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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Thu, 02 May 2013 11:38:23 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <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"/> <section><p type="head"> <s>ARCHIMEDES <lb/>HIS TRACT <lb/>De Incidentibus Humido, <lb/>OR OF THE <lb/>NATATION OF BODIES VPON, <lb/>OR SVBMERSION IN, <lb/>THE <lb/>WATER <lb/>OR OTHER LIQUIDS.</s></p><p type="head"> <s>IN TWO BOOKS.</s></p><p type="head"> <s>Tran&longs;lated from the Original Greek,</s></p><p type="head"> <s>Fir&longs;t into Latine, and afterwards into Italian, by <emph type="italics"/>NICOLO <lb/>TARTAGLIA,<emph.end type="italics"/> and by him familiarly demon­<lb/>&longs;trated by way of Dialogue, with <emph type="italics"/>Richard Wentworth,<emph.end type="italics"/><lb/>a Noble Engli&longs;h Gentleman, and his Friend.</s></p><p type="head"> <s>Together with the Learned Commentaries of <emph type="italics"/>Federico <lb/>Commandino,<emph.end type="italics"/> who hath Re&longs;tored &longs;uch of the Demon&longs;trations <lb/>as, thorow the Injury of Time, were obliterated.</s></p><p type="head"> <s>Now compared with the ORIGINAL, and Engli&longs;hed <lb/>By <emph type="italics"/>THOMAS SALVSBVRY,<emph.end type="italics"/> <expan abbr="E&longs;q.">E&longs;que</expan></s></p><p type="head"> <s><emph type="italics"/>LONDON,<emph.end type="italics"/> Printed by <emph type="italics"/>W. Leybourn,<emph.end type="italics"/> 1662.</s></p></section><section><pb xlink:href="073/01/002.jpg"/> <pb xlink:href="073/01/003.jpg" pagenum="335[333]"/><p type="head"> <s>ARCHIMEDES <lb/>HIS TRACT <lb/><emph type="italics"/>De <lb/>INCIDENTIBUS HUMIDO,<emph.end type="italics"/><lb/>OR OF <lb/>The Natation of Bodies upon, or Submer&longs;ion in, <lb/>the Water, or other Liquids.</s></p></section> </front> <body> <chap> <pb xlink:href="073/01/004.jpg"/><p type="head"> <s>BOOK I.</s></p><p type="head"> <s>RICARDO.</s></p><p type="main"> <s><emph type="italics"/>Dear Companion,<emph.end type="italics"/> I have peru&longs;ed your <emph type="italics"/>Indu&longs;trious Invention,<emph.end type="italics"/><lb/>in which I find not any thing that will not certainly hold <lb/>true; but, truth is, there are many of your Conclu&longs;ions <lb/>of which I under&longs;tand uot the Cau&longs;e, and therefore, if it <lb/>be not a trouble to you, I would de&longs;ire you to declare them <lb/>to me, for, indeed, nothing plea&longs;eth me, if the Cau&longs;e <lb/>thereof be hid from me.</s></p><p type="main"> <s>NICOLO. </s> <s>My obligations unto you are &longs;o many and <lb/>great, <emph type="italics"/>Honoured Campanion,<emph.end type="italics"/> that no reque&longs;t of yours ought <lb/>to be trouble&longs;ome to me, and therefore tell me what tho&longs;e Perticulars are of which <lb/>you know not the Cau&longs;e, for I &longs;hall endeavour with the utmo&longs;t of my power and <lb/>under&longs;tanding to &longs;atisfie you in all your demands.</s></p><p type="main"> <s>RIC. </s> <s>In the fir&longs;t <emph type="italics"/>Direction<emph.end type="italics"/> of the fir&longs;t Book of that your <emph type="italics"/>Indu&longs;trious Invention<emph.end type="italics"/><lb/>you conclude, That it is impo&longs;&longs;ible that the Water &longs;hould wholly receive into it <lb/>any material Solid Body that is lighter than it &longs;eif (as to <emph type="italics"/>&longs;peciæ<emph.end type="italics"/>) nay, you &longs;ay, That <lb/>there will alwaies a part of the Body &longs;tay or remain above the Waters Surface <lb/>(that is uncovered by it;) and, That as the whole Solid Body put into the Water <lb/>is in proportion to that part of it that &longs;hall be immerged, or received, into the Wa­<lb/>ter, &longs;o &longs;hall the Gravity of the Water be to the Gravity <emph type="italics"/>(in &longs;peciæ)<emph.end type="italics"/> of that &longs;ame <lb/>material Body: And that tho&longs;e Solid Bodies, that are by nature more Grave than the <lb/>Water, being put into the Water, &longs;hall pre&longs;ently make the &longs;aid Water give place; <lb/>and, That they do not only wholly enter or &longs;ubmerge in the &longs;ame, but go continu­<lb/>ally de&longs;cending untill they arrive at <emph type="italics"/>t<emph.end type="italics"/>he Bottom; and, That they &longs;ink to the Bot­<lb/>tom &longs;o much fa&longs;ter, by how much they are more Grave than the Water. </s> <s>And, <lb/>again, That tho&longs;e which are preci&longs;ely of the &longs;ame Gravity with the Water, being <lb/>put into the &longs;ame, are of nece&longs;&longs;ity wholly received into, or immerged by it, but <lb/>yet retained in the Surface of the &longs;aid Water, and much le&longs;s will the Water con­<lb/>&longs;ent that it do de&longs;cend to the Bottom: and, now, albeit that all the&longs;e things are <lb/>manife&longs;t to Sen&longs;e and Experience, yet neverthele&longs;s would I be very glad, if it be <lb/>po&longs;&longs;ible, that you would demon&longs;trate to me the mo&longs;t apt and proper Cau&longs;e of <lb/>the&longs;e Effects.</s></p> <pb xlink:href="073/01/005.jpg" pagenum="334"/><p type="main"> <s>NIC. </s> <s>The Cau&longs;e of all the&longs;e Effects is a&longs;&longs;igned by <emph type="italics"/>Archimedes,<emph.end type="italics"/> the <emph type="italics"/>Siracu&longs;an,<emph.end type="italics"/> in <lb/><arrow.to.target n="marg1126"></arrow.to.target><lb/>that Book <emph type="italics"/>De Incidentibus (^{*}) Aquæ,<emph.end type="italics"/> by me publi&longs;hed in Latine, and dedicated to <lb/>your &longs;elf, as I al&longs;o &longs;aid in the beginning of that my <emph type="italics"/>Indu&longs;trions Invention.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1126"></margin.target>* <emph type="italics"/>Aquæ,<emph.end type="italics"/> tan&longs;lated <lb/>by me <emph type="italics"/>Humido,<emph.end type="italics"/> as <lb/>the more Compre­<lb/>hen&longs;ive word, for <lb/>his Doctrine holds <lb/>true in all Liquids <lb/>as well as in Wa­<lb/>ter, <emph type="italics"/>&longs;oil.<emph.end type="italics"/> in Wine, <lb/>Oyl, Milk, <emph type="italics"/>&c.<emph.end type="italics"/></s></p><p type="main"> <s>RIC. </s> <s>I have &longs;een that &longs;ame <emph type="italics"/>Archimedes,<emph.end type="italics"/> and have very well under&longs;tood tho&longs;e <lb/>two Books in which he treateth <emph type="italics"/>De Centro Gravitatis æquerepentibus,<emph.end type="italics"/> or of the <lb/>Center of Gravity in Figures plain, or parallel to the Horizon; and likewi&longs;e tho&longs;e <lb/><emph type="italics"/>De Quadratura Parabolæ,<emph.end type="italics"/> or, of Squaring the Parabola; but ^{*}<emph type="italics"/>that<emph.end type="italics"/> in which he treat­<lb/>eth of Solids that Swim upon, or &longs;ink in Liquids, is &longs;o ob&longs;cure, that, to &longs;peak the <lb/>truth, there are many things in <emph type="italics"/>it<emph.end type="italics"/> which I do not under&longs;tand, and therefore before <lb/><arrow.to.target n="marg1127"></arrow.to.target><lb/>we proceed any farther, I &longs;hould take it for a favour if you would declare it to me <lb/>in your Vulgar Tongue, beginning with his fir&longs;t <emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> which &longs;peaketh in this <lb/>manner.</s></p><p type="margin"> <s><margin.target id="marg1127"></margin.target>* He &longs;peaks of but <lb/>one Book, <emph type="italics"/>Tartag­<lb/>lia<emph.end type="italics"/> having tran&longs;la­<lb/>ted no more.</s></p><p type="head"> <s>SVPPOSITION I.</s></p><p type="main"> <s><emph type="italics"/>It is &longs;uppo&longs;ed that the Liquid is of &longs;uch a nature, that <lb/>its parts being equi-jacent and contiguous, the le&longs;s <lb/>pre&longs;&longs;ed are repul&longs;ed by the more pre&longs;&longs;ed. </s> <s>And <lb/>that each of its parts is pre&longs;&longs;ed or repul&longs;ed by the <lb/>Liquor that lyeth over it, perpendicularly, if the <lb/>Liquid be de&longs;cending into any place, or pre&longs;&longs;ed any <lb/>whither by another.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>Every Science, Art, or Doctrine (as you know, <emph type="italics"/>Honoured Companion,<emph.end type="italics"/>) <lb/>hath its fir&longs;t undemon&longs;trable Principles, by which (they being <lb/>granted or &longs;uppo&longs;ed) the &longs;aid Science is proved, maintained, or de­<lb/>mon&longs;trated. </s> <s>And of the&longs;e Principles, &longs;ome are called <emph type="italics"/>Petitions,<emph.end type="italics"/><lb/>and others <emph type="italics"/>Demands,<emph.end type="italics"/> or <emph type="italics"/>Suppo&longs;itions.<emph.end type="italics"/> I &longs;ay, therefore, that the Science or Doctrine <lb/>of tho&longs;e Material Solids that Swim or Sink in Liquids, hath only two undemon­<lb/>&longs;trable <emph type="italics"/>Suppo&longs;itions,<emph.end type="italics"/> one of which is that above alledged, the which in compliance <lb/>with your de&longs;ire I have &longs;et 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 under&longs;tand the <lb/>parts of a Liquid to be <emph type="italics"/>Equijacent.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>When they are equidi&longs;tant from the Center of the World, or of the <lb/>Earth (which is the &longs;ame, although ^{*} &longs;ome hold that the Centers of the Earth <lb/>and Worldare different.)</s></p><p type="main"> <s>RIC. </s> <s>I under&longs;tand you not unle&longs;s you give me &longs;ome Example thereof in <lb/>Figure.<lb/><arrow.to.target n="marg1128"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1128"></margin.target>* The Coperni­<lb/>cans.</s></p><p type="main"> <s>NIC. </s> <s>To exemplifie this particular, Let us &longs;uppo&longs;e a quantity of Liquor (as <lb/>for in&longs;tance of Water) to be upon the Earth; then let us with the Imagination <lb/>cut the whole Earth together with that Water into two equal parts, in &longs;uch a <lb/>manner as that the &longs;aid Section may pa&longs;s ^{*} by the Center of the Earth: And let <lb/>us &longs;uppo&longs;e that one part of the Superficies of that Section, as well of the Water <lb/>as of the Earth, be the Superficies A B, and that the Center of the Earth be the <lb/>point K. </s> <s>This being done, let us in our Imagination de&longs;cribe a Circle upon the </s></p><p type="main"> <s><arrow.to.target n="marg1129"></arrow.to.target><lb/>&longs;aid Center K, of &longs;uch a bigne&longs;s as that the Circumference may pa&longs;s by the Super­<lb/>ficies of the Section of the Water: Now let this Circumference be E F G: and <lb/>let many Lines be drawn from the point K to the &longs;aid Circumference, cutting the <lb/>&longs;ame, as KE, KHO, KFQ KLP, KM. </s> <s>Now I &longs;ay, that all the&longs;e parts of <lb/>the &longs;aid Water, terminated in that Circumference, are Equijacent, as being all <pb xlink:href="073/01/006.jpg" pagenum="335"/>equidi&longs;tant from the point K, the Center of the World, which parts are G M, <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 under&longs;tand you very well, as to this particular: But tell me a little; he <lb/>&longs;aith that each of the parts of the Liquid is pre&longs;&longs;ed or repul&longs;ed by the Liquid that <lb/>is above it, according to the Perpendicular: I know not what that Liquid is that <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/>thorow &longs;ome Water, each part of the Water that is in that Line he &longs;uppo&longs;eth to <lb/>be pre&longs;&longs;ed or repul&longs;ed by the Water that lieth above it in that &longs;ame Line, and that <lb/>that repul&longs;e is made according to the &longs;ame Line, (that is, directly towards the <lb/>Center of the World) which Line is called a Perpendicular; becau&longs;e every <lb/>Right-Line that departeth from any point, and goeth directly towards the Worlds <lb/>Center is called a Perpendicular. </s> <s>And that you may the better under&longs;tand me, let <lb/><figure id="id.073.01.006.1.jpg" xlink:href="073/01/006/1.jpg"/><lb/>us imagine <lb/>the Line KHO, <lb/>and in that <lb/>let us imagine <lb/>&longs;everal parts, <lb/>as &longs;uppo&longs;e RS, <lb/>S T, T V, V H, <lb/>H O. </s> <s>I &longs;ay, <lb/>that he &longs;up­<lb/>po&longs;eth that <lb/>the part V H <lb/>is pre&longs;&longs;ed by <lb/>that placed a­<lb/>bove it, H O, <lb/>according to <lb/>the Line OK; <lb/>the which <lb/>O K, as hath been &longs;aid above, is called the Perpendicular pa&longs;&longs;ing thorow tho&longs;e two <lb/>parts. </s> <s>In like manner, I &longs;ay that the part T V is expul&longs;ed by the part V H, ac­<lb/>cording to the &longs;aid Line O K: and &longs;o the part S T to be pre&longs;&longs;ed by T V, according <lb/>to the &longs;aid Perpendicular O K, and R S by S T. </s> <s>And this you are to under&longs;tand <lb/>in all the other Lines that were protracted from the &longs;aid Point K, penetrating the <lb/>&longs;aid Water, As for Example, in <emph type="italics"/>K<emph.end type="italics"/> G, <emph type="italics"/>K<emph.end type="italics"/> M, <emph type="italics"/>K<emph.end type="italics"/> L, <emph type="italics"/>K<emph.end type="italics"/> F, <emph type="italics"/>K<emph.end type="italics"/> E, and infinite others of the <lb/>like kind.</s></p><p type="main"> <s>RIC. Indeed, <emph type="italics"/>Dear Companion,<emph.end type="italics"/> this your Explanation hath given megreat &longs;a­<lb/>tisfaction; for, in my Judgment, it &longs;eemeth that all the difficulty of this Suppo&longs;ition <lb/>con&longs;i&longs;ts in the&longs;e two particulars which you have declared to me.</s></p><p type="main"> <s>NIC. </s> <s>It doth &longs;o; for having under&longs;tood that the parts E H, H F, F L, L M, and <lb/>MG, determining in the Circumference of the &longs;aid Circle are equijacent, it is an <lb/>ea&longs;ie matter to under&longs;tand the fore&longs;aid <emph type="italics"/>Suppo&longs;ition<emph.end type="italics"/> in Order, which &longs;aith, <emph type="italics"/>That it is <lb/>&longs;uppo&longs;ed that the Liquid is of &longs;uch a nature, that the part thereof le&longs;s pre&longs;&longs;ed or thrust is re­<lb/>pul&longs;ed by the more thru&longs;t or pre&longs;&longs;ed.<emph.end type="italics"/> As for example, if the part E H were by chance <lb/>more thru&longs;t, crowded, or pre&longs;&longs;ed from above downwards by the Liquid, or &longs;ome <lb/>other matter that was over it, than the part H F, contiguous to it, it is &longs;uppo&longs;ed <lb/>that the &longs;aid part H F, le&longs;s pre&longs;&longs;ed, would be repul&longs;ed by the &longs;aid part E H. </s> <s>And <lb/>thus we ought to under&longs;tand of the other parts equijacent, in ca&longs;e that they be <lb/>contiguous, and not &longs;evered. </s> <s>That each of the parts thereof is pre&longs;&longs;ed and repul. <lb/></s> <s>&longs;ed by the <emph type="italics"/>L<emph.end type="italics"/>iquid that lieth over it Perpendicularly, is manife&longs;t by that which was <lb/>&longs;aid above, to wit, that it &longs;hould be repul&longs;ed, in ca&longs;e the <emph type="italics"/>L<emph.end type="italics"/>iquid be de&longs;cending into <lb/>any place, and thru&longs;t, or driven any whither by another.</s></p><p type="main"> <s>RIC. </s> <s>I under&longs;tand this Suppo&longs;ition very well, but yet me thinks that before <lb/>the Suppo&longs;ition, the Author ought to have defined tho&longs;e two particulars, which <lb/>you fir&longs;t declared to me, that is, how we are to under&longs;tand the parts of the <emph type="italics"/>L<emph.end type="italics"/>iquid <lb/>equijacent, and likewi&longs;e the Perpendicular.</s></p> <pb xlink:href="073/01/007.jpg" pagenum="336"/><p type="main"> <s>NIC. </s> <s>You &longs;ay truth.</s></p><p type="main"> <s>RIC. </s> <s>I have another que&longs;tion to aske you, which is this, Why the Author <lb/>u&longs;eth the word <emph type="italics"/>L<emph.end type="italics"/>iquid, or Humid, in&longs;tead of Water.</s></p><p type="main"> <s>NIC. </s> <s>It may be for two of the&longs;e two Cau&longs;es; the one is, that Water being the <lb/>principal of all <emph type="italics"/>L<emph.end type="italics"/>iquids, therefore &longs;aying <emph type="italics"/>Humidum<emph.end type="italics"/> he is to be under&longs;tood to mean <lb/>the chief Liquid, that is Water: The other, becau&longs;e that all the Propo&longs;itions of <lb/>this Book of his, do not only hold true in Water, but al&longs;o in every other <emph type="italics"/>L<emph.end type="italics"/>iquid, <lb/>as in Wine, Oyl, and the like: and therefore the Author might have u&longs;ed the word <lb/><emph type="italics"/>Humidum,<emph.end type="italics"/> as being a word more general than <emph type="italics"/>Aqua.<emph.end type="italics"/></s></p><p type="main"> <s>RIC. </s> <s>This I under&longs;tand, therefore let us come to the fir&longs;t <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> which, as <lb/>you know, in the Original &longs;peaks in this manner.</s></p><p type="head"> <s>PROP. I. THEOR. I.</s></p><p type="main"> <s><emph type="italics"/>If any Superficies &longs;hall be cut by a Plane thorough any <lb/>Point, and the Section be alwaies the Circumference <lb/>of a Circle, who&longs;e Center is the &longs;aid Point: that Su­<lb/>perficies &longs;hall be Spherical.<emph.end type="italics"/></s></p><p type="main"> <s>Let any Superficies be cut at plea&longs;ure by a Plane thorow the <lb/>Point K; and let the Section alwaies de&longs;cribe the Circumfe­<lb/>rence of a Circle that hath for its Center the Point K: I &longs;ay, <lb/>that that &longs;ame Superficies is Sphærical. </s> <s>For were it po&longs;&longs;ible that the <lb/>&longs;aid Superficies were not Sphærical, then all the Lines drawn <lb/>through the &longs;aid Point K unto that Superficies would not be equal, <lb/>Let therefore A and B be two <lb/>Points in the &longs;aid Superficies, &longs;o that <lb/><figure id="id.073.01.007.1.jpg" xlink:href="073/01/007/1.jpg"/><lb/>drawing the two Lines K A and <lb/>K B, let them, if po&longs;&longs;ible, be une­<lb/>qual: Then by the&longs;e two Lines let <lb/>a Plane be drawn cutting the &longs;aid <lb/>Superficies, and let the Section in <lb/>the Superficies make the Line <lb/>D A B G: Now this Line D A B G <lb/>is, by our pre-&longs;uppo&longs;al, a Circle, and <lb/>the Center thereof is the Point K, for &longs;uch the &longs;aid Superficies was <lb/>&longs;uppo&longs;ed to be. </s> <s>Therefore the two Lines K A and K B are equal: <lb/>But they were al&longs;o &longs;uppo&longs;ed to be unequal; which is impo&longs;&longs;ible: <lb/>It followeth therefore, of nece&longs;&longs;ity, that the &longs;aid Superficies be <lb/>Sphærical, that is, the Superficies of a Sphære.</s></p><p type="main"> <s>RIC. </s> <s>I under&longs;tand you very well; now let us proceed to the &longs;econd <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/><lb/>which, you know, runs thus.</s></p> <pb xlink:href="073/01/008.jpg" pagenum="337"/><p type="head"> <s>PROP. II. THEOR. II.</s></p><p type="main"> <s><emph type="italics"/>The Superficies of every Liquid that is con&longs;i&longs;tant and <lb/>&longs;etled &longs;hall be of a Sphærical Figure, which Figure <lb/>&longs;hall have the &longs;ame Center with the Earth.<emph.end type="italics"/></s></p><p type="main"> <s>Let us &longs;uppo&longs;e a Liquid that is of &longs;uch a con&longs;i&longs;tance as that it <lb/>is not moved, and that its Superficies be cut by a Plane along <lb/>by the Center of the Earth, and let the Center of the Earth <lb/>be the Point K: and let the Section of the Superficies be the Line <lb/>A B G D. </s> <s>I &longs;ay that the Line A B G D is the Circumference of a <lb/><figure id="id.073.01.008.1.jpg" xlink:href="073/01/008/1.jpg"/><lb/>Circle, and that the Center <lb/>thereof is the Point K And <lb/>if it be po&longs;&longs;ible that it may <lb/>not be the Circumference <lb/>of a Circle, the Right­<lb/><arrow.to.target n="marg1130"></arrow.to.target><lb/>Lines drawn ^{*} by the Point <lb/>K to the &longs;aid Line A B G D <lb/>&longs;hall not be equal. </s> <s>There­<lb/>fore let a Right-Line be <lb/>taken greater than &longs;ome of tho&longs;e produced from the Point K unto <lb/>the &longs;aid Line A B G D, and le&longs;&longs;er than &longs;ome other; and upon the <lb/>Point K let a Circle be de&longs;cribed at the length of that Line, <lb/>Now the Circumference of this Circle &longs;hall fall part without the <lb/>&longs;aid Line A B G D, and part within: it having been pre&longs;uppo&longs;ed <lb/>that its Semidiameter is greater than &longs;ome of tho&longs;e Lines that may <lb/>be drawn from the &longs;aid Point K unto the &longs;aid Line A B G D, and <lb/>le&longs;&longs;er than &longs;ome other. </s> <s>Let the Circumference of the de&longs;cribed <lb/>Circle be R B G H, and from B to K draw the Right-Line B K: and <lb/>drawn al&longs;o the two Lines K R, and K E L which make a Right­<lb/>Angle in the Point K: and upon the Center K de&longs;cribe the Circum­<lb/>ference X O P in the Plane and in the Liquid. </s> <s>The parts, there­<lb/>fore, of the Liquid that are ^{*} according to the Circumference <lb/><arrow.to.target n="marg1131"></arrow.to.target><lb/>X O P, for the rea&longs;ons alledged upon the fir&longs;t <emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> are equi­<lb/>jacent, or equipo&longs;ited, and contiguous to each other; and both <lb/>the&longs;e parts are pre&longs;t or thru&longs;t, according to the &longs;econd part of the <lb/><emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> by the Liquor which is above them. </s> <s>And becau&longs;e the <lb/>two Angles E K B and B K R are &longs;uppo&longs;ed equal [<emph type="italics"/>by the<emph.end type="italics"/> 26. <emph type="italics"/>of<emph.end type="italics"/> 3. <lb/><emph type="italics"/>of Euclid,<emph.end type="italics"/>] the two Circumferences or Arches B E and B R &longs;hall <lb/>be equal (fora&longs;much as R B G H was a Circle de&longs;cribed for &longs;atis­<lb/>faction of the Oponent, and K its Center:) And in like manner <lb/>the whole Triangle B E K &longs;hall be equal to the whole Triangle <lb/>B R K. </s> <s>And becau&longs;e al&longs;o the Triangle O P K for the &longs;ame rea&longs;on <pb xlink:href="073/01/009.jpg" pagenum="338"/>&longs;hall be equal to the Triangle O X K; Therefore (by common <lb/>Notion) &longs;ub&longs;tracting tho&longs;e two &longs;mall Triangles O P K and O X K <lb/>from the two others B E K and B R K, the two Remainders &longs;hall <lb/>be equal: one of which Remainders &longs;hall be the Quadrangle <lb/>B E O P, and the other B R X O. </s> <s>And becau&longs;e the whole Quadran­<lb/>gle B E O P is full of Liquor, and of the Quadrangle B R X O, <lb/>the part B A X O only is full, and the re&longs;idue B R A is wholly void <lb/>of Water: It followeth, therefore, that the Quadrangle B E O P <lb/>is more ponderous than the Quadrangle B R X O. </s> <s>And if the &longs;aid <lb/>Quadrangle B E O P be more Grave than the Quadrangle <lb/>B R X O, much more &longs;hall the Quadrangle B L O P exceed in Gra­<lb/>vity the &longs;aid Quadrangle B R X O: whence it followeth, that the <lb/>part O P is more pre&longs;&longs;ed than the part O X. But, by the fir&longs;t part <lb/>of the Suppo&longs;ition, the part le&longs;s pre&longs;&longs;ed &longs;hould be repul&longs;ed by the <lb/>part more pre&longs;&longs;ed: Therefore the part O X mu&longs;t be repul&longs;ed by <lb/>the part O P: But it was pre&longs;uppo&longs;ed that the Liquid did not <lb/>move: Wherefore it would follow that the le&longs;s pre&longs;&longs;ed would not <lb/>be repul&longs;ed by the more pre&longs;&longs;ed: And therefore it followeth of <lb/>nece&longs;&longs;ity that the Line A <emph type="italics"/>B<emph.end type="italics"/> G D is the Circumference of a Circle, <lb/>and that the Center of it is the point K. </s> <s>And in like manner &longs;hall <lb/>it be demon&longs;trated, if the Surface of the Liquid be cut by a Plane <lb/>thorow the Center of the Earth, that the Section &longs;hall be the Cir­<lb/>cumference of a Circle, and that the Center of the &longs;ame &longs;hall be <lb/>that very Point which is Center of the Earth. </s> <s>It is therefore mani­<lb/>fe&longs;t that the Superficies of a Liquid that is con&longs;i&longs;tant and &longs;etled <lb/>&longs;hall have the Figure of a Sphære, the Center of which &longs;hall be <lb/>the &longs;ame with that of the Earth, by the fir&longs;t <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/>; for it is <lb/>&longs;uch that being ever cut thorow the &longs;ame Point, the Section or Di­<lb/>vi&longs;ion de&longs;cribes the Circumference of a Circle which hath for Cen­<lb/>ter the &longs;elf-&longs;ame Point that is Center of the Earth: Which was to <lb/>be demon&longs;trated.</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"/>i.e.<emph.end type="italics"/> Parallel.</s></p><p type="main"> <s>RIC. </s> <s>I do thorowly under&longs;tand the&longs;e your Rea&longs;ons, and &longs;ince there is in them <lb/>no umbrage of Doubting, let us proceed to his third <emph type="italics"/>Propo&longs;ition.<emph.end type="italics"/></s></p><p type="head"> <s>PROP. III. THEOR. III.</s></p><p type="main"> <s><emph type="italics"/>Solid Magnitudes that being of equal Ma&longs;s with the <lb/>Liquid are al&longs;o equal to it in Gravity, being demit-<emph.end type="italics"/><lb/><arrow.to.target n="marg1132"></arrow.to.target><lb/><emph type="italics"/>ted into the [^{*} &longs;etled] Liquid do &longs;o &longs;ubmerge in the <lb/>&longs;ame as that they lie or appear not at all above the <lb/>Surface of the Liquid, nor yet do they &longs;ink to the <lb/>Bottom.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/010.jpg" pagenum="339"/><p type="margin"> <s><margin.target id="marg1132"></margin.target>* I add the word <lb/>&longs;etled, as nece&longs;&longs;ary <lb/>in making the Ex­<lb/>periment.</s></p><p type="main"> <s>NIC. </s> <s>In this <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> it is affirmed that tho&longs;e Solid Magnitules that hap­<lb/>pen to be equal in &longs;pecifical Gravity with the Liquid being lefeat liber­<lb/>ty in the &longs;aid Liquid do &longs;o &longs;ubmerge in the &longs;ame, as that they lie or ap­<lb/>pear not at all above the Surface of the Liquid, nor yet do they go or &longs;ink to the <lb/>Bottom.</s></p><p type="main"> <s>For &longs;uppo&longs;ing, on the contrary, that it were po&longs;&longs;ible for one of <lb/>tho&longs;e Solids being placed in the Liquid to lie in part without the <lb/>Liquid, that is above its Surface, (alwaies provided that the &longs;aid <lb/>Liquid be &longs;etled and undi&longs;turbed,) let us imagine any Plane pro­<lb/>duced thorow the Center of the Earth, thorow the Liquid, and <lb/>thorow that Solid Body: and let us imagine that the Section of the <lb/>Liquid is the Superficies A B G D, and the Section of the Solid <lb/>Body that is within it the Super&longs;icies E Z H T, and let us &longs;uppo&longs;e <lb/>the Center of the Earth to be the Point K: and let the part of the <lb/>&longs;aid Solid &longs;ubmerged in the Liquid be B G H T, and let that above <lb/>be B E Z G: and let the Solid Body be &longs;uppo&longs;ed to be comprized in <lb/>a Pyramid that hath its Parallelogram Ba&longs;e in the upper Surface of <lb/>the Liquid, and its Summity or Vertex in the Center of the Earth: <lb/>which Pyramid let us al&longs;o &longs;uppo&longs;e to be cut or divided by the &longs;ame <lb/>Plane in which is the Circumference A B G D, and let the Sections <lb/><figure id="id.073.01.010.1.jpg" xlink:href="073/01/010/1.jpg"/><lb/>of the Planes of the &longs;aid <lb/>Pyramid be K L and <lb/>K M: and in the Liquid <lb/>about the Center K let <lb/>there be de&longs;cribed a Su­<lb/>perficies of another <lb/>Sphære below E Z H T, <lb/>which let be X O P; <lb/>and let this be cut by <lb/>the Superficies of the Plane: And let there be another Pyramid ta­<lb/>ken or &longs;uppo&longs;ed equal and like to that which compri&longs;eth the &longs;aid <lb/>Solid Body, and contiguous and conjunct with the &longs;ame; and let <lb/>the Sections of its Superficies be K M and K N: and let us &longs;uppo&longs;e <lb/>another Solid to be taken or imagined, of Liquor, contained in that <lb/>&longs;ame Pyramid, which let be R S C Y, equal and like to the partial <lb/>Solid B H G T, which is immerged in the &longs;aid Liquid: But the <lb/>part of the Liquid which in the fir&longs;t Pyramid is under the Super­<lb/>ficies X O, and that, which in the other Pyramid is under the Su­<lb/>perficies O P, are equijacent or equipo&longs;ited and contiguous, but <lb/>are not pre&longs;&longs;ed equally; for that which is under the Superficies <lb/>X O is pre&longs;&longs;ed by the Solid T H E Z, and by the Liquor that is <lb/>contained between the two Spherical Superficies X O and L M <lb/>and the Planes of the Pyramid, but that which proceeds accord­<lb/>ing to F O is pre&longs;&longs;ed by the Solid R S C Y, and by the Liquid <pb xlink:href="073/01/011.jpg" pagenum="340"/>contained between the Sphærical Superficies that proceed accord­<lb/>ing to P O and M N and the Planes of the Pyramid; and the Gra­<lb/>vity of the Liquid, which is according to M N O P, &longs;hall be le&longs;&longs;er <lb/>than that which is according to L M X O; becau&longs;e that Solid of <lb/>Liquor which proceeds according to R S C Y is le&longs;s than the Solid <lb/>E Z H T (having been &longs;uppo&longs;ed to be equal in quantity to only <lb/>the part H B G T of that:) And the &longs;aid Solid E Z H T hath been <lb/>&longs;uppo&longs;ed to be equally grave with the Liquid: Therefore the Gra­<lb/>vity of the <emph type="italics"/>L<emph.end type="italics"/>iquid compri&longs;ed betwixt the two Sphærical Superfi­<lb/>cies L M and <emph type="italics"/>X<emph.end type="italics"/> O, and betwixt the &longs;ides L <emph type="italics"/>X<emph.end type="italics"/> and M O of the <lb/><figure id="id.073.01.011.1.jpg" xlink:href="073/01/011/1.jpg"/><lb/>Pyramid, together with <lb/>the whole Solid EZHT, <lb/>&longs;hall exceed the Gravity <lb/>of the Liquid compri­<lb/>&longs;ed betwixt the other <lb/>two Sphærical Superfi­<lb/>cies M N and O P, and <lb/>the Sides M O and N P <lb/>of the Pyramid, toge­<lb/>ther with the Solid of Liquor R S C Y by the quantity of the Gra­<lb/>vity of the part E B Z G, &longs;uppo&longs;ed to remain above the Surface of <lb/>the Liquid: And therefore it is manife&longs;t that the part which pro­<lb/>ceedeth according to the Circumference O P is pre&longs;&longs;ed, driven, and <lb/>repul&longs;ed, according to the <emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> by that which proceeds ac­<lb/>cording to the Circumference X O, by which means the Liquid <lb/>would not be &longs;etled and &longs;till: But we did pre&longs;uppo&longs;e that it was <lb/>&longs;etled, namely &longs;o, as to be without motion: It followeth, therefore, <lb/>that the &longs;aid Solid cannot in any part of it exceed or lie above the <lb/>Superficies of the Liquid: And al&longs;o that being dimerged in the Li­<lb/>quid it cannot de&longs;cend to the Bottom, for that all the parts of the <lb/>Liquid equijacent, or di&longs;po&longs;ed equally, are equally pre&longs;&longs;ed, becau&longs;e <lb/>the Solid is equally grave with the Liquid, by what we pre&longs;uppo&longs;ed.</s></p><p type="main"> <s>RIC. </s> <s>I do under&longs;tand your Argumentation, but I under&longs;tand not that Phra&longs;e <lb/><emph type="italics"/>Solid Magnitudes.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>I will declare this Term unto you. <emph type="italics"/>Magnitude<emph.end type="italics"/> is a general Word that <lb/>re&longs;pecteth all the Species of Continual Quantity; and the Species of Continual <lb/>Quantity are three, that is, the <emph type="italics"/>L<emph.end type="italics"/>ine, the Superficies, and the Body; which Body <lb/>is al&longs;o called a Solid, as having in it &longs;elf <emph type="italics"/>L<emph.end type="italics"/>ength, Breadth, and Thickne&longs;s, or Depth: <lb/>and therefore that none might equivocate or take that Term <emph type="italics"/>Magnitudes<emph.end type="italics"/> to be <lb/>meant of <emph type="italics"/>L<emph.end type="italics"/>ines, or Superficies, but only of Solid <emph type="italics"/>Magnitudes,<emph.end type="italics"/> that is, Bodies, he <lb/>did &longs;pecifie it by that manner of expre&longs;&longs;ion, as was &longs;aid. </s> <s>The truth is, that he <lb/>might have expre&longs;t that <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> in this manner: <emph type="italics"/>Solids (or Bodies) which being <lb/>of equal Gravity with an equal Ma&longs;s of the Liquid,<emph.end type="italics"/> &c. </s> <s>And this <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> would have <lb/>been more cleer and intelligible, for it is as &longs;ignificant to &longs;ay, a <emph type="italics"/>Solid,<emph.end type="italics"/> or, a <emph type="italics"/>Body,<emph.end type="italics"/> as <lb/>to &longs;ay, a <emph type="italics"/>Solid Magnitude:<emph.end type="italics"/> therefore wonder not if for the future I u&longs;e the&longs;e three <lb/>kinds of words indifferently.</s></p><p type="main"> <s>RIC. </s> <s>You have &longs;ufficiently &longs;atisfied me, wherefore that we may lo&longs;e no time <lb/>let us go forwards to the fourth <emph type="italics"/>Propo&longs;ition.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/012.jpg" pagenum="341"/><p type="head"> <s>PROP. IV. THEOR. IV.</s></p><p type="main"> <s><emph type="italics"/>Solid Magnitudes that are lighter than the Liquid, <lb/>being demitted into the &longs;etled Liquid, will not total­<lb/>ly &longs;ubmerge in the &longs;ame, but &longs;ome part thereof will <lb/>lie or &longs;tay above the Surface of the Liquid.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>In this fourth <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> it is concluded, that every Body or Solid that is <lb/>lighter (as to Specifical Gravity) than the <emph type="italics"/>L<emph.end type="italics"/>iquid, being put into the <lb/><emph type="italics"/>L<emph.end type="italics"/>iquid, will not totally &longs;ubmerge in the &longs;ame, but that &longs;ome part of it <lb/>will &longs;tay and appear without the <emph type="italics"/>L<emph.end type="italics"/>iquid, that is above its Surface.