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Removing DESpecs directory which deserted to git
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:stevi_stati_527_la_1605.xml</dcterms:identifier> <dcterms:creator identifier="GND:11875372X">Stevin, Simon</dcterms:creator> <dcterms:title xml:lang="la">Tomus quartus mathematicorum hypomnematum de statica</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1605</dcterms:date> <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> <dcterms:rights>CC-BY-SA</dcterms:rights> <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> <echodir>/permanent/archimedes_repository/large/stevi_stati_527_la_1605</echodir> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="527.01.001" n="1"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">TOMVS <lb/>QVARTVS</head> <head xml:id="echoid-head2" xml:space="preserve">MATHEMATICORVM <lb/>HYPOMNEMATVM <lb/>DE <lb/>STATICA.</head> <head xml:id="echoid-head3" style="it" xml:space="preserve">Quo comprehenduntur ea in quibus ſeſe exercuit</head> <head xml:id="echoid-head4" xml:space="preserve">ILLVSTRISSIMVS</head> <head xml:id="echoid-head5" xml:space="preserve">Illuſtriſsimo & antiquiſsimo ſtemmate ortus Princeps ac <lb/>Dominus M*AURITIUS* Princeps Auraicus, Comes <lb/>Naſſoviæ, Catti melibocorum, Viandę, Moerſii, & c. Marchio Veræ <lb/>& Vliſſingæ, & c. Dominus Civitatis Gravæ & ditionis Cuyc, <lb/>Civitatum Vyt, Daesburch, & c. Gubernator Geldriæ, <lb/>Hollandiæ, Zelandiæ, Weſ@friſiæ, Zutphaniæ, <lb/>Vltrajecti, Tranſiſalanæ, & c. Imperator exer-<lb/>citus Provinciarum fœdere conſociata-<lb/>rum Belgii, Archithalaſſus <lb/>Generalis, & c.</head> <head xml:id="echoid-head6" xml:space="preserve">Conſcriptus à S*IMONE* S*TEVINO* Brugenſi.</head> <figure> <image file="527.01.001-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.001-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head7" xml:space="preserve">LVGODINI BATAVORVM, <lb/>Ex Officinâ Ioannis Patii, Academiæ Typographi.</head> <head xml:id="echoid-head8" xml:space="preserve">Anno cI@ I@ cv.</head> <pb o="2" file="527.01.002" n="2" rhead="QVARTI TOMI"/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head9" xml:space="preserve">BREVIARIVM.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">CV*M* jam olim de S*STATICA* librum <lb/>conſcripſiſſem, & </s> <s xml:id="echoid-s2" xml:space="preserve">I*LLUSTRISS.</s> <s xml:id="echoid-s3" xml:space="preserve">* <lb/>P*RINCEPS,* ad varias res quæ in praxi <lb/>occurrunt, ejus notitiam neceſſariam <lb/>comperiſſet, magni illius deſiderio, & </s> <s xml:id="echoid-s4" xml:space="preserve"><lb/>cognoſcendiſtudio accenſus fuit, adeo ut <lb/>poſt alias Mathematicæ materia diſci-<lb/>plinas, naturâ & </s> <s xml:id="echoid-s5" xml:space="preserve">notitiâ priores, quàm <lb/>avidißimè ſtudioſiſsimeq́, ad S*TATICES* praxin ſeſe con-<lb/>tulerit: </s> <s xml:id="echoid-s6" xml:space="preserve">ut poſt primam editionem præter erratorum emenda-<lb/>tionem magna mult arum inventionum acceſsio facta ſit, quod <lb/>ex additamento liquebit: </s> <s xml:id="echoid-s7" xml:space="preserve">ut oper æprecium facturus mihi vi-<lb/>derer ſi S*TATICEN* ad I*LLUSTRISSIMI* P*RINCIPIS* <lb/>M*ATHEMATICA* H*YPOMNEMATA* adderem in ſex libros <lb/>it a partitam, ut I. </s> <s xml:id="echoid-s8" xml:space="preserve">de primis S*TATICES* elementis ſit. </s> <s xml:id="echoid-s9" xml:space="preserve">II. </s> <s xml:id="echoid-s10" xml:space="preserve">de <lb/> <anchor type="note" xlink:href="" symbol="*"/> ratione inveniendi centrum gravitatis. </s> <s xml:id="echoid-s11" xml:space="preserve">III. </s> <s