb><emph type="italics"></emph>P<emph.end type="italics"></emph.end>oint N upwards, and Q F Y P from Q: O G ſhall have to G X <lb></lb>a proportion compounded of the proportion, that I L hath to L A, <lb></lb>and of the proportion that A D hath to DI: But I L is to L A, <lb></lb>as two to five: And C B is to B D, as ſix to fifteen; that is, as two <lb></lb><arrow.to.target n="marg1307"></arrow.to.target><lb></lb>to five: And as C B is to B D, ſo is <emph type="italics"></emph>E B to B A<emph.end type="italics"></emph.end>; and D Z to <lb></lb><arrow.to.target n="marg1308"></arrow.to.target><lb></lb>D A: And of D Z and D A, L I and L A are double: and A D <lb></lb><arrow.to.target n="marg1309"></arrow.to.target><lb></lb>is to D I, as five to one: <emph type="italics"></emph>B<emph.end type="italics"></emph.end>ut the proportion compounded of the <lb></lb>proportion of two to five, and of the proportion of five to one, is <lb></lb><arrow.to.target n="marg1310"></arrow.to.target><lb></lb>the ſame with that of two to one: and two is to one, in double <lb></lb>proportion: Therefore, O G is double of GX: and, in the ſame <lb></lb>manner is P Y proved to be double of Y F: Therefore, ſince that <lb></lb>D S is Seſquialter of K R; <emph type="italics"></emph>B S<emph.end type="italics"></emph.end> ſhall be the Exceſs by which the <lb></lb>Axis is greater than Seſquialter of the Semi-parameter. </s> <s>If there<lb></lb>fore, the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion have the ſame proportion in Gravity unto the <lb></lb>Liquid, as the Square made of the Line <emph type="italics"></emph>B S,<emph.end type="italics"></emph.end> hath to the Square <lb></lb>made of <emph type="italics"></emph>B D,<emph.end type="italics"></emph.end> or greater, being demitted into the Liquid, ſo as hat <lb></lb>its <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe touch not the Liquid, it ſhall ſtand erect, or perpendicular: <lb></lb>For it hath been demonſtrated above, that the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion whoſe <lb></lb><arrow.to.target n="marg1311"></arrow.to.target><lb></lb>Axis is greater than Seſquialter of the Semi-parameter, if it have <lb></lb>not leſser proportion in Gravity unto the Liquid, than the Square <pb xlink:href="073/01/050.jpg" pagenum="380"></pb>made of the Exceſs by which the Axis is greater than Seſquialter <lb></lb>of the Semi-parameter, hath to the Square made of the Axis, being <lb></lb>demitted into the Liquid, ſo as hath been ſaid, it ſhall ſtand erect, <lb></lb>or <emph type="italics"></emph>P<emph.end type="italics"></emph.end>erpendicular.</s></p><p type="margin"> <s><margin.target id="marg1301"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1302"></margin.target>G</s></p><p type="margin"> <s><margin.target id="marg1303"></margin.target>H</s></p><p type="margin"> <s><margin.target id="marg1304"></margin.target>K</s></p><p type="margin"> <s><margin.target id="marg1305"></margin.target>L</s></p><p type="margin"> <s><margin.target id="marg1306"></margin.target>M</s></p><p type="margin"> <s><margin.target id="marg1307"></margin.target>N</s></p><p type="margin"> <s><margin.target id="marg1308"></margin.target>O</s></p><p type="margin"> <s><margin.target id="marg1309"></margin.target>P</s></p><p type="margin"> <s><margin.target id="marg1310"></margin.target>Q</s></p><p type="margin"> <s><margin.target id="marg1311"></margin.target>R</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"></emph>The particulars contained in this Tenth Propoſition, are divided by<emph.end type="italics"></emph.end> Archimedes <lb></lb><emph type="italics"></emph>into five Parts and Concluſions, each of which he proveth by a diſtinct Demonſtration.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1312"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1312"></margin.target>A</s></p><p type="main"> <s>It ſhall ſometimes ſtand perpendicular.] <emph type="italics"></emph>This is the firſt Concluſion, the <lb></lb>Demonstration of which he hath ſubjoyned to the Propoſition.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1313"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1313"></margin.target>B</s></p><p type="main"> <s>And ſometimes ſo inclined, as that its Baſe touch the Surface <lb></lb>of the Liquid, in one Point only.] <emph type="italics"></emph>This is demonſtrated in the third Con<lb></lb>cluſion.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Sometimes, ſo that its Baſe be moſt ſubmerged in the Liquid.] </s></p><p type="main"> <s><arrow.to.target n="marg1314"></arrow.to.target><lb></lb><emph type="italics"></emph>This pertaineth unto the fourth Concluſion.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1314"></margin.target>C</s></p><p type="main"> <s><emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd, ſometimes, ſo as that it doth not in the leaſt touch the Sur<lb></lb><arrow.to.target n="marg1315"></arrow.to.target><lb></lb>face of the Liquid.] <emph type="italics"></emph>This it doth hold true two wayes, one of which is explained is <lb></lb>the ſecond, and the other in the fifth Concluſion.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1315"></margin.target>D</s></p><p type="main"> <s>According to the proportion, that it hath to the Liquid in Gra<lb></lb><arrow.to.target n="marg1316"></arrow.to.target><lb></lb>vity. </s> <s>Every one of which Caſes ſhall be anon demonſtrated.] <lb></lb><emph type="italics"></emph>In<emph.end type="italics"></emph.end> Tartaglia's <emph type="italics"></emph>Verſion it is rendered, to the confuſion of the ſence,<emph.end type="italics"></emph.end> Quam autem pro<lb></lb>portionem habeant ad humidum in Gravitate fingula horum demonſtrabuntur.</s></p><p type="margin"> <s><margin.target id="marg1316"></margin.target>E</s></p><p type="main"> <s>It is manifeſt, therefore, that K C is greater than the Semi<lb></lb><arrow.to.target n="marg1317"></arrow.to.target><lb></lb>parameter] <emph type="italics"></emph>For, ſince B D hath to K C the ſame proportion, as fifteen to four, and <lb></lb>hath unto the Semi-parameter greater proportion; (a) the Semi-parameter ſhall be leſs<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1318"></arrow.to.target><lb></lb><emph type="italics"></emph>than K C.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1317"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1318"></margin.target>(a) <emph type="italics"></emph>By 10. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Let the Semi-parameter be equall to KR.] <emph type="italics"></emph>We have added theſe words,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1319"></arrow.to.target><lb></lb><emph type="italics"></emph>which are not to be found in<emph.end type="italics"></emph.end> Tartaglia.</s></p><p type="margin"> <s><margin.target id="marg1319"></margin.target>G</s></p><p type="main"> <s>But S B is alſo Seſquialter of BR.] <emph type="italics"></emph>For, D B is ſuppoſed Seſquialter of<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1320"></arrow.to.target><lb></lb><emph type="italics"></emph>B K; and D S alſo is Seſquialter of K R: Wherefore as<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>the whole D B, is to the whole <lb></lb>B K, ſo is the part D S to the part K R. Therefore, the Remainder S B, is alſo to the<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1321"></arrow.to.target><lb></lb><emph type="italics"></emph>Remainder B R, as D B is to B K.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1320"></margin.target>H</s></p><p type="margin"> <s><margin.target id="marg1321"></margin.target>(b) <emph type="italics"></emph>By 19 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd let them be like to the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion <emph type="italics"></emph>A B L.<emph.end type="italics"></emph.end>] Apollonius <emph type="italics"></emph>thus defineth<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1322"></arrow.to.target><lb></lb><emph type="italics"></emph>like Portions of the Sections of a Cone, in<emph.end type="italics"></emph.end> Lib. 6. Conicornm, <emph type="italics"></emph>as<emph.end type="italics"></emph.end> Eutocius <emph type="italics"></emph>writeth<emph.end type="italics"></emph.end> ^{*}; <lb></lb><arrow.to.target n="marg1323"></arrow.to.target><lb></lb><foreign lang="grc">ὄν οἱ̄ς ἀχ δεισω̄ν ὄν ἑχάσῳ ωαραλλήλων τη̄ <35>ὰσει, ἵσων τὸ πλη̄ο<34>, αἱ παράλληλος, καὶ ἁι <35>άσεις ωρὸς τάς αποτρμ<gap></gap><lb></lb>νομένας ἀπὸ <gap></gap> διαμέτσων τω̄ς κορυφαῑς ἐν τοῑς ἀντοῑς λόγοις εἰσι, καὶ αἱ ἀποτεμνόμεναι ωρὸς τὰς ἀ τεμνομίνασ<gap></gap></foreign><lb></lb><emph type="italics"></emph>that is,<emph.end type="italics"></emph.end> In both of which an equall number of Lines being drawn parallel to the <lb></lb>Baſe; the parallel and the Baſes have to the parts of the Diameters, cut off from <lb></lb>the Vertex, the ſameproportion: as alſo, the parts cut off, to the parts cut off. <lb></lb><emph type="italics"></emph>Now the Lines parallel to the Baſes are drawn, as I ſuppoſe, by making a Rectilineall Figure (cal-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1324"></arrow.to.target><lb></lb><emph type="italics"></emph>led)<emph.end type="italics"></emph.end> Signally inſcribed [<foreign lang="grc">χη̄μα γιωρίμως ἐγν̀<36>ρόμενον</foreign>] <emph type="italics"></emph>in both portions, having an equall num<lb></lb>ber of Sides in both. </s> <s>Therefore, like Portions are cut off from like Sections of a Cone; and <lb></lb>their Diameters, whether they be perpendicular to their Baſes, or making equall Angles with their <lb></lb>Baſes, have the ſame proportion unto their Baſes.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1325"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1322"></margin.target>K</s></p><p type="margin"> <s><margin.target id="marg1323"></margin.target>* <emph type="italics"></emph>Upon prop. 3 lib.<emph.end type="italics"></emph.end> 2 <lb></lb>Archim. <emph type="italics"></emph>Æqui<lb></lb>pond.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1324"></margin.target><emph type="italics"></emph>Vide<emph.end type="italics"></emph.end> Archim, <emph type="italics"></emph>ante <lb></lb>prop. 2. lib. 2. <lb></lb>Æquipond.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1325"></margin.target>L</s></p><p type="main"> <s>Now the Section of the Cone <emph type="italics"></emph>A E I<emph.end type="italics"></emph.end> ſhall paſs thorow K.] <lb></lb><emph type="italics"></emph>For, if it be poſſible, let it not paſs thorow K, but thorow ſome other Point of the Line D B, as <lb></lb>thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, whoſe <lb></lb>Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth <lb></lb>both A E and A I; A E in B, and A I in D; D B ſhall have to B V, the ſame proportion<emph.end type="italics"></emph.end> <pb xlink:href="073/01/051.jpg" pagenum="381"></pb><emph type="italics"></emph>that A Z hath to Z D; by the fourth Propoſition of<emph.end type="italics"></emph.end> Archimedes, De quadratura Para<lb></lb>bolæ: <emph type="italics"></emph>But A Z is Seſquialter of Z D; for it is as three to two, as we ſhallanon demon-<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1326"></arrow.to.target><lb></lb><emph type="italics"></emph>ſtrate: Therefore D B is Seſquialter of B V; but D B and B K are Seſquialter: <lb></lb>And, therefore, the Lines<emph.end type="italics"></emph.end> (c) <emph type="italics"></emph>B V and B K are equall: Which is imposſible: <lb></lb>Therefore the Section of the Right-angled Cone A E I, ſhall paſs thorow the Point K; which <lb></lb>we would demonstrate.