</s></p><p type="main"> <s>For &longs;uppo&longs;ing, on the contrary, that it were po&longs;&longs;ible for a Solid <lb/>more light than the Liquid, being demitted in the Liquid to &longs;ub­<lb/>merge totally in the &longs;ame, that is, &longs;o as that no part thereof re­<lb/>maineth above, or without the &longs;aid Liquid, (evermore &longs;uppo&longs;ing <lb/>that the Liquid be &longs;o con&longs;tituted as that it be not moved,) let us <lb/>imagine any Plane produced thorow the Center of the Earth, tho­<lb/>row the Liquid, and thorow that Solid Body: and that the Surface <lb/>of the Liquid is cut by this Plane according to the Circumference <lb/>A <emph type="italics"/>B<emph.end type="italics"/> G, and the Solid <emph type="italics"/>B<emph.end type="italics"/>ody according to the Figure R; and let the <lb/>Center of the Earth be K. </s> <s>And let there be imagined a Pyramid <lb/><figure id="id.073.01.012.1.jpg" xlink:href="073/01/012/1.jpg"/><lb/>that compri&longs;eth the Figure <lb/>R, as was done in the pre. <lb/></s> <s>cedent, that hath its Ver­<lb/>tex in the Point K, and let <lb/>the Superficies of that <lb/>Pyramid be cut by the <lb/>Superficies of the Plane <lb/>A <emph type="italics"/>B<emph.end type="italics"/> G, according to A K <lb/>and K <emph type="italics"/>B<emph.end type="italics"/>. </s> <s>And let us ima­<lb/>gine another Pyramid equal and like to this, and let its Superficies <lb/>be cut by the Superficies A <emph type="italics"/>B<emph.end type="italics"/> G according to K <emph type="italics"/>B<emph.end type="italics"/> and K <emph type="italics"/>G<emph.end type="italics"/>; and let <lb/>the Superficies of another Sphære be de&longs;cribed in the Liquid, upon <lb/>the Center K, and beneath the Solid R; and let that be cut by the <lb/>&longs;ame Plane according to <emph type="italics"/>X<emph.end type="italics"/> O P. And, la&longs;tly, let us &longs;uppo&longs;e ano­<lb/>ther Solid taken ^{*} from the Liquid, in this &longs;econd Pyramid, which <lb/><arrow.to.target n="marg1133"></arrow.to.target><lb/>let be H, equal to the Solid R. </s> <s>Now the parts of the Liquid, name­<lb/>ly, that which is under the Spherical Superficies that proceeds ac­<lb/>cording to the Superficies or Circumference <emph type="italics"/>X<emph.end type="italics"/> O, in the fir&longs;t Py­<lb/>ramid, and that which is under the Spherical Superficies that pro­<lb/>ceeds according to the Circumference O P, in the &longs;econd Pyramid, <lb/>are equijacent, and contiguous, but are not pre&longs;&longs;ed equally; for <pb xlink:href="073/01/013.jpg" pagenum="342"/>that of the fir&longs;t Pyramid is pre&longs;&longs;ed by the Solid R, and by the Liquid <lb/>which that containeth, that is, that which is in the place of the Py­<lb/>ramid according to A B O X: but that part which, in the other Py­<lb/>ramid, is pre&longs;&longs;ed by the Solid H, &longs;uppo&longs;ed to be of the &longs;ame Li­<lb/>quid, and by the Liquid which that containeth, that is, that which <lb/>is in the place of the &longs;aid Pyramid according to P O B G: and the <lb/>Gravity of the Solid R is le&longs;s than the Gravity of the Liquid <lb/>H, for that the&longs;e two Magnitudes were &longs;uppo&longs;ed to be equal in <lb/>Ma&longs;s, and the Solid R was &longs;uppo&longs;ed to be lighter than the Liquid: <lb/>and the Ma&longs;&longs;es of the two Pyramids of Liquor that containeth the&longs;e <lb/><arrow.to.target n="marg1134"></arrow.to.target><lb/>two Solids R and H are equal ^{*} by what was pre&longs;uppo&longs;ed: There­<lb/>fore the part of the Liquid that is under the Superficies that pro­<lb/>ceeds according to the Circumference O P is more pre&longs;&longs;ed; and, <lb/>therefore, by the <emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> it &longs;hall repul&longs;e that part which is le&longs;s <lb/>pre&longs;&longs;ed, whereby the &longs;aid Liquid will not be &longs;etled: But it was be­<lb/>fore &longs;uppo&longs;ed that it was &longs;etled: Therefore that Solid R &longs;hall not <lb/>totally &longs;ubmerge, but &longs;ome part thereof will remain without the <lb/>Liquid, that is, above its Surface, Which was the <emph type="italics"/>Propo&longs;ition.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1133"></margin.target>* That is a Ma&longs;s of <lb/>the Liquid.</s></p><p type="margin"> <s><margin.target id="marg1134"></margin.target>* For that the Py­<lb/>ramids were &longs;uppo­<lb/>&longs;ed equal.</s></p><p type="main"> <s>RIC. </s> <s>I have very well under&longs;tood you, therefore let us come to the fifth <emph type="italics"/>Pro­<lb/>po&longs;ition,<emph.end type="italics"/> which, as you know, doth thus &longs;peak.</s></p><p type="head"> <s>PROP. V. THEOR. V.</s></p><p type="main"> <s><emph type="italics"/>Solid Magnitudes that are lighter than the Liquid, <lb/>being demitted in the (&longs;etled) Liquid, will &longs;o far <lb/>&longs;ubmerge, till that a Ma&longs;s of Liquor, equal to the <lb/>Part &longs;ubmerged, doth in Gravity equalize the <lb/>whole Magnitude.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>It having, in the precedent, been demon&longs;trared that Solids lighter than <lb/>the Liquid, being demitted in the <emph type="italics"/>L<emph.end type="italics"/>iquid, alwaies a part of them remains <lb/>without the <emph type="italics"/>L<emph.end type="italics"/>iquid, that is above its Surface; In this fifth <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> it is <lb/>a&longs;&longs;erted, that &longs;o much of &longs;uch a Solid &longs;hall &longs;ubmerge, as that a Ma&longs;s of the <lb/><emph type="italics"/>L<emph.end type="italics"/>iquid equal to the part &longs;ubmerged, &longs;hall have equal Gravity with the whole <lb/>Solid.</s></p><p type="main"> <s>And to demon&longs;trate this, let us a&longs;&longs;ume all the &longs;ame Schemes <lb/>as before, in <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> 3. and likewi&longs;e let the Liquid be &longs;et­<lb/>led, and let the Solid E Z H T be lighter than the Liquid. <lb/></s> <s>Now if the &longs;aid Liquid be &longs;etled, the parts of it that are equija­<lb/>cent are equally pre&longs;&longs;ed: Therefore the Liquid that is beneath <pb xlink:href="073/01/014.jpg" pagenum="343"/>the Superficies that proceed according to the Circumferences X O <lb/>and P O are equally pre&longs;&longs;ed; whereby the Gravity pre&longs;&longs;ed is equal. <lb/><figure id="id.073.01.014.1.jpg" xlink:href="073/01/014/1.jpg"/><lb/>But the Gravity of the <lb/>Liquid which is in the <lb/><arrow.to.target n="marg1135"></arrow.to.target><lb/>fir&longs;t Pyramid ^{*} without <lb/>the Solid B H T G, is <lb/>equal to the Gravity of <lb/>the Liquid which is in <lb/>the other Pyramid with­<lb/>out the Liquid R S C Y: <lb/>It is manife&longs;t, therefore, <lb/>that the Gravity of the Solid E Z H T, is equal to the Gravity of <lb/>the Liquid R S C Y: Therefore it is manife&longs;t that a Ma&longs;s of Liquor <lb/>equal in Ma&longs;s to the part of the Solid &longs;ubmerged is equal in Gra­<lb/>vity to the whole Solid.</s></p><p type="margin"> <s><margin.target id="marg1135"></margin.target>* <emph type="italics"/>Without, i.e.<emph.end type="italics"/> that <lb/>being deducted.</s></p><p type="main"> <s>RIC. </s> <s>This was a pretty Demon&longs;tration, and becau&longs;e I very well under&longs;tand <lb/>it, let us lo&longs;e no time, but proceed to the &longs;ixth <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> &longs;peaking thus.</s></p><p type="head"> <s>PROP. VI. THEOR. VI.</s></p><p type="main"> <s><emph type="italics"/>Solid Magnitudes lighter than the Liquid being thru&longs;t <lb/>into the Liquid, are repul&longs;ed upwards with a Force <lb/>as great as is the exce&longs;s of the Gravity of a Ma&longs;s <lb/>of Liquor equal to the Magnitude above the Gra­<lb/>vity of the &longs;aid Magnitude.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>This &longs;ixth <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> &longs;aith, that the Solids lighter than the Liquid <lb/>demitted, thru&longs;t, or trodden by Force underneath the Liquids Sur­<lb/>face, are returned or driven upwards with &longs;o much Force, by <lb/>how much a quantity of the Liquid equal to the. </s> <s>Solid &longs;hall <lb/>exceed the &longs;aid Solid in Gravity.</s></p><p type="main"> <s>And to delucidate this <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> let the Solid A be lighter <lb/>than the <emph type="italics"/>L<emph.end type="italics"/>iquid, and let us &longs;uppo&longs;e that the Gravity of the &longs;aid <lb/>Solid A is B: and let the Gravity of a <emph type="italics"/>L<emph.end type="italics"/>iquid, equal in Ma&longs;s to A, <lb/>be B G. </s> <s>I &longs;ay, that the Solid A depre&longs;&longs;ed or demitted with Force <lb/>into the &longs;aid <emph type="italics"/>L<emph.end type="italics"/>iquid, &longs;hall be returned and repul&longs;ed upwards with <lb/>a Force equal to the Gravity G. </s> <s>And to demon&longs;trate this <emph type="italics"/>Propo­<lb/>&longs;ition,<emph.end type="italics"/> take the Solid D, equal in Gravity to the &longs;aid G. </s> <s>Now <lb/>the Solid compounded of the two Solids A and D will be lighter <lb/>than the <emph type="italics"/>L<emph.end type="italics"/>iquid: for the Gravity of the Solid compounded of <lb/>them both is BG, and the Gravity of as much Liquor as equal­<lb/>leth in greatne&longs;s the Solid A, is greater than the &longs;aid Gravity BG, <pb xlink:href="073/01/015.jpg" pagenum="344"/>for that B G is the Gravity of the Liquid equal in Ma&longs;s unto it: <lb/>Therefore the Solid compounded of tho&longs;e two Solids A and D <lb/>being dimerged, it &longs;hall, by the precedent, &longs;o much of it &longs;ubmerge, <lb/>as that a quantity of the Liquid equal to the &longs;aid &longs;ubmerged part <lb/>&longs;hall have equal Gravity with the &longs;aid compounded Solid. </s> <s>And <lb/><figure id="id.073.01.015.1.jpg" xlink:href="073/01/015/1.jpg"/><lb/>for an example of that <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> let the Su­<lb/>perficies of any Liquid be that which pro­<lb/>ceedeth according to the Circumference <lb/>A B G D: Becau&longs;e now a Ma&longs;s or quantity <lb/>of Liquor as big as the Ma&longs;s A hath equal <lb/>Gravity with the whole compounded Solid <lb/>A D: It is manife&longs;t that the &longs;ubmerged part <lb/>thereof &longs;hall be the Ma&longs;s A: and the remain­<lb/>der, namely, the part D, &longs;hall be wholly a­<lb/>top, that is, above the Surface of the Liquid. <lb/></s> <s>It is therefore evident, that the part A hath &longs;o much virtue or <lb/>Force to return upwards, that is, to ri&longs;e from below above the Li­<lb/>quid, as that which is upon it, to wit, the part D, hath to pre&longs;s it <lb/>downwards, for that neither part is repul&longs;ed by the other: But D <lb/>pre&longs;&longs;eth downwards with a Gravity equal to G, it having been &longs;up­<lb/>po&longs;ed that the Gravity of that part D was equal to G: Therefore <lb/>that is manife&longs;t which was to be demon&longs;trated.</s></p><p type="main"> <s>RIC. </s> <s>This was a fine Demon&longs;tration, and from this I perceive that you colle­<lb/>cted your <emph type="italics"/>Indu&longs;trious Invention<emph.end type="italics"/>; and e&longs;pecially that part of it which you in&longs;ert in <lb/>the fir&longs;t Book for the recovering of a Ship &longs;unk: and, indeed, I have many Que­<lb/>&longs;tions to ask you about that, but I will not now interrupt the Di&longs;cour&longs;e in hand, but <lb/>de&longs;ire that we may go on to the &longs;eventh <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> the purport whereof is this.</s></p><p type="head"> <s>PROP. VII. THEOR. VII.</s></p><p type="main"> <s><emph type="italics"/>Solid Magnitudes beavier than the Liquid, being de­<lb/>mitted into the [&longs;etled] Liquid, are boren down­<lb/>wards as far as they can de&longs;cend: and &longs;hall be lighter <lb/>in the Liquid by the Gravity of a Liquid Ma&longs;s of <lb/>the &longs;ame bigne&longs;s with the Solid Magnitude.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>This &longs;eventh <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> hath two parts to be demon&longs;trated.</s></p><p type="main"> <s>The fir&longs;t is, That all Solids heavier than the Liquid, being demit­<lb/>ted into the <emph type="italics"/>L<emph.end type="italics"/>iquid, are boren by their Gravities downwards as far <lb/>as they can de&longs;cend, that is untill they arrive at the Bottom. </s> <s>Which <lb/>fir&longs;t part is manife&longs;t, becau&longs;e the Parts of the <emph type="italics"/>L<emph.end type="italics"/>iquid, which &longs;till lie <lb/>under that Solid, are more pre&longs;&longs;ed than the others equijacent, <lb/>becau&longs;e that that Solid is &longs;uppo&longs;ed more grave than the Liquid. <pb xlink:href="073/01/016.jpg" pagenum="345"/>But now that that Solid is lighter in the Liquid than out of it, as <lb/>is affirmed in the &longs;econd part, &longs;hall be demon&longs;trated in this man­<lb/>ner. </s> <s>Take a Solid, as &longs;uppo&longs;e A, that is more grave than the Li­<lb/>quid, and &longs;uppo&longs;e the Gravity of that &longs;ame Solid A to be BG. <lb/></s> <s>And of a Ma&longs;s of <emph type="italics"/>L<emph.end type="italics"/>iquor of the &longs;ame bigne&longs;s with the Solid A, &longs;up­<lb/>po&longs;e the Gravity to be B: It is to be demon&longs;trated that the Solid <lb/>A, immerged in the Liquid, &longs;hall have a Gravity equal to G. </s> <s>And <lb/>to demon&longs;trate this, let us imagine another Solid, as &longs;uppo&longs;e D, <lb/>more light than the Liquid, but of &longs;uch a quality as that its Gravi­<lb/>ty is equal to B: and let this D be of &longs;uch a Magnitude, that a <lb/>Ma&longs;s of <emph type="italics"/>L<emph.end type="italics"/>iquor equal to it hath its Gravity equal to the Gravity <lb/>B G. </s> <s>Now the&longs;e two Solids D and A being compounded toge­<lb/>ther, all that Solid compounded of the&longs;e two &longs;hall be equally <lb/>Grave with the Water: becau&longs;e the Gravity of the&longs;e two Solids <lb/>together &longs;hall be equal to the&longs;e two Gravities, that is, to B G, and <lb/><figure id="id.073.01.016.1.jpg" xlink:href="073/01/016/1.jpg"/><lb/>to B; and the Gravity of a Liquid that hath its <lb/>Ma&longs;s equal to the&longs;e two Solids A and D, &longs;hall be <lb/>equal to the&longs;e two Gravities B G and B. <emph type="italics"/>L<emph.end type="italics"/>et <lb/>the&longs;e two Solids, therefore, be put in the <emph type="italics"/>L<emph.end type="italics"/>iquid, <lb/><arrow.to.target n="marg1136"></arrow.to.target><lb/>and they &longs;hall ^{*} remain in the Surface of that <emph type="italics"/>L<emph.end type="italics"/>i­<lb/>quid, (that is, they &longs;hall not be drawn or driven <lb/>upwards, nor yet downwards:) For if the Solid <lb/>A be more grave than the Liquid, it &longs;hall be <lb/>drawn or born by its Gravity downwards to­<lb/>wards the Bottom, with as much Force as by the Solid D it is thru&longs;t <lb/>upwards: And becau&longs;e the Solid D is lighter than the <emph type="italics"/>L<emph.end type="italics"/>iquid, it <lb/>&longs;hall rai&longs;e it upward with a Force as great as the Gravity G: Be­<lb/>cau&longs;e it hath been demon&longs;trated, in the &longs;ixth <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> That So­<lb/>lid Magnitudes that are lighter than the Water, being demitted in <lb/>the &longs;ame, are repul&longs;ed or driven upwards with a Force &longs;o much the <lb/>greater by how much a <emph type="italics"/>L<emph.end type="italics"/>iquid of equal Ma&longs;s with the Solid is more <lb/>Grave than the &longs;aid Solid: But the <emph type="italics"/>L<emph.end type="italics"/>iquid which is equal in Ma&longs;s <lb/>with the Solid D, is more grave than the &longs;aid Solid D, by the Gra­<lb/>vity G: Therefore it is manife&longs;t, that the Solid A is pre&longs;&longs;ed or <lb/>born downwards towards the Centre of the World, with a Force <lb/>as great as the Gravity G: Which was to be demon&longs;trated.</s></p><p type="margin"> <s><margin.target id="marg1136"></margin.target>* Or, according to <lb/><emph type="italics"/>Commandine,<emph.end type="italics"/> &longs;hall <lb/>be equall in Gravi­<lb/>ty to the Liquid, <lb/>neither moving up­<lb/>wards or down­<lb/>wards.</s></p><p type="main"> <s>RIC. </s> <s>This hath been an ingenuous Demon&longs;tration; and in regard I do &longs;uffici­<lb/>ently under&longs;tand it, that we may lo&longs;e no time, we will proceed to the &longs;econd <emph type="italics"/>Suppo­<lb/>&longs;ition,<emph.end type="italics"/> which, as I need not tell you, &longs;peaks thus.</s></p> <pb xlink:href="073/01/017.jpg" pagenum="346"/><p type="head"> <s>SVPPOSITION II.</s></p><p type="main"> <s><emph type="italics"/>It is &longs;uppo&longs;ed that tho&longs;e Solids which are moved up­<lb/>wards, do all a&longs;cend according to the Perpendicular <lb/>which is produced thorow their Centre of Gravity.<emph.end type="italics"/></s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"/>And tho&longs;e which are moved downwards, de&longs;cend, likewi&longs;e, according to the Perpendicular <lb/>that is produced thorow their Centre of Gravity, which he pretermitted either as known, <lb/>or as to be collected from what went before.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>For under&longs;tanding of this &longs;econd <emph type="italics"/>Suppo&longs;ition,<emph.end type="italics"/> it is requi&longs;ite to take notice <lb/>that every Solid that is lighter than the Liquid being by violence, or by &longs;ome other <lb/>occa&longs;ion, &longs;ubmerged in the Liquid, and then left at liberty, it &longs;hall, by that which <lb/>hath been proved in the &longs;ixth <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> be thru&longs;t or born up wards by the Liquid, <lb/>and that impul&longs;e or thru&longs;ting is &longs;uppo&longs;ed to be directly according to the Perpendi­<lb/>cular that is produced thorow the Centre of Gravity of that Solid; which Per­<lb/>pendicular, if you well remember, is that which is drawn in the Imagination <lb/>from the Centre of the World, or of the Earth, unto the Centre of Gravity of <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 &longs;heweth in that Book, intituled <emph type="italics"/>De Centris Gravium, vel de Æqui­<lb/>ponderantibus<emph.end type="italics"/>; and therefore repair thither and you &longs;hall be &longs;atisfied, for to declare <lb/>it to you in this place would cau&longs;e very great confu&longs;ion.</s></p><p type="main"> <s>RIC. </s> <s>I under&longs;tand you: &longs;ome other time we will talk of this, becau&longs;e I have <lb/>a mind at pre&longs;ent to proceed to the la&longs;t <emph type="italics"/>Propo&longs;ition,<emph.end type="italics"/> the Expo&longs;ition of which &longs;eemeth <lb/>to me very confu&longs;ed, and, as I conceive, the Author hath not therein &longs;hewn all <lb/>the Subject of that <emph type="italics"/>Propo&longs;ition<emph.end type="italics"/> in general, but only a part: which Propo&longs;ition <lb/>&longs;peaketh, as you know, in this form.</s></p><p type="head"> <s>PROP. VIII. THEOR. VIII.<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"/>If any Solid Magnitude, lighter than the Liquid, that <lb/>hath the Figure of a Portion of a Sphære, &longs;hall be<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1138"></arrow.to.target><lb/><emph type="italics"/>demitted into the Liquid in &longs;uch a manner as that <lb/>the Ba&longs;e of the Portion touch not the Liquid, the <lb/>Figure &longs;hall &longs;tand erectly, &longs;o, as that the Axis of <lb/>the &longs;aid Portion &longs;hall be according to the Perpen­<lb/>dicular. </s> <s>And if the Figure &longs;hall be inclined to any <lb/>&longs;ide, &longs;o, as that the Ba&longs;e of the Portion touch the <lb/>Liquid, it &longs;hall not continue &longs;o inclined as it was de­<lb/>mitted, but &longs;hall return to its uprightne&longs;s.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/018.jpg" pagenum="347"/><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"/>Propo&longs;ition,<emph.end type="italics"/> let a Solid Magnitude <lb/>that hath the Figure of a portion of a Sphære, as hath been &longs;aid, <lb/>be imagined to be de­<lb/><figure id="id.073.01.018.1.jpg" xlink:href="073/01/018/1.jpg"/><lb/>mitted into the Liquid; and <lb/>al&longs;o, let a Plain be &longs;uppo&longs;ed <lb/>to be produced thorow the <lb/>Axis of that portion, and <lb/>thorow the Center of the <lb/>Earth: and let the Section <lb/>of the Surface of the Liquid <lb/>be the Circumference A B <lb/>C D, and of the Figure, the <lb/>Circumference E F H, & let <lb/>E H be a right line, and F T <lb/>the Axis of the Portion. </s> <s>If now <lb/>it were po&longs;&longs;ible, for &longs;atisfact­<lb/>ion of the Adver&longs;ary, Let <lb/>it be &longs;uppo&longs;ed that the &longs;aid Axis were not according to the <emph type="italics"/>(a)<emph.end type="italics"/> Per­<lb/><arrow.to.target n="marg1139"></arrow.to.target><lb/>pendicular; we are then to demon&longs;trate, that the Figure will not <lb/>continue as it was con&longs;tituted by the Adver&longs;ary, but that it will re­<lb/>turn, as hath been &longs;aid, unto its former po&longs;ition, that is, that the <lb/>Axis F T &longs;hall be according to the Perpendicular. </s> <s>It is manife&longs;t, by <lb/>the <emph type="italics"/>Corollary<emph.end type="italics"/> of the 1. of 3. <emph type="italics"/>Euclide,<emph.end type="italics"/> that the Center of the Sphære <lb/>is in the Line F T, fora&longs;much as that is the Axis of that Figure. <lb/></s> <s>And in regard that the Por­<lb/><figure id="id.073.01.018.2.jpg" xlink:href="073/01/018/2.jpg"/><lb/>tion of a Sphære, may be <lb/>greater or le&longs;&longs;er than an He­<lb/>mi&longs;phære, and may al&longs;o be <lb/>an Hemi&longs;phære, let the Cen­<lb/>tre of the Sphære, in the He­<lb/>mi&longs;phære, be the Point T, <lb/>and in the le&longs;&longs;er Portion the <lb/>Point P, and in the greater, <lb/>the Point K, and let the Cen­<lb/>tre of the Earth be the Point <lb/>L. </s> <s>And &longs;peaking, fir&longs;t, of <lb/>that greater Portion which <lb/>hath its Ba&longs;e out of, or a­<lb/>bove, the Liquid, thorew the Points K and L, draw the Line KL <lb/>cutting the Circumference E F H in the Point N, Now, becau&longs;e <lb/><arrow.to.target n="marg1140"></arrow.to.target><lb/>every Portion of a Sphære, hath its Axis in the Line, that from the <lb/>Centre of the Sphære is drawn perpendicular unto its Ba&longs;e, and hath <lb/>its Centre of Gravity in the Axis; therefore that Portion of the Fi­<lb/>gure which is within the Liquid, which is compounded of two Por­ <pb xlink:href="073/01/019.jpg" pagenum="348"/>tions of a Sphære, &longs;hall have its Axis in the Perpendicular, that is <lb/>drawn through the point K; and its Centre of Gravity, for the &longs;ame <lb/>rea&longs;on, &longs;hall be in the Line N K: let us &longs;uppo&longs;e it to be the Point R: <lb/><arrow.to.target n="marg1141"></arrow.to.target><lb/>But the Centre of Gravity of the whole Portion is in the Line F T, <lb/>betwixt the Point R and <lb/><figure id="id.073.01.019.1.jpg" xlink:href="073/01/019/1.jpg"/><lb/>the Point F; let us &longs;uppo&longs;e <lb/>it to be the Point <emph type="italics"/>X<emph.end type="italics"/>: The re­<lb/>mainder, therefore, of that <lb/><arrow.to.target n="marg1142"></arrow.to.target><lb/>Figure elivated above the <lb/>Surface of the Liquid, hath <lb/>its Centre of Gravity in <lb/>the Line R X produced or <lb/>continued right out in the <lb/>Part towards X, taken &longs;o, <lb/>that the part prolonged may <lb/>have the &longs;ame proportion to <lb/>X R, that the Gravity of <lb/>that Portion that is demer­<lb/>ged in the Liquid hath to <lb/>the Gravity of that Figure which is above the Liquid; let us &longs;uppo&longs;e <lb/><arrow.to.target n="marg1143"></arrow.to.target><lb/>that ^{*} that Centre of the &longs;aid Figure be the Point S: and thorow that <lb/><arrow.to.target n="marg1144"></arrow.to.target><lb/>&longs;ame Centre S draw the Perpendicular L S. </s> <s>Now the Gravity of the Fi­<lb/>gure that is above the Liquid &longs;hall pre&longs;&longs;e from above downwards ac­<lb/>cording to the Perpendicular S L; & the Gravity of the Portion that <lb/>is &longs;ubmerged in the Liquid, &longs;hall pre&longs;&longs;e from below upwards, accor­<lb/>ding to the Perpendicular R L. </s> <s>Therefore that Figure will not conti­<lb/>nue according to our Adver&longs;aries Propo&longs;all, but tho&longs;e parts of the <lb/>&longs;aid Figure which are towards E, &longs;hall be born or drawn downwards, <lb/>& tho&longs;e which are towards H &longs;hall be born or driven upwards, and <lb/>this &longs;hall be &longs;o long untill that the Axis F T comes to be according <lb/>to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1139"></margin.target>(a) <emph type="italics"/>Perpendicular <lb/>is taken kere, as <lb/>in all other places, <lb/>by this Author for <lb/>the Line K L <lb/>drawn thorow the <lb/>Centre and Cir­<lb/>cumference of the <lb/>Earth.<emph.end type="italics"/></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"/>i. </s> <s>e,<emph.end type="italics"/> The Center <lb/>of Gravity.</s></p><p type="margin"> <s><margin.target id="marg1144"></margin.target>F</s></p><p type="main"> <s>And this &longs;ame Demon&longs;tration is in the &longs;ame manner verified in <lb/>the other <emph type="italics"/>P<emph.end type="italics"/>ortions. </s> <s>As, fir&longs;t, in the Hæmi&longs;phere that lieth with its <lb/>whole Ba&longs;e above or without the <emph type="italics"/>L<emph.end type="italics"/>iquid, the Centre of the Sphære <lb/>hath been &longs;uppo&longs;ed to be the <emph type="italics"/>P<emph.end type="italics"/>oint T; and therefore, imagining T <lb/>to be in the place, in which, in the other above mentioned, the <lb/><emph type="italics"/>P<emph.end type="italics"/>oint R was, arguing in all things el&longs;e as you did in that, you &longs;hall <lb/>find that the Figure which is above the <emph type="italics"/>L<emph.end type="italics"/>iquid &longs;hall pre&longs;s from <lb/>above downwards according to the <emph type="italics"/>P<emph.end type="italics"/>erpendicular S <emph type="italics"/>L<emph.end type="italics"/>; and the <lb/><emph type="italics"/>P<emph.end type="italics"/>ortion that is &longs;ubmerged in the <emph type="italics"/>L<emph.end type="italics"/>iquid &longs;hall pre&longs;s from below up­<lb/>wards according to the <emph type="italics"/>P<emph.end type="italics"/>erpendicular R <emph type="italics"/>L.<emph.end type="italics"/> And therefore it &longs;hall <lb/>follow, as in the other, namely, that the parts of the whole Figure <lb/>which are towards E, &longs;hall be born or pre&longs;&longs;ed downwards, and tho&longs;e <lb/><arrow.to.target n="marg1145"></arrow.to.target><lb/>that are towards H, &longs;hall be born or driven upwards: and this &longs;hall <lb/>be &longs;o long untill that the Axis F T come to &longs;tand ^{*} <emph type="italics"/>P<emph.end type="italics"/>erpendicular­ <pb xlink:href="073/01/020.jpg" pagenum="349"/>ly. </s> <s>The like &longs;hall al&longs;o hold true in the <emph type="italics"/>P<emph.end type="italics"/>ortion of the Sphære <lb/>le&longs;s than an Hemi&longs;phere that lieth with its whole Ba&longs;e above the <lb/>Liquid.</s></p><p type="margin"> <s><margin.target id="marg1145"></margin.target>* Or according <lb/>to the Perpendi­<lb/>cular.</s></p><p type="head"> <s>COMMANDINE.</s></p><p type="main"> <s><emph type="italics"/>The Demon&longs;tration of this Propo&longs;ition is defaced by the Injury of Time, which we have re­<lb/>&longs;tored, &longs;o far as by the Figures that remain, one may collect the Meaning of<emph.end type="italics"/> Archimedes, <lb/><emph type="italics"/>for we thought it not good to alter them: and what was wanting to their declaration and ex­<lb/>planation we have &longs;upplyed in our Commentaries, as we have al&longs;o determined to do in the &longs;e­<lb/>cond Propo&longs;ition of the &longs;econd Book.<emph.end type="italics"/></s></p><p type="main"> <s>If any Solid Magnitude lighter than the Liquid.] <emph type="italics"/>The&longs;e words, light-<emph.end type="italics"/><lb/><arrow.to.target n="marg1146"></arrow.to.target><lb/><emph type="italics"/>er than the Liquid, are added by us, and are not to be found in the Tran&longs;iation; for of the&longs;e <lb/>kind of Magnitudes doth<emph.end type="italics"/> Archimedes <emph type="italics"/>&longs;peak in this Propo&longs;ition.<emph.end type="italics"/></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 &longs;uch a manner as that the <lb/><arrow.to.target n="marg1147"></arrow.to.target><lb/>Ba&longs;e of the Portion touch not the Liquid.] <emph type="italics"/>That is, &longs;hall be &longs;o demitted into <lb/>the Liquid as that the Ba&longs;e &longs;hall be upwards, and the<emph.end type="italics"/> Vertex <emph type="italics"/>downwards, which he oppo&longs;eth <lb/>to that which he &longs;aith in the Propo&longs;ition following<emph.end type="italics"/>; Be demitted into the Liquid, &longs;o, as <lb/>that its Ba&longs;e be wholly within the Liquid; <emph type="italics"/>For the&longs;e words &longs;ignifie the Portion demit­<lb/>ted the contrary way, as namely, with the<emph.end type="italics"/> Vertex <emph type="italics"/>upwards and the Ba&longs;e downwards. </s> <s>The <lb/>&longs;ame manner of &longs;peech is frequently u&longs;ed in the &longs;econd Book; which treateth of the Portions <lb/>of Rectangle Conoids.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1147"></margin.target>B</s></p><p type="main"> <s>Now becau&longs;e every Portion of a Sphære hath its Axis in the Line <lb/><arrow.to.target n="marg1148"></arrow.to.target><lb/>that from the Center of the Sphære is drawn perpendicular to its <lb/>Ba&longs;e.] <emph type="italics"/>For draw a Line from B to C, and let K L cut the Circumference A B C D in the <lb/>Point G, and the Right Line B C in M<emph.end type="italics"/>: <lb/><figure id="id.073.01.020.1.jpg" xlink:href="073/01/020/1.jpg"/><lb/><emph type="italics"/>and becau&longs;e the two Circles A B C D, and <lb/>E F H do cut one another in the Points <lb/>B and C, the Right Line that conjoyneth <lb/>their Centers, namely, K L, doth cut the <lb/>Line B C in two equall parts, and at <lb/>Right Angles; as in our Commentaries <lb/>upon<emph.end type="italics"/> Prolomeys <emph type="italics"/>Plani&longs;phære we do <lb/>prove: But of the Portion of the Circle <lb/>B N C the Diameter is M N; and of the <lb/>Portion B G C the Diameter is M G;<emph.end type="italics"/><lb/><arrow.to.target n="marg1149"></arrow.to.target><lb/><emph type="italics"/>for the<emph.end type="italics"/> (a) <emph type="italics"/>Right Lines which are drawn <lb/>on both &longs;ides parallel to B C do make<emph.end type="italics"/><lb/><arrow.to.target n="marg1150"></arrow.to.target><lb/><emph type="italics"/>Right Angles with N G; and<emph.end type="italics"/> (b) <emph type="italics"/>for <lb/>that cau&longs;e are thereby cut in two equall <lb/>parts: Therefore the Axis of the Portion <lb/>of the Sphære B N C is N M; and the <lb/>Axis of the Portion B G C is M G: <lb/>from whence it followeth that the Axis of <lb/>the Portion demerged in the Liquid is <lb/>in the Line K L, namely N G. </s> <s>And &longs;ince the Center of Gravity of any Portion of a Sphære is <lb/>in the Axis, as we have demonstrated in our Book<emph.end type="italics"/> De Centro Gravitatis Solidorum, <emph type="italics"/>the <lb/>Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is, <lb/>of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra­<lb/>vity of tho&longs;e Portions of Sphæres. </s> <s>For &longs;uppo&longs;e, if po&longs;&longs;ible, that it be out of the Line N G, as <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> Becau&longs;e <lb/>therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha­<lb/>ving the &longs;ame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the <lb/>Portion B G C &longs;hall, by the 8 of the fir&longs;t Book of<emph.end type="italics"/> Archimedes, De Centro Gravitatis <pb xlink:href="073/01/021.jpg" pagenum="350"/>Planotum, <emph type="italics"/>be in the Line V Q prolonged: But that is impo&longs;&longs;ible; for it is in the Axis <lb/>G: It followeth, therefore, that the Center of Gravity of the Portion demerged in <lb/>Liquid be in the Line N K: which we propounded to be proved.<emph.end type="italics"/></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"/>(a)<emph.end type="italics"/> By 29. of the <lb/>fir&longs;t of <emph type="italics"/>Encl.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1150"></margin.target><emph type="italics"/>(b)<emph.end type="italics"/> By 3. of the <lb/>third.</s></p><p type="main"> <s>But the Centre of Gravity of the whole Portion is in the Line <lb/><arrow.to.target n="marg1151"></arrow.to.target><lb/>T, betwixt the Point R and the Point F; let us &longs;uppo&longs;e it to be<lb/>the Point X.] <emph type="italics"/>Let the Sphære becompleated, &longs;o as that there be added of that Portion<lb/>the Axis T Y, and the Center of Gravity Z. </s> <s>And becau&longs;e that from the whole Sphære,<lb/>who&longs;e Centre of Gravity is K, as we have al&longs;o demon&longs;trated in the (c) Book before named, the <lb/>is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind<emph.end type="italics"/><lb/><arrow.to.target n="marg1152"></arrow.to.target><lb/><emph type="italics"/>of the Portion E F H &longs;hall be in the Line Z K prolonged: And therefore it mu&longs;t of nece&longs;&longs;ity<lb/>fall betwixt K and F.<emph.end type="italics"/><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"/>By 8 of the <lb/>fir&longs;t<emph.end type="italics"/> 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/>face of the Liquid, hath its Center of Gravity in the <emph type="italics"/>L<emph.end type="italics"/>ine R X<lb/>prolonged.] <emph type="italics"/>By the &longs;ame 8 of the fir&longs;t Book of<emph.end type="italics"/> Archimedes, de Centro Gravita­<lb/>tis Planorum.</s></p><p type="main"> <s>Now the Gravity of the Figure that is above the <emph type="italics"/>L<emph.end type="italics"/>iquid &longs;hall<arrow.to.target n="marg1154"></arrow.to.target><lb/>pre&longs;s from above downwards according to S L; and the Gravit <lb/>of the Portion that is &longs;ubmerged in the <emph type="italics"/>L<emph.end type="italics"/>iquid &longs;hall pre&longs;s from be <lb/>low upwards, according to the Perpendicular R L.] <emph type="italics"/>By the &longs;econd Sup­<lb/>po&longs;ition of this. </s> <s>For the Magnitude that is demerged in the Liquid is moved upwards with as<lb/>much Force along R L, as that which is above the Liquid is moved downwards along S L; as<lb/>may be &longs;hewn by Propo&longs;ition 6. of this. </s> <s>And becau&longs;e they are moved along &longs;everall other Lines,<lb/>neither cau&longs;eth the others being le&longs;s moved; the which it continually doth when the Portion<lb/>is &longs;et according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes<lb/>do concur in one and the &longs;ame Perpendicular, namely, in the Axis of the Portion: and look<lb/>with what force or<emph.end type="italics"/> Impetus <emph type="italics"/>that which is in the Lipuid tendeth upwards, and with the like<lb/>doth that which is above or without the Liquid tend downwards along the &longs;ame Line: And<emph.end type="italics"/><lb/><arrow.to.target n="marg1155"></arrow.to.target><lb/><emph type="italics"/>therefore, in regard that the one doth not ^{*} exceed the other, the Portion &longs;hall no longer move <lb/>but &longs;hall &longs;tay and re&longs;t allwayes in one and the &longs;ame Po&longs;ition, unle&longs;s &longs;ome extrin&longs;ick Cau&longs;e<lb/>chance to intervene.