xml:id="echoid-s12" xml:space="preserve">de S*TATI-* <anchor type="note" xlink:label="note-527.01.002-01a" xlink:href="note-527.01.002-01"/> *CES* praxi. </s> <s xml:id="echoid-s13" xml:space="preserve">IV. </s> <s xml:id="echoid-s14" xml:space="preserve">de primis elementis <anchor type="note" xlink:href="" symbol="†"/> Hydrostatices. </s> <s xml:id="echoid-s15" xml:space="preserve">v. </s> <s xml:id="echoid-s16" xml:space="preserve">de <anchor type="note" xlink:label="note-527.01.002-02a" xlink:href="note-527.01.002-02"/> Hydrostatices praxi. </s> <s xml:id="echoid-s17" xml:space="preserve">VI. </s> <s xml:id="echoid-s18" xml:space="preserve">denique additamentum habeat.</s> <s xml:id="echoid-s19" xml:space="preserve"/> </p> <div xml:id="echoid-div4" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.002-01" xlink:href="note-527.01.002-01a" xml:space="preserve">centro ba-<lb/>rycâ.</note> <note symbol="†" position="left" xlink:label="note-527.01.002-02" xlink:href="note-527.01.002-02a" xml:space="preserve">aquam <lb/>ponderandi.</note> </div> <pb file="527.01.003" n="3"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="5"> <head xml:id="echoid-head10" xml:space="preserve">LIBER PRIMVS <lb/>STATIC AE <lb/>DE <lb/>STATICÆ ELEMENTIS.</head> <pb o="4" file="527.01.004" n="4" rhead="BREVIARIVM"/> </div> <div xml:id="echoid-div7" type="section" level="1" n="6"> <head xml:id="echoid-head11" xml:space="preserve">LIBRI I.</head> <p> <s xml:id="echoid-s20" xml:space="preserve">ST*ATICES* elementa, quæſunt degra-<lb/>vitate à phyſico corpore ſolâ cogita-<lb/>tione ſejunctâ, bipartitò ſunt diſtri-<lb/>buta. </s> <s xml:id="echoid-s21" xml:space="preserve">Prior pars 14 definitiones ha-<lb/>bet: </s> <s xml:id="echoid-s22" xml:space="preserve">poſterior 28 propoſitiones de <lb/>ponderum affectionibus, quorũ alia <lb/>recta, alia obliqua ſunt. </s> <s xml:id="echoid-s23" xml:space="preserve">Recta porro <lb/>duorum generum, elevantia ſcilicet, <lb/>& </s> <s xml:id="echoid-s24" xml:space="preserve">deprimentia, quæ primis octodecim propoſitionibus <lb/>ſunt comprehenſa. </s> <s xml:id="echoid-s25" xml:space="preserve">Obliqua quoque totidem ſunt, ele-<lb/>vantia item, & </s> <s xml:id="echoid-s26" xml:space="preserve">deprimentia, quæ reliquá propoſitionum <lb/>multitudine declarantur, ſed majoris evidentiæ cauſâ ta-<lb/>bula eſto, quæ primilibri breviarum ob oculos ponat.</s> <s xml:id="echoid-s27" xml:space="preserve"/> </p> <figure> <image file="527.01.004-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.004-01"/> </figure> <pb o="5" file="527.01.005" n="5"/> </div> <div xml:id="echoid-div8" type="section" level="1" n="7"> <head xml:id="echoid-head12" xml:space="preserve">PARS PRIOR</head> <head xml:id="echoid-head13" xml:space="preserve">DE DEFINITIONIBVS.</head> <head xml:id="echoid-head14" xml:space="preserve">I DEFINITIO.</head> <p> <s xml:id="echoid-s28" xml:space="preserve">Statica eſt quæ ponderis & </s> <s xml:id="echoid-s29" xml:space="preserve">gravitatis corporum ratio-<lb/>nes, proportiones, & </s> <s xml:id="echoid-s30" xml:space="preserve">qualitates interpretatur.