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1326"></margin.target>(c) <emph type="italics"></emph>By 9 of the <lb></lb>fifth,<emph.end type="italics"></emph.end></s></p><p type="main"> <s>In regard, therefore, that the three <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions A P O L, A E I <lb></lb><arrow.to.target n="marg1327"></arrow.to.target><lb></lb>and A T D are contained betwixt Right Lines and the Sections <lb></lb>of Right-angled Cones, and are Right, alike and unequall, <lb></lb>touching one another, upon one and the ſame Baſe.] <emph type="italics"></emph>After theſe words,<emph.end type="italics"></emph.end><lb></lb>upon one and the ſame Baſe, <emph type="italics"></emph>we may ſee that ſomething is obliterated, that is to be <lb></lb>deſired: and for the Demonſtration of theſe particulars, it is requiſite in this place to <lb></lb>premiſe ſome things: which will alſo be neceſſary unto the things that follow.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1327"></margin.target>M</s></p><p type="head"> <s>LEMMA. I.</s></p><p type="main"> <s>Let there be a Right <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine A B; and let it be cut by two <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines, <lb></lb>parallel to one another, A C and D E, ſo, that as <emph type="italics"></emph>A B<emph.end type="italics"></emph.end> is to <lb></lb>B D. ſo <emph type="italics"></emph>A C<emph.end type="italics"></emph.end> may be to D E. </s> <s>I ſay that the Line that con<lb></lb>joyneth the Points C and B ſhall likewiſe paſs by E.</s></p><figure id="id.073.01.051.1.jpg" xlink:href="073/01/051/1.jpg"></figure><p type="main"> <s><emph type="italics"></emph>For, if poſſible, let it not paſs by E, but either <lb></lb>above or below it. </s> <s>Let it first paſs below it, <lb></lb>as by F. </s> <s>The Triangles A B C and D B F ſhall <lb></lb>be alike: And, therefore, as<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>A B is to B D,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1328"></arrow.to.target><lb></lb><emph type="italics"></emph>ſo is A C to D F: But as A B is to B D, ſo was <lb></lb>A C to D E: Therefore<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>D F ſhall be equall to<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1329"></arrow.to.target><lb></lb><emph type="italics"></emph>D E: that is, the part to the whole: Which is <lb></lb>abſurd. </s> <s>The ſame abſurditie will follow, if the <lb></lb>Line C B be ſuppoſed to paſs above the Point E: <lb></lb>And, therefore, C B muſt of necesſity paſs thorow <lb></lb>E: Which was required to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1328"></margin.target>(a) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1329"></margin.target>(b) <emph type="italics"></emph>By 9. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA. II.</s></p><p type="main"> <s>Let there be two like <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions, contained betwixt Right Lines, <lb></lb>and the Sections of Right-angled Cones; A B C the great<lb></lb>er, whoſe Diameter let be B D; and E F C the leſser, whoſe <lb></lb>Diameter let be F G: and, let them be ſo applyed to one <lb></lb>another, that the greater include the leſser; and let their <lb></lb>Baſes A C and E C be in the ſame Right Line, that the ſame <lb></lb>Point C, may be the term or bound of them both: And, <lb></lb>then in the Section A B C, take any Point, as H; and draw <lb></lb>a Line from H to C. </s> <s>I ſay, that the Line H C, hath to that <lb></lb>part of it ſelf, that lyeth betwixt C and the Section E F C, the <lb></lb>ſame proportion that A C hath to C E.</s></p><p type="main"> <s><emph type="italics"></emph>Draw B C, which ſhall paſs thorow F, For, in regard, that the Portions are alike, the <lb></lb>Diameters with the Baſes contain equall Angles: And, therefore, B D and F G are parallel <lb></lb>to one another: and B D is to A C, as F G it to E C: and,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>B D is to F G, as <lb></lb>A C is to C E; that is,<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>as their halfes D C to C G; therefore, it followeth, by the<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1330"></arrow.to.target><lb></lb><emph type="italics"></emph>preceding Lemma, that the Line B C ſhall paſs by the Point F. Moreover, from the Point <lb></lb>H unto the Diameter B D, draw the Line H K, parallel to the Baſe A C: and, draw a Line<emph.end type="italics"></emph.end> <pb xlink:href="073/01/052.jpg" pagenum="382"></pb><figure id="id.073.01.052.1.jpg" xlink:href="073/01/052/1.jpg"></figure><lb></lb><emph type="italics"></emph>from K to C, cutting the Diameter F G in L: <lb></lb>and, thorow L, unto the Section E F. G, on the <lb></lb>part E, draw the Line L M, parallel unto the <lb></lb>ſame Baſe A C. And, of the Section A B C, <lb></lb>let the Line B N be the Parameter; and, of the <lb></lb>Section E F C, let F O be the Parameter. </s> <s>And, <lb></lb>becauſe the Triangles C B D and C F G are alike<emph.end type="italics"></emph.end>; <lb></lb>(b) <emph type="italics"></emph>therefore, as B C is to C F, ſo ſhall D C be<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1331"></arrow.to.target><lb></lb><emph type="italics"></emph>to C G, and B D to F G. Again, becauſe the <lb></lb>Triangles C K B and C L F, are alſo alike to <lb></lb>one another; therefore, as B C is to C F, that is, <lb></lb>as B D is to F G, ſo ſhall K C be to C L, and B K to F L: Wherefore, K C to C L, and,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1332"></arrow.to.target><lb></lb><emph type="italics"></emph>B K to F L, are as D C to C G; that is,<emph.end type="italics"></emph.end> (c) <emph type="italics"></emph>as their duplicates A C and C E: But as <lb></lb>B D is to F G, ſo is D C to C G; that is, A D to E G: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as B D is to <lb></lb>A D, ſo is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11 <lb></lb>of our firſt of<emph.end type="italics"></emph.end> Conicks: <emph type="italics"></emph>Therefore, the<emph.end type="italics"></emph.end> (d) <emph type="italics"></emph>three Lines B D, A D and B N are<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1333"></arrow.to.target><lb></lb><emph type="italics"></emph>Proportionalls. </s> <s>By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle <lb></lb>G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is <lb></lb>to A D, ſo is F G to E G: And, therefore, as A D is to B N, ſo is E G to F O:<emph.end type="italics"></emph.end> Ex equali, <lb></lb><emph type="italics"></emph>therefore, as D B is to B N, ſo is G F to F O: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as D B is to G F, ſo is <lb></lb>B N to F O: But as D B is to G F, ſo is B K to F L: Therefore, B K is to F L, as <lb></lb>B N is to F O: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as B K is to B N, ſo is F L to F O. Again, <lb></lb>becauſe the<emph.end type="italics"></emph.end> (e) <emph type="italics"></emph>Square H K is equall to the Rectangle B N; and the Square M L, equall<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1334"></arrow.to.target><lb></lb><emph type="italics"></emph>to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals: <lb></lb>and F L, L M, and F O ſhall alſo be Proportionals: And, therefore,<emph.end type="italics"></emph.end> (f) <emph type="italics"></emph>as the Line<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1335"></arrow.to.target><lb></lb><emph type="italics"></emph>B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the <lb></lb>Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M: <lb></lb>Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1336"></arrow.to.target><lb></lb><emph type="italics"></emph>B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,<emph.end type="italics"></emph.end><lb></lb>(g) <emph type="italics"></emph>as the Line B K is to the Line K H, ſo is the Line F L to L M: And,<emph.end type="italics"></emph.end> Permutando, <lb></lb><emph type="italics"></emph>as B K is to F L, ſo is K H to L M: But B K was to F L, as K C to C L: Therefore, <lb></lb>K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that <lb></lb>the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is, <lb></lb>as A C to C E, ſo is H C to C M; that is, to the ſame part of it ſelf, that lyeth betwixt C and <lb></lb>the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth <lb></lb>in other Lines, that are produced from the Point C, and the Sections E B C. And, that <lb></lb>B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G; <lb></lb>that is, as their Duplicates A C to C E.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1330"></margin.target>(a) <emph type="italics"></emph>By 15. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1331"></margin.target>(b) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1332"></margin.target>(c) <emph type="italics"></emph>By 15. of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1333"></margin.target>(d) <emph type="italics"></emph>By 17. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1334"></margin.target>(e) <emph type="italics"></emph>By 11 of our <lb></lb>firſt of<emph.end type="italics"></emph.end> Conicks.</s></p><p type="margin"> <s><margin.target id="marg1335"></margin.target>(f) <emph type="italics"></emph>By<emph.end type="italics"></emph.end> Cor. <emph type="italics"></emph>of 20. <lb></lb>of the ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1336"></margin.target>(g) <emph type="italics"></emph>By 23. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>From whence it is manifeſt, that all Lines ſo drawn, ſhall be cut by the <lb></lb>ſaid Section in the ſame proportion. </s> <s>For, by Diviſion and Converſion, <lb></lb>C M is to M H, and C F to F B, as C E to E A.</s></p><p type="head"> <s>LEMMA. III.</s></p><p type="main"> <s>And, hence it may alſo be proved, that the Lines which are <lb></lb>drawn in like Portions, ſo, as that with the Baſes, they con<lb></lb>tain equall Angles, ſhall alſo cut off like Portions; that is, <lb></lb>as in the foregoing Figure, the Portions H B C and M F C, <lb></lb>which the Lines C H and C M do cut off, are alſo alike to <lb></lb>each other.</s></p><p type="main"> <s><emph type="italics"></emph>For let C H and C M be divided in the midst in the Points P and <expan abbr="q;">que</expan> and thorow thoſe <lb></lb>Points draw the Lines R P S and T Q V parallel to the Diameters. </s> <s>Of the Portion <lb></lb>H S C the Diameter ſhall be P S, and of the Portion M V C the Diameter ſhall be<emph.end type="italics"></emph.end> <pb xlink:href="073/01/053.jpg" pagenum="383"></pb><emph type="italics"></emph>Q V. And, ſuppoſe that as the Square C R is to the Square C P, ſo is the Line B N unto <lb></lb>another Line; which let be S X: And, as the Square C T is to the Square C Q ſo let F O <lb></lb>be to V Y. </s> <s>Now it is manifeſt, by the things which we have demonſtrated, in our Commentaries, <lb></lb>upon the fourth Propoſition of<emph.end type="italics"></emph.end> Archimedes, De Conoidibus & Spheæroidibus, <emph type="italics"></emph>that the <lb></lb>Square C P is equall to the Rectangle P S X; and alſo, that the Square C Q is equall to <lb></lb>the Rectangle Q V Y; that is, the Lines S X and V Y, are the Parameters of the Sections H S C <lb></lb>and M V C: But ſince the Triangles C P R and C Q T are alike; C R ſhall have to C P, the <lb></lb>ſame Proportion that C T hath to C Q: And, therefore, the<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>Square C R ſhall have<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1337"></arrow.to.target><lb></lb><emph type="italics"></emph>to the Square C P, the ſame proportion that the<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.053.1.jpg" xlink:href="073/01/053/1.jpg"></figure><lb></lb><emph type="italics"></emph>Square C T hath to the Square C Q: There<lb></lb>fore, alſo, the Line B N ſhall be to the Line <lb></lb>S X, as the Line F O is to V Y: But H C was <lb></lb>to C M, as A C to C E: And, therefore, alſo, <lb></lb>their halves C P and C Q, are alſo to one <lb></lb>another, as A D and E G: And.