<emph.end type="italics"/></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"/>Or overcome.<emph.end type="italics"/></s></p><p type="head"> <s>PROP. IX. THEOR. IX.<lb/><arrow.to.target n="marg1156"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="marg1156"></margin.target>* In &longs;ome Greek <lb/>Coppies this is no <lb/>di&longs;tinct Propo&longs;i­<lb/>tion, but all <lb/>Commentators, <lb/>do divide it <lb/>from the Prece­<lb/>dent, as having a <lb/>di&longs;tinct demon­<lb/>&longs;tration in the <lb/>Originall.</s></p><p type="main"> <s>^{*} <emph type="italics"/>But if the Figure, lighter than the Liquid, be demit­<lb/>ted into the Liquid, &longs;o, as that its Ba&longs;e be wholly<lb/>within the &longs;aid Liquid, it &longs;hall continue in &longs;uch <lb/>manner erect, as that its Axis &longs;hall &longs;tand according <lb/>to the Perpendicular.<emph.end type="italics"/></s></p><p type="main"> <s>For &longs;uppo&longs;e, &longs;uch a Magnitude as that aforenamed to be de <lb/>mitted into the Liquid; and imagine a Plane to be produced<lb/>thorow the Axis of the Portion, and thorow the Center of the <lb/>Earth: And let the <emph type="italics"/>S<emph.end type="italics"/>ection of the Surface of the Liquid, be the Cir­<lb/>cumference A B C D, and of the Figure the Circumference E F <emph type="italics"/>H<emph.end type="italics"/><lb/>And let E H be a Right Line, and F T the Axis of the Portion. </s> <s>If<lb/>now it were po&longs;&longs;ible, for &longs;atisfaction of the Adver&longs;ary, let it be <lb/>&longs;uppo&longs;ed that the &longs;aid Axis were not according to the Perpendicu­<lb/>lar: we are now to demon&longs;trate that the Figure will not &longs;o conti­ <pb xlink:href="073/01/022.jpg" pagenum="351"/>nue, but will return to be according to the <lb/><figure id="id.073.01.022.1.jpg" xlink:href="073/01/022/1.jpg"/><lb/>Perpendieular. </s> <s>It is manife&longs;t that the Gen­<lb/>tre of the Sphære is in the Line F T. </s> <s>And <lb/>again, fora&longs;much as the Portion of a Sphære <lb/>may be greater or le&longs;&longs;er than an Hemi&longs;­<lb/>phære, and may al&longs;o be an Hemi&longs;phære, let <lb/>the Centre of the Sphære in the Hemi&longs;­<lb/>phære be the Point T, & in the le&longs;&longs;er Por­<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/>Point R. </s> <s>And &longs;peaking fir&longs;t of that greater <lb/>Portion which hath its Ba&longs;e within the <lb/>Liquid, thorow R and L, the Earths Cen­<lb/><figure id="id.073.01.022.2.jpg" xlink:href="073/01/022/2.jpg"/><lb/>tre, draw the line RL. </s> <s>The Portion that is <lb/>above the Liquid, hath its Axis in the Per­<lb/>pendicular pa&longs;&longs;ing thorow R; and by <lb/>what hath been &longs;aid before, its Centre of <lb/>Gravity &longs;hall be in the Line N R; let it <lb/>be the Point R: But the Centre of Gra­<lb/>vity of the whole Portion is in the line F <lb/>T, betwixt R and F; let it be X: The re­<lb/>mainder therefore of that Figure, which is <lb/>within the Liquid &longs;hall have its Centre in <lb/>the Right Line R <emph type="italics"/>X<emph.end type="italics"/> prolonged in the part <lb/><figure id="id.073.01.022.3.jpg" xlink:href="073/01/022/3.jpg"/><lb/>towards <emph type="italics"/>X,<emph.end type="italics"/> taken &longs;o, that the part pro­<lb/>longed may have the &longs;ame Proportion to <lb/>X R, that the Gravity of the Portion that <lb/>is above the Liquid hath to the Gravity <lb/>of the Figure that is within the Liquid. <lb/></s> <s>Let O be the Centre of that &longs;ame Figure: <lb/>and thorow O draw the Perpendicular L <lb/>O. </s> <s>Now the Gravity of the Portion that <lb/>is above the Liquid &longs;hall pre&longs;s according <lb/>to the Right Line R L downwards; and <lb/>the Gravity of the Figure that is in the <lb/>Liquid according to the Right Line O L upwards: There the Figure <lb/>&longs;hall not continue; but the parts of it towards H &longs;hall move down­<lb/>wards, and tho&longs;e towards E upwards: & <lb/><figure id="id.073.01.022.4.jpg" xlink:href="073/01/022/4.jpg"/><lb/>this &longs;hall ever be, &longs;o long as F T is accord­<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/><arrow.to.target n="marg1158"></arrow.to.target><lb/>hath its Axis in the Perpendicular pa&longs;&longs;ing <lb/>thorow K.] <emph type="italics"/>For draw B C cutting the Line N K in <lb/>M; and let N K out the Circumference<emph.end type="italics"/> A B <emph type="italics"/>C D in G. </s> <s>In <lb/>the &longs;ame manner as before me will demon&longs;trate, that the Axis<emph.end type="italics"/> <pb xlink:href="073/01/023.jpg" pagenum="352"/><emph type="italics"/>of the Portion of the Sphære is N M; and of the Portion B G C the Axis is G M: Wherefore <lb/>the Centre of Gravity of them both &longs;hall be in the Line N M: And becau&longs;e that from the Por­<lb/>tion B N C the Portion B G C, not having the &longs;ame Centre of Gravity, is cut off, the Centre <lb/>of Gravity of the remainder of the Magnitude that is above the Surface of the Liquid &longs;hall be <lb/>in the Line N K; namely, in the Line which conjoyneth the Centres of Gravity of the &longs;aid <lb/>Portions by the fore&longs;aid 8 of<emph.end type="italics"/> 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 &longs;ome of the&longs;e Figures C is put for X, and &longs;o it was in <lb/>the Greek Copy that I followed.</s></p><p type="main"> <s>RIC. </s> <s>This Demo&longs;tration is very difficult, to my thinking; but I believe that <lb/>it is becau&longs;e I have not in memory the Propo&longs;itions of that Book entituled <emph type="italics"/>De Cen­<lb/>tris Gravium.<emph.end type="italics"/></s></p><p type="main"> <s>NIC. </s> <s>It is &longs;o.</s></p><p type="main"> <s>RIC. </s> <s>We will take a more convenient time to di&longs;cour&longs;e of that, and now return <lb/><arrow.to.target n="marg1159"></arrow.to.target><lb/>to &longs;peak of the two la&longs;t Propo&longs;itions. </s> <s>And I &longs;ay that the Figures incerted in the <lb/>demon&longs;tration would in my opinion, have been better and more intelligble unto <lb/>me, drawing the Axis according to its proper Po&longs;ition; that is in the half Arch of <lb/>the&longs;e Figures, and then, to &longs;econd the Objection of the Adver&longs;ary, to &longs;uppo&longs;e <lb/>that the &longs;aid Figures &longs;tood &longs;omewhat Obliquely, to the end that the &longs;aid Axis, if it <lb/>were po&longs;&longs;ible, did not &longs;tand according to the Perpendicular &longs;o often mentioned, <lb/>which doing, the Propo&longs;ition would be proved in the &longs;ame manner as before: <lb/>and this way would be more naturall and clear.<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 becau&longs;e thus they were in the Greek Copy, <lb/>I thought not fit to alter them, although unto the better.</s></p><p type="main"> <s>RIC. Companion, you have thorowly &longs;atisfied me in all that in the beginning <lb/>of our Di&longs;cour&longs;e I asked of you, to morrow, God permitting, we will treat of <lb/>&longs;ome other ingenious Novelties.</s></p><p type="head"> <s>THE TRANSLATOR.</s></p><p type="main"> <s>I &longs;ay 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/>me, drawing the Axis Z T according to its proper Po&longs;ition, that <lb/>is in the half Arch of the&longs;e Figures.] <emph type="italics"/>And in this con&longs;ideration I have followed <lb/>the Schemes of<emph.end type="italics"/> Commandine, <emph type="italics"/>who being the Re&longs;torer of the Demon&longs;trations of the&longs;e two la&longs;t <lb/>Propo&longs;itions, hath well con&longs;idered what<emph.end type="italics"/> Ricardo <emph type="italics"/>here propo&longs;eth, and therefore hath drawn the <lb/>&longs;aid Axis (which in the Manu&longs;cripts that he had by him is lettered F T, and not as in that of<emph.end type="italics"/><lb/>Tartaylia <emph type="italics"/>Z T,) according to that its proper Po&longs;ition.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1161"></margin.target>A</s></p><p type="main"> <s>But becau&longs;e thus they were in the Greek Copy, I thought not <lb/><arrow.to.target n="marg1162"></arrow.to.target><lb/>fit to alter them although unto the better.] <emph type="italics"/>The Schemes of tho&longs;e Manu-<emph.end type="italics"/><lb/><figure id="id.073.01.023.1.jpg" xlink:href="073/01/023/1.jpg"/><lb/><emph type="italics"/>&longs;cripts that<emph.end type="italics"/> Tartaylia <emph type="italics"/>had &longs;een were more imperfect then tho&longs;e <lb/>in Commandines Copies; but for variety &longs;ake, take here one <lb/>of<emph.end type="italics"/> Tartaylia, <emph type="italics"/>it being that of the Portion of a Sphære, equall <lb/>to an Hemi&longs;phære, with its Axis oblique, and its Ba&longs;e dimitted <lb/>into the Liquid, and Lettered as in this Edition.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1162"></margin.target>B</s></p><p type="main"> <s><emph type="italics"/>Now Courteous Readers, I hope that you may, amid&longs;t the <lb/>great Ob&longs;curity of the Originall in the Demon&longs;trations of the&longs;e <lb/>two la&longs;t Propo&longs;itions, be able from the joynt light of the&longs;e two Famous Commentators of our <lb/>more famous Author, to di&longs;cern the truth of the Doctrine affirmed, namely, That Solids of the <lb/>Figure of Portions of Sphæres demitted into the Liquid with their Ba&longs;es upwards &longs;hall &longs;tand <lb/>erectly, that is, with their Axis according to the Perpendicular drawn from the Centre of the <lb/>Earth unto its Circumference: And that if the &longs;aid Portions be demitted with their Ba&longs;es <lb/>oblique and touching the Liquid in one Point, they &longs;hall not rest in that Obliquity, but &longs;hall <lb/>return to Rectitude: And that la&longs;tly, if the&longs;e Portions be demitted with their Ba&longs;es downwards, <lb/>they &longs;hall continue erect with their Axis according to the Perpendicular afore&longs;aid: &longs;o that no <lb/>more remains to be done, but that we&longs;et before you the 2 Books of this our Admirable Author.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/024.jpg" pagenum="353"/><p type="head"> <s>ARCHIMEDES, <lb/>HIS TRACT <lb/><emph type="italics"/>DE <lb/>INSIDENTIBUS HUMIDO,<emph.end type="italics"/><lb/>OR, <lb/>Of the NATATION of BODIES Upon, or <lb/>Submer&longs;ion In the WATER, or other LIQUIDS.</s></p><p type="head"> <s><emph type="italics"/>BOOK<emph.end type="italics"/> II.</s></p><p type="head"> <s>PROP. I. THEOR. I.</s></p><p type="main"> <s><emph type="italics"/>If any Magnitude lighter than the Liquid be demitted <lb/>into the &longs;aid Liquid, it &longs;hall have the &longs;ame proporti­<lb/>on in Gravity to a Liquid of equal Ma&longs;&longs;e, that the <lb/>part of the Magnitude demerged hath unto the <lb/>whole Magnitude.<emph.end type="italics"/></s></p><p type="main"> <s>For let any Solid Magnitude, as for in­<lb/>&longs;tance F A, lighter than the Liquid, be de­<lb/>merged in the Liquid, which let be F A: <lb/>And let the part thereof immerged be A, <lb/>and the part above the Liquid F, It is to <lb/>be demon&longs;trated that the Magnitude F A <lb/>hath the &longs;ame proportion in Gravity to a <lb/>Liquid of Equall Ma&longs;&longs;e that A hath to F <lb/>A. </s> <s>Take any Liquid Magnitude, as &longs;up­<lb/>po&longs;e N I, of equall Ma&longs;&longs;e with F A; and let F be equall to N, and <lb/>A to I: and let the Gravity of the whole Magnitude F A be B, and <lb/>let that of the Magnitude N I be O, <lb/>and let that of I be R. </s> <s>Now the <lb/><figure id="id.073.01.024.1.jpg" xlink:href="073/01/024/1.jpg"/><lb/>Magnitude F A hath the &longs;ame pro­<lb/>portion unto N I that the Gravity B <lb/>hath to the Gravity O R: But for <lb/>a&longs;much as the Magnitude F A demit­<lb/>ted into the Liquid is lighter than <lb/>the &longs;aid Liquid, it is manife&longs;t that a Ma&longs;&longs;e of the Liquid, I, equall <lb/>to the part of the Magnitude demerged, A, hath equall Gravity <lb/><arrow.to.target n="marg1163"></arrow.to.target><lb/>with the whole Magnitnde, F A: For this was <emph type="italics"/>(a)<emph.end type="italics"/> above demon­<lb/>&longs;trated: But B is the Gravity of the Magnitude F A, and R of I: <pb xlink:href="073/01/025.jpg" pagenum="354"/>Therefore B and R are equall. </s> <s>And becau&longs;e that of the Magni­<lb/>tude FA the <emph type="italics"/>G<emph.end type="italics"/>ravity is B: Therefore of the Liquid Body <emph type="italics"/>N<emph.end type="italics"/> I the <lb/>Gravity is O R. </s> <s>As F A is to N I, &longs;o is B to O R, or, &longs;o is R to <lb/>O R: But as R is to O R, &longs;o is I to N I, and A to F A: Therefore <lb/><arrow.to.target n="marg1164"></arrow.to.target><lb/>I is to N I, as F A to N I: And as I to N I &longs;o is <emph type="italics"/>(b)<emph.end type="italics"/> A to F A. <lb/></s> <s>Therefore F A is to N I, as A is to F A: Which was to be demon­<lb/>&longs;trated.</s></p><p type="margin"> <s><margin.target id="marg1163"></margin.target>(a) <emph type="italics"/>By 5. of the <lb/>fir&longs;t of this.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1164"></margin.target>(b) <emph type="italics"/>By 11. of the <lb/>fifth of<emph.end type="italics"/> Eucl.</s></p><p type="head"> <s>PROP. II. THEOR. II.<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"/>The Right Portion of a Right angled Conoide, when it <lb/>&longs;hall have its Axis le&longs;&longs;e than<emph.end type="italics"/> &longs;e&longs;quialter ejus quæ ad <lb/>Axem (<emph type="italics"/>or of its<emph.end type="italics"/> Semi-parameter) <emph type="italics"/>having any what <lb/>ever proportion to the Liquid in Gravity, being de­<lb/>mitted into the Liquid &longs;o as that its Ba&longs;e touch not <lb/>the &longs;aid Liquid, and being &longs;et &longs;tooping, it &longs;hall not <lb/>remain &longs;tooping, but &longs;hall be restored to uprightne&longs;&longs;e. <lb/></s> <s>I &longs;ay that the &longs;aid Portion &longs;hall &longs;tand upright when <lb/>the Plane that cuts it &longs;hall be parallel unto the Sur­<lb/>face of the Liquid.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be a Portion of a Rightangled Conoid, as hath been <lb/>&longs;aid; and let it lye &longs;tooping or inclining: It is to be demon­<lb/>&longs;trated that it will not &longs;o continue but &longs;hall be re&longs;tored to re­<lb/>ctitude. </s> <s>For let it be cut through the Axis by a plane erect upon <lb/>the Surface of the Liquid, and let the Section of the Portion be <lb/>A PO L, the Section of a Rightangled Cone, and let the Axis <lb/><figure id="id.073.01.025.1.jpg" xlink:href="073/01/025/1.jpg"/><lb/>of the Portion and Diameter of the <lb/>Section be N O: And let the Sect­<lb/>ion of the Surface of the Liquid be <lb/>I S. </s> <s>If now the Portion be not <lb/>erect, then neither &longs;hall A L be Pa­<lb/>rallel to I S: Wherefore N O will <lb/>not be at Right Angles with I S. </s></p><p type="main"> <s><arrow.to.target n="marg1166"></arrow.to.target><lb/>Draw therefore K <foreign lang="greek">w,</foreign> touching the Section of the Cone I, in the <lb/>Point P [that is parallel to I S: and from the Point P unto I S <lb/><arrow.to.target n="marg1167"></arrow.to.target><lb/>draw P F parallel unto O N, ^{*} which &longs;hall be the Diameter of the <lb/>Section I P O S, and the Axis of the Portion demerged in the <emph type="italics"/>L<emph.end type="italics"/>i­<lb/><arrow.to.target n="marg1168"></arrow.to.target><lb/>quid. </s> <s>In the next place take the Centres of Gravity: ^{*} and of <lb/>the Solid Magnitude A P O L, let the Centre of Gravity be R; and <lb/><arrow.to.target n="marg1169"></arrow.to.target><lb/>of I P O S let the Centre be B: ^{*} and draw a Line from B to R <lb/>prolonged unto G; which let be the Centre of Gravity of the <pb xlink:href="073/01/026.jpg" pagenum="355"/>remaining Figure I S L A. </s> <s>Becau&longs;e now that N O is <emph type="italics"/>Se&longs;quialter<emph.end type="italics"/><lb/>of R O, but le&longs;s than <emph type="italics"/>Se&longs;quialter ejus quæ u&longs;que ad Axem<emph.end type="italics"/> (or of its <lb/><emph type="italics"/>Semi-parameter<emph.end type="italics"/>;) ^{*} R O &longs;hall be le&longs;&longs;e than <emph type="italics"/>quæ u&longs;que ad Axem<emph.end type="italics"/> (or <lb/><arrow.to.target n="marg1170"></arrow.to.target><lb/>than the <emph type="italics"/>Semi-parameter<emph.end type="italics"/>;) ^{*} whereupon the Angle R P <foreign lang="greek">w</foreign> &longs;hall be <lb/><arrow.to.target n="marg1171"></arrow.to.target><lb/>acute. </s> <s>For &longs;ince the Line <emph type="italics"/>quæ u&longs;que ad Axem<emph.end type="italics"/> (or <emph type="italics"/>Semi-parameter<emph.end type="italics"/>) <lb/>is greater than R O, that Line which is drawn from the Point R, <lb/>and perpendicular to K <foreign lang="greek">w,</foreign> namely RT, meeteth with the line F P <lb/>without the Section, and for that cau&longs;e mu&longs;t of nece&longs;&longs;ity fall be­<lb/>tween the Points <emph type="italics"/>P<emph.end type="italics"/> and <foreign lang="greek">w;</foreign> Therefore if <emph type="italics"/>L<emph.end type="italics"/>ines be drawn through <lb/>B and G, parallel unto R T, they &longs;hall contain Right Angles with <lb/>the Surface of the Liquid: ^{*} and the part that is within the Li­<lb/><arrow.to.target n="marg1172"></arrow.to.target><lb/>quid &longs;hall move upwards according to the Perpendicular that is <lb/>drawn thorow B, parallel to R T, and the part that is above the Li­<lb/>quid &longs;hall move downwards according to that which is drawn tho­<lb/>row G; and the Solid A P O L &longs;hall not abide in this Po&longs;ition; for <lb/>that the parts towards A will move upwards, and tho&longs;e towards <lb/>B downwards; Wherefore N O &longs;hall be con&longs;tituted according to <lb/>the Perpendicular.]</s></p><p type="margin"> <s><margin.target id="marg1166"></margin.target>* <emph type="italics"/>Supplied by<emph.end type="italics"/> Fe­<lb/>derico Comman­<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"/>The Demon&longs;tration of this propo&longs;ition hath been much de&longs;ired; which we have (in like man­<lb/>ner as the 8 Prop. </s> <s>of the fir&longs;t Book) re&longs;tored according to<emph.end type="italics"/> Archimedes <emph type="italics"/>his own Schemes, and <lb/>illustrated it with Commentaries.<emph.end type="italics"/></s></p><p type="main"> <s>The Right Portion of a Rightangled Conoid, when it &longs;hall <lb/><arrow.to.target n="marg1173"></arrow.to.target><lb/>have its Axis le&longs;&longs;e than <emph type="italics"/>Se&longs;quialter ejus quæ u&longs;que ad Axem<emph.end type="italics"/> (or of <lb/>its <emph type="italics"/>Semi-parameter] In the Tran&longs;lation of<emph.end type="italics"/> Nicolo Tartaglia <emph type="italics"/>it is fal&longs;lyread<emph.end type="italics"/> great­<lb/>er then Se&longs;quialter, <emph type="italics"/>and &longs;o its rendered in the following Propo&longs;ition; but it is the Right <lb/>Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we &longs;ay <lb/>that Conoids are then con&longs;tituted erect when the cutting Plane, that is to &longs;ay, the Plane of the <lb/>Ba&longs;e, &longs;hall be parallel to the Surface of the Liquid.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1173"></margin.target>A</s></p><p type="main"> <s>Which &longs;hall be the Diameter of the Section I P O S, and the <lb/><arrow.to.target n="marg1174"></arrow.to.target><lb/>Axis of the Portion demerged in the Liquid.] <emph type="italics"/>By the 46 of the fir&longs;t of <lb/>the Conicks of<emph.end type="italics"/> Apollonious, <emph type="italics"/>or by the Corol­<lb/>lary of the 51 of the &longs;ame.<emph.end type="italics"/></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"/><p type="main"> <s>And of the Solid Magnitude A P <lb/><arrow.to.target n="marg1175"></arrow.to.target><lb/>O L, let the Centre of Gravity be R; <lb/>and of I P O S let the Centre be B.] <lb/><emph type="italics"/>For the Centre of Gravity of the Portion of a Right­<lb/>angled Conoid is in its Axis, which it &longs;o divideth <lb/>as that the part thereof terminating in the vertex, <lb/>be double to the other part terminating in the Ba&longs;e; as <lb/>in our Book<emph.end type="italics"/> De Centro Gravitatis Solidorum Propo. </s> <s>29. <emph type="italics"/>we have demon&longs;trated. </s> <s>And <lb/>&longs;ince the Centre of Gravity of the Portion A P O L is R, O R &longs;hall be double to RN and there­<lb/>fore N O &longs;hall be Se&longs;quialter of O R. </s> <s>And for the &longs;ame rea&longs;on, B the Centre of Gravity of the Por­<lb/>tion I P O S is in the Axis P F, &longs;o dividing it as that P B is double to B F;<emph.end type="italics"/></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/><arrow.to.target n="marg1176"></arrow.to.target><lb/>be the Centre of Gravity of the remaining Eigure I S L A.] <pb xlink:href="073/01/027.jpg" pagenum="356"/><emph type="italics"/>For if, the Line B R being prolonged unto G, G R hath the &longs;ame proportion to R B as the Por­<lb/>tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the <lb/>Liquid, the Toine G &longs;hall be its Centre of Gravity; by the 8 of the &longs;econd of<emph.end type="italics"/> Archimedes <lb/>de Centro Gravitatis Planorum, vel de <emph type="italics"/>Æ<emph.end type="italics"/>quiponderantibus.<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 &longs;hall be le&longs;s than <emph type="italics"/>quæ u&longs;que ad Axem<emph.end type="italics"/> (or than the Semi­<lb/>parameter.] <emph type="italics"/>By the 10 Propofit. </s> <s>of<emph.end type="italics"/> Euclids <emph type="italics"/>fifth Book of Elements. </s> <s>The Line<emph.end type="italics"/> quæ <lb/>u&longs;que ad Axem, <emph type="italics"/>(or the Semi-parameter) according to<emph.end type="italics"/> Archimedes, <emph type="italics"/>is the half of that<emph.end type="italics"/><lb/>juxta quam po&longs;&longs;unt, quæ á Sectione ducuntur, (<emph type="italics"/>or of the Parameter;) as appeareth <lb/>by the 4 Propo&longs;it of his Book<emph.end type="italics"/> De Conoidibus & Shpæroidibus: <emph type="italics"/>and for what rea&longs;on it is <lb/>&longs;o called, we have declared in the Commentaries upon him by us publi&longs;hed.<emph.end type="italics"/></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="greek">w</foreign> &longs;hall be acute.] <emph type="italics"/>Let the Line N O be <lb/>continued out to H, that &longs;o RH may be equall to <lb/>the Semi-parameter. </s> <s>If now from the Point H<emph.end type="italics"/><lb/><figure id="id.073.01.027.1.jpg" xlink:href="073/01/027/1.jpg"/><lb/><emph type="italics"/>a Line be drawn at Right Angles to N H, it &longs;hall <lb/>meet with FP without the Section; for being <lb/>drawn thorow O parallel to A L, it &longs;hall fall <lb/>without the Section, by the 17 of our &longs;irst Book of<emph.end type="italics"/><lb/>Conicks; <emph type="italics"/>Therefore let it meet in V: and <lb/>becau&longs;e F P is parallel to the Diameter, and H <lb/>V perpendicular to the &longs;ame Diameter, and R H <lb/>equall to the Semi-parameter, the Line drawn <lb/>from the Point R to V &longs;hall make Right Angles <lb/>with that Line which the Section toucheth in the Point P: that is with K<emph.end type="italics"/> <foreign lang="greek">w,</foreign> <emph type="italics"/>as &longs;hall anon be <lb/>demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and<emph.end type="italics"/> <foreign lang="greek">w;</foreign> <emph type="italics"/>and the Argle R<emph.end type="italics"/><lb/>P <foreign lang="greek">w</foreign> <emph type="italics"/>&longs;hall be an Acute Angle.<emph.end type="italics"/></s></p><p type="main"> <s>Let A B C be the Section of a Rightangled Cone, or a Parabola, <lb/>and its Diameter B D; and let the Line E F touch the <lb/>&longs;ame in the Point G: and in the Diameter B D take the Line <lb/>H K equall to the Semi-parameter: and thorow G, G L be­<lb/>ing drawn parallel to the Diameter, draw KM from the <lb/><emph type="italics"/>P<emph.end type="italics"/>oint K at Right Angles to B D cutting G L in M: I &longs;ay <lb/>that the Line prolonged thorow Hand Mis perpendicular to <lb/>E F, which it cutteth in N.</s></p><p type="main"> <s><emph type="italics"/>For from the Point G draw the Line G O at Right Angles to E F cutting the Diameter in <lb/>O: and again from the &longs;ame Point draw G P perpendicular to the Diameter: and let the <lb/>&longs;aid Diameter prolonged cut the Line E F in <expan abbr="q.">que</expan> P B &longs;hall be equall to B Q, by the 35 of<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1179"></arrow.to.target><lb/><emph type="italics"/>our fir&longs;t Book of<emph.end type="italics"/> Conick <emph type="italics"/>Sections,<emph.end type="italics"/> (a) <emph type="italics"/>and G<emph.end type="italics"/><lb/><figure id="id.073.01.027.2.jpg" xlink:href="073/01/027/2.jpg"/><lb/><emph type="italics"/>P a Mean-proportion all betmixt Q P and PO<emph.end type="italics"/>; <lb/><arrow.to.target n="marg1180"></arrow.to.target><lb/>(b) <emph type="italics"/>and therefore the Square of G P &longs;hall be e­<lb/>quall to the Rectangle of O P Q: But it is al&longs;o <lb/>equall to the Rectangle comprehended under P B <lb/>and the Line<emph.end type="italics"/> juxta quam po&longs;&longs;unt, <emph type="italics"/>or the Par­<lb/>ameter, by the 11 of our fir&longs;t Book of<emph.end type="italics"/> Conicks: <lb/><arrow.to.target n="marg1181"></arrow.to.target><lb/>(c) <emph type="italics"/>Therefore, look what proportion Q P hath to <lb/>P B, and the &longs;ame hath the Parameter unto P O: <lb/>But Q P is double unto<emph.end type="italics"/> P B, <emph type="italics"/>for that<emph.end type="italics"/> P B <emph type="italics"/>and B <lb/>Q are equall, as hath been &longs;aid: And therefore <lb/>the Parameter &longs;hall be double to the &longs;aid P O: <lb/>and by the &longs;ame Rea&longs;on P O is equall to that which we call the Semi-parameter, that is, to K H<emph.end type="italics"/>: <lb/><arrow.to.target n="marg1182"></arrow.to.target><lb/><emph type="italics"/>But<emph.end type="italics"/> (d) <emph type="italics"/>P G is equall to K M, and<emph.end type="italics"/> (e) <emph type="italics"/>the Angle O P G to the Angle H K M; for they are both<emph.end type="italics"/><lb/><arrow.to.target n="marg1183"></arrow.to.target><lb/><emph type="italics"/>Right Angles: And therefore O G al&longs;o is equall to H M, and the Angle P O G unto the<emph.end type="italics"/> <pb xlink:href="073/01/028.jpg" pagenum="357"/><figure id="id.073.01.028.1.jpg" xlink:href="073/01/028/1.jpg"/><lb/><emph type="italics"/>Angle K H M: Therefore<emph.end type="italics"/> (f) O G <emph type="italics"/>and H N are parallel,<emph.end type="italics"/><lb/><arrow.to.target n="marg1184"></arrow.to.target><lb/><emph type="italics"/>and the<emph.end type="italics"/> (g) <emph type="italics"/>Angle H N F equall to the Angle O G F; for <lb/>that G O being Perpendicular to E F, H N &longs;hall al&longs;o be per-<emph.end type="italics"/><lb/><arrow.to.target n="marg1185"></arrow.to.target><lb/><emph type="italics"/>pandicnlar to the &longs;ame: Which was to be demon &longs;trated.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1179"></margin.target>(a) <emph type="italics"/>By Cor. </s> <s>of 8. of <lb/>6. of<emph.end type="italics"/> Euclide.</s></p><p type="margin"> <s><margin.target id="marg1180"></margin.target>(b) <emph type="italics"/>By 17. of the<emph.end type="italics"/><lb/>6.</s></p><p type="margin"> <s><margin.target id="marg1181"></margin.target>(c) <emph type="italics"/>By 14. of the<emph.end type="italics"/><lb/>6.</s></p><p type="margin"> <s><margin.target id="marg1182"></margin.target>(d) <emph type="italics"/>By 33. of the<emph.end type="italics"/><lb/>1.</s></p><p type="margin"> <s><margin.target id="marg1183"></margin.target>(e) <emph type="italics"/>By 4. of the<emph.end type="italics"/> 1.</s></p><p type="margin"> <s><margin.target id="marg1184"></margin.target>(f) <emph type="italics"/>By 28. of the<emph.end type="italics"/><lb/>1.</s></p><p type="margin"> <s><margin.target id="marg1185"></margin.target>(g) <emph type="italics"/>By 29. of th<emph.end type="italics"/><lb/>1</s></p><p type="main"> <s>And the part which is within the Liquid <lb/><arrow.to.target n="marg1186"></arrow.to.target><lb/>doth move upwards according to the Per­<lb/>pendicular that is drawn thorow B parallel <lb/>to R T.] <emph type="italics"/>The rea&longs;on why this moveth upwards, and that <lb/>other downwards, along the Perpendicular Line, hath been &longs;hewn above in the 8 of the fir&longs;t <lb/>Book of this; &longs;o that we have judged it needle&longs;&longs;e to repeat it either in this, or in the re&longs;t <lb/>that follow.<emph.end type="italics"/></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"/>In the<emph.end type="italics"/> Antient <emph type="italics"/>Parabola (namely that a&longs;&longs;umed in a Rightangled <lb/>Cone) the Line<emph.end type="italics"/> juxta quam Po&longs;&longs;unt quæ in Sectione ordinatim du­<lb/>cuntur <emph type="italics"/>(which I, following<emph.end type="italics"/> Mydorgius, <emph type="italics"/>do call the<emph.end type="italics"/> Parameter<emph type="italics"/>) is<emph.end type="italics"/> (a) <lb/><arrow.to.target n="marg1187"></arrow.to.target><lb/><emph type="italics"/>double to that<emph.end type="italics"/> quæ ducta e&longs;t à Vertice Sectionis u&longs;que ad Axem, <emph type="italics"/>or in<emph.end type="italics"/><lb/>Archimedes <emph type="italics"/>phra&longs;e,<emph.end type="italics"/> <foreign lang="greek">ta_s us/xri tou_ a)/con<34>;</foreign> <emph type="italics"/>which I for that cau&longs;e, and <lb/>for want of a better word, name the<emph.end type="italics"/> Semiparameter: <emph type="italics"/>but in<emph.end type="italics"/> Modern <lb/><emph type="italics"/>Parabola's it is greater or le&longs;&longs;er then double. </s> <s>Now that throughout this <lb/>Book<emph.end type="italics"/> Archimedes <emph type="italics"/>&longs;peaketh of the Parabola in a Rectangled Cone, is mani­<lb/>fe&longs;t both by the fir&longs;t words of each Propo&longs;ition, & by this that no Parabola <lb/>hath its Parameter double to the Line<emph.end type="italics"/> quæ e&longs;t a Sectione ad Axem, <emph type="italics"/>&longs;ave <lb/>that which is taken in a Rightangled Cone. </s> <s>And in any other Parabola, for <lb/>the Line<emph.end type="italics"/> <foreign lang="greek">ta_s ms/xritou_ a)/eon<34></foreign> <emph type="italics"/>or<emph.end type="italics"/> quæ u&longs;que ad Axem <emph type="italics"/>to u&longs;urpe the Word<emph.end type="italics"/> Se­<lb/>miparameter <emph type="italics"/>would be neither proper nor true: but in this ca&longs;e it may pa&longs;s<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1187"></margin.target>(a) Rîvalt. <emph type="italics"/>in<emph.end type="italics"/> Ar­<lb/>chimed. <emph type="italics"/>de Cunoid <lb/>& Sphæroid.<emph.end type="italics"/> Prop. <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"/>The Right Portion of a Rightangled Conoid, when it <lb/>&longs;hall have its Axis le&longs;&longs;e than &longs;e&longs;quialter of the Se­<lb/>mi-parameter, the Axis having any what ever pro­<lb/>portion to the Liquid in Gravity, being demitted into <lb/>the Liquid &longs;o as that its Ba&longs;e be wholly within the <lb/>&longs;aid Liquid, and being &longs;et inclining, it &longs;hall not re­<lb/>main inclined, but &longs;hall be &longs;o re&longs;tored, as that its Ax­<lb/>is do &longs;tand upright, or according to the Perpendicular.