</s> <s xml:id="echoid-s31" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div9" type="section" level="1" n="8"> <head xml:id="echoid-head15" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s32" xml:space="preserve">QVemadmodum Geometria figurarum magnitudi-<lb/>nes non autem gravitates conſiderat, illas æ quales <lb/>vel inæquales ſolummodo judicans, quarum ma-<lb/>gnitudines æquales vel inæquales ſunt: </s> <s xml:id="echoid-s33" xml:space="preserve">Ita contra <lb/>Statica gravitates earundem nõ magnitudines ex-<lb/>pendit, easq́ue æquales vel inæquales habet, qua-<lb/>rum gravitates & </s> <s xml:id="echoid-s34" xml:space="preserve">pondera æqualia, vel inæqualia <lb/>ſunt. </s> <s xml:id="echoid-s35" xml:space="preserve">Et quemadmodum Geometriæ munus eſt in <lb/>Rationes, Proportiones, & </s> <s xml:id="echoid-s36" xml:space="preserve">affectiones Magnitu-<lb/>dinum inquirere: </s> <s xml:id="echoid-s37" xml:space="preserve">ita Statices eſt Rationes, Pro-<lb/>portiones, & </s> <s xml:id="echoid-s38" xml:space="preserve">affectiones gravitatum ſive ponde-<lb/>rum interpretari; </s> <s xml:id="echoid-s39" xml:space="preserve">quænoſtræ ſcriptionis finis eſt, & </s> <s xml:id="echoid-s40" xml:space="preserve">ſcopus.</s> <s xml:id="echoid-s41" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head16" xml:space="preserve">2 DEFINITIO.</head> <p> <s xml:id="echoid-s42" xml:space="preserve">Gravitas corporis eſt potentia deſcenſus in dato loco.</s> <s xml:id="echoid-s43" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div11" type="section" level="1" n="10"> <head xml:id="echoid-head17" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s44" xml:space="preserve">Gravitas & </s> <s xml:id="echoid-s45" xml:space="preserve">levitas, quam in corpore ineſſe vulgò dicimus, non eſt propria <lb/>& </s> <s xml:id="echoid-s46" xml:space="preserve">eſſentialis ejus forma, ſed ex relatione ad aliud nata, cujus pleniorem decla-<lb/>rationem alii loco ac tempori deſtinavimus. </s> <s xml:id="echoid-s47" xml:space="preserve">Nam nonnulla Materia, & </s> <s xml:id="echoid-s48" xml:space="preserve">cor-<lb/>pora in aëre gravia, in aquâ levia, in aëre vero levia, alibi eſſegravia depre-<lb/>henduntur. </s> <s xml:id="echoid-s49" xml:space="preserve">Cum itaque dicimus lignum centum pondo eſſe, potentiam <lb/>deſcenſus intelligi volumus in dato loco, hoceſt, in loco ſubjecto ubi ponde-<lb/>ratum eſt.</s> <s xml:id="echoid-s50" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s51" xml:space="preserve">Ex definitionis conſectario, levitatem corporum potentiam elationis in al-<lb/>tum eſſe intelligimus; </s> <s xml:id="echoid-s52" xml:space="preserve">ſed in dato loco, naturâ enim quodvis corpus grave eſt.</s> <s xml:id="echoid-s53" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div12" type="section" level="1" n="11"> <head xml:id="echoid-head18" xml:space="preserve">3 DEFINITIO.</head> <p> <s xml:id="echoid-s54" xml:space="preserve">Nota gravitas eſt quæ notâ ponderitate exprimitur.</s> <s xml:id="echoid-s55" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div13" type="section" level="1" n="12"> <head xml:id="echoid-head19" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s56" xml:space="preserve">Vt cum corpus vel gravitatem ſex librarum, octo marcarum, vel trium un-<lb/>ciarum, & </s> <s xml:id="echoid-s57" xml:space="preserve">c. </s> <s xml:id="echoid-s58" xml:space="preserve">eſſe dicimus; </s> <s xml:id="echoid-s59" xml:space="preserve">quod hujuſmodi notâ ponderitate ſit definita, no-<lb/>tam gravitatem appellamus.</s> <s xml:id="echoid-s60" xml:space="preserve"/> </p> <pb o="6" file="527.01.006" n="6" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div14" type="section" level="1" n="13"> <head xml:id="echoid-head20" xml:space="preserve">4 DEFINITIO.</head> <p> <s xml:id="echoid-s61" xml:space="preserve">Gravitati centrum eſt ex quo, vel ſola cogitatione, <lb/>ſuſpenſum corpus quemcumque ſitum dederis, illum re-<lb/>tinet.