<emph.end type="italics"></emph.end> Permu<lb></lb>tando, <emph type="italics"></emph>C P is to A D, as C Q is to E G: <lb></lb>But it hath been proved, that A D is to B N, <lb></lb>as E G to F O; and B N to S X, as F O to <lb></lb>V Y: Therefore,<emph.end type="italics"></emph.end> exæquali, <emph type="italics"></emph>C P ſhall be <lb></lb>to S X, as C Q is to V Y. And, ſince the <lb></lb>Square C P is equall to the Rectangle P S X, and the Square C Q to the Rectangle Q V Y, <lb></lb>the three Lines S P, PC and S X ſhall be proportionalls, and V Q, Q C and V Y ſhal be <lb></lb>Proportionalls alſo: And therefore alſo S P ſhall be to P C as V Q to Q C And as P C <lb></lb>is to C H, ſo ſhall Q C. be to C M: Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>as S P the Diameter of the <lb></lb>Portion H S C is to its Baſe C H, ſo is V Q the Diameter of the portion M V S the <lb></lb>Baſe C M; and the Angles which the Diameter with the Baſes do contain, are equall; and the <lb></lb>Lines S P and V Q are parallel: Therefore the Portions, alſo, H S C and M V C ſhall be alike: <lb></lb>Which was propoſed to be demonſtrated<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1337"></margin.target>(a) <emph type="italics"></emph>By 22. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA. IV.</s></p><p type="main"> <s><emph type="italics"></emph>L<emph.end type="italics"></emph.end>et there be two <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines A <emph type="italics"></emph>B<emph.end type="italics"></emph.end> and C D; and let them be cut in the <lb></lb>Points E and F, ſo that as A E is to E B, C F may be to F D: <lb></lb>and let them be cut again in two other Points G and H; and <lb></lb>let C H be to H D, as A G is to G B. </s> <s>I ſay that C F ſhall be to <lb></lb>F H as A E is E G.</s></p><p type="main"> <s><emph type="italics"></emph>For in regard that as A E is to E B, ſo is C F to F D; it followeth that, by Compounding, <lb></lb>as A B is to E B, ſo ſhall C D be to F D. Again, ſince that as A G is to G B, ſo is C H, to <lb></lb>H D; it followeth that, by Compounding and Converting, as G B is to A B, ſo ſhall H D be<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.053.2.jpg" xlink:href="073/01/053/2.jpg"></figure><lb></lb><emph type="italics"></emph>C D: Therefore,<emph.end type="italics"></emph.end> ex æquali, <emph type="italics"></emph>and Converting as E B <lb></lb>is to G B, ſo ſhall F D be to H D; And, by Conver<lb></lb>ſion of Propoſition, as E B is to E G, ſo ſhall F D <lb></lb>be to F H: But as A E is to E B, ſo is C F to F D:<emph.end type="italics"></emph.end><lb></lb>Ex æquali, <emph type="italics"></emph>therefore, as A E is to E G, ſo <lb></lb>ſhall CF be to F H.<emph.end type="italics"></emph.end> Again, another way. <emph type="italics"></emph>Let <lb></lb>the Lines A B and C D be applyed to one another, <lb></lb>ſo as that they doe make an Angle at the parts A and C; <lb></lb>and let A and C be in one and the ſame Point: then <lb></lb>draw Lines from D to B, from H to G, and from F to E. </s> <s>And ſince that as A E is to E B, <lb></lb>ſo is C F, that is A F to F D; therefore F E ſhall be parallel to D B<emph.end type="italics"></emph.end>; (a) <emph type="italics"></emph>and likewiſe<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1338"></arrow.to.target><lb></lb><emph type="italics"></emph>H G ſhall be parallel to D B; for that A H is to H D, as A G to G B<emph.end type="italics"></emph.end>: (b) <emph type="italics"></emph>Therefore F E <lb></lb>and H G are parallel to each other: And conſequently, as A E is to E G, ſo is A H, that is,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1339"></arrow.to.target><lb></lb><emph type="italics"></emph>C F to F H: Which was to be demonſtrated.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/054.jpg" pagenum="384"></pb><p type="margin"> <s><margin.target id="marg1338"></margin.target>(a) <emph type="italics"></emph>By 2. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1339"></margin.target>(b) <emph type="italics"></emph>By 30 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA. V.</s></p><p type="main"> <s>Again, let there be two like Portions, contained betwixt Right <lb></lb>Lines and the Sections of Right-angled Cones, as in the fore<lb></lb>going figure, A B C, whoſe Diameter is B D; and E F C, <lb></lb>whoſe Diameter is F G; and from the Point E, draw the <lb></lb>Line E H parallel to the Diameters B D and F G; and let it <lb></lb>cut the Section A B C in K: and from the Point C draw C H <lb></lb>touching the Section A B C in C, and meeting with the Line <lb></lb>E H in H; which alſo toucheth the Section E F C in the ſame <lb></lb>Point C, as ſhall be demonſtrated: I ſay that the Line drawn <lb></lb>from C <emph type="italics"></emph>H<emph.end type="italics"></emph.end> unto the Section E F C ſo as that it be parallel to <lb></lb>the Line E H, ſhall be divided in the ſame proportion by the <lb></lb>Section A B C, in which the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine C A is divided by the Section <lb></lb>E F C; and the part of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine C A which is betwixt the <lb></lb>two Sections, ſhall anſwer in proportion to the part of the Line <lb></lb>drawn, which alſo falleth betwixt the ſame Sections: that is, <lb></lb>as in the foregoing Figure, if D B be produced untill it meet <lb></lb>with C H in L, that it may interſect the Section E F C in the <lb></lb>Point M, the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine <emph type="italics"></emph>L<emph.end type="italics"></emph.end> B ſhall have to B M the ſame proportion <lb></lb>that C E hath to E A.</s></p><p type="main"> <s><emph type="italics"></emph>For let G F be prolonged untill it meet the ſame Line C H in N, cutting the Section A B C <lb></lb>in O; and drawing a Line from B to C, which ſhall paſſe by F, as hath been ſhewn, the<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.054.1.jpg" xlink:href="073/01/054/1.jpg"></figure><lb></lb><emph type="italics"></emph>Triangles C G F and C D B ſhall be alike; as <lb></lb>alſo the Triangles C F N and C B L: Wherefore<emph.end type="italics"></emph.end><lb></lb>(a) <emph type="italics"></emph>as G F is to D B, ſo ſhall C F b to C B:<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1340"></arrow.to.target><lb></lb><emph type="italics"></emph>And as<emph.end type="italics"></emph.end> (b) <emph type="italics"></emph>C F is to C B, ſo ſhall F N be <lb></lb>to B L: Therefore G F ſhall be to D B, as F N<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1341"></arrow.to.target><lb></lb><emph type="italics"></emph>to B L: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>G F ſhall be to <lb></lb>F N, as D B to B L: But D B is equall to <lb></lb>B L, by 35 of our Firſt Book of<emph.end type="italics"></emph.end> Conicks: <lb></lb><emph type="italics"></emph>Therefore<emph.end type="italics"></emph.end> (c) <emph type="italics"></emph>G F alſo ſhall be equall to F N:<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1342"></arrow.to.target><lb></lb><emph type="italics"></emph>And by 33 of the ſame, the Line C H touch<lb></lb>eth the Section E F C in the ſame Point. </s> <s>There<lb></lb>fore, drawing a Line from C to M, prolong it <lb></lb>untill it meet with the Section A B C in P; and <lb></lb>from P unto A C draw P Q parallel to B D. <lb></lb>Becauſe, now, that the Line C H toucheth the <lb></lb>Section E F C in the Point C; L M ſhall have <lb></lb>the ſame proportion to M D that C D hath to D E, <lb></lb>by the Fifth Propoſition of<emph.end type="italics"></emph.end> Archimedes <emph type="italics"></emph>in his <lb></lb>Book<emph.end type="italics"></emph.end> De Quadratura Patabolæ: <emph type="italics"></emph>And by <lb></lb>reaſon of the Similitude of the Triangles C M D <lb></lb>and C P Q, as C M is to C D, ſo ſhall C P <lb></lb>be to C Q: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as C M is to <lb></lb>C P, ſo ſhall C D be to C Q: But as C M is to C P, ſo is C E to C A,; as we have but <lb></lb>even now demonſtrated: And therefore, as C E is to C A, ſo is C D to C <expan abbr="q;">que</expan> that is as the <lb></lb>whole is to the whole, ſo is the part to the part: The remainder, therefore, D E is to the <lb></lb>Remainder Q A, as C E is to C A; that is, as C D is to C Q: And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>C D <lb></lb>is to D E, as C Q is to Q A: And L M is alſo to M D, as C D to D E: Therefore L M is<emph.end type="italics"></emph.end> <pb xlink:href="073/01/055.jpg" pagenum="385"></pb><emph type="italics"></emph>to M D, as C Q to Q A: But L B is to B D, by 5 of<emph.end type="italics"></emph.end> Archimedes, <emph type="italics"></emph>before recited, as C D <lb></lb>to D A: It is manifeſt therefore, by the precedent Lemma, that C D is to D Q, as L B is to <lb></lb>B M: But as C D is to D Q, ſo is C M to M P: Therefore L B is to B M, as C M to M P:<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1343"></arrow.to.target><lb></lb><emph type="italics"></emph>And it haveing been demonſtrated, that C M is to M P, as C E to E A; L B ſhall be to B M,<lb></lb>as C E to E A. </s> <s>And in like manner it ſhall be demonstrated that ſo is N O to O F; as alſo the <lb></lb>Remainders. </s> <s>And that alſo H K is to K E, as C E to E A, doth plainly appeare by the ſame<emph.end type="italics"></emph.end><lb></lb>5. <emph type="italics"></emph>of<emph.end type="italics"></emph.end> Archimedes<emph type="italics"></emph>: Which is that that we propounded to be demonſtrated.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1340"></margin.target>(a) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1341"></margin.target>(b) <emph type="italics"></emph>By 11 of the <lb></lb>fifth,<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1342"></margin.target>(c) <emph type="italics"></emph>By 14 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1343"></margin.target><emph type="italics"></emph>By 2. of the ſixth<emph.end type="italics"></emph.end></s></p><p type="head"> <s>LEMMA. VI.</s></p><p type="main"> <s>And, therefore, let the things ſtand as above; and deſcribe <lb></lb>yet another like Portion, contained betwixt a Right Line, and <lb></lb>the Section of the Rightangled Cone D R C, whoſe Diameter <lb></lb>is R S, that it may cut the Line F G in T; and prolong S R <lb></lb>unto the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine C H in V, which meeteth the Section A B C in <lb></lb>X, and E F C in Y. </s> <s>I ſay, that B M hath to M D, a propor<lb></lb>tion compounded of the proportion that E A hath to A C; <lb></lb>and of that which C D hath to D E.