<emph.end type="italics"/></s></p><p type="main"> <s>Let any Portion be demitted into the Liquid, as was &longs;aid; and <lb/>let its Ba&longs;e be in the <emph type="italics"/>L<emph.end type="italics"/>iquid; <lb/><figure id="id.073.01.028.2.jpg" xlink:href="073/01/028/2.jpg"/><lb/>and let it be cut thorow the <lb/>Axis, by a Plain erect upon the Sur­<lb/>face of the Liquid, and let the Se­<lb/>ction be A P O <emph type="italics"/>L,<emph.end type="italics"/> the Section of a <lb/>Right angled Cone: and let the Axis <lb/>of the Portion and Diameter of the <pb xlink:href="073/01/029.jpg" pagenum="356"/>Section of the Portion be A P O L, the Section of a Rightangled <lb/>Cone; and let the Axis of the Portion and Diameter of the Section <lb/>be N O, and the Section of the Surface of the Liquid I S. </s> <s>If now <lb/>the Portion be not erect, then N O &longs;hall not be at equall Angles with <lb/>I S. </s> <s>Draw R <foreign lang="greek">w</foreign> touching the Section of the Rightangled Conoid <lb/>in P, and parallel to I S: and from the Point P and parall to O N <lb/>draw <emph type="italics"/>P<emph.end type="italics"/> F: and take the Centers of Gravity; and of the Solid A <lb/><emph type="italics"/>P<emph.end type="italics"/> O L let the Centre be R; and of that which lyeth within the <lb/>Liquid let the Centre be B; and draw a Line from B to R pro­<lb/>longing it to G, that G may be the Centre of Gravity of the Solid <lb/>that is above the Liquid. </s> <s>And becau&longs;e N O is &longs;e&longs;quialter of R <lb/>O, and is greater than &longs;e&longs;quialter of the Semi-Parameter; it is ma­<lb/><arrow.to.target n="marg1188"></arrow.to.target><lb/>nife&longs;t that <emph type="italics"/>(a)<emph.end type="italics"/> R O is greater than the <lb/><figure id="id.073.01.029.1.jpg" xlink:href="073/01/029/1.jpg"/><lb/>Semi-parameter. ^{*}Let therefore R <lb/><arrow.to.target n="marg1189"></arrow.to.target><lb/>H be equall to the Semi-Parameter, <lb/><arrow.to.target n="marg1190"></arrow.to.target><lb/>^{*} and O <emph type="italics"/>H<emph.end type="italics"/> double to H M. </s> <s>Fora&longs;­<lb/>much therefore as N O is &longs;e&longs;quialter <lb/><arrow.to.target n="marg1191"></arrow.to.target><lb/>of R O, and M O of O H, <emph type="italics"/>(b)<emph.end type="italics"/> the <lb/>Remainder N M &longs;hall be &longs;e&longs;quialter <lb/>of the Remainder R H: Therefore <lb/>the Axis is greater than &longs;e&longs;quialter <lb/>of the Semi parameter by the quan­<lb/>tity of the Line M O. </s> <s>And let it be <lb/>&longs;uppo&longs;ed that the Portion hath not le&longs;&longs;e proportion in Gravity unto <lb/>the Liquid of equall Ma&longs;&longs;e, than the Square that is made of the <lb/>Exce&longs;&longs;e by which the Axis is greater than &longs;e&longs;quialter of the Semi­<lb/>parameter hath to the Square made of the Axis: It is therefore ma­<lb/>nife&longs;t that the Portion hath not le&longs;&longs;e proportion in Gravity to the <lb/>Liquid than the Square of the Line M O hath to the Square of N <lb/>O: But look what proportion the <emph type="italics"/>P<emph.end type="italics"/>ortion hath to the Liquid in <lb/>Gravity, the &longs;ame hath the <emph type="italics"/>P<emph.end type="italics"/>ortion &longs;ubmerged to the whole Solid: <lb/>for this hath been demon&longs;trated <emph type="italics"/>(c)<emph.end type="italics"/> above: ^{*}And look what pro­<lb/><arrow.to.target n="marg1192"></arrow.to.target><lb/>portion the &longs;ubmerged Portion hath to the whole <emph type="italics"/>P<emph.end type="italics"/>ortion, the <lb/><arrow.to.target n="marg1193"></arrow.to.target><lb/>&longs;ame hath the Square of <emph type="italics"/>P<emph.end type="italics"/> F unto the Square of N O: For it hath <lb/>been demon&longs;trated in <emph type="italics"/>(d) Lib. de Conoidibus,<emph.end type="italics"/> that if from a Right­<lb/><arrow.to.target n="marg1194"></arrow.to.target><lb/>angled Conoid two <emph type="italics"/>P<emph.end type="italics"/>ortions be cut by Planes in any fa&longs;hion pro­<lb/>duced, the&longs;e <emph type="italics"/>P<emph.end type="italics"/>ortions &longs;hall have the &longs;ame Proportion to each <lb/>other as the Squares of their Axes: The Square of P F, therefore, <lb/>hath not le&longs;&longs;e proportion to the Square of N O than the Square of <lb/>M O hath to the Square of N O: ^{*}Wherefore P F is not le&longs;&longs;e than <lb/><arrow.to.target n="marg1195"></arrow.to.target><lb/>M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn <lb/><arrow.to.target n="marg1196"></arrow.to.target><lb/>from H at Right Angles unto N O, it &longs;hall meet with B <emph type="italics"/>P,<emph.end type="italics"/> and &longs;hall <lb/><arrow.to.target n="marg1197"></arrow.to.target><lb/>fall betwixt B and P; let it fall in T: <emph type="italics"/>(e)<emph.end type="italics"/> And becau&longs;e <emph type="italics"/>P<emph.end type="italics"/> F is <lb/><arrow.to.target n="marg1198"></arrow.to.target><lb/>parallel to the Diameter, and H T is perpendicular unto the &longs;ame <lb/>Diameter, and R H equall to the Semi-parameter; a Line drawn <lb/>from R to T and prolonged, maketh Right Angles with the Line <pb xlink:href="073/01/030.jpg" pagenum="360"/>contingent unto the Section in the Point P: Wherefore it al&longs;o <lb/>maketh Right Angles with the Surface of the Liquid: and that <lb/>part of the Conoidall Solid which is within the Liquid &longs;hall move <lb/>upwards according to the Perpendicular drawn thorow B parallel <lb/>to R T; and that part which is above the Liquid &longs;hall move down­<lb/>wards according to that drawn thorow G, parallel to the &longs;aid R T: <lb/>And thus it &longs;hall continue to do &longs;o long untill that the Conoid be <lb/>re&longs;tored to uprightne&longs;&longs;e, or to &longs;tand according to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1188"></margin.target>(a) <emph type="italics"/>By 10. of the <lb/>fifth.<emph.end type="italics"/></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"/>By 19. of the <lb/>fifth.<emph.end type="italics"/></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"/>By 1. of this <lb/>&longs;econd Book.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1194"></margin.target>(d) <emph type="italics"/>By<emph.end type="italics"/> 6. De Co­<lb/>noilibus & <emph type="italics"/>S<emph.end type="italics"/>phæ­<lb/>roidibus <emph type="italics"/>of<emph.end type="italics"/> Archi­<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"/>By 2. of this <lb/>&longs;econd Book.<emph.end type="italics"/></s></p><p type="head"> <s>COMMANDINE.<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"/>So it is to be <lb/>read, and not R M, as<emph.end type="italics"/> Tartaglia's <emph type="italics"/>Tran&longs;lation hath is; which may be made appear from <lb/>that which followeth.<emph.end type="italics"/></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"/>In the Tran&longs;lation aforenamed it is fal&longs;ly render­<lb/>ed,<emph.end type="italics"/> O N <emph type="italics"/>double to<emph.end type="italics"/> 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/>Portion, the &longs;ame hath the Square of P F unto the Square of N O.] <lb/><emph type="italics"/>This place we have re&longs;tored in our Tran&longs;lation, at the reque&longs;t of &longs;ome friends: But it is demon­<lb/>&longs;trated by<emph.end type="italics"/> 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 le&longs;&longs;e than M O.] <emph type="italics"/>For by 10 of the fifth it followeth <lb/>that the Square of P F is not le&longs;&longs;e than the Square of M O: and therefore neither &longs;hall the <lb/>Line P F be leße than the Line M O, by 22 of the<emph.end type="italics"/></s></p><figure id="id.073.01.030.1.jpg" xlink:href="073/01/030/1.jpg"/><p type="main"> <s><arrow.to.target n="marg1203"></arrow.to.target><lb/><emph type="italics"/>&longs;ixth.<emph.end type="italics"/><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"/>By 14. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="main"> <s>Nor B P than H O,] <emph type="italics"/>For as P F is to <lb/>P B, &longs;o is M O to H O: and, by Permutation, as<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1205"></arrow.to.target><lb/><emph type="italics"/>P F is to M O, &longs;o is B P to H O; But P F is not <lb/>le&longs;&longs;e than M O as hath bin proved; (a) Therefore <lb/>neither &longs;hall B P be le&longs;&longs;e than H O.<emph.end type="italics"/></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/>from H at Right Angles unto N O, it <lb/>&longs;hall meet with B P, and &longs;hall fall be­<lb/>twixt B and P.] <emph type="italics"/>This Place was corrupt in the <lb/>Tran&longs;lation of<emph.end type="italics"/> Tartaglia<emph type="italics"/>: But it is thus demonstra­<lb/>ted. </s> <s>In regard that P F is not le&longs;&longs;e than O M, nor P B than O H, if we &longs;uppo&longs;e P F equall to <lb/>O M, P B &longs;hall be likewi&longs;e equall to O H: Wherefore the Line drawn thorow O, parallel to A L <lb/>&longs;hall fall without the Section, by 17 of the fir&longs;t of our Treati&longs;e of Conicks; And in regard that <lb/>B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth <lb/>al&longs;o meet with the &longs;ame beneath B, and it doth of nece&longs;&longs;ity fall betwixt B and P: But the <lb/>&longs;ame is much more to follow, if we &longs;uppo&longs;e P F to be greater than O M.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/031.jpg" pagenum="361"/><p type="head"> <s>PROP. V. THEOR. V.</s></p><p type="main"> <s><emph type="italics"/>The Right Portion of a Right-Angled Conoid lighter <lb/>than the Liquid, when it &longs;hall have its Axis great­<lb/>er than<emph.end type="italics"/> Se&longs;quialter <emph type="italics"/>of the Semi-parameter, if it have <lb/>not greater proportion in Gravity to the Liquid [of <lb/>equal Ma&longs;s] than the Exce&longs;&longs;e by which the Square <lb/>made of the Axis is greater than the Square made <lb/>of the Exce&longs;&longs;e by which the Axis is greater than<emph.end type="italics"/><lb/>&longs;e&longs;quialter <emph type="italics"/>of the Semi-Parameter hath to the <lb/>Square made of the Axis being demitted into the Li­<lb/>quid, &longs;o as that its Ba&longs;e be wholly within the Liquid, <lb/>and being &longs;et inclining, it &longs;hall not remain &longs;o inclined, <lb/>but &longs;hall turn about till that its Axis &longs;hall be accor­<lb/>ding to the Perpendicular.<emph.end type="italics"/></s></p><p type="main"> <s>For let any Portion be demitted into the Liquid, as hath been <lb/>&longs;aid; and let its Ba&longs;e be wholly within the Liquid, And being <lb/>cut thorow its Axis by a Plain erect upon the Surface of the <lb/>Liquid; its Section &longs;hall be the Section <lb/><figure id="id.073.01.031.1.jpg" xlink:href="073/01/031/1.jpg"/><lb/>of a Rightangled Cone: Let it be <lb/>A P O L, and let the Axis of the Por­<lb/>tion and Diameter of the Section be <lb/>N O; and the Section of the Surface of <lb/>the Liquid I S. </s> <s>And becau&longs;e the Axis <lb/>is not according to the Perpendicu­<lb/>lar, N O will not be at equall angles <lb/>with I S. </s> <s>Draw K <foreign lang="greek">w</foreign> touching the Se­<lb/>ction A P O L in P, and parallel unto <lb/>I S: and thorow P, draw P F parallel unto N O: and take the <lb/>Centres of Gravity; and of the Solid A P O L let the Centre be <lb/>R; and of that which lyeth above the Liquid let the Centre be B; <lb/>and draw a Line from B to R, prolonging it to G; which let be the <lb/>Centre of Gravity of the Solid demerged within the Liquid: and <lb/>moreover, take R H equall to the Semi-parameter, and let O H be <lb/>double to H M; and do in the re&longs;t as hath been &longs;aid <emph type="italics"/>(a)<emph.end type="italics"/> above. <lb/><arrow.to.target n="marg1206"></arrow.to.target><lb/>Now fora&longs;much as it was &longs;uppo&longs;ed that the Portion hath not greater <lb/>proportion in Gravity to the Liquid, than the Exce&longs;&longs;e by which <lb/>the Square N O is greater than the Square M O, hath to the &longs;aid <lb/>Square N O: And in regard that whatever proportion in Gravity <pb xlink:href="073/01/032.jpg" pagenum="362"/>the Portion hath to the Liquid of equall Ma&longs;&longs;e, the &longs;ame hath the <lb/>Magnitude of the Portion &longs;ubmerged unto the whole Portion; as <lb/>hath been demon&longs;trated in the fir&longs;t Propo&longs;ition; The Magnitude <lb/>&longs;ubmerged, therefore, &longs;hall not have greater proportion to the <lb/><arrow.to.target n="marg1207"></arrow.to.target><lb/>whole <emph type="italics"/>(b)<emph.end type="italics"/> Portion, than that which hath been mentioned: ^{*}And <lb/>therefore the whole Portion hath not greater proportion unto that <lb/><arrow.to.target n="marg1208"></arrow.to.target><lb/>which is above the Liquid, than the Square N O hath to the Square <lb/><arrow.to.target n="marg1209"></arrow.to.target><lb/>M O: But the <emph type="italics"/>(c)<emph.end type="italics"/> whole Portion hath the &longs;ame proportion unto <lb/>that which is above the Liquid that the Square N O hath to the <lb/>Square P F: Therefore the Square N O hath not greater propor­<lb/><arrow.to.target n="marg1210"></arrow.to.target><lb/>tion unto the Square P F, than it hath unto the Square M O: ^{*}And <lb/>hence it followeth that P F is not le&longs;&longs;e than O M, nor P B than O <lb/><arrow.to.target n="marg1211"></arrow.to.target><lb/>H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O <lb/>&longs;hall meet with B P betwixt P and B: Let it be in T: And be­<lb/>cau&longs;e that in the Section of the Rectangled Cone P F is parallel unto <lb/>the Diameter N O; and H T perpendicular unto the &longs;aid Diame­<lb/>ter; and R H equall to the Semi-parameter: It is manife&longs;t that <lb/>R T prolonged doth make Right Angles with K P <foreign lang="greek">w</foreign>: And there­<lb/>fore doth al&longs;o make Right Angles with I S: Therefore R T is per­<lb/>pendicular unto the Surface of the Liquid; And if thorow the <lb/>Points B and G Lines be drawn parallel unto R T, they &longs;hall be <lb/>perpendicular unto the Liquids Surface. </s> <s>The Portion, therefore, <lb/>which is above the Liquid &longs;hall move downwards in the Liquid ac­<lb/>cording to the Perpendicular drawn thorow B; and that part <lb/>which is within the Liquid &longs;hall move upwards according to the <lb/>Perpendicular drawn thorow G; and the Solid Portion A P O L <lb/>&longs;hall not continue &longs;o inclined, [<emph type="italics"/>as it was at its demer&longs;ion<emph.end type="italics"/>], but &longs;hall <lb/>move within the Liquid untill &longs;uch time that N O do &longs;tand accor­<lb/>ding to the Perpendicular.</s></p><p type="margin"> <s><margin.target id="marg1206"></margin.target>(a) <emph type="italics"/>In 4. Prop. of <lb/>this.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1207"></margin.target>(a) <emph type="italics"/>By 11. of the <lb/>fifth.<emph.end type="italics"/></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"/>By 26. of the <lb/>Book<emph.end type="italics"/> De Conoid. <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/><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/>unto that which is above the Liquid, than the Square N O hath to <lb/>the Square M O.] <emph type="italics"/>For in regard that the Magnitude of the Portion demerged <lb/>within the Liquid hath not greater proportion unto the whole Portion than the Exce&longs;&longs;e by which <lb/>the Square N O is greater than the Square M O hath to the &longs;aid Square N O; Converting of <lb/>the Proportion, by the 26. of the fifth of<emph.end type="italics"/> Euclid, <emph type="italics"/>of<emph.end type="italics"/> Campanus <emph type="italics"/>his Tran&longs;lation, the whole <lb/>Portion &longs;hall not have le&longs;&longs;er proportion unto the Magnitude &longs;ubmerged, than the Square N O <lb/>hath unto the Exce&longs;&longs;e by which N O is greater than the Square M O. </s> <s>Let a Portion be taken; <lb/>and let that part of it which is above the Liquid be the fir&longs;t Magnitude; the part of it which <lb/>is &longs;ubmerged the &longs;econd: and let the third Magnitude be the Square M O; and let the Exce&longs;&longs;e <lb/>by which the Square N O is greater than the Square M O be the fourth. </s> <s>Now of the&longs;e Mag­<lb/>nitudes, the proportion of the fir&longs;t and &longs;econd, unto the &longs;econd, is not le&longs;&longs;e than that of the third & <lb/>fourth unto the fourth: For the Square M O together with the Exce&longs;&longs;e by which the Square <lb/>N O exceedeth the Square M O is equall unto the &longs;aid Square N O: Wherefore, by Conver&longs;i­<lb/>on of Proportion, by 30 of the &longs;aid fifth Book, the proportion of the fir&longs;t and &longs;econd unto the <lb/>fir&longs;t, &longs;hall not be greater than that of the third and fourth unto the third: And, for the &longs;ame<emph.end type="italics"/> <pb xlink:href="073/01/033.jpg" pagenum="363"/><emph type="italics"/>the proportion of the whole Portion unto that part thereof which is above the Liquid &longs;hall not be <lb/>greater than that of the Square N O unto the Square M O: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s>And hence it followeth that P F is not le&longs;&longs;e than O M, nor P B </s></p><p type="main"> <s><arrow.to.target n="marg1213"></arrow.to.target><lb/>than O H.] <emph type="italics"/>This followeth by the 10 and 14 of the fifth, and by the 22 of the &longs;ixth of<emph.end type="italics"/><lb/>Euclid, <emph type="italics"/>as hath been &longs;aid above.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1213"></margin.target>B</s></p><p type="main"> <s>A <emph type="italics"/>L<emph.end type="italics"/>ine, therefore, drawn from Hat Right Angles unto N O &longs;hall <lb/><arrow.to.target n="marg1214"></arrow.to.target><lb/>meet with P B betwixt P and B.] <emph type="italics"/>Why this &longs;o falleth out, we will &longs;hew in the <lb/>next.<emph.end type="italics"/></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"/>The Right Portion of a Rightangled Conoid lighter <lb/>than the Liquid, when it &longs;hall have its Axis greater <lb/>than &longs;e&longs;quialter of the Semi-parameter, but le&longs;&longs;e than <lb/>to be unto the Semi-parameter in proportion as fifteen <lb/>to fower, being demitted into the Liquid &longs;o as that <lb/>its Ba&longs;e do touch the Liquid, it &longs;hall never stand &longs;o <lb/>enclined as that its Ba&longs;e toucheth the Liquid in one <lb/>Point only.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be a Portion, as was &longs;aid; and demit it into the Li­<lb/>quid in &longs;uch fa&longs;hion as that its Ba&longs;e do touch the Liquid in <lb/>one only Point: It is to be demon&longs;trated that the &longs;aid Portion <lb/><arrow.to.target n="marg1215"></arrow.to.target><lb/>&longs;hall not continue &longs;o, but &longs;hall turn about in &longs;uch manner as that <lb/>its Ba&longs;e do in no wi&longs;e touch the Surface of the Liquid. </s> <s>For let it be <lb/>cut thorow its Axis by a Plane erect <lb/><figure id="id.073.01.033.1.jpg" xlink:href="073/01/033/1.jpg"/><lb/>upon the <emph type="italics"/>L<emph.end type="italics"/>iquids Surface: and let <lb/>the Section of the Superficies of the <lb/>Portion be A P O L, the Section of <lb/>a Rightangled Cone; and the Sect­<lb/>ion of the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid be <lb/>A S; and the Axis of the Portion <lb/>and Diameter of the Section N O: <lb/>and let it be cut in F, &longs;o as that O <lb/>F be double to F N; and in <foreign lang="greek">w</foreign> &longs;o, as that N O may be to F <foreign lang="greek">w</foreign> in the <lb/>&longs;ame proportion as fifteen to four; and at Right Angles to N O <lb/>draw <foreign lang="greek">w</foreign> <emph type="italics"/>N<emph.end type="italics"/>ow becau&longs;e N O hath greater proportion unto F <foreign lang="greek">w</foreign> than <lb/>unto the Semi-parameter, let the Semi-parameter be equall to F B: <lb/><arrow.to.target n="marg1216"></arrow.to.target><lb/>and draw P C parallel unto A S, and touching the Section A P O L <lb/>in P; and P I parallel unto <emph type="italics"/>N O<emph.end type="italics"/>; and fir&longs;t let P I cut K<foreign lang="greek">w</foreign> in H. For­<lb/><arrow.to.target n="marg1217"></arrow.to.target><lb/>a&longs;much, therefore, as in the Portion A P O L, contained betwixt <lb/>the Right <emph type="italics"/>L<emph.end type="italics"/>ine and the Section of the Rightangled Cone, K <foreign lang="greek">w</foreign> is <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"/>&longs;aid K <foreign lang="greek">w</foreign> in H, and A S is parallel unto the <emph type="italics"/>L<emph.end type="italics"/>ine that toucheth in <lb/>P; It is nece&longs;&longs;ary that P I hath unto P H either the &longs;ame proportion <lb/>that <emph type="italics"/>N<emph.end type="italics"/> <foreign lang="greek">w</foreign> hath to <foreign lang="greek">w</foreign> O, or greater; for this hath already been de­<lb/>mon&longs;trated: But <emph type="italics"/>N<emph.end type="italics"/> <foreign lang="greek">w</foreign> is &longs;e&longs;quialter of <foreign lang="greek">w</foreign> O; and P I, therefore, is <lb/>either Se&longs;quialter of H P, or more than &longs;e&longs;quialter: Wherefore <lb/><arrow.to.target n="marg1218"></arrow.to.target><lb/>P H is to H I either double, or le&longs;&longs;e than double. <emph type="italics"/>L<emph.end type="italics"/>et P T be <lb/>double to T I: the Centre of Gravity of the part which is within <lb/>the <emph type="italics"/>L<emph.end type="italics"/>iquid &longs;hall be the Point T. </s> <s>Therefore draw a <emph type="italics"/>L<emph.end type="italics"/>ine from T <lb/>to F prolonging it; and let the Centre of <lb/><figure id="id.073.01.034.1.jpg" xlink:href="073/01/034/1.jpg"/><lb/>Gravity of the part which is above the <emph type="italics"/>L<emph.end type="italics"/>iquid <lb/>be G: and from the Point B at Right Angles <lb/>unto <emph type="italics"/>N O<emph.end type="italics"/> draw B R. </s> <s>And &longs;eeing that P I is <lb/>parallel unto the Diameter <emph type="italics"/>N O,<emph.end type="italics"/> and B R <lb/>perpendicular unto the &longs;aid Diameter, and F <lb/>B equall to the Semi-parameter; It is mani­<lb/>fe&longs;t that the <emph type="italics"/>L<emph.end type="italics"/>ine drawn thorow the Points <lb/>F and R being prolonged, maketh equall <lb/>Angles with that which toucheth the Section <lb/>A P O L in the Point P: and therefore doth al&longs;o make Right An­<lb/>gles with A S, and with the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid: and the <emph type="italics"/>L<emph.end type="italics"/>ines <lb/>drawn thorow T and G parallel unto F R &longs;hall be al&longs;o perpendicu­<lb/>lar to the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid: and of the Solid Magnitude A P <lb/>O L, the part which is within the <emph type="italics"/>L<emph.end type="italics"/>iquid moveth upwards according <lb/>to the Perpendicular drawn thorow T; and the part which is above <lb/>the <emph type="italics"/>L<emph.end type="italics"/>iquid moveth downwards according to that drawn thorow G: <lb/><arrow.to.target n="marg1219"></arrow.to.target><lb/>The Solid A <emph type="italics"/>P<emph.end type="italics"/> O L, therefore, &longs;hall turn about, and its Ba&longs;e &longs;hall <lb/>not in the lea&longs;t touch the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid, And if <emph type="italics"/>P<emph.end type="italics"/> I do not <lb/>cut the <emph type="italics"/>L<emph.end type="italics"/>ine K <foreign lang="greek">w,</foreign> as in the &longs;econd Figure, it is manife&longs;t that the <lb/><emph type="italics"/>P<emph.end type="italics"/>oint T, which is the Centre of Gravity of the &longs;ubmerged <emph type="italics"/>P<emph.end type="italics"/>ortion, <lb/>falleth betwixt <emph type="italics"/>P<emph.end type="italics"/> and I: And for the other particulars remaining, <lb/>they are demon&longs;trated 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/><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 demon&longs;trated that the &longs;aid <emph type="italics"/>P<emph.end type="italics"/>ortion &longs;hall not continue <lb/>&longs;o, but &longs;hall turn about in &longs;uch manner as that its Ba&longs;e do in no wi&longs;e <lb/>touch the Surface of the Liquid.] <emph type="italics"/>The&longs;e words are added by us, as having been <lb/>omitted by<emph.end type="italics"/> Tartaglia.</s></p><p type="main"> <s><emph type="italics"/>N<emph.end type="italics"/>ow becau&longs;e N O hath greater proportion to F <foreign lang="greek">w</foreign> than unto </s></p><p type="main"> <s><arrow.to.target n="marg1221"></arrow.to.target><lb/>the Semi parameter.] <emph type="italics"/>For the Diameter of the Portion N O hath unto F<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>the <lb/>&longs;ame proportion as fifteen to fower: But it was &longs;uppo&longs;ed to have le&longs;&longs;e proportion unto the <lb/>Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F<emph.end type="italics"/> <foreign lang="greek">w</foreign><lb/><emph type="italics"/>than unto the Semi-parameter: And therefore<emph.end type="italics"/> (a) <emph type="italics"/>the Semi-parameter &longs;hall be greater<emph.end type="italics"/><lb/><arrow.to.target n="marg1222"></arrow.to.target><lb/><emph type="italics"/>than the &longs;aid F<emph.end type="italics"/> <foreign lang="greek">w.</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"/>By 10. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>Fora&longs;much, therefore, as in the <emph type="italics"/>P<emph.end type="italics"/>ortion <emph type="italics"/>A P O L,<emph.end type="italics"/> contained, be­<lb/><arrow.to.target n="marg1223"></arrow.to.target><lb/>twixt the Right <emph type="italics"/>L<emph.end type="italics"/>ine and the Section of the Rightangled Cone K <lb/><foreign lang="greek">w</foreign> is parallel to A L, and <emph type="italics"/>P I<emph.end type="italics"/> parallel unto the Diameter, and cut by <pb xlink:href="073/01/035.jpg" pagenum="365"/>the &longs;aid K <foreign lang="greek">w</foreign> in H, and A S is parallel unto the Line that toucheth <lb/>in P; It is nece&longs;&longs;ary that P I hath unto P H either the &longs;ame propor­<lb/>tion that N <foreign lang="greek">w</foreign> hath to <foreign lang="greek">w</foreign> O, or greater; for this hath already been <lb/>demon&longs;trated.] <emph type="italics"/>Where this is demon&longs;trated either by<emph.end type="italics"/> Archimedes <emph type="italics"/>him&longs;elf, or by <lb/>any other, doth not appear; touching which we will here in&longs;ert a Demon&longs;tration, after that <lb/>we have explained &longs;ome things that pertaine thereto.<emph.end type="italics"/></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/>the point D, taken in the Line A C, draw D E and D F at <lb/>plea&longs;ure unto A B: and in the &longs;ame Line any Points G and L <lb/>being taken, draw G H & L M parallel to D E, & G K and <lb/>L N parallel unto F D: Then from the Points D & G as farre <lb/>as to the Line M L draw D O P, cutting G H in O, and G Q <lb/>parallel unto B A. </s> <s>I &longs;ay that the Lines that lye betwixt the Pa­<lb/>rallels unto F D have unto tho&longs;e that lye betwixt the Par­<lb/>allels unto D E (namely K N to G Q or to O P; F K to D O; <lb/>and F N to D P) the &longs;ame mutuall proportion: that is to &longs;ay, <lb/>the &longs;ame that A F hath to A E.</s></p><p type="main"> <s><emph type="italics"/>For in regard that the Triangles A F D, A K G, and A N L<emph.end type="italics"/><lb/><figure id="id.073.01.035.1.jpg" xlink:href="073/01/035/1.jpg"/><lb/><emph type="italics"/>are alike, and E F D, H K G, and M N L are al&longs;o alike: There-<emph.end type="italics"/><lb/><arrow.to.target n="marg1224"></arrow.to.target><lb/><emph type="italics"/>fore,<emph.end type="italics"/> (a) <emph type="italics"/>as A F is to F D, &longs;o &longs;hall A K be to K G; and as F D is to <lb/>F E, &longs;o &longs;hall K G be to K H: Wherefore,<emph.end type="italics"/> ex equali, <emph type="italics"/>as A F is to F <lb/>E, &longs;o &longs;hall A K be to K H: And, by Conver&longs;ion of proportion, as <lb/>A F is to A E, &longs;o &longs;hall A K be to K H. </s> <s>It is in the &longs;ame manner <lb/>proved that, as A F is to A E, &longs;o &longs;hall A N be to A M. </s> <s>Now A<emph.end type="italics"/><lb/><arrow.to.target n="marg1225"></arrow.to.target><lb/><emph type="italics"/>N being to A M, as A K is to A H; The<emph.end type="italics"/> (b) <emph type="italics"/>Remainder K N &longs;hall <lb/>be unto the Remainder H M, that is unto G Q, or unto O P, as <lb/>A N is to A M; that is, as A F is to A E: Again, A K is to <lb/>A H, as A F is to A E; Therefore the Remainder F K &longs;hall be to <lb/>the Remainder E H, namely to D O, as A F is to A E. </s> <s>We might in <lb/>like manner demonstrate that &longs;o is F N to D P: Which is that that <lb/>was required to be demonstrated.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1224"></margin.target>(a) <emph type="italics"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1225"></margin.target>(b) <emph type="italics"/>By 5. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA II.</s></p><p type="main"> <s>In the &longs;ame Line A B let there be two Points R and S, &longs;o di&longs;po­<lb/>&longs;ed, that A S may have the &longs;ame Proportion to A R that <lb/>A F hath to A E; and thorow R draw R T parallel to E D, <lb/>and thorow S draw S T parallel to F D, &longs;o, as that it may <lb/>meet with R T in the Point T. </s> <s>I &longs;ay that the Point T fall­<lb/>eth in the Line A C.</s></p> <pb xlink:href="073/01/036.jpg" pagenum="366"/><figure id="id.073.01.036.1.jpg" xlink:href="073/01/036/1.jpg"/><p type="main"> <s><emph type="italics"/>For if it be po&longs;&longs;ible, let it fall &longs;hort of it: and let R T be pro­<lb/>longed as farre as to A C in V: and then thorow V draw V X pa­<lb/>rallel to F D. Now, by the thing we have last demon&longs;trated, A X <lb/>&longs;hall have the &longs;ame proportion unto A R, as A F hath to A E. <lb/></s> <s>But A S hath al&longs;o the &longs;ame proportion to A R: Wherefore<emph.end type="italics"/> (a) <lb/><arrow.to.target n="marg1226"></arrow.to.target><lb/>A S <emph type="italics"/>is equall to A X, the part to the whole, which is impo&longs;&longs;i­<lb/>ble. </s> <s>The &longs;ame ab&longs;urdity will follow if we &longs;uppo&longs;e the Toint <lb/>T to fall beyond the Line A C: It is therefore nece&longs;&longs;ary that <lb/>it do fall in the &longs;aid A C. </s> <s>Which we propounded to be demonstrated.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1226"></margin.target>(a) <emph type="italics"/>By 9. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA III.</s></p><p type="main"> <s>Let there be a Parabola, who&longs;e Diameter <lb/><arrow.to.target n="marg1227"></arrow.to.target><lb/>let be A B; and let the Right Lines A C and B D be ^{*} con­<lb/>tingent to it, A C in the Point C, and B D in B: And two <lb/>Lines being drawn thorow C, the one C E, parallel unto <lb/>the Diameter; the other C F, parallel to B D; take any <lb/>Point in the Diameter, as G; and as F B is to B G, &longs;o let B <lb/>G be to B H: and thorow G and H draw G K L, and H E <lb/>M, parallel unto B D; and thorow M draw M N O parallel <lb/>to <emph type="italics"/>A C,<emph.end type="italics"/> and cutting the Diameter in O: and the Line <emph type="italics"/>N P<emph.end type="italics"/><lb/>being drawn thorow <emph type="italics"/>N<emph.end type="italics"/> unto the Diameter let it be parallel <lb/>to B D. </s> <s>I &longs;ay 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"/>For the Line M N O cutteth the Diameter either in G, or in other Points: and if it do <lb/>cut it in G, one and the &longs;ame Point &longs;hall be noted by the two letters G and O. </s> <s>Therfore F C, <lb/>P N, and H E M being Parallels, and A C being Parallels to M N O, they &longs;hall make the<emph.end type="italics"/><lb/><figure id="id.073.01.036.2.jpg" xlink:href="073/01/036/2.jpg"/><lb/><emph type="italics"/>Triangles A F C, O P N and O H M like to<emph.end type="italics"/><lb/><arrow.to.target n="marg1228"></arrow.to.target><lb/><emph type="italics"/>each other: Wherefore<emph.end type="italics"/> (a) <emph type="italics"/>O H &longs;hall be to <lb/>H M, as A F to FC: and<emph.end type="italics"/> ^{*} Permutando, <lb/><arrow.to.target n="marg1229"></arrow.to.target><lb/><emph type="italics"/>O H &longs;hall be to A F, as H M to F C: But <lb/>the Square H M is to the Square G L as the Line <lb/>H B is to the Line B G, by 20. of our fir&longs;t Book <lb/>of<emph.end type="italics"/> Conicks; <emph type="italics"/>and the Square G L is unto the <lb/>Square F C, as the Line G B is to the Line B F: <lb/>and the Lines H B, B G and B F are thereupon<emph.end type="italics"/><lb/><arrow.to.target n="marg1230"></arrow.to.target><lb/><emph type="italics"/>Proportionals: Therefore the<emph.end type="italics"/> (b) <emph type="italics"/>Squares <lb/>H M, G L and F C and there Sides, &longs;hall al&longs;o be <lb/>Proportionals: And, therefore, as the (c) <lb/>Square H M is to the Square G L, &longs;o is the Line<emph.end type="italics"/><lb/><arrow.to.target n="marg1231"></arrow.to.target><lb/><emph type="italics"/>H M to the Line F C: But as H M is to F C, &longs;o <lb/>is O H to A F; and as the Square H M is to <lb/>the Square G L, &longs;o is the Line H B to B G; that <lb/>is, B G to B F: From whence it followeth that <lb/>O H is to A F, as B G to B F: And<emph.end type="italics"/> Permu­<lb/>tando, <emph type="italics"/>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/>are equall, by 35. of our fir&longs;t Book of<emph.end type="italics"/> Conicks: <emph type="italics"/>And therefore N O is double to G B: <lb/>Which was to be demon&longs;trated.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/037.jpg" pagenum="367"/><p type="margin"> <s><margin.target id="marg1228"></margin.target>(a) <emph type="italics"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></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"/>By 22. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1231"></margin.target>(c) <emph type="italics"/>By<emph.end type="italics"/> Cor. <emph type="italics"/>of 20. <lb/>of the &longs;ixth.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA IV.</s></p><p type="main"> <s>The &longs;ame things a&longs;&longs;umed again, and M Q being drawn from the <lb/>Point M unto the Diameter, let it touch the Section in the <lb/>Point M. </s> <s>I &longs;ay that H Q hath to Q O, the &longs;ame proportion <lb/>that G H hath to C N.</s></p><p type="main"> <s><emph type="italics"/>For make H R equall to G F; and &longs;eeing that<emph.end type="italics"/><lb/><figure id="id.073.01.037.1.jpg" xlink:href="073/01/037/1.jpg"/><lb/><emph type="italics"/>the Triangles A F C and O P N are alike, and <lb/>P N equall to F C, we might in like manner de­<lb/>mon&longs;trate P O and F A to be equall to each other: <lb/>Wherefore P O &longs;hall be double to F B: But H O <lb/>is double to G B: Therefore the Remainder P H <lb/>is al&longs;o double to the Remainder F G; that is, to <lb/>R H: And therefore is followeth that P R, R H <lb/>and F G are equall to one another; as al&longs;o that <lb/>R G and P F are equall: For P G is common to <lb/>both R P and G F. </s> <s>Since therefore, that H B is <lb/>to B G, as G B is to B F, by Conver&longs;ion of Pro­<lb/>portion, B H &longs;hall be to H G, as B G is to G F: <lb/>But Q H is to H B, as H O to B G. </s> <s>For by 35 <lb/>of our fir&longs;t Book of<emph.end type="italics"/> Conicks, <emph type="italics"/>in regard that Q <lb/>M toucheth the Section in the Point M, H B and <lb/>B Q &longs;hall be equall, and Q H double to H B: <lb/>Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>Q H &longs;hall be to H G, as <lb/>H O to G F; that is, to H R: and,<emph.end type="italics"/> Permu­<lb/>tando, <emph type="italics"/>Q H &longs;hall be to H O, as H G to H R: again, by Conver&longs;ion, H Q &longs;hall be to Q <lb/>O, as H G to G R; that is, to P F; and, by the &longs;ame rea&longs;on, to C N: Whichwas to be de­<lb/>mon&longs;trated.<emph.end type="italics"/></s></p><p type="main"> <s>The&longs;e things therefore being explained, we come now to that <lb/>which was propounded. </s> <s>I &longs;ay, therefore, fir&longs;t that <emph type="italics"/>N C<emph.end type="italics"/> hath <lb/>to C K the &longs;ame proportion that H G hath to G B.