</s></p><p type="main"> <s><emph type="italics"></emph>For, we ſhall firſt demonſtrate, that the Line C H toucheth the Section D R C in the <lb></lb>Point C; and that L M is to M D, as alſo N F to F T, and V Y to Y R, as C D is to E D. <lb></lb>And, becauſe now that L B is to B M, as C E is to E A; therefore, Compounding and Conver<lb></lb>ting, B M ſhall be to L M, as E A to A C: And, as L M is to M D, ſo ſhall C D be to <lb></lb>D E: The proportion, therefore, of B M to M D, is compounded of the proportion that <lb></lb>B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion <lb></lb>of B M to M D, ſhall alſo be compounded of the proportion that E A hath to A C, and of <lb></lb>that which C D hath to D E. </s> <s>In the ſame manner it ſhal be demonſtrated, that O F hath to <lb></lb>F T, and alſo X Y to Y R, a proportion compounded of thoſe ſame proportions; and ſo in <lb></lb>the reſt: Which was to be demonstrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>By which it appeareth that the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ines ſo drawn; which fall betwixt <lb></lb>the Sections A B C and D R C, ſhall be divided by the Section E F C <lb></lb>in the ſame Proportion.</s></p><p type="main"> <s>And C B is to B D, as ſix to fifteen.] <emph type="italics"></emph>For we have ſuppoſed that B K is<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1344"></arrow.to.target><lb></lb><emph type="italics"></emph>double of K D: Wherefore, by Compoſition B D ſhall be to K D as three to one; that is, as <lb></lb>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: <lb></lb>And, by Converſion of proportion and Convert<lb></lb>ing, C B is to B D, as ſix to ſifteen.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1344"></margin.target>N</s></p><figure id="id.073.01.055.1.jpg" xlink:href="073/01/055/1.jpg"></figure><p type="main"> <s>And as C B is to B D, ſo is <lb></lb><arrow.to.target n="marg1345"></arrow.to.target><lb></lb>E B to B A; and D Z to D A.] <lb></lb><emph type="italics"></emph>For the Triangles C B E and D B A being <lb></lb>alike; As C B is to B E, ſo ſhall D B be to B A: <lb></lb>And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as C B is to B D, ſo ſhall <lb></lb>E B be to B A: Againe, as B C is to C E ſo <lb></lb>ſhall B D be to D A, And,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as <lb></lb>C B is to B D, ſo ſhall C E, that is, D Z <lb></lb>equall to it, be to D A.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1345"></margin.target>O</s></p><p type="main"> <s>And of D Z and D A, L I and <lb></lb><arrow.to.target n="marg1346"></arrow.to.target><lb></lb>L A are double.] <emph type="italics"></emph>That the Line L A is <lb></lb>double of D A, is manifeſt, for that B D is the Diameter of the Portion. </s> <s>And that L I is <lb></lb>dovble to D Z ſhall be thus demonſtrated. </s> <s>For as much as ZD is to D A, as two to five: <lb></lb>therefore, Converting and Dividing, A Z, that is, I Z, ſhall be to Z D, as three to two:<emph.end type="italics"></emph.end> <pb xlink:href="073/01/056.jpg" pagenum="386"></pb><emph type="italics"></emph>Again, by dividing, I D ſhall be to D Z, as one to two: But Z D was to D A, that is, to D L, <lb></lb>as two to five: Therefore,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>and Converting, L D is to D I, as five to one: and, by <lb></lb>Converſion of Proportion, D L is to D I, as five to four: But D Z was to D L, as two to <lb></lb>five: Therefore, again,<emph.end type="italics"></emph.end> ex equali, <emph type="italics"></emph>D Z is to L I, as two to four: Therefort L I is double <lb></lb>of D Z: Which was to be demonſtrated.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1347"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1346"></margin.target>P</s></p><p type="margin"> <s><margin.target id="marg1347"></margin.target>Q</s></p><p type="main"> <s>And, A D is to D I, as five to one.] <emph type="italics"></emph>This we have but juſt now demon<lb></lb>ſtrated.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1348"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1348"></margin.target>R</s></p><p type="main"> <s>For it hath been demonſtrated, above, that the Portion whoſe <lb></lb>Axis is greater than Seſquialter of the Semi-parameter, if it have <lb></lb>not leſſer proportion in Gravity to the Liquid, &c.] <emph type="italics"></emph>He hath demonstra<lb></lb>ted this in the fourth Propoſition of this Book.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>CONCLVSION II.</s></p><p type="main"> <s><emph type="italics"></emph>If the Portion have leſſer proportion in Gravity to the<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1349"></arrow.to.target><lb></lb><emph type="italics"></emph>Liquid, than the Square S B hath to the Square <lb></lb>B D, but greater than the Square X O hath to the <lb></lb>Square B D, being demitted into the Liquid, ſo in<lb></lb>clined, as that its Baſe touch not the Liquid, it ſhall <lb></lb>continue inclined, ſo, as that its Baſe ſhall not in the <lb></lb>leaſt touch the Surface of the Liquid, and its Axis <lb></lb>ſhall make an Angle with the Liquids Surface, greater <lb></lb>than the Angle X.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1349"></margin.target>A</s></p><p type="main"> <s>Therfore repeating the firſt figure, let the Portion have unto <lb></lb>the Liquid in Gravitie a proportion greater than the Square <lb></lb>X O hath to the ſquare B D, but leſſer than the Square made of <lb></lb>the Exceſſe by which the Axis is greater than Seſquialter of the Semi<lb></lb><figure id="id.073.01.056.1.jpg" xlink:href="073/01/056/1.jpg"></figure><lb></lb>Parameter, that is, of S B, hath to <lb></lb>the Square B D: and as the Portion <lb></lb>is to the Liquid in Gravity, ſo let <lb></lb>the Square made of the Line <foreign lang="grc">ψ</foreign> be <lb></lb>to the Square B D: <foreign lang="grc">ψ</foreign> ſhall be great<lb></lb><arrow.to.target n="marg1350"></arrow.to.target><lb></lb>er than X O, but leſſer than the <lb></lb>Exceſſe by which the Axis is grea<lb></lb>ter than Seſquialter of the Semi<lb></lb>parameter, that is, than S B. </s> <s>Let <lb></lb>a Right Line M N be applyed to <lb></lb>fall between the Conick-Sections <lb></lb>A M Q L and A <emph type="italics"></emph>X<emph.end type="italics"></emph.end> D, [<emph type="italics"></emph>parallel to <lb></lb>B D falling betwixt O X and B D,<emph.end type="italics"></emph.end>] and equall to the Line <foreign lang="grc">ψ</foreign>: and let <lb></lb>it cut the remaining Conick Section A H I in the point H, and the <lb></lb><arrow.to.target n="marg1351"></arrow.to.target><lb></lb>Right Line R G in V. </s> <s>It ſhall be demonſtrated that M H is double to <lb></lb>H N, like as it was demonſtrated that O G is double to G X. <pb xlink:href="073/01/057.jpg" pagenum="387"></pb><figure id="id.073.01.057.1.jpg" xlink:href="073/01/057/1.jpg"></figure><lb></lb>And from the Point M draw M Y <lb></lb>touching the Section A M Q L in M; <lb></lb>and M C perpendicular to B D: and <lb></lb>laſtly, having drawn A N & prolong<lb></lb>ed it to Q, the Lines A N & N Q ſhall <lb></lb>be equall to each other. </s> <s>For in <lb></lb>regard that in the Like Portions <lb></lb><arrow.to.target n="marg1352"></arrow.to.target><lb></lb>A M Q L and A <emph type="italics"></emph>X<emph.end type="italics"></emph.end> D the Lines A Q <lb></lb>and A N are drawn from the Baſes <lb></lb>unto the Portions, which Lines <lb></lb>contain equall Angles with the ſaid <lb></lb>Baſes, Q A ſhall have the ſame proportion to A M that L A hath <lb></lb>to A D: Therefore A N is equall to N Q, and A Q parallel to M Y. <lb></lb><arrow.to.target n="marg1353"></arrow.to.target><lb></lb>It is to be demonſtrated that the Portion being demitted into the <lb></lb>Liquid, and ſo inclined as that its Baſe touch not the Liquid, it <lb></lb>ſhall continue inclined ſo as that its Baſe ſhall not in the leaſt touch <lb></lb>the Surface of the Liquid, and its Axis ſhall make an Angle with <lb></lb>the Liquids Surface greater than the Angle X. </s> <s>Let it be demitted <lb></lb>into the Liquid, and let it ſtand, ſo, as that its Baſe do touch the <lb></lb>Surface of the Liquid in one Point only; and let the Portion be cut <lb></lb>thorow the Axis by a Plane erect unto the Surface of the Liquid, <lb></lb><figure id="id.073.01.057.2.jpg" xlink:href="073/01/057/2.jpg"></figure><lb></lb>and Let the Section of the Super<lb></lb>ficies of the Portion be A P O L, <lb></lb>the Section of a Rightangled Cone, <lb></lb>and let the Section of the Liquids <lb></lb>Surface be A O; And let the Axis <lb></lb>of the Portion and Diameter of the <lb></lb>Section be <emph type="italics"></emph>B<emph.end type="italics"></emph.end> D: and let B D be <lb></lb><arrow.to.target n="marg1354"></arrow.to.target><lb></lb>cut in the Points K and R as hath <lb></lb>been ſaid; alſo draw P G Parallel to <lb></lb>A O and touching the Section <lb></lb>A P O L in P; and from that Point <lb></lb>draw P T Parallel to B D, and P S perpendicular to the ſame B D. <lb></lb>Now, foraſmuch as the Portion is unto the Liquid in Gravity, as <lb></lb>the Square made of the Line <foreign lang="grc">ψ</foreign> is to the Square B D; and ſince that <lb></lb>as the portion is unto the Liquid in Gravitie, ſo is the part thereof <lb></lb>ſubmerged unto the whole Portion; and that as the part ſubmerged <lb></lb>is to the whole, ſo is the Square T P to the Square B D; It follow<lb></lb>eth that the Line <foreign lang="grc">ψ</foreign> ſhall be equall to T P: And therefore the Lines <lb></lb>M N and P T, as alſo the Portions A M Q and A P O ſhall like<lb></lb>wiſe be equall to each other. </s> <s>And ſeeing that in the Equall and <lb></lb>Like Portions A P O L and A M Q L the Lines A O and A Q <lb></lb><arrow.to.target n="marg1355"></arrow.to.target><lb></lb>are drawn from the extremites of their Baſes, ſo, as that the Portions <lb></lb>cut off do make Equall Angles with their Diameters; as alſo the <pb xlink:href="073/01/058.jpg" pagenum="388"></pb>Angles at Y and G being equall; therefore the Lines Y B and G B, <lb></lb>and B C and B S ſhall alſo be equall: And therefore C R and S R, <lb></lb>and M V and P Z, and V N and Z T, ſhall be equall likewiſe. <lb></lb><arrow.to.target n="marg1356"></arrow.to.target><lb></lb>Since therefore M V is Leſſer than double of V N, it is manifeſt that <lb></lb>P Z is leſſer than double of Z T. <emph type="italics"></emph>L<emph.end type="italics"></emph.end>et P <foreign lang="grc">ω</foreign> be double of <foreign lang="grc">ω</foreign> T; and <lb></lb>drawing a <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine from <foreign lang="grc">ω</foreign> to K, prolong it to E. </s> <s>Now the Centre of <lb></lb>Gravity of the whole Portion ſhall be the point K; and the Centre <lb></lb>of that part which is in the Liquid ſhall be <foreign lang="grc">ω,</foreign> and of that which is <lb></lb>above the Liquid ſhall be in the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine K E, which let be E: But the <lb></lb><emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine K Z ſhall be perpendicular unto the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid: <lb></lb>And therefore alſo the Lines drawn thorow the Points E and <foreign lang="grc">ω</foreign> parall<lb></lb><arrow.to.target n="marg1357"></arrow.to.target><lb></lb>lell unto K Z, ſhall be perpendicular sunto the ſame: Therefore the <lb></lb>Portion ſhall not abide, but ſhall turn about ſo, as that its <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe <lb></lb>do not in the leaſt touch the Surface of the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquid; in regard that <lb></lb>now when it toucheth in but one Point only, it moveth upwards, on <lb></lb><arrow.to.target n="marg1358"></arrow.to.target><lb></lb>the part towards A: It is therefore perſpicuous, that the Portion <lb></lb>ſhall conſiſt ſo, as that its Axis ſhall make an Angle with the <emph type="italics"></emph>L<emph.end type="italics"></emph.end>iquids <lb></lb>Surface greater than the Angle X.