</s></p><p type="main"> <s><emph type="italics"/>For &longs;ince that H Q is to Q O, as H G to C N<emph.end type="italics"/>; <lb/><figure id="id.073.01.037.2.jpg" xlink:href="073/01/037/2.jpg"/><lb/><emph type="italics"/>that is, to A O, equall to the &longs;aid C N: The Re­<lb/>mainder G Q &longs;hall be to the Remainder Q A, as <lb/>H Q to Q O: and, for the &longs;ame cau&longs;e, the Lines <lb/>A C and G L prolonged, by the things that wee <lb/>have above demonstrated, &longs;hall inter&longs;ect or meet <lb/>in the Line Q M. Again, G Q is to Q A, <lb/>as H Q to Q O: that is, as H G to F P; as<emph.end type="italics"/><lb/><arrow.to.target n="marg1232"></arrow.to.target><lb/>(a) <emph type="italics"/>was bnt now demonstrated, But unto<emph.end type="italics"/> (b) <emph type="italics"/>G<emph.end type="italics"/><lb/><arrow.to.target n="marg1233"></arrow.to.target><lb/><emph type="italics"/>Q two Lines taken together, Q B that is H B, and <lb/>B G are equall: and to Q A H F is equall; for <lb/>if from the equall Magnitudes H B and B Q there <lb/>be taken the equall Magnitudes F B and B A, the <lb/>Re mainder &longs;hall be equall; Therefore taking H <lb/>G from the two Lines H B and B G, there &longs;hall re­<lb/>main a Magnitude double to B G; that is, O H: <lb/>and P F taken from F H, the Remainder is H P: <lb/>Wherefore<emph.end type="italics"/> (c) <emph type="italics"/>O H is to H P, as G Q to Q A:<emph.end type="italics"/><lb/><arrow.to.target n="marg1234"></arrow.to.target><lb/><emph type="italics"/>But as G Q is to Q A, &longs;o is H Q to Q O;<emph.end type="italics"/> <pb xlink:href="073/01/038.jpg" pagenum="368"/><arrow.to.target n="marg1235"></arrow.to.target><lb/><emph type="italics"/>that is, H G to N C: and as<emph.end type="italics"/> (d) <emph type="italics"/>O H is to H P, &longs;o is G B to C K; For O H is double <lb/>to G B, and H P al&longs;o double to G F; that is, to C K; Therefore H G hath the &longs;ame propor­<lb/>tion to N C, that G B hath to C K: And<emph.end type="italics"/> Permutando, <emph type="italics"/>N C hath to C K the &longs;ame proportion <lb/>that H G hath to G B.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1232"></margin.target>(a) <emph type="italics"/>By<emph.end type="italics"/> 2. Lemma.</s></p><p type="margin"> <s><margin.target id="marg1233"></margin.target>(b) <emph type="italics"/>By<emph.end type="italics"/> 4. Lemma.</s></p><p type="margin"> <s><margin.target id="marg1234"></margin.target>(b) <emph type="italics"/>By 19. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1235"></margin.target>(d) <emph type="italics"/>By 15. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>Then take &longs;ome other Point at plea&longs;ure in the Section, which <lb/>let be S: and thorow S draw two Lines, the one S T paral­<lb/>lel to D B, and cutting the Diameter in the Point T; the <lb/>other S V parallel to A C, and cutting C E in V. </s> <s>I &longs;ay <lb/>that V C hath greater proportion to C K, than T G hath <lb/>to G B.</s></p><p type="main"> <s><emph type="italics"/>For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the <lb/>Diameter parallel to B D: G T &longs;hall be le&longs;&longs;e than G Y, in regard that V S is leße than V X: <lb/>And, by the fir&longs;t Lemma, Y G &longs;hall be to V C, as H G to N C; that is, as G B to C K, which <lb/>was demon&longs;trated but now: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>Y G &longs;hall be to G B, as V C to C K: But <lb/>T G, for that it is le&longs;&longs;e than Y G, hath le&longs;&longs;e proportion to G B, than Y G hath to the &longs;ame; <lb/>Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de­<lb/>mon&longs;trated. </s> <s>Therefore a Po&longs;ition given G K, there &longs;hall be in the Section one only Point, to <lb/>wit M, from which two Lines M E H and M N O being drawn, N C &longs;hall have the &longs;ame pro­<lb/>portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall­<lb/>eth betwixt A C, and the Line parallel unto it &longs;hall alwayes have greater proportion to C K, <lb/>than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there­<lb/>fore, is manife&longs;t which was affirmed by<emph.end type="italics"/> Archimedes, <emph type="italics"/>to wit, that the Line P I hath unto P H, <lb/>either the &longs;ame proportion that N<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>hath to<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>O, or greater.<emph.end type="italics"/><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 le&longs;&longs;e than double.] <lb/><emph type="italics"/>If le&longs;&longs;e than double, let P T be double to T I: The Centre of Gravity of that part of the <lb/>Portion that is within the Liquid &longs;hall be the<emph.end type="italics"/><lb/><figure id="id.073.01.038.1.jpg" xlink:href="073/01/038/1.jpg"/><lb/><emph type="italics"/>Point T: But if P H be double to H I, H &longs;hall <lb/>be the Centre of Gravity; And draw H F, and <lb/>prolong it unto the Centre of that part of the Por­<lb/>tion which is above the Liquid, namely, unto G, <lb/>and the re&longs;t is demon&longs;trated as before. </s> <s>And the <lb/>&longs;ame is to be under&longs;tood in the Propo&longs;ition that <lb/>followeth.<emph.end type="italics"/></s></p><p type="main"> <s>The Solid A P O L, therefore, <lb/>&longs;hall turn about, and its Ba&longs;e &longs;hall <lb/>not in the lea&longs;t touch the Surface <lb/>of the Liquid.] <emph type="italics"/>In<emph.end type="italics"/> Tartaglia's <emph type="italics"/>Tran&longs;lation it is rendered<emph.end type="italics"/> ut Ba&longs;is ip&longs;ius non tangent <lb/>&longs;uperficiem humidi &longs;ecundum unum &longs;ignum; <emph type="italics"/>but we have cho&longs;en to read<emph.end type="italics"/> ut Ba&longs;is ip&longs;ius <lb/>nullo modo humidi &longs;uperficiem contingent, <emph type="italics"/>both here, and in the following Propo&longs;itions, <lb/>becau&longs;e the Greekes frequently u&longs;e<emph.end type="italics"/> <foreign lang="greek">w(de\ei)=s, w(de\<gap/></foreign> <emph type="italics"/>pro<emph.end type="italics"/> <foreign lang="greek">w)dei\s<gap/> & ou)di\n</foreign>: <emph type="italics"/>&longs;o that<emph.end type="italics"/> <foreign lang="greek">ou)de)/sinoudei/s,</foreign> nullus <lb/>e&longs;t; <foreign lang="greek">ou)d<gap/>u(p)e(ro\s</foreign> à nullo, <emph type="italics"/>and &longs;o of others of the like nature.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/039.jpg" pagenum="369"/><p type="head"> <s>PROP. VII. THE OR. VII.</s></p><p type="main"> <s><emph type="italics"/>The Right Portion of a Rightangled Conoid lighter <lb/>than the Liquid, when it &longs;hall have its Axis greater <lb/>than Se&longs;quialter of the Semi-parameter, but le&longs;&longs;e <lb/>than to be unto the &longs;aid Semi-parameter in proportion <lb/>as fi&longs;teen to fower, being demitted into the Liquid &longs;o <lb/>as that its Ba&longs;e be wholly within the Liquid, it &longs;hall <lb/>never &longs;tand &longs;o as that its Ba&longs;e do touch the Surface <lb/>of the Liquid, but &longs;o, that it be wholly within the <lb/>Liquid, and &longs;hall not in the lea&longs;t touch its Surface.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be a Portion as hath been &longs;aid; and let it be de­<lb/>mitted into the Liquid, as we have &longs;uppo&longs;ed, &longs;o as that its <lb/>Ba&longs;e do touch the Surface in one Point only: It is to be de­<lb/>mon&longs;trated that the &longs;ame &longs;hall not &longs;o <lb/><figure id="id.073.01.039.1.jpg" xlink:href="073/01/039/1.jpg"/><lb/>continue, but &longs;hall turn about in <lb/>&longs;uch manner as that its Ba&longs;e do in no <lb/>wi&longs;e touch the Surface of the Liquid. <lb/></s> <s>For let it be cut thorow its Axis by <lb/>a Plane erect upon the Liquids Sur­<lb/>face: and let the Section be A P O L, <lb/>the Section of a Rightangled <lb/>Cone; the Section of the Liquids <lb/>Surface S L; and the Axis of the <lb/>Portion and Diameter of the Section P F: and let P F be cut in <lb/>R, &longs;o, as that R P may be double to R F, and in <foreign lang="greek">w</foreign> &longs;o as that P F <lb/>may be to R <foreign lang="greek">w</foreign> as fifteen to fower: and draw <foreign lang="greek">w</foreign> K at Right Angles </s></p><p type="main"> <s><arrow.to.target n="marg1237"></arrow.to.target><lb/>to P F: <emph type="italics"/>(a)<emph.end type="italics"/> R <foreign lang="greek">w</foreign> &longs;hall be le&longs;&longs;e than the Semi-parameter. </s> <s>There­<lb/>fore let R H be &longs;uppo&longs;ed equall to the Semi-parameter: and <lb/>draw C O touching the Section in O and parallel unto S L; and <lb/>let N O be parallel unto P F; and fir&longs;t let N O cut K <foreign lang="greek">w</foreign> in the Point <lb/>I, as in the former Schemes: It &longs;hall be demon&longs;trated that N O is <lb/>to O I either &longs;e&longs;quialter, or greater than &longs;e&longs;quialter. </s> <s>Let O I be <lb/>le&longs;&longs;e than double to I N; and let O B be double to B N: and let <lb/>them be di&longs;po&longs;ed like as before. </s> <s>We might likewi&longs;e demon&longs;trate <lb/>that if a Line be drawn thorow R and T it will make Right Angles <lb/>with the Line C O, and with the Surface of the Liquid: Where­<lb/>fore Lines being drawn from the Points B and G parallels unto <lb/>R T, they al&longs;o &longs;hall be Perpendiculars to the Surface of the Liquid: <lb/>The Portion therefore which is above the Liquid &longs;hall move down­ <pb xlink:href="073/01/040.jpg" pagenum="370"/><figure id="id.073.01.040.1.jpg" xlink:href="073/01/040/1.jpg"/><lb/>wards according to that &longs;ame Perpendicular <lb/>which pa&longs;&longs;eth thorow B; and the Portion <lb/>which is within the Liquid &longs;hall move up­<lb/>wards acording to that pa&longs;&longs;ing thorow G: <lb/>From whence it is manife&longs;t that the Solid <lb/>&longs;hall turn about in &longs;uch manner, as that <lb/>its Ba&longs;e &longs;hall in no wi&longs;e touch the Surface <lb/>of the Liquid; for that now when it touch­<lb/>eth but in one Point only, it moveth down­<lb/>wards on the part towards L. </s> <s>And though <lb/>N O &longs;hould not cut <foreign lang="greek">w</foreign> K, yet &longs;hall the &longs;ame hold true.</s></p><p type="margin"> <s><margin.target id="marg1237"></margin.target>(a) <emph type="italics"/>By 10 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="head"> <s>PROP. VIII. THE OR. VIII.</s></p><p type="main"> <s><emph type="italics"/>The Right Portion of a Rightangled Conoid, when it <lb/>&longs;hall have its Axis greater than &longs;e&longs;quialter of the Se­<lb/>mi-parameter, but le&longs;&longs;e than to be unto the &longs;aid Semi­<lb/>parameter, in proportion as fifteen to fower, if it <lb/>have a le&longs;&longs;er proportion in Gravity to the Liquid, than <lb/>the Square made of the Exce&longs;&longs;e by which the Axis is <lb/>greater than Se&longs;quialter of the Semi-parameter hath <lb/>to the Square made of the Axis, being demitted into <lb/>the Liquid, &longs;o as that its Ba&longs;e touch not the Liquid, <lb/>it &longs;hall neither return to Perpendicularity, nor conti­<lb/>nue inclined, &longs;ave only when the Axis makes an <lb/>Angle with the Surface of the Liquid, equall to that <lb/>which we &longs;hall pre&longs;ently &longs;peak of.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be a Portion as hath been &longs;aid; and let B D be equall <lb/>to the Axis: and let B K be double to K D; and R K equall <lb/><arrow.to.target n="marg1238"></arrow.to.target><lb/>to the Semi-parameter: and let C B be Se&longs;quialter of B R: <lb/>C D &longs;hall be al&longs;o Sefquialter of K R. </s> <s>And as the Portion is to the <lb/>Liquid in Gravity, &longs;o let the Square F Q be to the Square D B; <lb/>and let F be double to Q: It is manife&longs;t, therefore, that F Q hath <lb/>to D B, le&longs;s proportion than C B hath to B D; For C B is the <lb/>Exce&longs;s by which the Axis is greater than Se&longs;quialter of the Semi­<lb/><arrow.to.target n="marg1239"></arrow.to.target><lb/>parameter: And, therefore, F Q is le&longs;s than B C; and, for the <lb/><arrow.to.target n="marg1240"></arrow.to.target><lb/>&longs;ame rea&longs;on, F is le&longs;s than B R. </s> <s>Let R <foreign lang="greek">y</foreign> be equall to F; and draw <lb/><foreign lang="greek">y</foreign> E perpendicular to B D; which let be in power or contence the <lb/>half of that which the Lines K R and <foreign lang="greek">y</foreign> B containeth; and <lb/>draw a Line from B to E: It is to be demon&longs;trated, that the <pb xlink:href="073/01/041.jpg" pagenum="371"/>Portion demitted into the Liquid, like as hath been &longs;aid, &longs;hall &longs;tand <lb/>enclined &longs;o as that its Axis do make an Angle with the Surface of <lb/>the Liquid equall unto the Angle E B <foreign lang="greek">*y.</foreign> For demit any Portion <lb/>into the Liquid &longs;o as that its Ba&longs;e <lb/><figure id="id.073.01.041.1.jpg" xlink:href="073/01/041/1.jpg"/><lb/>touch not the Liquids Surface; <lb/>and, if it can be done, let the <lb/>Axis not make an Angle with the <lb/>Liquids Surface equall to the <lb/>Angle E B <foreign lang="greek">*y</foreign>; but fir&longs;t, let it be <lb/>greater: and the Portion being <lb/>cut thorow the Axis by a Plane e­<lb/>rect unto [<emph type="italics"/>or upon<emph.end type="italics"/>] the Surface of <lb/>the Liquid, let the Section be A P <lb/>O L the Section of a Rightangled <lb/>Cone; the Section of the Surface of the Liquid X S; and let the <lb/>Axis of the Portion and Diameter of the Section be N O: and <lb/>draw P Y parallel to X S, and touching the Section A P O L in P; <lb/>and P M parallel to N O; and P I perpendicular to N O: and <lb/>moreover, let B R be equall to O <foreign lang="greek">w,</foreign> and R K to T <foreign lang="greek">w;</foreign> and let <foreign lang="greek">w</foreign> H <lb/>be perpendicular to the Axis. </s> <s>Now becau&longs;e it hath been &longs;uppo&longs;ed <lb/><arrow.to.target n="marg1241"></arrow.to.target><lb/>that the Axis of the Portion doth make an Angle with the Surface <lb/>of the Liquid greater than the Angle B, the Angle P Y I &longs;hall be <lb/>greater than the Angle B: Therefore the Square P I hath greater <lb/><arrow.to.target n="marg1242"></arrow.to.target><lb/>proportion to the Square Y I, than the Square E <foreign lang="greek">*y</foreign> hath to the <lb/>Square <foreign lang="greek">*y</foreign> B: But as the Square P I is to the Square Y I, &longs;o is the <lb/><arrow.to.target n="marg1243"></arrow.to.target><lb/>Line K R unto the Line I Y; and as the Square E <foreign lang="greek">*y</foreign> is to the Square <lb/><arrow.to.target n="marg1244"></arrow.to.target><lb/><foreign lang="greek">*y</foreign> B, &longs;o is half of the Line K R unto the Line <foreign lang="greek">*y</foreign> B: Wherefore <lb/><emph type="italics"/>(a)<emph.end type="italics"/> K R hath greater proportion to I Y, than the half of K R hath <lb/><arrow.to.target n="marg1245"></arrow.to.target><lb/>to <foreign lang="greek">*y</foreign> B: And, con&longs;equently, I Y isle&longs;&longs;e than the double of <foreign lang="greek">*y</foreign> B, <lb/>and is the double of O I: Therefore O I is le&longs;&longs;e than <foreign lang="greek">*y</foreign> B; and I <foreign lang="greek">w</foreign><lb/><arrow.to.target n="marg1246"></arrow.to.target><lb/>greater than <foreign lang="greek">*y</foreign> R: but <foreign lang="greek">*y</foreign> R is equall to F: Therefore I <foreign lang="greek">w</foreign> is greater <lb/><arrow.to.target n="marg1247"></arrow.to.target><lb/>than F. </s> <s>And becau&longs;e that the Portion is &longs;uppo&longs;ed to be in Gra­<lb/>vity unto the Liquid, as the Square F Q is to the Square B D; and <lb/>&longs;ince that as the Portion is to the Liquid in Gravity, &longs;o is the part <lb/>thereof &longs;ubmerged unto the whole Portion; and in regard that as <lb/>the part thereof &longs;ubmerged is to the whole, &longs;o is the Square P M to <lb/>the Square O N; It followeth, that the Square P M is to the Square <lb/>N O, as the Square F Q is to the Square B D: And therefore F <lb/><arrow.to.target n="marg1248"></arrow.to.target><lb/>Q is equall to P M: But it hath been demon&longs;trated that P H is <lb/><arrow.to.target n="marg1249"></arrow.to.target><lb/>greater than F: It is manife&longs;t, therefore, that P M is le&longs;&longs;e than <lb/>&longs;e&longs;quialter of P H: And con&longs;equently that P H is greater than <lb/>the double of H M. </s> <s>Let P Z be double to Z M: T &longs;hall be the Cen­<lb/>tre of Gravity of the whole Solid; the Centre of that part of it <lb/>which is within the Liquid, the Point Z; and of the remaining <lb/><arrow.to.target n="marg1250"></arrow.to.target><lb/>part the Centre &longs;hall be in the Line Z T prolonged unto G. </s> <s>In <pb xlink:href="073/01/042.jpg" pagenum="372"/>the &longs;ame manner we might demon­<lb/><figure id="id.073.01.042.1.jpg" xlink:href="073/01/042/1.jpg"/><lb/>&longs;trate the <emph type="italics"/>L<emph.end type="italics"/>ine T H to be perpendi­<lb/>cular unto the Surface of the Liquid: <lb/>and that the Portion demerged with­<lb/>in the <emph type="italics"/>L<emph.end type="italics"/>iquid moveth or a&longs;cend­<lb/>eth out of the <emph type="italics"/>L<emph.end type="italics"/>iquid according to <lb/>the Perpendicular that &longs;hall be <lb/>drawn thorow Z unto the Surface <lb/>of the Liquid; and that the part <lb/>that is above the Liquid de&longs;cendeth <lb/>into the Liquid according to that <lb/>drawn thorow G: therefore the Portion will not continue &longs;o inclined <lb/>as was &longs;uppo&longs;ed: But neither &longs;hall it return to Rectitude or Per­<lb/>pendicularity; For that of the Perpendiculars drawn thorow Z and <lb/>G, that pa&longs;&longs;ing thorow Z doth fall on tho&longs;e parts which are to­<lb/>wards L; and that that pa&longs;&longs;eth thorow G on tho&longs;e towards A: <lb/>Wherefore it followeth that the Centre Z do move upwards, <lb/>and G downwards: Therefore the parts of the whole Solid which <lb/>are towards A &longs;hall move downwards, and tho&longs;e towards L up­<lb/>wards. </s> <s>Again let the Propo&longs;ition run in other termes; and let <lb/>the Axis of the Portion make an Angle with the Surface of the <lb/><arrow.to.target n="marg1251"></arrow.to.target><lb/>Liquid le&longs;&longs;e than that which is at B. </s> <s>Therefore the Square P I <lb/>hath le&longs;&longs;er Proportion unto the Square <lb/><figure id="id.073.01.042.2.jpg" xlink:href="073/01/042/2.jpg"/><lb/>I Y, than the Square E <foreign lang="greek">*y</foreign> hath to the <lb/>Square <foreign lang="greek">*y</foreign> B: Wherefore K R hath <lb/>le&longs;&longs;er proportion to I Y, than the half <lb/>of K R hath to <foreign lang="greek">*y</foreign> B: And, for the <lb/>&longs;ame rea&longs;on, I Y is greater than dou­<lb/>ble of <foreign lang="greek">*y</foreign> B: but it is double of O I: <lb/>Therefore O I &longs;hall be greater than <lb/><foreign lang="greek">*y</foreign> B: But the Totall O <foreign lang="greek">w</foreign> is equall <lb/>to R B, and the Remainder <foreign lang="greek">w</foreign> I le&longs;&longs;e <lb/>than <foreign lang="greek">y</foreign> R: Wherefore P H &longs;hall al&longs;o <lb/>be le&longs;&longs;e than F. And, in regard that <lb/>M P is equall to F Q, it is manife&longs;t that P M is greater than &longs;e&longs;qui­<lb/>alter of P H; and that P H is le&longs;&longs;e than double of <emph type="italics"/>H<emph.end type="italics"/> M. <emph type="italics"/>L<emph.end type="italics"/>et P Z <lb/>be double to Z M. </s> <s>The Centre of Gravity of the whole Solid &longs;hall <lb/>again be T; that of the part which is within the Liquid Z; and <lb/>drawing a Line from Z to T, the Centre of Gravity of that which <lb/>is above the Liquid &longs;hall be found in that Line portracted, that is <lb/>in G: Therefore, Perpendiculars being drawn thorow Z and G <lb/><arrow.to.target n="marg1252"></arrow.to.target><lb/>unto the Surface of the Liquid that are parallel to T H, it followeth <lb/>that the &longs;aid Portion &longs;hall not &longs;tay, but &longs;hall turn about till <lb/>that its Axis do make an Angle with the Waters Surface greater than <lb/>that which it now maketh. </s> <s>And becau&longs;e that when before we <pb xlink:href="073/01/043.jpg" pagenum="373"/>did &longs;uppo&longs;e that it made an Angle greater than the Angle B, the <lb/>Poriton did not re&longs;t then neither; It is manife&longs;t that it &longs;hall &longs;tay <lb/><arrow.to.target n="marg1253"></arrow.to.target><lb/>or re&longs;t when it &longs;hall make an Angle eqnall to B. </s> <s>For &longs;o &longs;hall I O <lb/>be equall to <foreign lang="greek">*y</foreign> <emph type="italics"/>B<emph.end type="italics"/>; and <foreign lang="greek">w</foreign> I equall to <lb/><figure id="id.073.01.043.1.jpg" xlink:href="073/01/043/1.jpg"/><lb/><foreign lang="greek">*y</foreign> R; and P H equall to F: There­<lb/>fore <emph type="italics"/>M P<emph.end type="italics"/> &longs;hall be &longs;e&longs;quialter of <emph type="italics"/>P H,<emph.end type="italics"/><lb/>and <emph type="italics"/>P H<emph.end type="italics"/> double of H M: And there­<lb/>fore &longs;ince H is the Centre of Gravity <lb/>of that part of it which is within the <lb/>Liquid, it &longs;hall move upwards along <lb/>the &longs;ame <emph type="italics"/>P<emph.end type="italics"/>erpendicular according to <lb/>which the whole <emph type="italics"/>P<emph.end type="italics"/>ortion moveth; <lb/>and along the &longs;ame al&longs;o &longs;hall the part <lb/>which is above move downwards: <lb/>The <emph type="italics"/>P<emph.end type="italics"/>ortion therefore &longs;hall re&longs;t; for­<lb/>a&longs;much as the parts are not repul&longs;ed 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"/>By 13. of the <lb/>fifth.<emph.end type="italics"/></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"/>C B<emph.end type="italics"/> be &longs;e&longs;quialter of <emph type="italics"/>B R<emph.end type="italics"/>: C D &longs;hall al&longs;o be &longs;e&longs;quialter <lb/><arrow.to.target n="marg1254"></arrow.to.target><lb/>of K R.] <emph type="italics"/>In the Tran&longs;lation it is read thus:<emph.end type="italics"/> Sit autem & CB quidem hemeolia <lb/>ip&longs;ius B R: C D autem ip&longs;ius K R. <emph type="italics"/>But we at the reading of this pa&longs;&longs;age have thought <lb/>fit thus to correctit; for it is not &longs;uppo&longs;ed &longs;o to be, but from the things &longs;uppo&longs;ed is proved to <lb/>be &longs;o. </s> <s>For if B<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>be double of<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>D, D B &longs;hall be &longs;e&longs;quialter of B<emph.end type="italics"/> <foreign lang="greek">y.</foreign> <emph type="italics"/>And becau&longs;e E B is <lb/>&longs;e&longs;quialter of B R, it followeth that the<emph.end type="italics"/> (a) <emph type="italics"/>Remainder C D is &longs;e&longs;quialter of<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>R; that is, of<emph.end type="italics"/><lb/><arrow.to.target n="marg1255"></arrow.to.target><lb/><emph type="italics"/>the Semi-parameter: Wherefore B C &longs;hall be the Exce&longs;&longs;e by which the Axis is greater than <lb/>&longs;e&longs;quialter of the Semi-parameter.<emph.end type="italics"/></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"/>By 19. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>And therefore F Q is le&longs;&longs;e than <emph type="italics"/>B C.] For in regard that the Portion hath<emph.end type="italics"/><lb/><arrow.to.target n="marg1256"></arrow.to.target><lb/><emph type="italics"/>the &longs;ame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B; <lb/>and hath le&longs;&longs;er proportion than the Square made of the Exce&longs;&longs;e by which the Axis <lb/>is greater than Se&longs;quialter of the Semi parameter, hath to the Square made of the Axis; that <lb/>is, leßer than the Square C B hath to the Square B D; for the Line B D was &longs;uppo&longs;ed to be <lb/>equall unto the Axis: Therefore the Square F Q &longs;hall have to the Square D B le&longs;&longs;er proporti­<lb/>on than the Sqnare C B to the &longs;ame Square B D: And therefore the Square<emph.end type="italics"/> (b) <emph type="italics"/>F Q &longs;hall be<emph.end type="italics"/><lb/><arrow.to.target n="marg1257"></arrow.to.target><lb/><emph type="italics"/>leße than the Square C B: And, for that rea&longs;on, the Line F Q &longs;hall be leße than B C.<emph.end type="italics"/></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"/>By 8 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>And, for the &longs;ame rea&longs;on, F is le&longs;&longs;e than <emph type="italics"/>B R.] For C B being &longs;e&longs;qui-<emph.end type="italics"/><lb/><arrow.to.target n="marg1258"></arrow.to.target><lb/><emph type="italics"/>alter of B R, and F Q &longs;e&longs;quialter of F<emph.end type="italics"/>: (c) F <emph type="italics"/>Q &longs;hall be likewi&longs;e le&longs;&longs;e than B C; and F<emph.end type="italics"/><lb/><arrow.to.target n="marg1259"></arrow.to.target><lb/><emph type="italics"/>leße than B R.<emph.end type="italics"/></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"/>By 14 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>Now becau&longs;e it hath been &longs;uppo&longs;ed that the Axis of the <emph type="italics"/>P<emph.end type="italics"/>ortion <lb/><arrow.to.target n="marg1260"></arrow.to.target><lb/>doth make an Angle with the Surface of the Liquid greater than <lb/>the Angle <emph type="italics"/>B,<emph.end type="italics"/> the Angle <emph type="italics"/>P Y I<emph.end type="italics"/> &longs;hall be greater than the Angle <emph type="italics"/>B.] <lb/>For the Line P Y being parallel to the Surface of the Liquid, that is, to XS<emph.end type="italics"/>; (d) <emph type="italics"/>the Angle<emph.end type="italics"/><lb/><arrow.to.target n="marg1261"></arrow.to.target><lb/><emph type="italics"/>P Y I &longs;hall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the <lb/>Line X S: And therefore &longs;hall be greater than the Angle B.<emph.end type="italics"/></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"/>By 29 of the <lb/>fir&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s>Therefore the Square <emph type="italics"/>P I<emph.end type="italics"/> hath greater proportion to the Square <lb/><arrow.to.target n="marg1262"></arrow.to.target><lb/>Y I, than the Square E <foreign lang="greek">*y</foreign> hath to the Square <foreign lang="greek">*y</foreign> <emph type="italics"/>B] Let the Triangles P I Y <lb/>and E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B, be de&longs;cribed apart: And &longs;eeing that the Angle P Y I is greater <lb/>than the Angle E B<emph.end type="italics"/> <foreign lang="greek">y,</foreign> <emph type="italics"/>unto the Line I Y, and at the Point Y a&longs;&longs;igned in<emph.end type="italics"/><lb/><figure id="id.073.01.043.2.jpg" xlink:href="073/01/043/2.jpg"/><lb/><emph type="italics"/>the &longs;ame, make the Angle V Y I equall to the Angle E B<emph.end type="italics"/> <foreign lang="greek">y</foreign>; <emph type="italics"/>But <lb/>the Right Angle at I, is equall unto the Right Angle at<emph.end type="italics"/> <foreign lang="greek">y;</foreign> <emph type="italics"/>therefore the<emph.end type="italics"/> <pb xlink:href="073/01/044.jpg" pagenum="374"/><emph type="italics"/>Remaining Angle Y V I is equall to the Remaining Angle B E<emph.end type="italics"/> <foreign lang="greek">y.</foreign> <emph type="italics"/>And therefore the<emph.end type="italics"/><lb/><arrow.to.target n="marg1263"></arrow.to.target><lb/>(e) <emph type="italics"/>Line V I hath to the Line I Y the &longs;ame proportion that the Line E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>hath to<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B: But <lb/>the<emph.end type="italics"/> (f) <emph type="italics"/>Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-<emph.end type="italics"/><lb/><arrow.to.target n="marg1264"></arrow.to.target><lb/><emph type="italics"/>to the &longs;ame: Therefore<emph.end type="italics"/> (g) <emph type="italics"/>T I &longs;hall have greater proportion unto I Y, than E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>hath to<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B: <lb/>And, by the &longs;ame rea&longs;on, the Square T I &longs;hall have greater proportion to the Square I Y, than<emph.end type="italics"/><lb/><arrow.to.target n="marg1265"></arrow.to.target><lb/><emph type="italics"/>the Square E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>hath to the Square<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B.<emph.end type="italics"/><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"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1264"></margin.target>(f) <emph type="italics"/>By 8. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1265"></margin.target>(g) <emph type="italics"/>By 13 of the <lb/>fifth.<emph.end type="italics"/></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, &longs;o is the Line K R unto <lb/>the Line I Y] <emph type="italics"/>For by 11. of the fir&longs;t of our<emph.end type="italics"/> Conicks, <emph type="italics"/>the Square P I is equall <lb/>to the Rectangle contained under the Line I O, and under the Parameter; which <lb/>we &longs;uppo&longs;ed to be eqnall to the Semi-parameter; that is, the double of K R<emph.end type="italics"/>: </s></p><p type="main"> <s><arrow.to.target n="marg1267"></arrow.to.target><lb/><emph type="italics"/>But I Y is double of I O, by 33 of the &longs;ame: And, therefore, the<emph.end type="italics"/> (h) <emph type="italics"/>Rectangle made of K R <lb/>and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;<emph.end type="italics"/><lb/><arrow.to.target n="marg1268"></arrow.to.target><lb/><emph type="italics"/>that is, to the Square P I: But as the<emph.end type="italics"/> (i) <emph type="italics"/>Rectangle compounded of K R and I Y is to the <lb/>Square I Y, &longs;o is the Line K R unto the Line I Y: Therefore the Line K R &longs;hall have unto I <lb/>Y, the &longs;ame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I <lb/>hath to the Square I Y.<emph.end type="italics"/><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"/>By 26. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1268"></margin.target>(i) <emph type="italics"/>By<emph.end type="italics"/> Lem. </s> <s>22 <emph type="italics"/>of <lb/>the tenth.<emph.end type="italics"/></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="greek">*y</foreign> is to the Square <foreign lang="greek">*y</foreign> B, &longs;o is half of the <lb/>Line K R unto the Line <foreign lang="greek">y</foreign> B.] <emph type="italics"/>For the Square E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>having been &longs;uppo&longs;ed equall <lb/>to half the Rectangle contained under the Line K R and<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B; that is, to that contained under <lb/>the half of K R and the Line<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B; and &longs;eeing that as the<emph.end type="italics"/> (k) <emph type="italics"/>Rectangle made of half K R<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1270"></arrow.to.target><lb/><emph type="italics"/>and of B<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>is to the Square<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B, &longs;o is half K R unto the Line<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B; the half of K R &longs;hall have <lb/>the &longs;ame proportion to<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B, as the Square E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>hath to the Square<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B.<emph.end type="italics"/><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"/>By<emph.end type="italics"/> Lem. </s> <s>22 <emph type="italics"/>of <lb/>the tenth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1271"></margin.target>H</s></p><p type="main"> <s>And, con&longs;equently, I Y is le&longs;&longs;e than the double of <foreign lang="greek">y</foreign> B.] <lb/><emph type="italics"/>For, as half K R is to<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B, &longs;o is K R to another Line: it &longs;hall be<emph.end type="italics"/> (1) <emph type="italics"/>greater than I Y; that<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1272"></arrow.to.target><lb/><emph type="italics"/>is, than that to which K R hath le&longs;&longs;er proportion; and it &longs;hall be double of<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B: Therefore <lb/>I Y is le&longs;&longs;e than the double of<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B.<emph.end type="italics"/><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"/>By 10 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1273"></margin.target>K</s></p><p type="main"> <s>And I <foreign lang="greek">w</foreign> greater than <foreign lang="greek">y</foreign> R.] <emph type="italics"/>For O having been &longs;uppo&longs;ed equall to B R, <lb/>if from B R,<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B be taken, and from O<emph.end type="italics"/> <foreign lang="greek">w,</foreign> <emph type="italics"/>O I, which is le&longs;&longs;er than B, be taken; the <lb/>Remainder I<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>&longs;hall be greater than the Remainder<emph.end type="italics"/> <foreign lang="greek">*y</foreign> <emph type="italics"/>R.<emph.end type="italics"/></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"/>By the fourteenth of the fifth of<emph.end type="italics"/><lb/>Euclids <emph type="italics"/>Elements: For the Line O N is equall to B D.<emph.end type="italics"/></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 demon&longs;trated that P H is greater than F.] <lb/><emph type="italics"/>For it was demon&longs;trated that I<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>is greater than F: And P H is equall to I<emph.end type="italics"/> <foreign lang="greek">w.</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 &longs;ame manner we might demon&longs;trate the Line T H <lb/>to be Perpendicular unto the Surface of the Liquid.] <emph type="italics"/>For T<emph.end type="italics"/> <foreign lang="greek">a</foreign> <emph type="italics"/>is equall <lb/>to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated, <lb/>the Line T H &longs;hall be drawn Perpendicular unto the Liquids Surface.<emph.end type="italics"/></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 le&longs;&longs;er proportion unto the <lb/>Square I Y, than the Square E <foreign lang="greek"><gap/></foreign> hath to the Square <foreign lang="greek">y</foreign> B.] <lb/><emph type="italics"/>The&longs;e, and other particulars of the like nature, that follow both in this and the following <lb/>Propo&longs;itions, &longs;hall be demon&longs;trated by us no otherwi&longs;e than we have done above.<emph.end type="italics"/></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/>the Surface of the Liquid, that are parallel to T H, it followeth <lb/>that the &longs;aid Portion &longs;hall not &longs;tay, but &longs;hall turn about till that its <lb/>Axis do make an Angle with the Waters Surface greater than that <lb/>which it now maketh.] <emph type="italics"/>For in that the Line drawn thorow G, doth fall perpendicu­<lb/>larly towards tho&longs;e parts which are next to L; but that thorow Z, towards tho&longs;e next to A; <lb/>It is nece&longs;&longs;ary that the Centre G do move downwards, and Z upwards: and, therefore, the <lb/>parts of the Solid next to L &longs;hall move downwards, and tho&longs;e towards A upwards, that the <lb/>Axis may makea greater Angle with the Surface of the Liquid.