</s></p><p type="margin"> <s><margin.target id="marg1350"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1351"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1352"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1353"></margin.target>E F</s></p><p type="margin"> <s><margin.target id="marg1354"></margin.target>G</s></p><p type="margin"> <s><margin.target id="marg1355"></margin.target>H</s></p><p type="margin"> <s><margin.target id="marg1356"></margin.target>K</s></p><p type="margin"> <s><margin.target id="marg1357"></margin.target>L</s></p><p type="margin"> <s><margin.target id="marg1358"></margin.target>M</s></p><p type="head"> <s>COMMANDINE.<lb></lb><arrow.to.target n="marg1359"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1359"></margin.target>A</s></p><p type="main"> <s>If the Portion have leſſer proportion in Gravity to the Liquid, <lb></lb>than the Square S B hath to the Square B D, but greater than the <lb></lb>Square X O hath to the Square B D.] <emph type="italics"></emph>This is the ſecond part of the Tenth <lb></lb>propoſition; and the other pat is with their Demonſtrations, ſhall hereafter follow in the ſame Order.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><foreign lang="grc">Ψ</foreign> ſhall be greater than <emph type="italics"></emph>X<emph.end type="italics"></emph.end> O, but leſſer than the Exceſs by </s></p><p type="main"> <s><arrow.to.target n="marg1360"></arrow.to.target><lb></lb>which the Axis is greater than Seſquialter of the Semi-parameter, <lb></lb>that is than S B.] <emph type="italics"></emph>This followeth from the 10 of the fifth Book of<emph.end type="italics"></emph.end> Euclids <emph type="italics"></emph>Elements.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1361"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1360"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1361"></margin.target>C</s></p><p type="main"> <s>It ſhall be demonſtrated, that M H is double to H N, like as it <lb></lb>was demonſtrated, that O G is double to G X.] <emph type="italics"></emph>As in the firſt Concluſion <lb></lb>of this Propoſition, and from what we have but even now written, thereupon appeareth:<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1362"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1362"></margin.target>D</s></p><p type="main"> <s>For in regard that in the like Portions A M Q L and A X D, the <lb></lb>Lines A Q and A N are drawn from the Baſes unto the Portions, <lb></lb>which Lines contain equall Angles with the ſaid Baſes, Q A ſhall <lb></lb>have the ſame proportion to A N, that L A hath to A D.] <lb></lb><emph type="italics"></emph>This we have demonstrated above.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1363"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1363"></margin.target>E</s></p><p type="main"> <s>Therefore A N is equall to N Q] <emph type="italics"></emph>For ſince that Q A is to A N, as L A to <lb></lb>A D; Dividing and Converting, A N ſhall be to N Q as A D to D L: But A D <lb></lb>is equall to D L; for that D B is ſuppoſed to be the Diameter of the Portion: Therefore<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1364"></arrow.to.target><lb></lb><emph type="italics"></emph>alſo<emph.end type="italics"></emph.end> (a) <emph type="italics"></emph>A N is equall to N <expan abbr="q.">que</expan><emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1364"></margin.target>(a) <emph type="italics"></emph>By 14 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>And A Q parallel to M Y.] <emph type="italics"></emph>By the fifth of the ſecond Book of<emph.end type="italics"></emph.end> Apollonius <emph type="italics"></emph>his Conicks.<emph.end type="italics"></emph.end><lb></lb> </s><s><arrow.to.target n="marg1365"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1365"></margin.target>F</s></p><p type="main"> <s>And let B D be cut in the Points K and R as hath been ſaid.] </s></p><p type="main"> <s><arrow.to.target n="marg1366"></arrow.to.target><lb></lb><emph type="italics"></emph>In the firſt Conciuſion of this Propoſition: And let it be cut in K, ſo, as that B K be double to <lb></lb>K D, and in R ſo, as that K R may be equall to the Semi-parameter.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1366"></margin.target>G</s></p><p type="main"> <s>And, ſeeing that in the Equall and Like Portions A P O L and <lb></lb><arrow.to.target n="marg1367"></arrow.to.target><lb></lb>A <emph type="italics"></emph>M<emph.end type="italics"></emph.end> Q L, the Lines A O and A Q are drawn from the Extremities <lb></lb>of their Baſes, ſo, as that the Portions cut off, do make equall Angles <pb xlink:href="073/01/059.jpg" pagenum="389"></pb>with their Diameters; as alſo, the Angles at Y and G being equall; <lb></lb>Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.] <lb></lb><emph type="italics"></emph>Let the Line A Q cut the Diameter D B in<emph.end type="italics"></emph.end> <foreign lang="grc">γ,</foreign> <emph type="italics"></emph>and let it cut A O in<emph.end type="italics"></emph.end> <foreign lang="grc">δ.</foreign> <emph type="italics"></emph>Now becauſe that in<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.059.1.jpg" xlink:href="073/01/059/1.jpg"></figure><lb></lb><emph type="italics"></emph>the equall and like Portions A P O L & A M Q L, <lb></lb>from the Extremities of their Baſes, A O and <lb></lb>A Q are drawn, that contain equall Angles with <lb></lb>thoſe Baſes; and ſince the Angles at D, are both <lb></lb>Right; Therefore, the Remaining Angles A<emph.end type="italics"></emph.end> <foreign lang="grc">δ</foreign> <emph type="italics"></emph>D <lb></lb>and A<emph.end type="italics"></emph.end> <foreign lang="grc">γ</foreign> D <emph type="italics"></emph>ſhall be equall to one another: But <lb></lb>the Line P G is parallel unto the Line A O; alſo <lb></lb>M Y is parallel to A <expan abbr="q;">que</expan> and P S and M C to <lb></lb>A D: Therefore the Triangles P G S and M Y C, <lb></lb>as alſo the Triangles A<emph.end type="italics"></emph.end> <foreign lang="grc">δ</foreign> <emph type="italics"></emph>D and A<emph.end type="italics"></emph.end> <foreign lang="grc">γ</foreign> <emph type="italics"></emph>D, are all <lb></lb>alike to each other<emph.end type="italics"></emph.end>: (b) <emph type="italics"></emph>And as A D is to A<emph.end type="italics"></emph.end> <foreign lang="grc">δ,</foreign><lb></lb><arrow.to.target n="marg1368"></arrow.to.target><lb></lb><emph type="italics"></emph>ſo is A D to A<emph.end type="italics"></emph.end> <foreign lang="grc">γ</foreign><emph type="italics"></emph>: and,<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>the Lines <lb></lb>A D and A D are equall to each other: Therefore, <lb></lb>A<emph.end type="italics"></emph.end> <foreign lang="grc">δ</foreign> <emph type="italics"></emph>and A<emph.end type="italics"></emph.end> <foreign lang="grc">γ</foreign> <emph type="italics"></emph>are alſo equall: But A O and <lb></lb>A Q are equall to each other; as alſo their halves <lb></lb>A T and A N: Therefore the Remainders T<emph.end type="italics"></emph.end> <foreign lang="grc">δ</foreign> <emph type="italics"></emph>and N<emph.end type="italics"></emph.end> <foreign lang="grc">γ</foreign><emph type="italics"></emph>; that is, TG and MY, are alſo<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1369"></arrow.to.target><lb></lb><figure id="id.073.01.059.2.jpg" xlink:href="073/01/059/2.jpg"></figure><lb></lb><emph type="italics"></emph>equall. </s> <s>And, as<emph.end type="italics"></emph.end> (c) <emph type="italics"></emph>P G is to G S, ſo is M Y to <lb></lb>Y C: and<emph.end type="italics"></emph.end> Permutando, <emph type="italics"></emph>as P G is to M Y, ſo is <lb></lb>G S to Y C: And, therefore, G S and Y C are <lb></lb>equall; as alſo their halves B S and B C: From <lb></lb>whence it followeth, that the Remainders S R and C R <lb></lb>are alſo equall: And, conſequently, that P Z and <lb></lb>M V, and V N and Z T, are lkiewiſe equall to one <lb></lb>another.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1367"></margin.target>H</s></p><p type="margin"> <s><margin.target id="marg1368"></margin.target>(b) <emph type="italics"></emph>By 4. of the <lb></lb>ſixth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1369"></margin.target>(c) <emph type="italics"></emph>By 34 of the <lb></lb>firſt,<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Since, therefore, that N V is leſſer <lb></lb><arrow.to.target n="marg1370"></arrow.to.target><lb></lb>than double of V N.] <emph type="italics"></emph>For M H is double of <lb></lb>H N, and M V is leſſer than M H: Therefore, M V <lb></lb>is leſſer than double of H N, and much leſſer than <lb></lb>double of V N.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1370"></margin.target>K</s></p><p type="main"> <s>Therefore, the Portion ſhall not abide, but ſhall turn about, <lb></lb><arrow.to.target n="marg1371"></arrow.to.target><lb></lb>ſo, as that its Baſe do not in the leaſt touch the Surface of <lb></lb>the Liquid; in regard that now when it toucheth in but one Point <lb></lb>only, it moveth upwards on the part towards A.] Tartaglia's <emph type="italics"></emph>his Tranſla<lb></lb>tion hath it thus,<emph.end type="italics"></emph.end> Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum <lb></lb>unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina<lb></lb>tur<emph type="italics"></emph>: Which we have thought fit in this manner to correct, from other Places of<emph.end type="italics"></emph.end><lb></lb>Archimedes, <emph type="italics"></emph>that the ſenſe might be the more perſpicuous. </s> <s>For in the ſixth Propoſition of this, <lb></lb>he thus writeth (as we alſo have it in the Tranſlation,)<emph.end type="italics"></emph.end> The Solid A P O L, therefore, ſhall <lb></lb>turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid. <emph type="italics"></emph>Again, <lb></lb>in the ſeventh Propoſition<emph.end type="italics"></emph.end>; From whence it is manifeſt, that its Baſe ſhall turn about in <lb></lb>ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For <lb></lb>that now when it toucheth but in one Point only, it moveth downwards on the part <lb></lb>towards L. <emph type="italics"></emph>And that the Portion moveth upwards, on the part towards A, doth plainly ap<lb></lb>pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow <foreign lang="grc">ω</foreign>, de <lb></lb>fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary <lb></lb>that the Centre<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>do move upwards, and the Centre E downwards.