<emph.end type="italics"/></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 &longs;o &longs;hall I O be equall to <foreign lang="greek">y</foreign> B; and <foreign lang="greek">w</foreign> I equall to I R; and <lb/>P H equall to F.] <emph type="italics"/>This plainly appeareth in the third Figure, which is added by us.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/045.jpg" pagenum="375"/><p type="head"> <s>PROP. IX. THE OR. IX.</s></p><p type="main"> <s><emph type="italics"/>The Right Portion of a Rightangled Conoid, when it <lb/>&longs;hall have its Axis greater than Se&longs;quialter of the <lb/>Semi-parameter, but le&longs;&longs;er than to be unto the &longs;aid <lb/>Semi-parameter in proportion as fifteen to four, and <lb/>hath greater proportion in Gravity to the Liquid, than <lb/>the exce&longs;s by which the Square made of the Axis is <lb/>greater than the Square made of the Exce&longs;s, by which <lb/>the Axis is greater than Se&longs;quialter of the Semi­<lb/>parameter, hath to the Square made of the Axis, <lb/>being demitted into the Liquid, &longs;o as that its Ba&longs;e <lb/>be wholly within the Liquid, and being &longs;et inclining<lb/>it &longs;hall neither turn about, &longs;o as that its Axis &longs;tand <lb/>according to the Perpendicular, nor remain inclined, <lb/>&longs;ave only when the Axis makes an Angle with <lb/>the Surface of the Liquid, equall to that aßigned <lb/>as before.<emph.end type="italics"/></s></p><p type="main"> <s>Let there be a Portion as was &longs;aid; and &longs;uppo&longs;e D B equall to <lb/>the Axis of the <emph type="italics"/>P<emph.end type="italics"/>ortion: and let B K be double to K D; and <lb/>K R equall to the Semi-parameter: and C B Se&longs;quialter of <lb/>B R. </s> <s>And as the Portion is to the Liquid in Gravity, &longs;o let the Ex­<lb/>ce&longs;&longs;e by which the Square B D exceeds the Square F Q be to the <lb/>Square B D: and let F be double to Q: It is manife&longs;t, therefore, <lb/>that the Exce&longs;&longs;e by which the <lb/><figure id="id.073.01.045.1.jpg" xlink:href="073/01/045/1.jpg"/><lb/>Square B D is greater than the <lb/>Square B C hath le&longs;ser proportion <lb/>to the Square B D, than the Exce&longs;s <lb/>by which the Square B D is greater <lb/>than the Square F Q hath to the <lb/>Square B D; for B C is the Exce&longs;s <lb/>by which the Axis of the Portion is <lb/>greater than Se&longs;quialter of the <lb/>Semi-parameter: And, therefore, </s></p><p type="main"> <s><arrow.to.target n="marg1280"></arrow.to.target><lb/>the Square B D doth more exceed <lb/>the Square F Q, than doth the <lb/>Square B C: And, con&longs;equently, the Line F Q is le&longs;s than B C; <pb xlink:href="073/01/046.jpg" pagenum="376"/>and F le&longs;s than B R. </s> <s>Let R <foreign lang="greek">*y</foreign> be equall to F; and draw <foreign lang="greek">*y</foreign> E <lb/>perpendicular to B D; which let be in power the half of that <lb/>which the Lines K R and <foreign lang="greek">*y</foreign> B containeth; and draw a Line from <lb/>B to E: I &longs;ay that the Portion demitted into the Liquid, &longs;o as that <lb/>its Ba&longs;e be wholly within the Liquid, &longs;hall &longs;o &longs;tand, as that its Axis <lb/>do make an Angle with the Liquids Surface, equall to the Angle B. <lb/></s> <s>For let the <emph type="italics"/>P<emph.end type="italics"/>ortion be demitted into the Liquid, as hath been &longs;aid; <lb/>and let the Axis not make an Angle with the Liquids Surface, equall <lb/>to B, but fir&longs;t a greater: and the &longs;ame being cut thorow the Axis <lb/>by a Plane erect unto the Surface of the Liquid, let the Section of <lb/>the Portion be A P O L, the Section of a Rightangled Cone; the <lb/>Section of the Surface of the Liquid <foreign lang="greek">*g</foreign> I; and the Axis of the <lb/>Portion and Diameter of the Section N O; which let be cut in <lb/>the Points <foreign lang="greek">w</foreign> and T, as before: and draw Y P, parallelto <foreign lang="greek">*g</foreign> I, and <lb/>touching the Section in P, and MP parallel to N O, and P S perpen­<lb/>dicular to the Axis. </s> <s>And becau&longs;e now that the Axis of the Portion <lb/>maketh an <emph type="italics"/>A<emph.end type="italics"/>ngle with the Liquids Surface greater than the Angle <lb/>B, the Angle S Y P &longs;hall al&longs;o be greater than the Angle B: And, <lb/>therefore, the Square P S hath greater proportion to the Square <lb/><arrow.to.target n="marg1281"></arrow.to.target><lb/>S Y, than the Square <foreign lang="greek">*y</foreign> E hath to the Square <foreign lang="greek">*y</foreign> B: And, for that <lb/>cau&longs;e, K R hath greater proportion to S Y, than the half of K R <lb/>hath to <foreign lang="greek">*y</foreign> B: Therefore, S Y is le&longs;s than the double of <foreign lang="greek">*y</foreign> B; and <lb/><arrow.to.target n="marg1282"></arrow.to.target><lb/>S O le&longs;s than <foreign lang="greek">y</foreign> B: <emph type="italics"/>A<emph.end type="italics"/>nd, therefore, S <foreign lang="greek">w</foreign> is greater than R <foreign lang="greek">y</foreign>; and <lb/><arrow.to.target n="marg1283"></arrow.to.target><lb/>P H greater than F. <emph type="italics"/>A<emph.end type="italics"/>nd, becau&longs;e that the <emph type="italics"/>P<emph.end type="italics"/>ortion hath the <lb/>&longs;ame proportion in Gravity unto the Liquid, that the Exce&longs;s by <lb/>which the Square B D, is greater than the Square F Q, hath unto <lb/>the Square B D; and that as the Portion is in proportion to the <lb/>Liquid in Gravity, &longs;o is the part thereof &longs;ubmerged unto the whole <lb/>Portion; It followeth that the part &longs;ubmerged, hath the &longs;ame <lb/>proportion to the whole <emph type="italics"/>P<emph.end type="italics"/>ortion, that the Exce&longs;s by which the <lb/>Square B D is greater than the Square F Q hath unto the Square <lb/>B D: <emph type="italics"/>A<emph.end type="italics"/>nd, therefore, the whole <emph type="italics"/>P<emph.end type="italics"/>ortion &longs;hall have the &longs;ame propor­<lb/><arrow.to.target n="marg1284"></arrow.to.target><lb/>tion to that part which is above the <lb/><figure id="id.073.01.046.1.jpg" xlink:href="073/01/046/1.jpg"/><lb/>Liquid, that the Square B D hath to <lb/>the Square F Q: But as the whole <lb/>Portion is to that part which is above <lb/>the Liquid, &longs;o is the Square N O unto <lb/>the Square P M: Therefore, P M <lb/>&longs;hall be equall to F Q: But it <lb/>hath been demon&longs;trated, that P H is <lb/>greater than F. And, therefore, <lb/>MH &longs;hall be le&longs;s than <expan abbr="q;">que</expan> and P H <lb/>greater than double of H M. </s> <s>Let <lb/>therefore, P Z be double to Z M: <pb xlink:href="073/01/047.jpg" pagenum="377"/>and drawing a Line from Z to T pro­<lb/><figure id="id.073.01.047.1.jpg" xlink:href="073/01/047/1.jpg"/><lb/>long it unto G. </s> <s>The Centre of <lb/>Gravity of the whole Portion &longs;hall <lb/>be T; of that part which is above <lb/>the Liquid Z; and of the Remain­<lb/>der which is within the Liquid, the <lb/>Centre &longs;hall be in the Line Z T pro­<lb/>longed; let it be in G: It &longs;hall be <lb/>demon&longs;trated, as before, that T H <lb/>is perpendicular to the Surface of <lb/>the Liquid, and that the Lines <lb/>drawn thorow Z and G parallel to the &longs;aid T H, are al&longs;o perpen­<lb/>diculars unto the &longs;ame: Therefore, the Part which is above the <lb/>Liquid &longs;hall move downwards, along that which pa&longs;seth thorow Z; <lb/>and that which is within it, &longs;hall move upwards, along that which <lb/>pa&longs;seth thorow G: And, therefore, the Portion &longs;hall not remain <lb/>&longs;o inclined, nor &longs;hall &longs;o turn about, as that its Axis be perpendicular <lb/><arrow.to.target n="marg1285"></arrow.to.target><lb/>unto the Surface of the Liquid; for the parts towards L &longs;hall move <lb/>downwards, and tho&longs;e towards <emph type="italics"/>A<emph.end type="italics"/> upwards; as may appear by <lb/>the things already demon&longs;trated. </s> <s>And, if the Axis &longs;hould make <lb/>an Angle with the Surface of the Liquid, le&longs;s than the Angle B; <lb/>it &longs;hall in like manner be demon&longs;trated, that the Portion will not <lb/><arrow.to.target n="marg1286"></arrow.to.target><lb/>re&longs;t, but incline untill that its Axis do make an Angle with the <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/><arrow.to.target n="marg1287"></arrow.to.target><lb/>F Q, than doth the Square B C: And, con&longs;equently, the Line <lb/>F Q, is le&longs;s than B C; and F le&longs;s than B R.] <emph type="italics"/>Becau&longs;e the Exce&longs;s by <lb/>which the Square B D exceedeth the Square B C; having le&longs;s proportion unto the Square B D, <lb/>than the Exce&longs;s by which the Square B D exceedeth the Square F Q, hath to the &longs;aid Square<emph.end type="italics"/>; <lb/>(a) <emph type="italics"/>the Exce&longs;s by which the Square B D exceedeth the Square B C &longs;hall be le&longs;s than the Exce&longs;s<emph.end type="italics"/><lb/><arrow.to.target n="marg1288"></arrow.to.target><lb/><emph type="italics"/>by which it exceedeth the Square F Q: Therefore, the Square F Q is le&longs;s than the Square B C: <lb/>and, con&longs;quently, the Line F Q le&longs;s than the Line BC: But F Q hath the &longs;ameproportion <lb/>to F, that B C hath to B R; for the Antecedents are each Se&longs;quialter of their con&longs;equents: <lb/>And<emph.end type="italics"/> (b) <emph type="italics"/>F Q being le&longs;s than B C, F &longs;hall al&longs;o be le&longs;s than B R.<emph.end type="italics"/><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"/>By 8. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1289"></margin.target>(b) <emph type="italics"/>By 14. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>And, for that cau&longs;e, K R hath greater proportion to S Y, than <lb/>the half of K R hath to <foreign lang="greek">y</foreign> B.] <emph type="italics"/>For K R is to S Y, as the Square P S is to the Square<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1290"></arrow.to.target><lb/><emph type="italics"/>S Y: and the half of the Line K R is to the Line<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B, as the Square E<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>is to the Square<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>B.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1290"></margin.target>B</s></p><p type="main"> <s>And S O le&longs;s than <foreign lang="greek">y</foreign> B.] <emph type="italics"/>For S Y is double of S O.<emph.end type="italics"/><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"/>For P H is equall to S<emph.end type="italics"/> <foreign lang="greek">w,</foreign> <emph type="italics"/>and R<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>equall to F.<emph.end type="italics"/></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 &longs;hall have the &longs;ame propor­</s></p><p type="main"> <s><arrow.to.target n="marg1293"></arrow.to.target><lb/>tion to that part which is above the Liquid, that the Square B D <lb/>hath to the Square F Q] <emph type="italics"/>Becau&longs;e that the part &longs;ubmerged, being to the whole Portion <lb/>as the Exce&longs;s by which the Square B D is greater than the Square F Q, is to the Square B D; <lb/>the whole Portion, Converting, &longs;hall be to the part thereof &longs;ubmerged, as the Square B D is to<emph.end type="italics"/> <pb xlink:href="073/01/048.jpg" pagenum="378"/><emph type="italics"/>the Exce&longs;s by which it exceedeth the Square F Q: And, therefore, by Conver&longs;ion of Proportion, <lb/>the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square, <lb/>F <expan abbr="q;">que</expan> for the Square B D is &longs;o much greater than the Exce&longs;s by which it exceedeth the Squar, <lb/>F Q as is the &longs;aid Square F <expan abbr="q.">que</expan><emph.end type="italics"/><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 &longs;hall move downwards, and tho&longs;e to­<lb/>wards A upwards.] <emph type="italics"/>We thus carrect the&longs;e words, for in<emph.end type="italics"/> Tartaglia's <emph type="italics"/>Tran&longs;lation it <lb/>is fal&longs;ly, as I conceive, read<emph.end type="italics"/> Quoniam quæ ex parte L ad &longs;uperiora ferentur, <emph type="italics"/>becau&longs;e <lb/>the Line thàt pa&longs;&longs;eth thorow Z falls perpendicularly on the parts towards L, and that thorow<lb/>G falleth perpendicularly on the parts towards A: Whereupon the Centre Z, together with tho&longs;e <lb/>parts which are towards L &longs;hall move downwards; and the Centre G, together with the parts <lb/>which are towards A upwards.<emph.end type="italics"/></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 &longs;hall in like manner be demon&longs;trated that the Portion &longs;hall not <lb/>re&longs;t, but incline untill that its Axis do make an Angle with the <lb/>Surface of the Liquid, equall to the Angle B.] <emph type="italics"/>This may be ea&longs;ily demon­<lb/>&longs;tratred, as nell from what hath been &longs;aid in the precedent Propo&longs;ition, as al&longs;o from the two <lb/>latter Figures, by us in&longs;erted<emph.end type="italics"/></s></p><p type="head"> <s>PROP. X. THEOR. X.</s></p><p type="main"> <s><emph type="italics"/>The Right Portion of a Rightangled Conoid, lighter <lb/>than the Liquid, when it &longs;hall have its Axis greater <lb/>than to be unto the Semiparameter, in proportion as <lb/>fifteen to four, being demitted into the Liquid, &longs;o as<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1296"></arrow.to.target><lb/><emph type="italics"/>that its Ba&longs;e touch not the &longs;ame, it &longs;hall &longs;ometimes<emph.end type="italics"/><lb/><arrow.to.target n="marg1297"></arrow.to.target><lb/><emph type="italics"/>&longs;tand perpendicular; &longs;ometimes inclined; and &longs;ome­<lb/>times &longs;o inclined, as that its Ba&longs;e touch the Surface <lb/>of the Liquid in one Point only, and that in two Po-<emph.end type="italics"/><lb/><arrow.to.target n="marg1298"></arrow.to.target><lb/><emph type="italics"/>&longs;itions; &longs;ometimes &longs;o that its Ba&longs;e be more &longs;ubmer-<emph.end type="italics"/><lb/><arrow.to.target n="marg1299"></arrow.to.target><lb/><emph type="italics"/>ged in the Liquid; and &longs;ometimes &longs;o as that it doth <lb/>not in the lea&longs;t touch the Surface of the Liquid;<emph.end type="italics"/><lb/><arrow.to.target n="marg1300"></arrow.to.target><lb/><emph type="italics"/>according to the proportion that it hath to the Liquid <lb/>in Gravity. </s> <s>Every one of which Ca&longs;es &longs;hall be anon <lb/>demon&longs;trated.<emph.end type="italics"/></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 &longs;aid; and it being cut <lb/>thorow its Axis, by a Plane erect unto the Superficies of the <lb/>Liquid, let the Section be A P O L, the Section of a Right <lb/>angled Cone; and the Axis of the Portion and Diameter of the <lb/>Section B D: and let B D be cut in the Point K, &longs;o as that B K <lb/>be double of K D; and in C, &longs;o as that B D may have the &longs;ame <lb/><arrow.to.target n="marg1301"></arrow.to.target><lb/>proportion to K C, as fifteen to four: It is manife&longs;t, therefore, <lb/><arrow.to.target n="marg1302"></arrow.to.target><lb/>that K C is greater than the Semi-parameter: Let the Semi­ <pb xlink:href="073/01/049.jpg" pagenum="379"/>parameter be equall to K R: and <lb/><figure id="id.073.01.049.1.jpg" xlink:href="073/01/049/1.jpg"/><lb/><arrow.to.target n="marg1303"></arrow.to.target><lb/>let D S be Se&longs;quialter of K R: but <lb/>S B is al&longs;o Se&longs;quialter of B R: <lb/>Therefore, draw a Line from A to <lb/>B; and thorow C draw C E Per­<lb/>pendicular to B D, cutting the Line <lb/>A B in the Point E; and thorow E <lb/>draw E Z parallel unto B D. Again, <lb/>A B being divided into two equall <lb/>parts in T, draw T H parallel to the <lb/>&longs;ame B D: and let Sections of <lb/>Rightangled Cones be de&longs;cribed, A E I about the Diameter E Z; <lb/>and A T D about the Diameter T H; and let them be like to the <lb/><arrow.to.target n="marg1304"></arrow.to.target><lb/>Portion A B L: Now the Section of the Cone A E I, &longs;hall pa&longs;s <lb/><arrow.to.target n="marg1305"></arrow.to.target><lb/>thorow K; and the Line drawn from R perpendicular unto B D, <lb/>&longs;hall cut the &longs;aid A E I; let it cut it in the Points Y G: and <lb/>thorow Y and G draw P Y Q and O G N parallels unto B D, and <lb/>cutting A T D in the Points F and X: la&longs;tly, draw P <foreign lang="greek">*f</foreign> and O X <lb/>touching the Section A P O L in the <emph type="italics"/>P<emph.end type="italics"/>oints P and O. </s> <s>In regard, <lb/><arrow.to.target n="marg1306"></arrow.to.target><lb/>therefore, that the three <emph type="italics"/>P<emph.end type="italics"/>ortions A P O L, A E I, and A T D are <lb/>contained betwixt Right Lines, and the Sections of Rightangled <lb/>Cones, and are right alike and unequall, touching one another, upon <lb/>one and the &longs;ame Ba&longs;e; and N X G O being drawn from the <lb/><emph type="italics"/>P<emph.end type="italics"/>oint N upwards, and Q F Y P from Q: O G &longs;hall have to G X <lb/>a proportion compounded of the proportion, that I L hath to L A, <lb/>and of the proportion that A D hath to DI: But I L is to L A, <lb/>as two to five: And C B is to B D, as &longs;ix to fifteen; that is, as two <lb/><arrow.to.target n="marg1307"></arrow.to.target><lb/>to five: And as C B is to B D, &longs;o is <emph type="italics"/>E B to B A<emph.end type="italics"/>; and D Z to <lb/><arrow.to.target n="marg1308"></arrow.to.target><lb/>D A: And of D Z and D A, L I and L A are double: and A D <lb/><arrow.to.target n="marg1309"></arrow.to.target><lb/>is to D I, as five to one: <emph type="italics"/>B<emph.end type="italics"/>ut the proportion compounded of the <lb/>proportion of two to five, and of the proportion of five to one, is <lb/><arrow.to.target n="marg1310"></arrow.to.target><lb/>the &longs;ame with that of two to one: and two is to one, in double <lb/>proportion: Therefore, O G is double of GX: and, in the &longs;ame <lb/>manner is P Y proved to be double of Y F: Therefore, &longs;ince that <lb/>D S is Se&longs;quialter of K R; <emph type="italics"/>B S<emph.end type="italics"/> &longs;hall be the Exce&longs;s by which the <lb/>Axis is greater than Se&longs;quialter of the Semi-parameter. </s> <s>If there­<lb/>fore, the <emph type="italics"/>P<emph.end type="italics"/>ortion have the &longs;ame proportion in Gravity unto the <lb/>Liquid, as the Square made of the Line <emph type="italics"/>B S,<emph.end type="italics"/> hath to the Square <lb/>made of <emph type="italics"/>B D,<emph.end type="italics"/> or greater, being demitted into the Liquid, &longs;o as hat <lb/>its <emph type="italics"/>B<emph.end type="italics"/>a&longs;e touch not the Liquid, it &longs;hall &longs;tand erect, or perpendicular: <lb/>For it hath been demon&longs;trated above, that the <emph type="italics"/>P<emph.end type="italics"/>ortion who&longs;e <lb/><arrow.to.target n="marg1311"></arrow.to.target><lb/>Axis is greater than Se&longs;quialter of the Semi-parameter, if it have <lb/>not le&longs;ser proportion in Gravity unto the Liquid, than the Square <pb xlink:href="073/01/050.jpg" pagenum="380"/>made of the Exce&longs;s by which the Axis is greater than Se&longs;quialter <lb/>of the Semi-parameter, hath to the Square made of the Axis, being <lb/>demitted into the Liquid, &longs;o as hath been &longs;aid, it &longs;hall &longs;tand erect, <lb/>or <emph type="italics"/>P<emph.end type="italics"/>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"/>The particulars contained in this Tenth Propo&longs;ition, are divided by<emph.end type="italics"/> Archimedes <lb/><emph type="italics"/>into five Parts and Conclu&longs;ions, each of which he proveth by a di&longs;tinct Demon&longs;tration.<emph.end type="italics"/><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 &longs;hall &longs;ometimes &longs;tand perpendicular.] <emph type="italics"/>This is the fir&longs;t Conclu&longs;ion, the <lb/>Demonstration of which he hath &longs;ubjoyned to the Propo&longs;ition.<emph.end type="italics"/></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 &longs;ometimes &longs;o inclined, as that its Ba&longs;e touch the Surface <lb/>of the Liquid, in one Point only.] <emph type="italics"/>This is demon&longs;trated in the third Con­<lb/>clu&longs;ion.<emph.end type="italics"/></s></p><p type="main"> <s>Sometimes, &longs;o that its Ba&longs;e be mo&longs;t &longs;ubmerged in the Liquid.] </s></p><p type="main"> <s><arrow.to.target n="marg1314"></arrow.to.target><lb/><emph type="italics"/>This pertaineth unto the fourth Conclu&longs;ion.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1314"></margin.target>C</s></p><p type="main"> <s><emph type="italics"/>A<emph.end type="italics"/>nd, &longs;ometimes, &longs;o as that it doth not in the lea&longs;t touch the Sur­<lb/><arrow.to.target n="marg1315"></arrow.to.target><lb/>face of the Liquid.] <emph type="italics"/>This it doth hold true two wayes, one of which is explained is <lb/>the &longs;econd, and the other in the fifth Conclu&longs;ion.<emph.end type="italics"/></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/><arrow.to.target n="marg1316"></arrow.to.target><lb/>vity. </s> <s>Every one of which Ca&longs;es &longs;hall be anon demon&longs;trated.] <lb/><emph type="italics"/>In<emph.end type="italics"/> Tartaglia's <emph type="italics"/>Ver&longs;ion it is rendered, to the confu&longs;ion of the &longs;ence,<emph.end type="italics"/> Quam autem pro­<lb/>portionem habeant ad humidum in Gravitate fingula horum demon&longs;trabuntur.</s></p><p type="margin"> <s><margin.target id="marg1316"></margin.target>E</s></p><p type="main"> <s>It is manife&longs;t, therefore, that K C is greater than the Semi­<lb/><arrow.to.target n="marg1317"></arrow.to.target><lb/>parameter] <emph type="italics"/>For, &longs;ince B D hath to K C the &longs;ame proportion, as fifteen to four, and <lb/>hath unto the Semi-parameter greater proportion; (a) the Semi-parameter &longs;hall be le&longs;s<emph.end type="italics"/><lb/><arrow.to.target n="marg1318"></arrow.to.target><lb/><emph type="italics"/>than K C.<emph.end type="italics"/></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"/>By 10. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>Let the Semi-parameter be equall to KR.] <emph type="italics"/>We have added the&longs;e words,<emph.end type="italics"/><lb/><arrow.to.target n="marg1319"></arrow.to.target><lb/><emph type="italics"/>which are not to be found in<emph.end type="italics"/> Tartaglia.</s></p><p type="margin"> <s><margin.target id="marg1319"></margin.target>G</s></p><p type="main"> <s>But S B is al&longs;o Se&longs;quialter of BR.] <emph type="italics"/>For, D B is &longs;uppo&longs;ed Se&longs;quialter of<emph.end type="italics"/><lb/><arrow.to.target n="marg1320"></arrow.to.target><lb/><emph type="italics"/>B K; and D S al&longs;o is Se&longs;quialter of K R: Wherefore as<emph.end type="italics"/> (b) <emph type="italics"/>the whole D B, is to the whole <lb/>B K, &longs;o is the part D S to the part K R. Therefore, the Remainder S B, is al&longs;o to the<emph.end type="italics"/><lb/><arrow.to.target n="marg1321"></arrow.to.target><lb/><emph type="italics"/>Remainder B R, as D B is to B K.<emph.end type="italics"/></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"/>By 19 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>A<emph.end type="italics"/>nd let them be like to the <emph type="italics"/>P<emph.end type="italics"/>ortion <emph type="italics"/>A B L.<emph.end type="italics"/>] Apollonius <emph type="italics"/>thus defineth<emph.end type="italics"/><lb/><arrow.to.target n="marg1322"></arrow.to.target><lb/><emph type="italics"/>like Portions of the Sections of a Cone, in<emph.end type="italics"/> Lib. 6. Conicornm, <emph type="italics"/>as<emph.end type="italics"/> Eutocius <emph type="italics"/>writeth<emph.end type="italics"/> ^{*}; <lb/><arrow.to.target n="marg1323"></arrow.to.target><lb/><foreign lang="greek">o)/n oi(_s a)x deisw_n o)/n e(xa/sw| warallh/lwn th_ <35>a\sei, i(/swn to\ plh_o<34>, ai( para/llhlos, kai\ a(i <35>a/seis wro\s ta/s apotrm<gap/><lb/>nome/nas a)po\ <gap/> diame/tswn tw_s korufai_s e)n toi_s a)ntoi_s lo/gois ei)si, kai\ ai( a)potemno/menai wro\s ta\s a) temnomi/nas<gap/></foreign><lb/><emph type="italics"/>that is,<emph.end type="italics"/> In both of which an equall number of Lines being drawn parallel to the <lb/>Ba&longs;e; the parallel and the Ba&longs;es have to the parts of the Diameters, cut off from <lb/>the Vertex, the &longs;ameproportion: as al&longs;o, the parts cut off, to the parts cut off. <lb/><emph type="italics"/>Now the Lines parallel to the Ba&longs;es are drawn, as I &longs;uppo&longs;e, by making a Rectilineall Figure (cal-<emph.end type="italics"/><lb/><arrow.to.target n="marg1324"></arrow.to.target><lb/><emph type="italics"/>led)<emph.end type="italics"/> Signally in&longs;cribed [<foreign lang="greek">xh_ma giwri/mws e)gn\<36>ro/menon</foreign>] <emph type="italics"/>in both portions, having an equall num­<lb/>ber of Sides in both. </s> <s>Therefore, like Portions are cut off from like Sections of a Cone; and <lb/>their Diameters, whether they be perpendicular to their Ba&longs;es, or making equall Angles with their <lb/>Ba&longs;es, have the &longs;ame proportion unto their Ba&longs;es.<emph.end type="italics"/><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"/>Upon prop. 3 lib.<emph.end type="italics"/> 2 <lb/>Archim. <emph type="italics"/>Æqui­<lb/>pond.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1324"></margin.target><emph type="italics"/>Vide<emph.end type="italics"/> Archim, <emph type="italics"/>ante <lb/>prop. 2. lib. 2. <lb/>Æquipond.<emph.end type="italics"/></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"/>A E I<emph.end type="italics"/> &longs;hall pa&longs;s thorow K.] <lb/><emph type="italics"/>For, if it be po&longs;&longs;ible, let it not pa&longs;s thorow K, but thorow &longs;ome other Point of the Line D B, as <lb/>thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, who&longs;e <lb/>Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth <lb/>both A E and A I; A E in B, and A I in D; D B &longs;hall have to B V, the &longs;ame proportion<emph.end type="italics"/> <pb xlink:href="073/01/051.jpg" pagenum="381"/><emph type="italics"/>that A Z hath to Z D; by the fourth Propo&longs;ition of<emph.end type="italics"/> Archimedes, De quadratura Para­<lb/>bolæ: <emph type="italics"/>But A Z is Se&longs;quialter of Z D; for it is as three to two, as we &longs;hallanon demon-<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1326"></arrow.to.target><lb/><emph type="italics"/>&longs;trate: Therefore D B is Se&longs;quialter of B V; but D B and B K are Se&longs;quialter: <lb/>And, therefore, the Lines<emph.end type="italics"/> (c) <emph type="italics"/>B V and B K are equall: Which is impos&longs;ible: <lb/>Therefore the Section of the Right-angled Cone A E I, &longs;hall pa&longs;s thorow the Point K; which <lb/>we would demonstrate.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1326"></margin.target>(c) <emph type="italics"/>By 9 of the <lb/>fifth,<emph.end type="italics"/></s></p><p type="main"> <s>In regard, therefore, that the three <emph type="italics"/>P<emph.end type="italics"/>ortions A P O L, A E I <lb/><arrow.to.target n="marg1327"></arrow.to.target><lb/>and A T D are contained betwixt Right Lines and the Sections <lb/>of Right-angled Cones, and are Right, alike and unequall, <lb/>touching one another, upon one and the &longs;ame Ba&longs;e.] <emph type="italics"/>After the&longs;e words,<emph.end type="italics"/><lb/>upon one and the &longs;ame Ba&longs;e, <emph type="italics"/>we may &longs;ee that &longs;omething is obliterated, that is to be <lb/>de&longs;ired: and for the Demon&longs;tration of the&longs;e particulars, it is requi&longs;ite in this place to <lb/>premi&longs;e &longs;ome things: which will al&longs;o be nece&longs;&longs;ary unto the things that follow.<emph.end type="italics"/></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"/>L<emph.end type="italics"/>ine A B; and let it be cut by two <emph type="italics"/>L<emph.end type="italics"/>ines, <lb/>parallel to one another, A C and D E, &longs;o, that as <emph type="italics"/>A B<emph.end type="italics"/> is to <lb/>B D. &longs;o <emph type="italics"/>A C<emph.end type="italics"/> may be to D E. </s> <s>I &longs;ay that the Line that con­<lb/>joyneth the Points C and B &longs;hall likewi&longs;e pa&longs;s by E.</s></p><figure id="id.073.01.051.1.jpg" xlink:href="073/01/051/1.jpg"/><p type="main"> <s><emph type="italics"/>For, if po&longs;&longs;ible, let it not pa&longs;s by E, but either <lb/>above or below it. </s> <s>Let it first pa&longs;s below it, <lb/>as by F. </s> <s>The Triangles A B C and D B F &longs;hall <lb/>be alike: And, therefore, as<emph.end type="italics"/> (a) <emph type="italics"/>A B is to B D,<emph.end type="italics"/><lb/><arrow.to.target n="marg1328"></arrow.to.target><lb/><emph type="italics"/>&longs;o is A C to D F: But as A B is to B D, &longs;o was <lb/>A C to D E: Therefore<emph.end type="italics"/> (b) <emph type="italics"/>D F &longs;hall be equall to<emph.end type="italics"/><lb/><arrow.to.target n="marg1329"></arrow.to.target><lb/><emph type="italics"/>D E: that is, the part to the whole: Which is <lb/>ab&longs;urd. </s> <s>The &longs;ame ab&longs;urditie will follow, if the <lb/>Line C B be &longs;uppo&longs;ed to pa&longs;s above the Point E: <lb/>And, therefore, C B mu&longs;t of neces&longs;ity pa&longs;s thorow <lb/>E: Which was required to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1328"></margin.target>(a) <emph type="italics"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1329"></margin.target>(b) <emph type="italics"/>By 9. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA. II.</s></p><p type="main"> <s>Let there be two like <emph type="italics"/>P<emph.end type="italics"/>ortions, contained betwixt Right Lines, <lb/>and the Sections of Right-angled Cones; A B C the great­<lb/>er, who&longs;e Diameter let be B D; and E F C the le&longs;ser, who&longs;e <lb/>Diameter let be F G: and, let them be &longs;o applyed to one <lb/>another, that the greater include the le&longs;ser; and let their <lb/>Ba&longs;es A C and E C be in the &longs;ame Right Line, that the &longs;ame <lb/>Point C, may be the term or bound of them both: And, <lb/>then in the Section A B C, take any Point, as H; and draw <lb/>a Line from H to C. </s> <s>I &longs;ay, that the Line H C, hath to that <lb/>part of it &longs;elf, that lyeth betwixt C and the Section E F C, the <lb/>&longs;ame proportion that A C hath to C E.</s></p><p type="main"> <s><emph type="italics"/>Draw B C, which &longs;hall pa&longs;s thorow F, For, in regard, that the Portions are alike, the <lb/>Diameters with the Ba&longs;es contain equall Angles: And, therefore, B D and F G are parallel <lb/>to one another: and B D is to A C, as F G it to E C: and,<emph.end type="italics"/> Permutando, <emph type="italics"/>B D is to F G, as <lb/>A C is to C E; that is,<emph.end type="italics"/> (a) <emph type="italics"/>as their halfes D C to C G; therefore, it followeth, by the<emph.end type="italics"/><lb/><arrow.to.target n="marg1330"></arrow.to.target><lb/><emph type="italics"/>preceding Lemma, that the Line B C &longs;hall pa&longs;s by the Point F. Moreover, from the Point <lb/>H unto the Diameter B D, draw the Line H K, parallel to the Ba&longs;e A C: and, draw a Line<emph.end type="italics"/> <pb xlink:href="073/01/052.jpg" pagenum="382"/><figure id="id.073.01.052.1.jpg" xlink:href="073/01/052/1.jpg"/><lb/><emph type="italics"/>from K to C, cutting the Diameter F G in L: <lb/>and, thorow L, unto the Section E F. G, on the <lb/>part E, draw the Line L M, parallel unto the <lb/>&longs;ame Ba&longs;e A C. And, of the Section A B C, <lb/>let the Line B N be the Parameter; and, of the <lb/>Section E F C, let F O be the Parameter. </s> <s>And, <lb/>becau&longs;e the Triangles C B D and C F G are alike<emph.end type="italics"/>; <lb/>(b) <emph type="italics"/>therefore, as B C is to C F, &longs;o &longs;hall D C be<emph.end type="italics"/><lb/><arrow.to.target n="marg1331"></arrow.to.target><lb/><emph type="italics"/>to C G, and B D to F G. Again, becau&longs;e the <lb/>Triangles C K B and C L F, are al&longs;o alike to <lb/>one another; therefore, as B C is to C F, that is, <lb/>as B D is to F G, &longs;o &longs;hall K C be to C L, and B K to F L: Wherefore, K C to C L, and,<emph.end type="italics"/><lb/><arrow.to.target n="marg1332"></arrow.to.target><lb/><emph type="italics"/>B K to F L, are as D C to C G; that is,<emph.end type="italics"/> (c) <emph type="italics"/>as their duplicates A C and C E: But as <lb/>B D is to F G, &longs;o is D C to C G; that is, A D to E G: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as B D is to <lb/>A D, &longs;o is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11 <lb/>of our fir&longs;t of<emph.end type="italics"/> Conicks: <emph type="italics"/>Therefore, the<emph.end type="italics"/> (d) <emph type="italics"/>three Lines B D, A D and B N are<emph.end type="italics"/><lb/><arrow.to.target n="marg1333"></arrow.to.target><lb/><emph type="italics"/>Proportionalls. </s> <s>By the &longs;ame rea&longs;on, likewi&longs;e, the Square E G being equall to the Rectangle <lb/>G F O, the three other Lines F G, E G and F O, &longs;hall be al&longs;o Proportionals: And, as B D is <lb/>to A D, &longs;o is F G to E G: And, therefore, as A D is to B N, &longs;o is E G to F O:<emph.end type="italics"/> Ex equali, <lb/><emph type="italics"/>therefore, as D B is to B N, &longs;o is G F to F O: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as D B is to G F, &longs;o is <lb/>B N to F O: But as D B is to G F, &longs;o is B K to F L: Therefore, B K is to F L, as <lb/>B N is to F O: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as B K is to B N, &longs;o is F L to F O. Again, <lb/>becau&longs;e the<emph.end type="italics"/> (e) <emph type="italics"/>Square H K is equall to the Rectangle B N; and the Square M L, equall<emph.end type="italics"/><lb/><arrow.to.target n="marg1334"></arrow.to.target><lb/><emph type="italics"/>to the Rectangle L F O, therefore, the three Lines B K, K H and B N &longs;hall be Proportionals: <lb/>and F L, L M, and F O &longs;hall al&longs;o be Proportionals: And, therefore,<emph.end type="italics"/> (f) <emph type="italics"/>as the Line<emph.end type="italics"/><lb/><arrow.to.target n="marg1335"></arrow.to.target><lb/><emph type="italics"/>B K is to the Line B N, &longs;o &longs;hall the Square B K, be to the Square H K: And, as the <lb/>Line F L is to the Line F O, &longs;o &longs;hall the Square F L be to the Square L M: <lb/>Therefore, becau&longs;e that as B K is to B N, &longs;o is F L to F O; as the Square<emph.end type="italics"/><lb/><arrow.to.target n="marg1336"></arrow.to.target><lb/><emph type="italics"/>B K is to the Square K H, &longs;o &longs;hall the Square F L be to the Square L M: Therefore,<emph.end type="italics"/><lb/>(g) <emph type="italics"/>as the Line B K is to the Line K H, &longs;o is the Line F L to L M: And,<emph.end type="italics"/> Permutando, <lb/><emph type="italics"/>as B K is to F L, &longs;o is K H to L M: But B K was to F L, as K C to C L: Therefore, <lb/>K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manife&longs;t that <lb/>the Line H C al&longs;o &longs;hall pa&longs;s thorow the Point M: As K C, therefore, is to C L, that is, <lb/>as A C to C E, &longs;o is H C to C M; that is, to the &longs;ame part of it &longs;elf, that lyeth betwixt C and <lb/>the Section E F C. And, in like manner might we demon&longs;trate, that the &longs;ame happeneth <lb/>in other Lines, that are produced from the Point C, and the Sections E B C. And, that <lb/>B C hath the &longs;ame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G; <lb/>that is, as their Duplicates A C to C E.