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1371"></margin.target>L</s></p><p type="main"> <s>It is therefore perſpicuous, that the Portion ſhall conſiſt, ſo, as that <lb></lb>its Axis ſhall make an Angle with the Liquids Surface greater than <lb></lb>the Angle <emph type="italics"></emph>X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter<emph.end type="italics"></emph.end> <pb xlink:href="073/01/060.jpg" pagenum="390"></pb><figure id="id.073.01.060.1.jpg" xlink:href="073/01/060/1.jpg"></figure><lb></lb><emph type="italics"></emph>B D in<emph.end type="italics"></emph.end> <foreign lang="grc">λ</foreign><emph type="italics"></emph>; and from the Point O, and parallel to <lb></lb>A<emph.end type="italics"></emph.end> <foreign lang="grc">λ,</foreign> <emph type="italics"></emph>draw O X; and let it touch the Section in O, <lb></lb>as in the first Figure: And the<emph.end type="italics"></emph.end> (d) <emph type="italics"></emph>Angle at X,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1372"></arrow.to.target><lb></lb><emph type="italics"></emph>ſhall be equall alſo to the angle<emph.end type="italics"></emph.end> <foreign lang="grc">λ</foreign><emph type="italics"></emph>: But the angle at Y <lb></lb>is equall to the Angle at<emph.end type="italics"></emph.end> <foreign lang="grc">γ;</foreign> <emph type="italics"></emph>and the<emph.end type="italics"></emph.end> (e) <emph type="italics"></emph>Angle<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1373"></arrow.to.target><lb></lb>A <foreign lang="grc">Γ</foreign> D <emph type="italics"></emph>greater than the Angle A<emph.end type="italics"></emph.end> <foreign lang="grc">λ</foreign> <emph type="italics"></emph>D, which falleth <lb></lb>without it: Therefore the Angle at Y ſhall be great<lb></lb>er than that at X. </s> <s>And becauſe now the Portion <lb></lb>turneth about, ſo, as that the Baſe doth not touch <lb></lb>the Liquid, the Axis ſhall make an Angle with its <lb></lb>Surface greater than the Angle G; that is, than the <lb></lb>Angle Y: And, for that reaſon, much greater than <lb></lb>the Angle X.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1372"></margin.target>(d) <emph type="italics"></emph>By 29 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1373"></margin.target>(e) <emph type="italics"></emph>By 16. of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>CONCLUSION III.</s></p><p type="main"> <s><emph type="italics"></emph>If the Portion have the ſame proportion in Gravity to the <lb></lb>Liquid, that the Square X O hath to the Square<emph.end type="italics"></emph.end><lb></lb>BD, <emph type="italics"></emph>being demitted into the Liquid, ſo inclined, as that <lb></lb>its Baſe touch not the Liquid, it ſhall ſtand and <lb></lb>continue inclined, ſo, as that its Baſe touch the Sur<lb></lb>face of the Liquid, in one Point only, and its Axis ſhall <lb></lb>make an Angle with the Liquids Surface equall to the <lb></lb>Angle X. And, if the Portion have the ſame proportion <lb></lb>in Gravity to the Liquid, that the Square P F hath <lb></lb>to the Square B D, being demitted into the Liquid, <lb></lb>& ſet ſo inclined, as that its Baſe touch not the Liquid, <lb></lb>it ſhall ſtand inclined, ſo, as that its Baſe touch the <lb></lb>Surface of the Liquid in one Point only, & its Axis ſhall <lb></lb>make an Angle with it, equall to the Angle<emph.end type="italics"></emph.end> <foreign lang="grc">Φ.</foreign></s></p><p type="main"> <s>Let the Portion have the ſame proportion in Gravity to tho <lb></lb>Liquid that the Square <emph type="italics"></emph>X<emph.end type="italics"></emph.end>O hath to the Square B D; and let <lb></lb>it be demitted into the Liquid ſo inclined, as that its Baſe touch <lb></lb><figure id="id.073.01.060.2.jpg" xlink:href="073/01/060/2.jpg"></figure><lb></lb>not the Liquid. </s> <s>And cutting it by <lb></lb>a Plane thorow the Axis, erect unto <lb></lb>the Surface of the Liquid, let the <lb></lb>Section of the Solid, be the Section <lb></lb>of a Right-angled Cone, A P M L; <lb></lb>let the Section of the Surface of the <lb></lb>Liquid be I M; and the Axis of the <lb></lb>Portion and Diameter of the Section <lb></lb>B D; and let B D be divided as be<lb></lb>fore; and draw PN parallel to IM <pb xlink:href="073/01/061.jpg" pagenum="391"></pb>and touching the Section in P, and T P parallel to B D; and P S perpen<lb></lb>dicular unto B D. </s> <s>It is to be demonſtrated that the Portion ſhall <lb></lb><figure id="id.073.01.061.1.jpg" xlink:href="073/01/061/1.jpg"></figure><lb></lb>not ſtand ſo, but ſhall encline until <lb></lb>that the Baſe touch the Surface of <lb></lb>the Liquid, in one Point only, for let <lb></lb>the ſuperior figure ſtand as it was, <lb></lb>and draw O C, Perpendicular to B D; <lb></lb>and drawing a <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ine from A to <emph type="italics"></emph>X,<emph.end type="italics"></emph.end><lb></lb>prolong it to Q: A X ſhalbe equall <lb></lb>to <emph type="italics"></emph>X<emph.end type="italics"></emph.end> <expan abbr="q.">que</expan> Then draw O X parallel <lb></lb>to A <expan abbr="q.">que</expan> And becauſe the Portion <lb></lb>is ſuppoſed to have the ſame pro<lb></lb>portion in Gravity to the Liquid <lb></lb>that the ſquare X O hath to the <lb></lb>Square B D; the part thereof ſubmerged ſhall alſo have the ſame <lb></lb>proportion to the whole; that is, the Square T P to the Square <lb></lb><arrow.to.target n="marg1374"></arrow.to.target><lb></lb>B D; and ſo T P ſhall be equal to <emph type="italics"></emph>X<emph.end type="italics"></emph.end> O: And ſince that of the <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions <lb></lb>I P M and A O Q the Diameters are equall, the portions ſhall alſo be <lb></lb><arrow.to.target n="marg1375"></arrow.to.target><lb></lb>equall. <emph type="italics"></emph>A<emph.end type="italics"></emph.end>gain, becauſe that in the Equall and <emph type="italics"></emph>L<emph.end type="italics"></emph.end>ike <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortions A O Q L <lb></lb><arrow.to.target n="marg1376"></arrow.to.target><lb></lb>and AP ML the Lines A Q and I M, which cut off equall <emph type="italics"></emph>P<emph.end type="italics"></emph.end>or<lb></lb>tions, are drawn, that, from the Extremity of the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe, and this <lb></lb>not from the Extremity; it appeareth that that which is drawn from <lb></lb>the end or Extremity of the <emph type="italics"></emph>B<emph.end type="italics"></emph.end>aſe, ſhall make the Acute Angle with <lb></lb>the Diameter of the whole <emph type="italics"></emph>P<emph.end type="italics"></emph.end>ortion leſset. <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd the Angle at <emph type="italics"></emph>X<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1377"></arrow.to.target><lb></lb>being leſſe than the Angle at N, B C ſhall be greater than B S; and <lb></lb>C R leſſer than S R: <emph type="italics"></emph>A<emph.end type="italics"></emph.end>nd, therfore O G ſhall be leſſer than P Z; <lb></lb>and G <emph type="italics"></emph>X<emph.end type="italics"></emph.end> greater than Z T: Therfore P Z is greater than double of <lb></lb>Z T; being that O G is double of G X. </s> <s>Let P H be double to H T; <lb></lb>and drawing a Line from H to K, prolong it to <foreign lang="grc">ω.</foreign> The Center of <lb></lb>Gravity of the whole Portion ſhall be K; the Center of the part <lb></lb>which is within the Liquid H, and that of the part which is above <lb></lb>the Liquid in the Line K <foreign lang="grc">ω</foreign>; which ſuppoſed to be <foreign lang="grc">ω.</foreign> Therefore it <lb></lb>ſhall be demonſtrated, both, that K H is perpendicular to the Surface <lb></lb>of the Liquid, and thoſe Lines alſo that are drawn thorow the Points <lb></lb>Hand <foreign lang="grc">ω</foreign> parallel to K H: And therfore the Portion ſhall not reſt, but <lb></lb>ſhall encline untill that its Baſe do touch the Surface of the Liquid <lb></lb>in one Point; and ſo it ſhall continue. </s> <s>For in the Equall Portions <lb></lb>A O Q L and A P M L, the <lb></lb><figure id="id.073.01.061.2.jpg" xlink:href="073/01/061/2.jpg"></figure><lb></lb>Lines A Q and A M, that cut off <lb></lb>equall Portions, ſhall be dawn <lb></lb>from the Ends or Terms of the Baſes; <lb></lb>and A O Q and A P M ſhall be <lb></lb>demonſtrated, as in the former, to <lb></lb><arrow.to.target n="marg1378"></arrow.to.target><lb></lb>be equall: Therfore A Q and A M, <lb></lb>do make equall Acute Angles with <lb></lb>the Diameters of the Portions; and <pb xlink:href="073/01/062.jpg" pagenum="392"></pb>the Angles at X and N are equall. </s> <s>And, therefore, if drawing HK, <lb></lb>it be prolonged to <foreign lang="grc">ω,</foreign> the Centre of Gravity of the whole Portion ſhall <lb></lb>be K; of the part which is within the Liquid H; and of the part which <lb></lb>is above the Liquid in K <foreign lang="grc">ὠ</foreign> as ſuppoſe in <foreign lang="grc">ω;</foreign> and H K perpendicular to <lb></lb><figure id="id.073.01.062.1.jpg" xlink:href="073/01/062/1.jpg"></figure><lb></lb>the Surface of the Liquid. </s> <s>Therfore <lb></lb>along the ſame Right Lines ſhall the <lb></lb>part which is within the Liquid move <lb></lb>upwards, and the part above it down<lb></lb>wards: And therfore the Portion <lb></lb>ſhall reſt with one of its Points <lb></lb>touching the Surface of the Liquid, <lb></lb>and its Axis ſhall make with the <lb></lb><arrow.to.target n="marg1379"></arrow.to.target><lb></lb>ſame an Angle equall to X. </s> <s>It is <lb></lb>to be demonſtrated in the ſame <lb></lb>manner that the Portion that hath <lb></lb>the ſame proportion in Gravity to the Liquid, that the Square P F hath <lb></lb>to the Square B D, being demitted into the Liquid, ſo, as that its <lb></lb>Baſe touch not the Liquid, it ſhall ſtand inclined, ſo, as that its Baſe <lb></lb>touch the Surface of the Liquid in one Point only; and its Axis ſhall <lb></lb>make therwith an Angle equall to the Angle <foreign lang="grc">φ.</foreign></s></p><p type="margin"> <s><margin.target id="marg1374"></margin.target>A</s></p><p type="margin"> <s><margin.target id="marg1375"></margin.target>B</s></p><p type="margin"> <s><margin.target id="marg1376"></margin.target>C</s></p><p type="margin"> <s><margin.target id="marg1377"></margin.target>D</s></p><p type="margin"> <s><margin.target id="marg1378"></margin.target>E</s></p><p type="margin"> <s><margin.target id="marg1379"></margin.target>F</s></p><p type="head"> <s>COMMANDINE.<lb></lb><arrow.to.target n="marg1380"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1380"></margin.target>A</s></p><p type="main"> <s>That is the Square T P to the Square B D.] <emph type="italics"></emph>By the twenty ſixth of the Book<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1381"></arrow.to.target><lb></lb><emph type="italics"></emph>of<emph.end type="italics"></emph.end> Archimedes, De Conoidibus & Sphæroidibus: <emph type="italics"></emph>Therefore, (a) the Square T P <lb></lb>ſhall be equall to the Square X O: And for that reaſon, the Line T P equall to the <lb></lb>Line X O.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1382"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1381"></margin.target>(a) <emph type="italics"></emph>By 9 of the <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1382"></margin.target>B</s></p><p type="main"> <s>The Portions ſhall alſo be equall.] <emph type="italics"></emph>By the twenty fifth of the ſame Book.<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1383"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1383"></margin.target>C</s></p><p type="main"> <s>Again, becauſe that in the Equall and Like Portions, A O Q L <lb></lb>and A P M L.] <emph type="italics"></emph>For, in the Portion A P M L, deſcribe the Portion A O Q equall <lb></lb>to the Portion I P M: The Point Q falleth beneath M; for otherwiſe, the Whole would be <lb></lb>equall to the Part. </s> <s>Then draw I V parallel to A Q, and cutting the Diameter is<emph.end type="italics"></emph.end> <foreign lang="grc">ψ;</foreign> <emph type="italics"></emph>and <lb></lb>let I M cut the ſame<emph.end type="italics"></emph.end> <foreign lang="grc">ς;</foreign> <emph type="italics"></emph>and A Q in<emph.end type="italics"></emph.end> <foreign lang="grc">ς.</foreign> <emph type="italics"></emph>I ſay <lb></lb>that the Angle A<emph.end type="italics"></emph.end> <foreign lang="grc">υ</foreign> <emph type="italics"></emph>D, is leſſer than the Angle<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.062.2.jpg" xlink:href="073/01/062/2.jpg"></figure><lb></lb><emph type="italics"></emph>I<emph.end type="italics"></emph.end> <foreign lang="grc">σ</foreign> <emph type="italics"></emph>D. </s> <s>For the Angle I<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>D is equall to the <lb></lb>Angle A<emph.end type="italics"></emph.end> <foreign lang="grc">υ</foreign> <emph type="italics"></emph>D: (b) But the interiour Angle<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1384"></arrow.to.target><lb></lb><emph type="italics"></emph>I<emph.end type="italics"></emph.end> <foreign lang="grc">ψ</foreign> <emph type="italics"></emph>D is leſſer than the exteriour I<emph.end type="italics"></emph.end> <foreign lang="grc">σ</foreign> <emph type="italics"></emph>D: There-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1385"></arrow.to.target><lb></lb><emph type="italics"></emph>fore, (c) A<emph.end type="italics"></emph.end> <foreign lang="grc">υ</foreign> <emph type="italics"></emph>D ſhall alſo be lefter than I<emph.end type="italics"></emph.end> <foreign lang="grc">σ</foreign> <emph type="italics"></emph>D.<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1386"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1384"></margin.target>(b) <emph type="italics"></emph>By 29 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1385"></margin.target><emph type="italics"></emph>(c) By 16 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1386"></margin.target>D</s></p><p type="main"> <s>And the Angle at X, being leſſe <lb></lb>than the Angle at N.] <emph type="italics"></emph>Thorow O draw twe <lb></lb>Lines, O C perpendicular to the Diameter B D, and <lb></lb>O X touching the Section in the Point O, and cutting<emph.end type="italics"></emph.end></s></p><p type="main"> <s><arrow.to.target n="marg1387"></arrow.to.target><lb></lb><emph type="italics"></emph>the Diameter in X: (d) O X ſhall be parallel <lb></lb>to A <expan abbr="q;">que</expan> and the<emph.end type="italics"></emph.end> (e) <emph type="italics"></emph>Angle at X, ſhall be equall to<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1388"></arrow.to.target><lb></lb><emph type="italics"></emph>that at<emph.end type="italics"></emph.end> <foreign lang="grc">υ</foreign>: <emph type="italics"></emph>Therefore, the<emph.end type="italics"></emph.end> (f) <emph type="italics"></emph>Angle at X,<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1389"></arrow.to.target><lb></lb><emph type="italics"></emph>ſhall be leſſer than the Angle at<emph.end type="italics"></emph.end> <foreign lang="grc">ς;</foreign> <emph type="italics"></emph>that is, to <lb></lb>that at N: And, conſequently, X ſhall fall beneath N: Therefore, the Line X B is greater than <lb></lb>N B. And, ſince B C is equall to X B, and B S equall to N B; B C ſhall be greater than B S.<emph.end type="italics"></emph.end></s></p> <pb xlink:href="073/01/063.jpg" pagenum="397"></pb><p type="margin"> <s><margin.target id="marg1387"></margin.target><emph type="italics"></emph>(d) By 5 of our ſe<lb></lb>cond of<emph.end type="italics"></emph.end> Conicks.</s></p><p type="margin"> <s><margin.target id="marg1388"></margin.target>(e) <emph type="italics"></emph>By 29 of the <lb></lb>firſt.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1389"></margin.target>(f) <emph type="italics"></emph>By 39 of our <lb></lb>firſt of<emph.end type="italics"></emph.end> Conicks.</s></p><p type="main"> <s>Therefore, A Q and A M do make equall Acute Angles with <lb></lb><arrow.to.target n="marg1390"></arrow.to.target><lb></lb>the Diameters of the Portions.] <emph type="italics"></emph>We demonſtrate this as in the Commentaries <lb></lb>upon the ſecond Concluſion.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1390"></margin.target>E</s></p><p type="main"> <s>It is to be demonſtrated in the ſame manner, that the Portion <lb></lb><arrow.to.target n="marg1391"></arrow.to.target><lb></lb>that hath the ſame proportion in Gravity to the Liquid, that the <lb></lb>Square P F hath to the Square B D, <lb></lb>being demitted into the Liquid, ſo, <lb></lb><figure id="id.073.01.063.1.jpg" xlink:href="073/01/063/1.jpg"></figure><lb></lb>as that its Baſe touch not the Li<lb></lb>quid, it ſhall ſtand inclined, ſo, as <lb></lb>that its Baſe touch the Surface of the <lb></lb>Liquid in one point only; and its Axis <lb></lb>ſhall make therewith an angle equall <lb></lb>to the Angle <foreign lang="grc">φ.</foreign>] <emph type="italics"></emph>Let the Portion be to the <lb></lb>Liquid in Gravity, as the Square P F to the <lb></lb>Square B D: and being demitted into the <lb></lb>Liquid, ſo inclined, as that its Baſe touch not <lb></lb>the Liquid, let it be cut thorow the Axis by a <lb></lb>Plane erect to the Surface of the Liquid, that <lb></lb>that the Section may be A M O L, the Section <lb></lb>of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit <lb></lb>of the Portion and Diameter of the Section B D; which let be cut into the ſame parts as <lb></lb>we ſaid before, and draw M N parallel to I O, that it may touch the Section in the Point <lb></lb>M; and M T parallel to B D, and P M S perpe ndicular to the ſame. </s> <s>It is to be demon<lb></lb>strated, that the Portion ſhall not reſt, but ſhall incline, ſo, as that it touch the Liquids <lb></lb>Surface, in one Point of its Baſe only. </s> <s>For,<emph.end type="italics"></emph.end><lb></lb><figure id="id.073.01.063.2.jpg" xlink:href="073/01/063/2.jpg"></figure><lb></lb><emph type="italics"></emph>draw P C perpendicular to B D; and drawing <lb></lb>a Line from A to F, prolong it till it meet with <lb></lb>the Section in <expan abbr="q;">que</expan> and thorow P draw P<emph.end type="italics"></emph.end> <foreign lang="grc">φ</foreign> <emph type="italics"></emph>pa<lb></lb>rallel to A Q: Now, by the things allready de<lb></lb>monſtrated by us, A F and F Q ſhall be equall <lb></lb>to one another. </s> <s>And being that the Portion hath <lb></lb>the ſame proportion in Gravity unto the Liquid, <lb></lb>that the Square P F hath to the Square B D; and <lb></lb>ſeeing that the part ſubmerged, hath the ſame pro-<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="marg1392"></arrow.to.target><lb></lb><emph type="italics"></emph>partion to the whole Portion; that is, the Squàre <lb></lb>M T to the Square B D; (g) the Square M T <lb></lb>ſhall be equall to the Square P F; and, by the <lb></lb>ſame reaſon, the Line M T equall to the Line <lb></lb>P F. </s> <s>So that there being drawn in the equall & like <lb></lb>portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the <lb></lb>firſt from the Extreme term of the Baſe, the laſt not from the Extremity; it followeth, that <lb></lb>A Q drawn from the Extremity, containeth a leſſer Acute Angle with the Diameter of the <lb></lb>Portion, than I O: But the Line P<emph.end type="italics"></emph.end> <foreign lang="grc">φ</foreign> <emph type="italics"></emph>is parallel to the Line A Q, and M N to I O: There<lb></lb>fore, the Angle at<emph.end type="italics"></emph.end> <foreign lang="grc">φ</foreign> <emph type="italics"></emph>ſhall be leſſer than the Angle at N; but the Line B C greater than B S; <lb></lb>and S R, that is, M X, greater than C R, that is, than P Y: and, by the ſame reaſon, X T <lb></lb>leſſer than Y F. And, ſince P Y is double to Y F, M X ſhall be greater than double to <lb></lb>Y F, and much greater than double of X T. </s> <s>Let M H be double to H T, and draw a <lb></lb>Line from H to K, prolonging it. </s> <s>Now, the Centre of Gravity of the whole Portion <lb></lb>ſhall be the Point K; of the part within the Liquid H; and of the Remaining part above <lb></lb>the Liquid in the Line H K produced, as ſuppoſe in<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>It ſhall be demonſtrated in the ſame <lb></lb>manner, as before, that both the Line K H and thoſe that are drawn thorow the Points H <lb></lb>and<emph.end type="italics"></emph.end> <foreign lang="grc">ω</foreign> <emph type="italics"></emph>parallel to the ſaid K H, are perpendicular to the Surface of the Liquid: The <lb></lb>Portion therefore, ſhall not reſt; but when it ſhall be enclined ſo far as to touch the Sur<lb></lb>face of the Liquid in one Point and no more, then it ſhall ſtay. </s> <s>For the Angle at N<emph.end type="italics"></emph.end> <pb xlink:href="073/01/064.jpg" pagenum="398"></pb><figure id="id.073.01.064.1.jpg" xlink:href="073/01/064/1.jpg"></figure><lb></lb><emph type="italics"></emph>ſhall be equall to the Angle at<emph.end type="italics"></emph.end> <foreign lang="grc">φ;</foreign> <emph type="italics"></emph>and the Line B S <lb></lb>equall to the Line B C; and S R to C R: Where<lb></lb>fore, M H ſhall be likewiſe equall to P Y. There<lb></lb>fore, having drawn HK and prolonged it; the <lb></lb>Centre of Gravity of the whole Portion ſhall be <lb></lb>K; of that which is in the Liquid H; and of <lb></lb>that which is above it, the Centre ſhall be in <lb></lb>the Line prolonged: let it be in<emph.end type="italics"></emph.end> <foreign lang="grc">ω.</foreign> <emph type="italics"></emph>There<lb></lb>fore, along that ſame Line K H, which is per<lb></lb>pendicular to the Surface of the Liquid, ſhall <lb></lb>the part which is within the Liquid move up<lb></lb>wards, and that which is above the Liquld <lb></lb>downwards: And, for this cauſe, the Portion, <lb></lb>ſhall be no longer moved, but ſhall ſtay, and <lb></lb>reſt, ſo, as that its Baſe do touch the Liquids Surface in but one Point; and its Axis <lb></lb>maketh an Angle therewith equall to the Angle<emph.end type="italics"></emph.end> <foreign lang="grc">φ</foreign><emph type="italics"></emph>; And, this is that which we were to <lb></lb>demonſtrate.<emph.end type="italics"></emph.end></s></p><p type="margin"> <s><margin.target id="marg1391"></margin.target>F</s></p><p type="margin"> <s><margin.target id="marg1392"></margin.target>(g) <emph type="italics"></emph>By 9 of t <lb></lb>fifth.<emph.end type="italics"></emph.end></s></p><p type="head"> <s>CONCLVSION IV.</s></p><p type="main"> <s><emph type="italics"></emph>If the Portion have greater proportion in Gravity <lb></lb>to the Liquid, than the Square F P to the Square <lb></lb>B D, but leſſer than that of the Square X O to the <lb></lb>Square B D, being demitted into the Liquid, <lb></lb>and inclined, ſo, as that its Baſe touch not the <lb></lb>Liquid, it ſhall ſtand and reſt, ſo, as that its Baſe <lb></lb>ſhall be more ſubmerged in the Liquid.