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1330"></margin.target>(a) <emph type="italics"/>By 15. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1331"></margin.target>(b) <emph type="italics"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1332"></margin.target>(c) <emph type="italics"/>By 15. of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1333"></margin.target>(d) <emph type="italics"/>By 17. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1334"></margin.target>(e) <emph type="italics"/>By 11 of our <lb/>fir&longs;t of<emph.end type="italics"/> Conicks.</s></p><p type="margin"> <s><margin.target id="marg1335"></margin.target>(f) <emph type="italics"/>By<emph.end type="italics"/> Cor. <emph type="italics"/>of 20. <lb/>of the &longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1336"></margin.target>(g) <emph type="italics"/>By 23. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="main"> <s>From whence it is manife&longs;t, that all Lines &longs;o drawn, &longs;hall be cut by the <lb/>&longs;aid Section in the &longs;ame proportion. </s> <s>For, by Divi&longs;ion and Conver&longs;ion, <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 al&longs;o be proved, that the Lines which are <lb/>drawn in like Portions, &longs;o, as that with the Ba&longs;es, they con­<lb/>tain equall Angles, &longs;hall al&longs;o cut off like Portions; that is, <lb/>as in the foregoing Figure, the Portions H B C and M F C, <lb/>which the Lines C H and C M do cut off, are al&longs;o alike to <lb/>each other.</s></p><p type="main"> <s><emph type="italics"/>For let C H and C M be divided in the midst in the Points P and <expan abbr="q;">que</expan> and thorow tho&longs;e <lb/>Points draw the Lines R P S and T Q V parallel to the Diameters. </s> <s>Of the Portion <lb/>H S C the Diameter &longs;hall be P S, and of the Portion M V C the Diameter &longs;hall be<emph.end type="italics"/> <pb xlink:href="073/01/053.jpg" pagenum="383"/><emph type="italics"/>Q V. And, &longs;uppo&longs;e that as the Square C R is to the Square C P, &longs;o is the Line B N unto <lb/>another Line; which let be S X: And, as the Square C T is to the Square C Q &longs;o let F O <lb/>be to V Y. </s> <s>Now it is manife&longs;t, by the things which we have demon&longs;trated, in our Commentaries, <lb/>upon the fourth Propo&longs;ition of<emph.end type="italics"/> Archimedes, De Conoidibus & Spheæroidibus, <emph type="italics"/>that the <lb/>Square C P is equall to the Rectangle P S X; and al&longs;o, that the Square C Q is equall to <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/>and M V C: But &longs;ince the Triangles C P R and C Q T are alike; C R &longs;hall have to C P, the <lb/>&longs;ame Proportion that C T hath to C Q: And, therefore, the<emph.end type="italics"/> (a) <emph type="italics"/>Square C R &longs;hall have<emph.end type="italics"/><lb/><arrow.to.target n="marg1337"></arrow.to.target><lb/><emph type="italics"/>to the Square C P, the &longs;ame proportion that the<emph.end type="italics"/><lb/><figure id="id.073.01.053.1.jpg" xlink:href="073/01/053/1.jpg"/><lb/><emph type="italics"/>Square C T hath to the Square C Q: There­<lb/>fore, al&longs;o, the Line B N &longs;hall be to the Line <lb/>S X, as the Line F O is to V Y: But H C was <lb/>to C M, as A C to C E: And, therefore, al&longs;o, <lb/>their halves C P and C Q, are al&longs;o to one <lb/>another, as A D and E G: And.<emph.end type="italics"/> Permu­<lb/>tando, <emph type="italics"/>C P is to A D, as C Q is to E G: <lb/>But it hath been proved, that A D is to B N, <lb/>as E G to F O; and B N to S X, as F O to <lb/>V Y: Therefore,<emph.end type="italics"/> exæquali, <emph type="italics"/>C P &longs;hall be <lb/>to S X, as C Q is to V Y. And, &longs;ince the <lb/>Square C P is equall to the Rectangle P S X, and the Square C Q to the Rectangle Q V Y, <lb/>the three Lines S P, PC and S X &longs;hall be proportionalls, and V Q, Q C and V Y &longs;hal be <lb/>Proportionalls al&longs;o: And therefore al&longs;o S P &longs;hall be to P C as V Q to Q C And as P C <lb/>is to C H, &longs;o &longs;hall Q C. be to C M: Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>as S P the Diameter of the <lb/>Portion H S C is to its Ba&longs;e C H, &longs;o is V Q the Diameter of the portion M V S the <lb/>Ba&longs;e C M; and the Angles which the Diameter with the Ba&longs;es do contain, are equall; and the <lb/>Lines S P and V Q are parallel: Therefore the Portions, al&longs;o, H S C and M V C &longs;hall be alike: <lb/>Which was propo&longs;ed to be demon&longs;trated<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1337"></margin.target>(a) <emph type="italics"/>By 22. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA. IV.</s></p><p type="main"> <s><emph type="italics"/>L<emph.end type="italics"/>et there be two <emph type="italics"/>L<emph.end type="italics"/>ines A <emph type="italics"/>B<emph.end type="italics"/> and C D; and let them be cut in the <lb/>Points E and F, &longs;o that as A E is to E B, C F may be to F D: <lb/>and let them be cut again in two other Points G and H; and <lb/>let C H be to H D, as A G is to G B. </s> <s>I &longs;ay that C F &longs;hall be to <lb/>F H as A E is E G.</s></p><p type="main"> <s><emph type="italics"/>For in regard that as A E is to E B, &longs;o is C F to F D; it followeth that, by Compounding, <lb/>as A B is to E B, &longs;o &longs;hall C D be to F D. Again, &longs;ince that as A G is to G B, &longs;o is C H, to <lb/>H D; it followeth that, by Compounding and Converting, as G B is to A B, &longs;o &longs;hall H D be<emph.end type="italics"/><lb/><figure id="id.073.01.053.2.jpg" xlink:href="073/01/053/2.jpg"/><lb/><emph type="italics"/>C D: Therefore,<emph.end type="italics"/> ex æquali, <emph type="italics"/>and Converting as E B <lb/>is to G B, &longs;o &longs;hall F D be to H D; And, by Conver­<lb/>&longs;ion of Propo&longs;ition, as E B is to E G, &longs;o &longs;hall F D <lb/>be to F H: But as A E is to E B, &longs;o is C F to F D:<emph.end type="italics"/><lb/>Ex æquali, <emph type="italics"/>therefore, as A E is to E G, &longs;o <lb/>&longs;hall CF be to F H.<emph.end type="italics"/> Again, another way. <emph type="italics"/>Let <lb/>the Lines A B and C D be applyed to one another, <lb/>&longs;o as that they doe make an Angle at the parts A and C; <lb/>and let A and C be in one and the &longs;ame Point: then <lb/>draw Lines from D to B, from H to G, and from F to E. </s> <s>And &longs;ince that as A E is to E B, <lb/>&longs;o is C F, that is A F to F D; therefore F E &longs;hall be parallel to D B<emph.end type="italics"/>; (a) <emph type="italics"/>and likewi&longs;e<emph.end type="italics"/><lb/><arrow.to.target n="marg1338"></arrow.to.target><lb/><emph type="italics"/>H G &longs;hall be parallel to D B; for that A H is to H D, as A G to G B<emph.end type="italics"/>: (b) <emph type="italics"/>Therefore F E <lb/>and H G are parallel to each other: And con&longs;equently, as A E is to E G, &longs;o is A H, that is,<emph.end type="italics"/><lb/><arrow.to.target n="marg1339"></arrow.to.target><lb/><emph type="italics"/>C F to F H: Which was to be demon&longs;trated.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/054.jpg" pagenum="384"/><p type="margin"> <s><margin.target id="marg1338"></margin.target>(a) <emph type="italics"/>By 2. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1339"></margin.target>(b) <emph type="italics"/>By 30 of the <lb/>fir&longs;t.<emph.end type="italics"/></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/>Lines and the Sections of Right-angled Cones, as in the fore­<lb/>going figure, A B C, who&longs;e Diameter is B D; and E F C, <lb/>who&longs;e Diameter is F G; and from the Point E, draw the <lb/>Line E H parallel to the Diameters B D and F G; and let it <lb/>cut the Section A B C in K: and from the Point C draw C H <lb/>touching the Section A B C in C, and meeting with the Line <lb/>E H in H; which al&longs;o toucheth the Section E F C in the &longs;ame <lb/>Point C, as &longs;hall be demon&longs;trated: I &longs;ay that the Line drawn <lb/>from C <emph type="italics"/>H<emph.end type="italics"/> unto the Section E F C &longs;o as that it be parallel to <lb/>the Line E H, &longs;hall be divided in the &longs;ame proportion by the <lb/>Section A B C, in which the <emph type="italics"/>L<emph.end type="italics"/>ine C A is divided by the Section <lb/>E F C; and the part of the <emph type="italics"/>L<emph.end type="italics"/>ine C A which is betwixt the <lb/>two Sections, &longs;hall an&longs;wer in proportion to the part of the Line <lb/>drawn, which al&longs;o falleth betwixt the &longs;ame Sections: that is, <lb/>as in the foregoing Figure, if D B be produced untill it meet <lb/>with C H in L, that it may inter&longs;ect the Section E F C in the <lb/>Point M, the <emph type="italics"/>L<emph.end type="italics"/>ine <emph type="italics"/>L<emph.end type="italics"/> B &longs;hall have to B M the &longs;ame proportion <lb/>that C E hath to E A.</s></p><p type="main"> <s><emph type="italics"/>For let G F be prolonged untill it meet the &longs;ame Line C H in N, cutting the Section A B C <lb/>in O; and drawing a Line from B to C, which &longs;hall pa&longs;&longs;e by F, as hath been &longs;hewn, the<emph.end type="italics"/><lb/><figure id="id.073.01.054.1.jpg" xlink:href="073/01/054/1.jpg"/><lb/><emph type="italics"/>Triangles C G F and C D B &longs;hall be alike; as <lb/>al&longs;o the Triangles C F N and C B L: Wherefore<emph.end type="italics"/><lb/>(a) <emph type="italics"/>as G F is to D B, &longs;o &longs;hall C F b to C B:<emph.end type="italics"/><lb/><arrow.to.target n="marg1340"></arrow.to.target><lb/><emph type="italics"/>And as<emph.end type="italics"/> (b) <emph type="italics"/>C F is to C B, &longs;o &longs;hall F N be <lb/>to B L: Therefore G F &longs;hall be to D B, as F N<emph.end type="italics"/><lb/><arrow.to.target n="marg1341"></arrow.to.target><lb/><emph type="italics"/>to B L: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>G F &longs;hall be to <lb/>F N, as D B to B L: But D B is equall to <lb/>B L, by 35 of our Fir&longs;t Book of<emph.end type="italics"/> Conicks: <lb/><emph type="italics"/>Therefore<emph.end type="italics"/> (c) <emph type="italics"/>G F al&longs;o &longs;hall be equall to F N:<emph.end type="italics"/><lb/><arrow.to.target n="marg1342"></arrow.to.target><lb/><emph type="italics"/>And by 33 of the &longs;ame, the Line C H touch­<lb/>eth the Section E F C in the &longs;ame Point. </s> <s>There­<lb/>fore, drawing a Line from C to M, prolong it <lb/>untill it meet with the Section A B C in P; and <lb/>from P unto A C draw P Q parallel to B D. <lb/>Becau&longs;e, now, that the Line C H toucheth the <lb/>Section E F C in the Point C; L M &longs;hall have <lb/>the &longs;ame proportion to M D that C D hath to D E, <lb/>by the Fifth Propo&longs;ition of<emph.end type="italics"/> Archimedes <emph type="italics"/>in his <lb/>Book<emph.end type="italics"/> De Quadratura Patabolæ: <emph type="italics"/>And by <lb/>rea&longs;on of the Similitude of the Triangles C M D <lb/>and C P Q, as C M is to C D, &longs;o &longs;hall C P <lb/>be to C Q: And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as C M is to <lb/>C P, &longs;o &longs;hall C D be to C Q: But as C M is to C P, &longs;o is C E to C A,; as we have but <lb/>even now demon&longs;trated: And therefore, as C E is to C A, &longs;o is C D to C <expan abbr="q;">que</expan> that is as the <lb/>whole is to the whole, &longs;o is the part to the part: The remainder, therefore, D E is to the <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"/> Permutando, <emph type="italics"/>C D <lb/>is to D E, as C Q is to Q A: And L M is al&longs;o to M D, as C D to D E: Therefore L M is<emph.end type="italics"/> <pb xlink:href="073/01/055.jpg" pagenum="385"/><emph type="italics"/>to M D, as C Q to Q A: But L B is to B D, by 5 of<emph.end type="italics"/> Archimedes, <emph type="italics"/>before recited, as C D <lb/>to D A: It is manife&longs;t therefore, by the precedent Lemma, that C D is to D Q, as L B is to <lb/>B M: But as C D is to D Q, &longs;o is C M to M P: Therefore L B is to B M, as C M to M P:<emph.end type="italics"/><lb/><arrow.to.target n="marg1343"></arrow.to.target><lb/><emph type="italics"/>And it haveing been demon&longs;trated, that C M is to M P, as C E to E A; L B &longs;hall be to B M,<lb/>as C E to E A. </s> <s>And in like manner it &longs;hall be demonstrated that &longs;o is N O to O F; as al&longs;o the <lb/>Remainders. </s> <s>And that al&longs;o H K is to K E, as C E to E A, doth plainly appeare by the &longs;ame<emph.end type="italics"/><lb/>5. <emph type="italics"/>of<emph.end type="italics"/> Archimedes<emph type="italics"/>: Which is that that we propounded to be demon&longs;trated.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1340"></margin.target>(a) <emph type="italics"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1341"></margin.target>(b) <emph type="italics"/>By 11 of the <lb/>fifth,<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1342"></margin.target>(c) <emph type="italics"/>By 14 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1343"></margin.target><emph type="italics"/>By 2. of the &longs;ixth<emph.end type="italics"/></s></p><p type="head"> <s>LEMMA. VI.</s></p><p type="main"> <s>And, therefore, let the things &longs;tand as above; and de&longs;cribe <lb/>yet another like Portion, contained betwixt a Right Line, and <lb/>the Section of the Rightangled Cone D R C, who&longs;e Diameter <lb/>is R S, that it may cut the Line F G in T; and prolong S R <lb/>unto the <emph type="italics"/>L<emph.end type="italics"/>ine C H in V, which meeteth the Section A B C in <lb/>X, and E F C in Y. </s> <s>I &longs;ay, that B M hath to M D, a propor­<lb/>tion compounded of the proportion that E A hath to A C; <lb/>and of that which C D hath to D E.</s></p><p type="main"> <s><emph type="italics"/>For, we &longs;hall fir&longs;t demon&longs;trate, that the Line C H toucheth the Section D R C in the <lb/>Point C; and that L M is to M D, as al&longs;o N F to F T, and V Y to Y R, as C D is to E D. <lb/>And, becau&longs;e now that L B is to B M, as C E is to E A; therefore, Compounding and Conver­<lb/>ting, B M &longs;hall be to L M, as E A to A C: And, as L M is to M D, &longs;o &longs;hall C D be to <lb/>D E: The proportion, therefore, of B M to M D, is compounded of the proportion that <lb/>B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion <lb/>of B M to M D, &longs;hall al&longs;o be compounded of the proportion that E A hath to A C, and of <lb/>that which C D hath to D E. </s> <s>In the &longs;ame manner it &longs;hal be demon&longs;trated, that O F hath to <lb/>F T, and al&longs;o X Y to Y R, a proportion compounded of tho&longs;e &longs;ame proportions; and &longs;o in <lb/>the re&longs;t: Which was to be demonstrated.<emph.end type="italics"/></s></p><p type="main"> <s>By which it appeareth that the <emph type="italics"/>L<emph.end type="italics"/>ines &longs;o drawn; which fall betwixt <lb/>the Sections A B C and D R C, &longs;hall be divided by the Section E F C <lb/>in the &longs;ame Proportion.</s></p><p type="main"> <s>And C B is to B D, as &longs;ix to fifteen.] <emph type="italics"/>For we have &longs;uppo&longs;ed that B K is<emph.end type="italics"/><lb/><arrow.to.target n="marg1344"></arrow.to.target><lb/><emph type="italics"/>double of K D: Wherefore, by Compo&longs;ition B D &longs;hall be to K D as three to one; that is, as <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/>And, by Conver&longs;ion of proportion and Convert­<lb/>ing, C B is to B D, as &longs;ix to &longs;ifteen.<emph.end type="italics"/></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"/><p type="main"> <s>And as C B is to B D, &longs;o is <lb/><arrow.to.target n="marg1345"></arrow.to.target><lb/>E B to B A; and D Z to D A.] <lb/><emph type="italics"/>For the Triangles C B E and D B A being <lb/>alike; As C B is to B E, &longs;o &longs;hall D B be to B A: <lb/>And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as C B is to B D, &longs;o &longs;hall <lb/>E B be to B A: Againe, as B C is to C E &longs;o <lb/>&longs;hall B D be to D A, And,<emph.end type="italics"/> Permutando, <emph type="italics"/>as <lb/>C B is to B D, &longs;o &longs;hall C E, that is, D Z <lb/>equall to it, be to D A.<emph.end type="italics"/></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/><arrow.to.target n="marg1346"></arrow.to.target><lb/>L A are double.] <emph type="italics"/>That the Line L A is <lb/>double of D A, is manife&longs;t, for that B D is the Diameter of the Portion. </s> <s>And that L I is <lb/>dovble to D Z &longs;hall be thus demon&longs;trated. </s> <s>For as much as ZD is to D A, as two to five: <lb/>therefore, Converting and Dividing, A Z, that is, I Z, &longs;hall be to Z D, as three to two:<emph.end type="italics"/> <pb xlink:href="073/01/056.jpg" pagenum="386"/><emph type="italics"/>Again, by dividing, I D &longs;hall be to D Z, as one to two: But Z D was to D A, that is, to D L, <lb/>as two to five: Therefore,<emph.end type="italics"/> ex equali, <emph type="italics"/>and Converting, L D is to D I, as five to one: and, by <lb/>Conver&longs;ion of Proportion, D L is to D I, as five to four: But D Z was to D L, as two to <lb/>five: Therefore, again,<emph.end type="italics"/> ex equali, <emph type="italics"/>D Z is to L I, as two to four: Therefort L I is double <lb/>of D Z: Which was to be demon&longs;trated.<emph.end type="italics"/><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"/>This we have but ju&longs;t now demon­<lb/>&longs;trated.<emph.end type="italics"/></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 demon&longs;trated, above, that the Portion who&longs;e <lb/>Axis is greater than Se&longs;quialter of the Semi-parameter, if it have <lb/>not le&longs;&longs;er proportion in Gravity to the Liquid, &c.] <emph type="italics"/>He hath demonstra­<lb/>ted this in the fourth Propo&longs;ition of this Book.<emph.end type="italics"/></s></p><p type="head"> <s>CONCLVSION II.</s></p><p type="main"> <s><emph type="italics"/>If the Portion have le&longs;&longs;er proportion in Gravity to the<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1349"></arrow.to.target><lb/><emph type="italics"/>Liquid, than the Square S B hath to the Square <lb/>B D, but greater than the Square X O hath to the <lb/>Square B D, being demitted into the Liquid, &longs;o in­<lb/>clined, as that its Ba&longs;e touch not the Liquid, it &longs;hall <lb/>continue inclined, &longs;o, as that its Ba&longs;e &longs;hall not in the <lb/>lea&longs;t touch the Surface of the Liquid, and its Axis <lb/>&longs;hall make an Angle with the Liquids Surface, greater <lb/>than the Angle X.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1349"></margin.target>A</s></p><p type="main"> <s>Therfore repeating the fir&longs;t figure, let the Portion have unto <lb/>the Liquid in Gravitie a proportion greater than the Square <lb/>X O hath to the &longs;quare B D, but le&longs;&longs;er than the Square made of <lb/>the Exce&longs;&longs;e by which the Axis is greater than Se&longs;quialter of the Semi­<lb/><figure id="id.073.01.056.1.jpg" xlink:href="073/01/056/1.jpg"/><lb/>Parameter, that is, of S B, hath to <lb/>the Square B D: and as the Portion <lb/>is to the Liquid in Gravity, &longs;o let <lb/>the Square made of the Line <foreign lang="greek">y</foreign> be <lb/>to the Square B D: <foreign lang="greek">y</foreign> &longs;hall be great­<lb/><arrow.to.target n="marg1350"></arrow.to.target><lb/>er than X O, but le&longs;&longs;er than the <lb/>Exce&longs;&longs;e by which the Axis is grea­<lb/>ter than Se&longs;quialter of the Semi­<lb/>parameter, that is, than S B. </s> <s>Let <lb/>a Right Line M N be applyed to <lb/>fall between the Conick-Sections <lb/>A M Q L and A <emph type="italics"/>X<emph.end type="italics"/> D, [<emph type="italics"/>parallel to <lb/>B D falling betwixt O X and B D,<emph.end type="italics"/>] and equall to the Line <foreign lang="greek">y</foreign>: and let <lb/>it cut the remaining Conick Section A H I in the point H, and the <lb/><arrow.to.target n="marg1351"></arrow.to.target><lb/>Right Line R G in V. </s> <s>It &longs;hall be demon&longs;trated that M H is double to <lb/>H N, like as it was demon&longs;trated that O G is double to G X. <pb xlink:href="073/01/057.jpg" pagenum="387"/><figure id="id.073.01.057.1.jpg" xlink:href="073/01/057/1.jpg"/><lb/>And from the Point M draw M Y <lb/>touching the Section A M Q L in M; <lb/>and M C perpendicular to B D: and <lb/>la&longs;tly, having drawn A N & prolong­<lb/>ed it to Q, the Lines A N & N Q &longs;hall <lb/>be equall to each other. </s> <s>For in <lb/>regard that in the Like Portions <lb/><arrow.to.target n="marg1352"></arrow.to.target><lb/>A M Q L and A <emph type="italics"/>X<emph.end type="italics"/> D the Lines A Q <lb/>and A N are drawn from the Ba&longs;es <lb/>unto the Portions, which Lines <lb/>contain equall Angles with the &longs;aid <lb/>Ba&longs;es, Q A &longs;hall have the &longs;ame proportion to A M that L A hath <lb/>to A D: Therefore A N is equall to N Q, and A Q parallel to M Y. <lb/><arrow.to.target n="marg1353"></arrow.to.target><lb/>It is to be demon&longs;trated that the Portion being demitted into the <lb/>Liquid, and &longs;o inclined as that its Ba&longs;e touch not the Liquid, it <lb/>&longs;hall continue inclined &longs;o as that its Ba&longs;e &longs;hall not in the lea&longs;t touch <lb/>the Surface of the Liquid, and its Axis &longs;hall make an Angle with <lb/>the Liquids Surface greater than the Angle X. </s> <s>Let it be demitted <lb/>into the Liquid, and let it &longs;tand, &longs;o, as that its Ba&longs;e do touch the <lb/>Surface of the Liquid in one Point only; and let the Portion be cut <lb/>thorow the Axis by a Plane erect unto the Surface of the Liquid, <lb/><figure id="id.073.01.057.2.jpg" xlink:href="073/01/057/2.jpg"/><lb/>and Let the Section of the Super­<lb/>ficies of the Portion be A P O L, <lb/>the Section of a Rightangled Cone, <lb/>and let the Section of the Liquids <lb/>Surface be A O; And let the Axis <lb/>of the Portion and Diameter of the <lb/>Section be <emph type="italics"/>B<emph.end type="italics"/> D: and let B D be <lb/><arrow.to.target n="marg1354"></arrow.to.target><lb/>cut in the Points K and R as hath <lb/>been &longs;aid; al&longs;o draw P G Parallel to <lb/>A O and touching the Section <lb/>A P O L in P; and from that Point <lb/>draw P T Parallel to B D, and P S perpendicular to the &longs;ame B D. <lb/>Now, fora&longs;much as the Portion is unto the Liquid in Gravity, as <lb/>the Square made of the Line <foreign lang="greek">y</foreign> is to the Square B D; and &longs;ince that <lb/>as the portion is unto the Liquid in Gravitie, &longs;o is the part thereof <lb/>&longs;ubmerged unto the whole Portion; and that as the part &longs;ubmerged <lb/>is to the whole, &longs;o is the Square T P to the Square B D; It follow­<lb/>eth that the Line <foreign lang="greek">y</foreign> &longs;hall be equall to T P: And therefore the Lines <lb/>M N and P T, as al&longs;o the Portions A M Q and A P O &longs;hall like­<lb/>wi&longs;e be equall to each other. </s> <s>And &longs;eeing that in the Equall and <lb/>Like Portions A P O L and A M Q L the Lines A O and A Q <lb/><arrow.to.target n="marg1355"></arrow.to.target><lb/>are drawn from the extremites of their Ba&longs;es, &longs;o, as that the Portions <lb/>cut off do make Equall Angles with their Diameters; as al&longs;o the <pb xlink:href="073/01/058.jpg" pagenum="388"/>Angles at Y and G being equall; therefore the Lines Y B and G B, <lb/>and B C and B S &longs;hall al&longs;o be equall: And therefore C R and S R, <lb/>and M V and P Z, and V N and Z T, &longs;hall be equall likewi&longs;e. <lb/><arrow.to.target n="marg1356"></arrow.to.target><lb/>Since therefore M V is Le&longs;&longs;er than double of V N, it is manife&longs;t that <lb/>P Z is le&longs;&longs;er than double of Z T. <emph type="italics"/>L<emph.end type="italics"/>et P <foreign lang="greek">w</foreign> be double of <foreign lang="greek">w</foreign> T; and <lb/>drawing a <emph type="italics"/>L<emph.end type="italics"/>ine from <foreign lang="greek">w</foreign> to K, prolong it to E. </s> <s>Now the Centre of <lb/>Gravity of the whole Portion &longs;hall be the point K; and the Centre <lb/>of that part which is in the Liquid &longs;hall be <foreign lang="greek">w,</foreign> and of that which is <lb/>above the Liquid &longs;hall be in the <emph type="italics"/>L<emph.end type="italics"/>ine K E, which let be E: But the <lb/><emph type="italics"/>L<emph.end type="italics"/>ine K Z &longs;hall be perpendicular unto the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid: <lb/>And therefore al&longs;o the Lines drawn thorow the Points E and <foreign lang="greek">w</foreign> parall­<lb/><arrow.to.target n="marg1357"></arrow.to.target><lb/>lell unto K Z, &longs;hall be perpendicular sunto the &longs;ame: Therefore the <lb/>Portion &longs;hall not abide, but &longs;hall turn about &longs;o, as that its <emph type="italics"/>B<emph.end type="italics"/>a&longs;e <lb/>do not in the lea&longs;t touch the Surface of the <emph type="italics"/>L<emph.end type="italics"/>iquid; in regard that <lb/>now when it toucheth in but one Point only, it moveth upwards, on <lb/><arrow.to.target n="marg1358"></arrow.to.target><lb/>the part towards A: It is therefore per&longs;picuous, that the Portion <lb/>&longs;hall con&longs;i&longs;t &longs;o, as that its Axis &longs;hall make an Angle with the <emph type="italics"/>L<emph.end type="italics"/>iquids <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/><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 le&longs;&longs;er proportion in Gravity to the Liquid, <lb/>than the Square S B hath to the Square B D, but greater than the <lb/>Square X O hath to the Square B D.] <emph type="italics"/>This is the &longs;econd part of the Tenth <lb/>propo&longs;ition; and the other pat is with their Demon&longs;trations, &longs;hall hereafter follow in the &longs;ame Order.<emph.end type="italics"/></s></p><p type="main"> <s><foreign lang="greek">*y</foreign> &longs;hall be greater than <emph type="italics"/>X<emph.end type="italics"/> O, but le&longs;&longs;er than the Exce&longs;s by </s></p><p type="main"> <s><arrow.to.target n="marg1360"></arrow.to.target><lb/>which the Axis is greater than Se&longs;quialter of the Semi-parameter, <lb/>that is than S B.] <emph type="italics"/>This followeth from the 10 of the fifth Book of<emph.end type="italics"/> Euclids <emph type="italics"/>Elements.<emph.end type="italics"/><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 &longs;hall be demon&longs;trated, that M H is double to H N, like as it <lb/>was demon&longs;trated, that O G is double to G X.] <emph type="italics"/>As in the fir&longs;t Conclu&longs;ion <lb/>of this Propo&longs;ition, and from what we have but even now written, thereupon appeareth:<emph.end type="italics"/></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/>Lines A Q and A N are drawn from the Ba&longs;es unto the Portions, <lb/>which Lines contain equall Angles with the &longs;aid Ba&longs;es, Q A &longs;hall <lb/>have the &longs;ame proportion to A N, that L A hath to A D.] <lb/><emph type="italics"/>This we have demonstrated above.<emph.end type="italics"/></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"/>For &longs;ince that Q A is to A N, as L A to <lb/>A D; Dividing and Converting, A N &longs;hall be to N Q as A D to D L: But A D <lb/>is equall to D L; for that D B is &longs;uppo&longs;ed to be the Diameter of the Portion: Therefore<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1364"></arrow.to.target><lb/><emph type="italics"/>al&longs;o<emph.end type="italics"/> (a) <emph type="italics"/>A N is equall to N <expan abbr="q.">que</expan><emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1364"></margin.target>(a) <emph type="italics"/>By 14 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="main"> <s>And A Q parallel to M Y.] <emph type="italics"/>By the fifth of the &longs;econd Book of<emph.end type="italics"/> Apollonius <emph type="italics"/>his Conicks.<emph.end type="italics"/><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 &longs;aid.] </s></p><p type="main"> <s><arrow.to.target n="marg1366"></arrow.to.target><lb/><emph type="italics"/>In the fir&longs;t Conciu&longs;ion of this Propo&longs;ition: And let it be cut in K, &longs;o, as that B K be double to <lb/>K D, and in R &longs;o, as that K R may be equall to the Semi-parameter.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1366"></margin.target>G</s></p><p type="main"> <s>And, &longs;eeing that in the Equall and Like Portions A P O L and <lb/><arrow.to.target n="marg1367"></arrow.to.target><lb/>A <emph type="italics"/>M<emph.end type="italics"/> Q L, the Lines A O and A Q are drawn from the Extremities <lb/>of their Ba&longs;es, &longs;o, as that the Portions cut off, do make equall Angles <pb xlink:href="073/01/059.jpg" pagenum="389"/>with their Diameters; as al&longs;o, the Angles at Y and G being equall; <lb/>Therefore, the Lines Y B and G B, & B C & B S, &longs;hall al&longs;o be equall.] <lb/><emph type="italics"/>Let the Line A Q cut the Diameter D B in<emph.end type="italics"/> <foreign lang="greek">g,</foreign> <emph type="italics"/>and let it cut A O in<emph.end type="italics"/> <foreign lang="greek">d.</foreign> <emph type="italics"/>Now becau&longs;e that in<emph.end type="italics"/><lb/><figure id="id.073.01.059.1.jpg" xlink:href="073/01/059/1.jpg"/><lb/><emph type="italics"/>the equall and like Portions A P O L & A M Q L, <lb/>from the Extremities of their Ba&longs;es, A O and <lb/>A Q are drawn, that contain equall Angles with <lb/>tho&longs;e Ba&longs;es; and &longs;ince the Angles at D, are both <lb/>Right; Therefore, the Remaining Angles A<emph.end type="italics"/> <foreign lang="greek">d</foreign> <emph type="italics"/>D <lb/>and A<emph.end type="italics"/> <foreign lang="greek">g</foreign> D <emph type="italics"/>&longs;hall be equall to one another: But <lb/>the Line P G is parallel unto the Line A O; al&longs;o <lb/>M Y is parallel to A <expan abbr="q;">que</expan> and P S and M C to <lb/>A D: Therefore the Triangles P G S and M Y C, <lb/>as al&longs;o the Triangles A<emph.end type="italics"/> <foreign lang="greek">d</foreign> <emph type="italics"/>D and A<emph.end type="italics"/> <foreign lang="greek">g</foreign> <emph type="italics"/>D, are all <lb/>alike to each other<emph.end type="italics"/>: (b) <emph type="italics"/>And as A D is to A<emph.end type="italics"/> <foreign lang="greek">d,</foreign><lb/><arrow.to.target n="marg1368"></arrow.to.target><lb/><emph type="italics"/>&longs;o is A D to A<emph.end type="italics"/> <foreign lang="greek">g</foreign><emph type="italics"/>: and,<emph.end type="italics"/> Permutando, <emph type="italics"/>the Lines <lb/>A D and A D are equall to each other: Therefore, <lb/>A<emph.end type="italics"/> <foreign lang="greek">d</foreign> <emph type="italics"/>and A<emph.end type="italics"/> <foreign lang="greek">g</foreign> <emph type="italics"/>are al&longs;o equall: But A O and <lb/>A Q are equall to each other; as al&longs;o their halves <lb/>A T and A N: Therefore the Remainders T<emph.end type="italics"/> <foreign lang="greek">d</foreign> <emph type="italics"/>and N<emph.end type="italics"/> <foreign lang="greek">g</foreign><emph type="italics"/>; that is, TG and MY, are al&longs;o<emph.end type="italics"/><lb/><arrow.to.target n="marg1369"></arrow.to.target><lb/><figure id="id.073.01.059.2.jpg" xlink:href="073/01/059/2.jpg"/><lb/><emph type="italics"/>equall. </s> <s>And, as<emph.end type="italics"/> (c) <emph type="italics"/>P G is to G S, &longs;o is M Y to <lb/>Y C: and<emph.end type="italics"/> Permutando, <emph type="italics"/>as P G is to M Y, &longs;o is <lb/>G S to Y C: And, therefore, G S and Y C are <lb/>equall; as al&longs;o their halves B S and B C: From <lb/>whence it followeth, that the Remainders S R and C R <lb/>are al&longs;o equall: And, con&longs;equently, that P Z and <lb/>M V, and V N and Z T, are lkiewi&longs;e equall to one <lb/>another.<emph.end type="italics"/></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"/>By 4. of the <lb/>&longs;ixth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1369"></margin.target>(c) <emph type="italics"/>By 34 of the <lb/>fir&longs;t,<emph.end type="italics"/></s></p><p type="main"> <s>Since, therefore, that N V is le&longs;&longs;er <lb/><arrow.to.target n="marg1370"></arrow.to.target><lb/>than double of V N.] <emph type="italics"/>For M H is double of <lb/>H N, and M V is le&longs;&longs;er than M H: Therefore, M V <lb/>is le&longs;&longs;er than double of H N, and much le&longs;&longs;er than <lb/>double of V N.