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Again, let the Portion have greater proportion in <lb></lb>Gravity to the Liquid, than the Square F P to the <lb></lb>Square B D, but leſſer than that of the Square X O to <lb></lb>the Square B D; and as the Portion is in Gravity to the Liquid, <lb></lb>ſo let the Square made of the Line <foreign lang="grc">ψ</foreign> be to the Square B D. <foreign lang="grc">Ψ</foreign><lb></lb>ſhall be greater than F P, and leſſer than X O. Apply, therefore, <lb></lb>the right Line I V to fall betwixt the Portions A V Q L and A X D; <lb></lb>and let it be equall to <foreign lang="grc">ψ,</foreign> and parallel to B D; and let it meet <lb></lb>the Remaining Section in Y: V Y ſhall alſo be proved double <lb></lb>to Y I, like as it hath been demonſtrated, that O G is double off <lb></lb>G X. And, draw from V, the Line V <foreign lang="grc">ω,</foreign> touching the Section <lb></lb>A V Q L in V; and drawing a Line from A to I, prolong it unto <lb></lb><expan abbr="q.">que</expan> We prove in the ſame manner, that the Line A I is equall <lb></lb>to I <expan abbr="q;">que</expan> and that A Q is parallel to V <foreign lang="grc">ω.</foreign> It is to be demonſtrated, <lb></lb>that the Portion being demitted into the Liquid, and ſo inclined, <lb></lb>as that its Baſe touch not the Liquid, ſhall ſtand, ſo, that its Baſe <lb></lb>ſhall be more ſubmerged in the Liquid, than to touch it Surface in <pb xlink:href="073/01/065.jpg" pagenum="399"></pb>but one Point only. </s> <s>For let it be de<lb></lb><figure id="id.073.01.065.1.jpg" xlink:href="073/01/065/1.jpg"></figure><lb></lb>mitted into the Liquid, as hath been <lb></lb>ſaid; and let it firſt be ſo inclined, as <lb></lb>that its Baſe do not in the leaſt <lb></lb>touch the Surface of the Liquid. </s> <s>And <lb></lb>then it being cut thorow the Axis, <lb></lb>by a Plane erect unto the Surface of <lb></lb>the Liquid, let the Section of the <lb></lb>Portion be A N Z G; that of the <lb></lb>Liquids Surface E Z; the Axis of <lb></lb>the Portion and Diameter of the <lb></lb>Section B D; and let B D be cut in <lb></lb>the Points K and R, as before; and <lb></lb>draw N L parallel to E Z, and touching the Section A N Z G <lb></lb>in N, and N S perpendicular to <lb></lb><figure id="id.073.01.065.2.jpg" xlink:href="073/01/065/2.jpg"></figure><lb></lb>B D. Now, ſeeing that the Por<lb></lb>tion is in Gravity unto the Liquid, <lb></lb>as the Square made of the Line <lb></lb>is to the Square B D; <foreign lang="grc">ψ</foreign> ſhall <lb></lb>be equall to N T: Which is to <lb></lb>be demonſtrated as above: And, <lb></lb>therefore, N T is alſo equall to <lb></lb>V I: The Portions, therefore, <lb></lb>A V Q and E N Z are equall to <lb></lb>one another. </s> <s>And, ſince that in <lb></lb>the Equall and like Portions A V <lb></lb>Q L and A N Z G, there are drawn A Q and E Z, cutting off <lb></lb>equall Portions, that from the <lb></lb><figure id="id.073.01.065.3.jpg" xlink:href="073/01/065/3.jpg"></figure><lb></lb>Extremity of the Baſe, this not <lb></lb>from the Extreme, that which is <lb></lb>drawn from the Extremity of the <lb></lb>Baſe, ſhall make the Acute Angle <lb></lb>with the Diameter of the Portion <lb></lb>leſſer: and in the Triangles N L S <lb></lb>and V <foreign lang="grc">ω</foreign> C, the Angle at L is <lb></lb>greater than the Angle at <foreign lang="grc">ω</foreign>: <lb></lb>Therefore, B S ſhall be leſſer <lb></lb>than B C; and S R leſſer than <lb></lb>C R: and, conſequently, N X <lb></lb>greater than V H; and X T leſſer than H I. Seeing, therefore, <lb></lb>that V Y is double to Y I; It is manifeſt, that N X is greater than <lb></lb>double to X T. </s> <s>Let N M be double to M T: It is manifeſt, from what <lb></lb>hath been ſaid, that the Portion ſhall not reſt, but will incline, untill <lb></lb>that its Bafe do touch the Surface of the Liquid: and it toucheth it in <lb></lb>one Point only, as appeareth in the Figure: And other things <pb xlink:href="073/01/066.jpg" pagenum="400"></pb><figure id="id.073.01.066.1.jpg" xlink:href="073/01/066/1.jpg"></figure><lb></lb>ſtanding as before, we will again <lb></lb>demonſtrate, that N T is equall to <lb></lb>V I; and that the Portions A V Q <lb></lb>and A N Z are equall to each other. <lb></lb></s> <s>Therefore, in regard, that in the <lb></lb>Equall and Like Portions A V Q L <lb></lb>and A N Z G, there are drawn <lb></lb>A Q and A Z cutting off equall Por<lb></lb>tions, they ſhall with the Diameters <lb></lb>of the Portions, contain equall <lb></lb>Angles. </s> <s>Therefore, in the Triangles <lb></lb>N L S and V <foreign lang="grc">ω</foreign> C, the Angles at <lb></lb>the Points <emph type="italics"></emph>L<emph.end type="italics"></emph.end> and <foreign lang="grc">ω</foreign> are equall; and the Right Line B S equall to <lb></lb>B C; S R to C R; N X to V H; and X T to H I: And, ſince <lb></lb>V Y is double to Y I, N X ſhall be greater than double of X T. <lb></lb></s> <s>Let therefore, N M be double to M T. </s> <s>It is hence again manifeſt, <lb></lb>that the Portion will not remain, but ſhall incline on the part <lb></lb>towards A: But it was ſuppoſed, that the ſaid Portion did <lb></lb>touch the Surface of the Liquid in one ſole Point: Therefore, <lb></lb>its Baſe muſt of neceſſity ſubmerge farther into the Liquid.</s></p><p type="head"> <s>CONCLVSION V.</s></p><p type="main"> <s><emph type="italics"></emph>If the Portion have leſſer proportion in Gravity to <lb></lb>the Liquid, than the Square F P to the Square <lb></lb>B D, being demitted into the Liquid, and in<lb></lb>clined, ſo, as that its Baſe touch not the Liquid, <lb></lb>it ſhall ſtand ſo inclined, as that its Axis ſhall <lb></lb>make an Angle with the Surface of the Liquid, <lb></lb>leſſe than the Angle<emph.end type="italics"></emph.end> <foreign lang="grc">ψ;</foreign> <emph type="italics"></emph>And its Baſe ſhall <lb></lb>not in the leaſt touch the Liquids Surface.<emph.end type="italics"></emph.end></s></p><p type="main"> <s>Finally, let the Portion have leſſer proportion to the Liquid <lb></lb>in Gravity, than the Square F P hath to the Square B D; and <lb></lb>as the Portion is in Gravity to the Liquid, ſo let the <lb></lb>Square made of the Line <foreign lang="grc">ψ</foreign> be to the Square B D. <foreign lang="grc">ψ</foreign> ſhall be <lb></lb>leſſer than P F. Again, apply any Right Line as G I, falling <lb></lb>betwixt the Sections A G Q L and A X D, and parallel to B D; <lb></lb>and let it cut the Middle Conick Section in the Point H, and <pb xlink:href="073/01/067.jpg" pagenum="401"></pb>the Right Line R Y in Y. </s> <s>We <lb></lb><figure id="id.073.01.067.1.jpg" xlink:href="073/01/067/1.jpg"></figure><lb></lb>ſhall demonſtrate G H to be double <lb></lb>to H I, as it hathbeen demonſtra<lb></lb>ted, that O G is double to G X. <lb></lb></s> <s>Then draw G <foreign lang="grc">ω</foreign> touching the Section <lb></lb>A G Q L in G; and G C perpen di<lb></lb>cular to B D; and drawing a Line <lb></lb>from A to I, prolong it to <expan abbr="q.">que</expan> Now <lb></lb>A I ſhall be equall to I <expan abbr="q;">que</expan> and <lb></lb>A Q parallel to G <foreign lang="grc">ω.</foreign> It is to be <lb></lb>demonſtrated, that the Portion being <lb></lb>demitted into the Liquid, and inclined, ſo, as that its Baſe touch <lb></lb>the Liquid, it ſhall ſtand ſo incli<lb></lb><figure id="id.073.01.067.2.jpg" xlink:href="073/01/067/2.jpg"></figure><lb></lb>ned, as that its Axis ſhall make <lb></lb>an Angle with the Surface of the <lb></lb>Liquid leſſe than the Angle <foreign lang="grc">φ;</foreign><lb></lb>and its Baſe ſhall not in the leaſt <lb></lb>touch the Liquids Surface. </s> <s>For <lb></lb>let it be demitted into the Liquid, <lb></lb>and let it ſtand, ſo, as that its Baſe <lb></lb>do touch the Surface of the Liquid <lb></lb>in one Point only: and the Portion <lb></lb>being cut thorow the Axis by a <lb></lb>Plane erect unto the Surface of the Liquid, let the Section of <lb></lb><figure id="id.073.01.067.3.jpg" xlink:href="073/01/067/3.jpg"></figure><lb></lb>the Portion be A N Z L, the Section <lb></lb>of a Rightangled Cone; that of <lb></lb>the Surface of the Liquid A Z; and <lb></lb>the Axis of the Portion and Dia<lb></lb>meter of the Section B D; and let <lb></lb>B D be cut in the Points K and R <lb></lb>as hath been ſaid above; and draw <lb></lb>N F parallel to A Z, and touching <lb></lb>the Section of the Cone in the Point <lb></lb>N; and N T parallel to B D; and <lb></lb>N S perpendicular to the ſame. </s> <s>Be<lb></lb>cauſe, now, that the Portion is in Gravity to the Liquid, as <lb></lb>the Square made of <foreign lang="grc">ψ</foreign> is to the Square B D; and ſince that as the <lb></lb>Portion is to the Liquid in Gravity, ſo is the Square N T to the <lb></lb>Square B D, by the things that have been ſaid; it is plain, that <lb></lb>N T is equall to the Line <foreign lang="grc">ψ</foreign>: And, therefore, alſo, the Portions <lb></lb>A N Z and A G Q are equall. </s> <s>And, ſeeing that in the Equall and <lb></lb>Like Portions A G Q L and A N Z L; there are drawn from the <lb></lb>Extremities of their Baſes, A Q and A Z which cut off equall Porti<lb></lb>ons: It is obvious, that with the Diameters of the Portions they <pb xlink:href="073/01/068.jpg" pagenum="402"></pb>make equall Angles; and that in the Triangles N F S and G <foreign lang="grc">ω</foreign> C <lb></lb>the Angles at F and <foreign lang="grc">ω</foreign> are equall; as alſo, that S B and B C, and<lb></lb>S R and C R are equall to one another: And, therefore, N X and<lb></lb>G Y are alſo equall; and X T and Y I. </s> <s>And ſince G H is double<lb></lb>to H I, N X ſhall be leſſer than double of X T. </s> <s>Let N M therefore<lb></lb>be double to M T; and drawing a Line from M to K, prolong it<lb></lb>unto E. </s> <s>Now the Centre of Gravity of the whole ſhall be the<lb></lb>Point K; of the part which is in the Liquid the Point M; and<lb></lb>that of the part which is above the Liquid in the Line prolonged <lb></lb>as ſuppoſe in E. Therefore, by what was even now demonſtrated <lb></lb>it is manifeſt that the Portion ſhall not ſtay thus, but ſhall incline, ſo <lb></lb>as that its Baſe do in no wiſe touch the Surface of the Liquid <lb></lb>And that the Portion will ſtand, ſo, as to make an Angle with the<lb></lb>Surface of the Liquid leſſer than<lb></lb><figure id="id.073.01.068.1.jpg" xlink:href="073/01/068/1.jpg"></figure><lb></lb>the Angle <foreign lang="grc">φ,</foreign> ſhall thus be demon <lb></lb>ſtrated. </s> <s>Let it, if poſſible, ſtand,<lb></lb>ſo, as that it do not make an Angle<lb></lb>leſſer than the Angle <foreign lang="grc">φ;</foreign> and diſpoſe<lb></lb>all things elſe in the ſame manner a <lb></lb>before; as is done in the preſet <lb></lb>Figure. </s> <s>We are to demonſtrat <lb></lb>in the ſame method, that N T is e<lb></lb>quall to <foreign lang="grc">ψ;</foreign> and by the ſame reaſor <lb></lb>equall alſo to G I. </s> <s>And ſince that in<lb></lb>the Triangles P <foreign lang="grc">φ</foreign> C and N F S, the Angle F is not leſſer than the<lb></lb>Angle <foreign lang="grc">φ,</foreign> B F ſhall not be greater than B C: And, therefore, neither<lb></lb>ſhall S R be leſſer than C R; nor N X than P Y: But ſince P F is<lb></lb>greater than N T, let P F be Seſquialter of P Y: N T ſhall be leſſer<lb></lb>than Seſquialter of N X: And, therefore, N X ſhall be greate <lb></lb>than double of X T. </s> <s>Let N M be double of M T; and drawing <lb></lb>Line from M to K prolong it. </s> <s>It is manifeſt, now, by what hath<lb></lb>been ſaid, that the Portion ſhall not continue in this poſition, but ſhall<lb></lb>turn about, ſo, as that its Axis do make an Angle with the Surface<lb></lb>of the Liquid, leſſer than the Angle <foreign lang="grc">φ.</foreign> </s></p></chap> </body> <back></back> </text></archimedes>