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1370"></margin.target>K</s></p><p type="main"> <s>Therefore, the Portion &longs;hall not abide, but &longs;hall turn about, <lb/><arrow.to.target n="marg1371"></arrow.to.target><lb/>&longs;o, as that its Ba&longs;e do not in the lea&longs;t touch the Surface of <lb/>the Liquid; in regard that now when it toucheth in but one Point <lb/>only, it moveth upwards on the part towards A.] Tartaglia's <emph type="italics"/>his Tran&longs;la­<lb/>tion hath it thus,<emph.end type="italics"/> Non ergo manet Portio &longs;ed inclinabitur ut Ba&longs;is ip&longs;ius, nec &longs;ecundum <lb/>unum tangat Superficiem Humidi, quon am nunc &longs;ecundum unum tacta ip&longs;a reclina­<lb/>tur<emph type="italics"/>: Which we have thought fit in this manner to correct, from other Places of<emph.end type="italics"/><lb/>Archimedes, <emph type="italics"/>that the &longs;en&longs;e might be the more per&longs;picuous. </s> <s>For in the &longs;ixth Propo&longs;ition of this, <lb/>he thus writeth (as we al&longs;o have it in the Tran&longs;lation,)<emph.end type="italics"/> The Solid A P O L, therefore, &longs;hall <lb/>turn about, and its Ba&longs;e &longs;hall not in the lea&longs;t touch the Surface of the Liquid. <emph type="italics"/>Again, <lb/>in the &longs;eventh Propo&longs;ition<emph.end type="italics"/>; From whence it is manife&longs;t, that its Ba&longs;e &longs;hall turn about in <lb/>&longs;uch manner, a that its Ba&longs;e doth in no wi&longs;e touch the Surface of the Liquid; For <lb/>that now when it toucheth but in one Point only, it moveth downwards on the part <lb/>towards L. <emph type="italics"/>And that the Portion moveth upwards, on the part towards A, doth plainly ap­<lb/>pear: For &longs;ince that the Perpendiculars unto the Surface of the Liquid, that pa&longs;s thorow <foreign lang="greek">w</foreign>, de <lb/>fall on the part towards A, and tho&longs;e that pa&longs;s thorow E, on the part towards L; it is nece&longs;&longs;ary <lb/>that the Centre<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>do move upwards, and the Centre E downwards.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1371"></margin.target>L</s></p><p type="main"> <s>It is therefore per&longs;picuous, that the Portion &longs;hall con&longs;i&longs;t, &longs;o, as that <lb/>its Axis &longs;hall make an Angle with the Liquids Surface greater than <lb/>the Angle <emph type="italics"/>X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter<emph.end type="italics"/> <pb xlink:href="073/01/060.jpg" pagenum="390"/><figure id="id.073.01.060.1.jpg" xlink:href="073/01/060/1.jpg"/><lb/><emph type="italics"/>B D in<emph.end type="italics"/> <foreign lang="greek">l</foreign><emph type="italics"/>; and from the Point O, and parallel to <lb/>A<emph.end type="italics"/> <foreign lang="greek">l,</foreign> <emph type="italics"/>draw O X; and let it touch the Section in O, <lb/>as in the first Figure: And the<emph.end type="italics"/> (d) <emph type="italics"/>Angle at X,<emph.end type="italics"/><lb/><arrow.to.target n="marg1372"></arrow.to.target><lb/><emph type="italics"/>&longs;hall be equall al&longs;o to the angle<emph.end type="italics"/> <foreign lang="greek">l</foreign><emph type="italics"/>: But the angle at Y <lb/>is equall to the Angle at<emph.end type="italics"/> <foreign lang="greek">g;</foreign> <emph type="italics"/>and the<emph.end type="italics"/> (e) <emph type="italics"/>Angle<emph.end type="italics"/><lb/><arrow.to.target n="marg1373"></arrow.to.target><lb/>A <foreign lang="greek">*g</foreign> D <emph type="italics"/>greater than the Angle A<emph.end type="italics"/> <foreign lang="greek">l</foreign> <emph type="italics"/>D, which falleth <lb/>without it: Therefore the Angle at Y &longs;hall be great­<lb/>er than that at X. </s> <s>And becau&longs;e now the Portion <lb/>turneth about, &longs;o, as that the Ba&longs;e doth not touch <lb/>the Liquid, the Axis &longs;hall make an Angle with its <lb/>Surface greater than the Angle G; that is, than the <lb/>Angle Y: And, for that rea&longs;on, much greater than <lb/>the Angle X.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1372"></margin.target>(d) <emph type="italics"/>By 29 of the <lb/>fir&longs;t.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1373"></margin.target>(e) <emph type="italics"/>By 16. of the <lb/>fir&longs;t.<emph.end type="italics"/></s></p><p type="head"> <s>CONCLUSION III.</s></p><p type="main"> <s><emph type="italics"/>If the Portion have the &longs;ame proportion in Gravity to the <lb/>Liquid, that the Square X O hath to the Square<emph.end type="italics"/><lb/>BD, <emph type="italics"/>being demitted into the Liquid, &longs;o inclined, as that <lb/>its Ba&longs;e touch not the Liquid, it &longs;hall &longs;tand and <lb/>continue inclined, &longs;o, as that its Ba&longs;e touch the Sur­<lb/>face of the Liquid, in one Point only, and its Axis &longs;hall <lb/>make an Angle with the Liquids Surface equall to the <lb/>Angle X. And, if the Portion have the &longs;ame proportion <lb/>in Gravity to the Liquid, that the Square P F hath <lb/>to the Square B D, being demitted into the Liquid, <lb/>& &longs;et &longs;o inclined, as that its Ba&longs;e touch not the Liquid, <lb/>it &longs;hall &longs;tand inclined, &longs;o, as that its Ba&longs;e touch the <lb/>Surface of the Liquid in one Point only, & its Axis &longs;hall <lb/>make an Angle with it, equall to the Angle<emph.end type="italics"/> <foreign lang="greek">*f.</foreign></s></p><p type="main"> <s>Let the Portion have the &longs;ame proportion in Gravity to tho <lb/>Liquid that the Square <emph type="italics"/>X<emph.end type="italics"/>O hath to the Square B D; and let <lb/>it be demitted into the Liquid &longs;o inclined, as that its Ba&longs;e touch <lb/><figure id="id.073.01.060.2.jpg" xlink:href="073/01/060/2.jpg"/><lb/>not the Liquid. </s> <s>And cutting it by <lb/>a Plane thorow the Axis, erect unto <lb/>the Surface of the Liquid, let the <lb/>Section of the Solid, be the Section <lb/>of a Right-angled Cone, A P M L; <lb/>let the Section of the Surface of the <lb/>Liquid be I M; and the Axis of the <lb/>Portion and Diameter of the Section <lb/>B D; and let B D be divided as be­<lb/>fore; and draw PN parallel to IM <pb xlink:href="073/01/061.jpg" pagenum="391"/>and touching the Section in P, and T P parallel to B D; and P S perpen­<lb/>dicular unto B D. </s> <s>It is to be demon&longs;trated that the Portion &longs;hall <lb/><figure id="id.073.01.061.1.jpg" xlink:href="073/01/061/1.jpg"/><lb/>not &longs;tand &longs;o, but &longs;hall encline until <lb/>that the Ba&longs;e touch the Surface of <lb/>the Liquid, in one Point only, for let <lb/>the &longs;uperior figure &longs;tand as it was, <lb/>and draw O C, Perpendicular to B D; <lb/>and drawing a <emph type="italics"/>L<emph.end type="italics"/>ine from A to <emph type="italics"/>X,<emph.end type="italics"/><lb/>prolong it to Q: A X &longs;halbe equall <lb/>to <emph type="italics"/>X<emph.end type="italics"/> <expan abbr="q.">que</expan> Then draw O X parallel <lb/>to A <expan abbr="q.">que</expan> And becau&longs;e the Portion <lb/>is &longs;uppo&longs;ed to have the &longs;ame pro­<lb/>portion in Gravity to the Liquid <lb/>that the &longs;quare X O hath to the <lb/>Square B D; the part thereof &longs;ubmerged &longs;hall al&longs;o have the &longs;ame <lb/>proportion to the whole; that is, the Square T P to the Square <lb/><arrow.to.target n="marg1374"></arrow.to.target><lb/>B D; and &longs;o T P &longs;hall be equal to <emph type="italics"/>X<emph.end type="italics"/> O: And &longs;ince that of the <emph type="italics"/>P<emph.end type="italics"/>ortions <lb/>I P M and A O Q the Diameters are equall, the portions &longs;hall al&longs;o be <lb/><arrow.to.target n="marg1375"></arrow.to.target><lb/>equall. <emph type="italics"/>A<emph.end type="italics"/>gain, becau&longs;e that in the Equall and <emph type="italics"/>L<emph.end type="italics"/>ike <emph type="italics"/>P<emph.end type="italics"/>ortions A O Q L <lb/><arrow.to.target n="marg1376"></arrow.to.target><lb/>and AP ML the Lines A Q and I M, which cut off equall <emph type="italics"/>P<emph.end type="italics"/>or­<lb/>tions, are drawn, that, from the Extremity of the <emph type="italics"/>B<emph.end type="italics"/>a&longs;e, and this <lb/>not from the Extremity; it appeareth that that which is drawn from <lb/>the end or Extremity of the <emph type="italics"/>B<emph.end type="italics"/>a&longs;e, &longs;hall make the Acute Angle with <lb/>the Diameter of the whole <emph type="italics"/>P<emph.end type="italics"/>ortion le&longs;set. <emph type="italics"/>A<emph.end type="italics"/>nd the Angle at <emph type="italics"/>X<emph.end type="italics"/><lb/><arrow.to.target n="marg1377"></arrow.to.target><lb/>being le&longs;&longs;e than the Angle at N, B C &longs;hall be greater than B S; and <lb/>C R le&longs;&longs;er than S R: <emph type="italics"/>A<emph.end type="italics"/>nd, therfore O G &longs;hall be le&longs;&longs;er than P Z; <lb/>and G <emph type="italics"/>X<emph.end type="italics"/> greater than Z T: Therfore P Z is greater than double of <lb/>Z T; being that O G is double of G X. </s> <s>Let P H be double to H T; <lb/>and drawing a Line from H to K, prolong it to <foreign lang="greek">w.</foreign> The Center of <lb/>Gravity of the whole Portion &longs;hall be K; the Center of the part <lb/>which is within the Liquid H, and that of the part which is above <lb/>the Liquid in the Line K <foreign lang="greek">w</foreign>; which &longs;uppo&longs;ed to be <foreign lang="greek">w.</foreign> Therefore it <lb/>&longs;hall be demon&longs;trated, both, that K H is perpendicular to the Surface <lb/>of the Liquid, and tho&longs;e Lines al&longs;o that are drawn thorow the Points <lb/>Hand <foreign lang="greek">w</foreign> parallel to K H: And therfore the Portion &longs;hall not re&longs;t, but <lb/>&longs;hall encline untill that its Ba&longs;e do touch the Surface of the Liquid <lb/>in one Point; and &longs;o it &longs;hall continue. </s> <s>For in the Equall Portions <lb/>A O Q L and A P M L, the <lb/><figure id="id.073.01.061.2.jpg" xlink:href="073/01/061/2.jpg"/><lb/>Lines A Q and A M, that cut off <lb/>equall Portions, &longs;hall be dawn <lb/>from the Ends or Terms of the Ba&longs;es; <lb/>and A O Q and A P M &longs;hall be <lb/>demon&longs;trated, as in the former, to <lb/><arrow.to.target n="marg1378"></arrow.to.target><lb/>be equall: Therfore A Q and A M, <lb/>do make equall Acute Angles with <lb/>the Diameters of the Portions; and <pb xlink:href="073/01/062.jpg" pagenum="392"/>the Angles at X and N are equall. </s> <s>And, therefore, if drawing HK, <lb/>it be prolonged to <foreign lang="greek">w,</foreign> the Centre of Gravity of the whole Portion &longs;hall <lb/>be K; of the part which is within the Liquid H; and of the part which <lb/>is above the Liquid in K <foreign lang="greek">w)</foreign> as &longs;uppo&longs;e in <foreign lang="greek">w;</foreign> and H K perpendicular to <lb/><figure id="id.073.01.062.1.jpg" xlink:href="073/01/062/1.jpg"/><lb/>the Surface of the Liquid. </s> <s>Therfore <lb/>along the &longs;ame Right Lines &longs;hall the <lb/>part which is within the Liquid move <lb/>upwards, and the part above it down­<lb/>wards: And therfore the Portion <lb/>&longs;hall re&longs;t with one of its Points <lb/>touching the Surface of the Liquid, <lb/>and its Axis &longs;hall make with the <lb/><arrow.to.target n="marg1379"></arrow.to.target><lb/>&longs;ame an Angle equall to X. </s> <s>It is <lb/>to be demon&longs;trated in the &longs;ame <lb/>manner that the Portion that hath <lb/>the &longs;ame proportion in Gravity to the Liquid, that the Square P F hath <lb/>to the Square B D, being demitted into the Liquid, &longs;o, as that its <lb/>Ba&longs;e touch not the Liquid, it &longs;hall &longs;tand inclined, &longs;o, as that its Ba&longs;e <lb/>touch the Surface of the Liquid in one Point only; and its Axis &longs;hall <lb/>make therwith an Angle equall to the Angle <foreign lang="greek">f.</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/><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"/>By the twenty &longs;ixth of the Book<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1381"></arrow.to.target><lb/><emph type="italics"/>of<emph.end type="italics"/> Archimedes, De Conoidibus & Sphæroidibus: <emph type="italics"/>Therefore, (a) the Square T P <lb/>&longs;hall be equall to the Square X O: And for that rea&longs;on, the Line T P equall to the <lb/>Line X O.<emph.end type="italics"/><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"/>By 9 of the <lb/>fifth.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1382"></margin.target>B</s></p><p type="main"> <s>The Portions &longs;hall al&longs;o be equall.] <emph type="italics"/>By the twenty fifth of the &longs;ame Book.<emph.end type="italics"/></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, becau&longs;e that in the Equall and Like Portions, A O Q L <lb/>and A P M L.] <emph type="italics"/>For, in the Portion A P M L, de&longs;cribe the Portion A O Q equall <lb/>to the Portion I P M: The Point Q falleth beneath M; for otherwi&longs;e, the Whole would be <lb/>equall to the Part. </s> <s>Then draw I V parallel to A Q, and cutting the Diameter is<emph.end type="italics"/> <foreign lang="greek">y;</foreign> <emph type="italics"/>and <lb/>let I M cut the &longs;ame<emph.end type="italics"/> <foreign lang="greek">s;</foreign> <emph type="italics"/>and A Q in<emph.end type="italics"/> <foreign lang="greek">s.</foreign> <emph type="italics"/>I &longs;ay <lb/>that the Angle A<emph.end type="italics"/> <foreign lang="greek">u</foreign> <emph type="italics"/>D, is le&longs;&longs;er than the Angle<emph.end type="italics"/><lb/><figure id="id.073.01.062.2.jpg" xlink:href="073/01/062/2.jpg"/><lb/><emph type="italics"/>I<emph.end type="italics"/> <foreign lang="greek">s</foreign> <emph type="italics"/>D. </s> <s>For the Angle I<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>D is equall to the <lb/>Angle A<emph.end type="italics"/> <foreign lang="greek">u</foreign> <emph type="italics"/>D: (b) But the interiour Angle<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1384"></arrow.to.target><lb/><emph type="italics"/>I<emph.end type="italics"/> <foreign lang="greek">y</foreign> <emph type="italics"/>D is le&longs;&longs;er than the exteriour I<emph.end type="italics"/> <foreign lang="greek">s</foreign> <emph type="italics"/>D: There-<emph.end type="italics"/><lb/><arrow.to.target n="marg1385"></arrow.to.target><lb/><emph type="italics"/>fore, (c) A<emph.end type="italics"/> <foreign lang="greek">u</foreign> <emph type="italics"/>D &longs;hall al&longs;o be lefter than I<emph.end type="italics"/> <foreign lang="greek">s</foreign> <emph type="italics"/>D.<emph.end type="italics"/><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"/>By 29 of the <lb/>fir&longs;t.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1385"></margin.target><emph type="italics"/>(c) By 16 of the <lb/>fir&longs;t.<emph.end type="italics"/></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 le&longs;&longs;e <lb/>than the Angle at N.] <emph type="italics"/>Thorow O draw twe <lb/>Lines, O C perpendicular to the Diameter B D, and <lb/>O X touching the Section in the Point O, and cutting<emph.end type="italics"/></s></p><p type="main"> <s><arrow.to.target n="marg1387"></arrow.to.target><lb/><emph type="italics"/>the Diameter in X: (d) O X &longs;hall be parallel <lb/>to A <expan abbr="q;">que</expan> and the<emph.end type="italics"/> (e) <emph type="italics"/>Angle at X, &longs;hall be equall to<emph.end type="italics"/><lb/><arrow.to.target n="marg1388"></arrow.to.target><lb/><emph type="italics"/>that at<emph.end type="italics"/> <foreign lang="greek">u</foreign>: <emph type="italics"/>Therefore, the<emph.end type="italics"/> (f) <emph type="italics"/>Angle at X,<emph.end type="italics"/><lb/><arrow.to.target n="marg1389"></arrow.to.target><lb/><emph type="italics"/>&longs;hall be le&longs;&longs;er than the Angle at<emph.end type="italics"/> <foreign lang="greek">s;</foreign> <emph type="italics"/>that is, to <lb/>that at N: And, con&longs;equently, X &longs;hall fall beneath N: Therefore, the Line X B is greater than <lb/>N B. And, &longs;ince B C is equall to X B, and B S equall to N B; B C &longs;hall be greater than B S.<emph.end type="italics"/></s></p> <pb xlink:href="073/01/063.jpg" pagenum="397"/><p type="margin"> <s><margin.target id="marg1387"></margin.target><emph type="italics"/>(d) By 5 of our &longs;e­<lb/>cond of<emph.end type="italics"/> Conicks.</s></p><p type="margin"> <s><margin.target id="marg1388"></margin.target>(e) <emph type="italics"/>By 29 of the <lb/>fir&longs;t.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1389"></margin.target>(f) <emph type="italics"/>By 39 of our <lb/>fir&longs;t of<emph.end type="italics"/> Conicks.</s></p><p type="main"> <s>Therefore, A Q and A M do make equall Acute Angles with <lb/><arrow.to.target n="marg1390"></arrow.to.target><lb/>the Diameters of the Portions.] <emph type="italics"/>We demon&longs;trate this as in the Commentaries <lb/>upon the &longs;econd Conclu&longs;ion.<emph.end type="italics"/></s></p><p type="margin"> <s><margin.target id="marg1390"></margin.target>E</s></p><p type="main"> <s>It is to be demon&longs;trated in the &longs;ame manner, that the Portion <lb/><arrow.to.target n="marg1391"></arrow.to.target><lb/>that hath the &longs;ame proportion in Gravity to the Liquid, that the <lb/>Square P F hath to the Square B D, <lb/>being demitted into the Liquid, &longs;o, <lb/><figure id="id.073.01.063.1.jpg" xlink:href="073/01/063/1.jpg"/><lb/>as that its Ba&longs;e touch not the Li­<lb/>quid, it &longs;hall &longs;tand inclined, &longs;o, as <lb/>that its Ba&longs;e touch the Surface of the <lb/>Liquid in one point only; and its Axis <lb/>&longs;hall make therewith an angle equall <lb/>to the Angle <foreign lang="greek">f.</foreign>] <emph type="italics"/>Let the Portion be to the <lb/>Liquid in Gravity, as the Square P F to the <lb/>Square B D: and being demitted into the <lb/>Liquid, &longs;o inclined, as that its Ba&longs;e touch not <lb/>the Liquid, let it be cut thorow the Axis by a <lb/>Plane erect to the Surface of the Liquid, that <lb/>that the Section may be A M O L, the Section <lb/>of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit <lb/>of the Portion and Diameter of the Section B D; which let be cut into the &longs;ame parts as <lb/>we &longs;aid before, and draw M N parallel to I O, that it may touch the Section in the Point <lb/>M; and M T parallel to B D, and P M S perpe ndicular to the &longs;ame. </s> <s>It is to be demon­<lb/>strated, that the Portion &longs;hall not re&longs;t, but &longs;hall incline, &longs;o, as that it touch the Liquids <lb/>Surface, in one Point of its Ba&longs;e only. </s> <s>For,<emph.end type="italics"/><lb/><figure id="id.073.01.063.2.jpg" xlink:href="073/01/063/2.jpg"/><lb/><emph type="italics"/>draw P C perpendicular to B D; and drawing <lb/>a Line from A to F, prolong it till it meet with <lb/>the Section in <expan abbr="q;">que</expan> and thorow P draw P<emph.end type="italics"/> <foreign lang="greek">f</foreign> <emph type="italics"/>pa­<lb/>rallel to A Q: Now, by the things allready de­<lb/>mon&longs;trated by us, A F and F Q &longs;hall be equall <lb/>to one another. </s> <s>And being that the Portion hath <lb/>the &longs;ame proportion in Gravity unto the Liquid, <lb/>that the Square P F hath to the Square B D; and <lb/>&longs;eeing that the part &longs;ubmerged, hath the &longs;ame pro-<emph.end type="italics"/><lb/><arrow.to.target n="marg1392"></arrow.to.target><lb/><emph type="italics"/>partion to the whole Portion; that is, the Squàre <lb/>M T to the Square B D; (g) the Square M T <lb/>&longs;hall be equall to the Square P F; and, by the <lb/>&longs;ame rea&longs;on, the Line M T equall to the Line <lb/>P F. </s> <s>So that there being drawn in the equall & like <lb/>portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the <lb/>fir&longs;t from the Extreme term of the Ba&longs;e, the la&longs;t not from the Extremity; it followeth, that <lb/>A Q drawn from the Extremity, containeth a le&longs;&longs;er Acute Angle with the Diameter of the <lb/>Portion, than I O: But the Line P<emph.end type="italics"/> <foreign lang="greek">f</foreign> <emph type="italics"/>is parallel to the Line A Q, and M N to I O: There­<lb/>fore, the Angle at<emph.end type="italics"/> <foreign lang="greek">f</foreign> <emph type="italics"/>&longs;hall be le&longs;&longs;er than the Angle at N; but the Line B C greater than B S; <lb/>and S R, that is, M X, greater than C R, that is, than P Y: and, by the &longs;ame rea&longs;on, X T <lb/>le&longs;&longs;er than Y F. And, &longs;ince P Y is double to Y F, M X &longs;hall be greater than double to <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/>Line from H to K, prolonging it. </s> <s>Now, the Centre of Gravity of the whole Portion <lb/>&longs;hall be the Point K; of the part within the Liquid H; and of the Remaining part above <lb/>the Liquid in the Line H K produced, as &longs;uppo&longs;e in<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>It &longs;hall be demon&longs;trated in the &longs;ame <lb/>manner, as before, that both the Line K H and tho&longs;e that are drawn thorow the Points H <lb/>and<emph.end type="italics"/> <foreign lang="greek">w</foreign> <emph type="italics"/>parallel to the &longs;aid K H, are perpendicular to the Surface of the Liquid: The <lb/>Portion therefore, &longs;hall not re&longs;t; but when it &longs;hall be enclined &longs;o far as to touch the Sur­<lb/>face of the Liquid in one Point and no more, then it &longs;hall &longs;tay. </s> <s>For the Angle at N<emph.end type="italics"/> <pb xlink:href="073/01/064.jpg" pagenum="398"/><figure id="id.073.01.064.1.jpg" xlink:href="073/01/064/1.jpg"/><lb/><emph type="italics"/>&longs;hall be equall to the Angle at<emph.end type="italics"/> <foreign lang="greek">f;</foreign> <emph type="italics"/>and the Line B S <lb/>equall to the Line B C; and S R to C R: Where­<lb/>fore, M H &longs;hall be likewi&longs;e equall to P Y. There­<lb/>fore, having drawn HK and prolonged it; the <lb/>Centre of Gravity of the whole Portion &longs;hall be <lb/>K; of that which is in the Liquid H; and of <lb/>that which is above it, the Centre &longs;hall be in <lb/>the Line prolonged: let it be in<emph.end type="italics"/> <foreign lang="greek">w.</foreign> <emph type="italics"/>There­<lb/>fore, along that &longs;ame Line K H, which is per­<lb/>pendicular to the Surface of the Liquid, &longs;hall <lb/>the part which is within the Liquid move up­<lb/>wards, and that which is above the Liquld <lb/>downwards: And, for this cau&longs;e, the Portion, <lb/>&longs;hall be no longer moved, but &longs;hall &longs;tay, and <lb/>re&longs;t, &longs;o, as that its Ba&longs;e do touch the Liquids Surface in but one Point; and its Axis <lb/>maketh an Angle therewith equall to the Angle<emph.end type="italics"/> <foreign lang="greek">f</foreign><emph type="italics"/>; And, this is that which we were to <lb/>demon&longs;trate.<emph.end type="italics"/></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"/>By 9 of t <lb/>fifth.<emph.end type="italics"/></s></p><p type="head"> <s>CONCLVSION IV.</s></p><p type="main"> <s><emph type="italics"/>If the Portion have greater proportion in Gravity <lb/>to the Liquid, than the Square F P to the Square <lb/>B D, but le&longs;&longs;er than that of the Square X O to the <lb/>Square B D, being demitted into the Liquid, <lb/>and inclined, &longs;o, as that its Ba&longs;e touch not the <lb/>Liquid, it &longs;hall &longs;tand and re&longs;t, &longs;o, as that its Ba&longs;e <lb/>&longs;hall be more &longs;ubmerged in the Liquid.<emph.end type="italics"/></s></p><p type="main"> <s>Again, let the Portion have greater proportion in <lb/>Gravity to the Liquid, than the Square F P to the <lb/>Square B D, but le&longs;&longs;er than that of the Square X O to <lb/>the Square B D; and as the Portion is in Gravity to the Liquid, <lb/>&longs;o let the Square made of the Line <foreign lang="greek">y</foreign> be to the Square B D. <foreign lang="greek">*y</foreign><lb/>&longs;hall be greater than F P, and le&longs;&longs;er than X O. Apply, therefore, <lb/>the right Line I V to fall betwixt the Portions A V Q L and A X D; <lb/>and let it be equall to <foreign lang="greek">y,</foreign> and parallel to B D; and let it meet <lb/>the Remaining Section in Y: V Y &longs;hall al&longs;o be proved double <lb/>to Y I, like as it hath been demon&longs;trated, that O G is double off <lb/>G X. And, draw from V, the Line V <foreign lang="greek">w,</foreign> touching the Section <lb/>A V Q L in V; and drawing a Line from A to I, prolong it unto <lb/><expan abbr="q.">que</expan> We prove in the &longs;ame manner, that the Line A I is equall <lb/>to I <expan abbr="q;">que</expan> and that A Q is parallel to V <foreign lang="greek">w.</foreign> It is to be demon&longs;trated, <lb/>that the Portion being demitted into the Liquid, and &longs;o inclined, <lb/>as that its Ba&longs;e touch not the Liquid, &longs;hall &longs;tand, &longs;o, that its Ba&longs;e <lb/>&longs;hall be more &longs;ubmerged in the Liquid, than to touch it Surface in <pb xlink:href="073/01/065.jpg" pagenum="399"/>but one Point only. </s> <s>For let it be de­<lb/><figure id="id.073.01.065.1.jpg" xlink:href="073/01/065/1.jpg"/><lb/>mitted into the Liquid, as hath been <lb/>&longs;aid; and let it fir&longs;t be &longs;o inclined, as <lb/>that its Ba&longs;e do not in the lea&longs;t <lb/>touch the Surface of the Liquid. </s> <s>And <lb/>then it being cut thorow the Axis, <lb/>by a Plane erect unto the Surface of <lb/>the Liquid, let the Section of the <lb/>Portion be A N Z G; that of the <lb/>Liquids Surface E Z; the Axis of <lb/>the Portion and Diameter of the <lb/>Section B D; and let B D be cut in <lb/>the Points K and R, as before; and <lb/>draw N L parallel to E Z, and touching the Section A N Z G <lb/>in N, and N S perpendicular to <lb/><figure id="id.073.01.065.2.jpg" xlink:href="073/01/065/2.jpg"/><lb/>B D. Now, &longs;eeing that the Por­<lb/>tion is in Gravity unto the Liquid, <lb/>as the Square made of the Line <lb/>is to the Square B D; <foreign lang="greek">y</foreign> &longs;hall <lb/>be equall to N T: Which is to <lb/>be demon&longs;trated as above: And, <lb/>therefore, N T is al&longs;o equall to <lb/>V I: The Portions, therefore, <lb/>A V Q and E N Z are equall to <lb/>one another. </s> <s>And, &longs;ince that in <lb/>the Equall and like Portions A V <lb/>Q L and A N Z G, there are drawn A Q and E Z, cutting off <lb/>equall Portions, that from the <lb/><figure id="id.073.01.065.3.jpg" xlink:href="073/01/065/3.jpg"/><lb/>Extremity of the Ba&longs;e, this not <lb/>from the Extreme, that which is <lb/>drawn from the Extremity of the <lb/>Ba&longs;e, &longs;hall make the Acute Angle <lb/>with the Diameter of the Portion <lb/>le&longs;&longs;er: and in the Triangles N L S <lb/>and V <foreign lang="greek">w</foreign> C, the Angle at L is <lb/>greater than the Angle at <foreign lang="greek">w</foreign>: <lb/>Therefore, B S &longs;hall be le&longs;&longs;er <lb/>than B C; and S R le&longs;&longs;er than <lb/>C R: and, con&longs;equently, N X <lb/>greater than V H; and X T le&longs;&longs;er than H I. Seeing, therefore, <lb/>that V Y is double to Y I; It is manife&longs;t, that N X is greater than <lb/>double to X T. </s> <s>Let N M be double to M T: It is manife&longs;t, from what <lb/>hath been &longs;aid, that the Portion &longs;hall not re&longs;t, but will incline, untill <lb/>that its Bafe do touch the Surface of the Liquid: and it toucheth it in <lb/>one Point only, as appeareth in the Figure: And other things <pb xlink:href="073/01/066.jpg" pagenum="400"/><figure id="id.073.01.066.1.jpg" xlink:href="073/01/066/1.jpg"/><lb/>&longs;tanding as before, we will again <lb/>demon&longs;trate, that N T is equall to <lb/>V I; and that the Portions A V Q <lb/>and A N Z are equall to each other. <lb/></s> <s>Therefore, in regard, that in the <lb/>Equall and Like Portions A V Q L <lb/>and A N Z G, there are drawn <lb/>A Q and A Z cutting off equall Por­<lb/>tions, they &longs;hall with the Diameters <lb/>of the Portions, contain equall <lb/>Angles. </s> <s>Therefore, in the Triangles <lb/>N L S and V <foreign lang="greek">w</foreign> C, the Angles at <lb/>the Points <emph type="italics"/>L<emph.end type="italics"/> and <foreign lang="greek">w</foreign> are equall; and the Right Line B S equall to <lb/>B C; S R to C R; N X to V H; and X T to H I: And, &longs;ince <lb/>V Y is double to Y I, N X &longs;hall be greater than double of X T. <lb/></s> <s>Let therefore, N M be double to M T. </s> <s>It is hence again manife&longs;t, <lb/>that the Portion will not remain, but &longs;hall incline on the part <lb/>towards A: But it was &longs;uppo&longs;ed, that the &longs;aid Portion did <lb/>touch the Surface of the Liquid in one &longs;ole Point: Therefore, <lb/>its Ba&longs;e mu&longs;t of nece&longs;&longs;ity &longs;ubmerge farther into the Liquid.</s></p><p type="head"> <s>CONCLVSION V.</s></p><p type="main"> <s><emph type="italics"/>If the Portion have le&longs;&longs;er proportion in Gravity to <lb/>the Liquid, than the Square F P to the Square <lb/>B D, being demitted into the Liquid, and in­<lb/>clined, &longs;o, as that its Ba&longs;e touch not the Liquid, <lb/>it &longs;hall &longs;tand &longs;o inclined, as that its Axis &longs;hall <lb/>make an Angle with the Surface of the Liquid, <lb/>le&longs;&longs;e than the Angle<emph.end type="italics"/> <foreign lang="greek">y;</foreign> <emph type="italics"/>And its Ba&longs;e &longs;hall <lb/>not in the lea&longs;t touch the Liquids Surface.<emph.end type="italics"/></s></p><p type="main"> <s>Finally, let the Portion have le&longs;&longs;er proportion to the Liquid <lb/>in Gravity, than the Square F P hath to the Square B D; and <lb/>as the Portion is in Gravity to the Liquid, &longs;o let the <lb/>Square made of the Line <foreign lang="greek">y</foreign> be to the Square B D. <foreign lang="greek">y</foreign> &longs;hall be <lb/>le&longs;&longs;er than P F. Again, apply any Right Line as G I, falling <lb/>betwixt the Sections A G Q L and A X D, and parallel to B D; <lb/>and let it cut the Middle Conick Section in the Point H, and <pb xlink:href="073/01/067.jpg" pagenum="401"/>the Right Line R Y in Y. </s> <s>We <lb/><figure id="id.073.01.067.1.jpg" xlink:href="073/01/067/1.jpg"/><lb/>&longs;hall demon&longs;trate G H to be double <lb/>to H I, as it hathbeen demon&longs;tra­<lb/>ted, that O G is double to G X. <lb/></s> <s>Then draw G <foreign lang="greek">w</foreign> touching the Section <lb/>A G Q L in G; and G C perpen di­<lb/>cular to B D; and drawing a Line <lb/>from A to I, prolong it to <expan abbr="q.">que</expan> Now <lb/>A I &longs;hall be equall to I <expan abbr="q;">que</expan> and <lb/>A Q parallel to G <foreign lang="greek">w.</foreign> It is to be <lb/>demon&longs;trated, that the Portion being <lb/>demitted into the Liquid, and inclined, &longs;o, as that its Ba&longs;e touch <lb/>the Liquid, it &longs;hall &longs;tand &longs;o incli­<lb/><figure id="id.073.01.067.2.jpg" xlink:href="073/01/067/2.jpg"/><lb/>ned, as that its Axis &longs;hall make <lb/>an Angle with the Surface of the <lb/>Liquid le&longs;&longs;e than the Angle <foreign lang="greek">f;</foreign><lb/>and its Ba&longs;e &longs;hall not in the lea&longs;t <lb/>touch the Liquids Surface. </s> <s>For <lb/>let it be demitted into the Liquid, <lb/>and let it &longs;tand, &longs;o, as that its Ba&longs;e <lb/>do touch the Surface of the Liquid <lb/>in one Point only: and the Portion <lb/>being cut thorow the Axis by a <lb/>Plane erect unto the Surface of the Liquid, let the Section of <lb/><figure id="id.073.01.067.3.jpg" xlink:href="073/01/067/3.jpg"/><lb/>the Portion be A N Z L, the Section <lb/>of a Rightangled Cone; that of <lb/>the Surface of the Liquid A Z; and <lb/>the Axis of the Portion and Dia­<lb/>meter of the Section B D; and let <lb/>B D be cut in the Points K and R <lb/>as hath been &longs;aid above; and draw <lb/>N F parallel to A Z, and touching <lb/>the Section of the Cone in the Point <lb/>N; and N T parallel to B D; and <lb/>N S perpendicular to the &longs;ame. </s> <s>Be­<lb/>cau&longs;e, now, that the Portion is in Gravity to the Liquid, as <lb/>the Square made of <foreign lang="greek">y</foreign> is to the Square B D; and &longs;ince that as the <lb/>Portion is to the Liquid in Gravity, &longs;o is the Square N T to the <lb/>Square B D, by the things that have been &longs;aid; it is plain, that <lb/>N T is equall to the Line <foreign lang="greek">y</foreign>: And, therefore, al&longs;o, the Portions <lb/>A N Z and A G Q are equall. </s> <s>And, &longs;eeing that in the Equall and <lb/>Like Portions A G Q L and A N Z L; there are drawn from the <lb/>Extremities of their Ba&longs;es, A Q and A Z which cut off equall Porti­<lb/>ons: It is obvious, that with the Diameters of the Portions they <pb xlink:href="073/01/068.jpg" pagenum="402"/>make equall Angles; and that in the Triangles N F S and G <foreign lang="greek">w</foreign> C <lb/>the Angles at F and <foreign lang="greek">w</foreign> are equall; as al&longs;o, that S B and B C, and<lb/>S R and C R are equall to one another: And, therefore, N X and<lb/>G Y are al&longs;o equall; and X T and Y I. </s> <s>And &longs;ince G H is double<lb/>to H I, N X &longs;hall be le&longs;&longs;er than double of X T. </s> <s>Let N M therefore<lb/>be double to M T; and drawing a Line from M to K, prolong it<lb/>unto E. </s> <s>Now the Centre of Gravity of the whole &longs;hall be the<lb/>Point K; of the part which is in the Liquid the Point M; and<lb/>that of the part which is above the Liquid in the Line prolonged <lb/>as &longs;uppo&longs;e in E. Therefore, by what was even now demon&longs;trated <lb/>it is manife&longs;t that the Portion &longs;hall not &longs;tay thus, but &longs;hall incline, &longs;o <lb/>as that its Ba&longs;e do in no wi&longs;e touch the Surface of the Liquid <lb/>And that the Portion will &longs;tand, &longs;o, as to make an Angle with the<lb/>Surface of the Liquid le&longs;&longs;er than<lb/><figure id="id.073.01.068.1.jpg" xlink:href="073/01/068/1.jpg"/><lb/>the Angle <foreign lang="greek">f,</foreign> &longs;hall thus be demon <lb/>&longs;trated. </s> <s>Let it, if po&longs;&longs;ible, &longs;tand,<lb/>&longs;o, as that it do not make an Angle<lb/>le&longs;&longs;er than the Angle <foreign lang="greek">f;</foreign> and di&longs;po&longs;e<lb/>all things el&longs;e in the &longs;ame manner a <lb/>before; as is done in the pre&longs;et <lb/>Figure. </s> <s>We are to demon&longs;trat <lb/>in the &longs;ame method, that N T is e­<lb/>quall to <foreign lang="greek">y;</foreign> and by the &longs;ame rea&longs;or <lb/>equall al&longs;o to G I. </s> <s>And &longs;ince that in<lb/>the Triangles P <foreign lang="greek">f</foreign> C and N F S, the Angle F is not le&longs;&longs;er than the<lb/>Angle <foreign lang="greek">f,</foreign> B F &longs;hall not be greater than B C: And, therefore, neither<lb/>&longs;hall S R be le&longs;&longs;er than C R; nor N X than P Y: But &longs;ince P F is<lb/>greater than N T, let P F be Se&longs;quialter of P Y: N T &longs;hall be le&longs;&longs;er<lb/>than Se&longs;quialter of N X: And, therefore, N X &longs;hall be greate <lb/>than double of X T. </s> <s>Let N M be double of M T; and drawing <lb/>Line from M to K prolong it. </s> <s>It is manife&longs;t, now, by what hath<lb/>been &longs;aid, that the Portion &longs;hall not continue in this po&longs;ition, but &longs;hall<lb/>turn about, &longs;o, as that its Axis do make an Angle with the Surface<lb/>of the Liquid, le&longs;&longs;er than the Angle <foreign lang="greek">f.</foreign> </s></p></chap> </body> <back></back> </text></archimedes>