</s> <s xml:id="echoid-s62" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div15" type="section" level="1" n="14"> <head xml:id="echoid-head21" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s63" xml:space="preserve">ABC globus eſto, æquabili ubique & </s> <s xml:id="echoid-s64" xml:space="preserve">materiâ & </s> <s xml:id="echoid-s65" xml:space="preserve">pondere, <lb/> <anchor type="figure" xlink:label="fig-527.01.006-01a" xlink:href="fig-527.01.006-01"/> quem cogitatione noſtra ex centro D, lineâ E D ſuſpenſum <lb/>fingamus, qui quoquo modo verſatus, motusq́ue, quem de-<lb/>deris ſitum, retinebit, ſi enim B ad locum A aliæq́ue partes <lb/>alio transferantur immotæ manebunt, ſecus materia inæqua-<lb/>bilis eſſet, & </s> <s xml:id="echoid-s66" xml:space="preserve">alio loco denſior graviorq́ue, alio verò rarior & </s> <s xml:id="echoid-s67" xml:space="preserve"><lb/>levior, quod contra theſin eſſet. </s> <s xml:id="echoid-s68" xml:space="preserve">D itaque, ex definitionis ſen-<lb/>tentia, centrum gravitatis fuerit globi A B C. </s> <s xml:id="echoid-s69" xml:space="preserve">Idem judicium <lb/>deomnibus eſto, nullum enim non corpus inordinatæ figuræ <lb/>& </s> <s xml:id="echoid-s70" xml:space="preserve">materiæ inæquabilis gravitatis ſit ſive figuræ ordinatæ, & </s> <s xml:id="echoid-s71" xml:space="preserve"><lb/>æquabilis gravitatis, hujuſmodi punctum habet, à quo ſuſpenſum eandem po-<lb/>ſitionem ſervat quæ data fuit, quod gravitatis centrum appellatur. </s> <s xml:id="echoid-s72" xml:space="preserve">Vt autem <lb/>ſuis proprietatibus magis innoteſcat hoc addemus. </s> <s xml:id="echoid-s73" xml:space="preserve">Gravitatis centrum in cor-<lb/>poribus, ut columnis, ſphæris ſphæroïdibus, & </s> <s xml:id="echoid-s74" xml:space="preserve">quinque ordinatis, & </s> <s xml:id="echoid-s75" xml:space="preserve">c. </s> <s xml:id="echoid-s76" xml:space="preserve">ſi ſunt <lb/>ex materia æquabiliter ubique ponderoſa, idem eſt cum figuræ vel magnitu-<lb/>dinis puncto quod Geometricè centrum appellatur. </s> <s xml:id="echoid-s77" xml:space="preserve">Corporum vero inæqua-<lb/>biliter ponderosâ hæc puncta magnitudinis & </s> <s xml:id="echoid-s78" xml:space="preserve">gravitatis eodem loco non ha-<lb/>bent. </s> <s xml:id="echoid-s79" xml:space="preserve">In pyramidibus enim, & </s> <s xml:id="echoid-s80" xml:space="preserve">inordinatis ſolidis non magnitudinis centrum, <lb/>ſed gravitatis tantum eſt. </s> <s xml:id="echoid-s81" xml:space="preserve">Multa etiam corpora ſunt, ut annuli, unci, pelves, & </s> <s xml:id="echoid-s82" xml:space="preserve"><lb/>alia hujuſmodi, quæ gravitatis centrum, non intra verũ extra materiam habĕt.</s> <s xml:id="echoid-s83" xml:space="preserve"/> </p> <div xml:id="echoid-div15" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.006-01" xlink:href="fig-527.01.006-01a"> <image file="527.01.006-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.006-01"/> </figure> </div> <p> <s xml:id="echoid-s84" xml:space="preserve">In definitione, vel ſolâ cogitatione, dicitur, quod in definitioneilla poni de-<lb/>bent, quæ definiti naturam maximè declarant, quod & </s> <s xml:id="echoid-s85" xml:space="preserve">in Pappus 8 lib. </s> <s xml:id="echoid-s86" xml:space="preserve">ubi <lb/>gravitatis centrum definit, cogitatione commodiſſime fecit. </s> <s xml:id="echoid-s87" xml:space="preserve">Etiam iſto pacto <lb/>definire licet: </s> <s xml:id="echoid-s88" xml:space="preserve">Gravitatis centrum eſt, per quod plana quavis ducta corpus in duas <lb/>partes aquilibres dividunt. </s> <s xml:id="echoid-s89" xml:space="preserve">Quid autem æquilibritas ſive ęquipondium ſit 11 de-<lb/>finitione dicitur.</s> <s xml:id="echoid-s90" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div17" type="section" level="1" n="15"> <head xml:id="echoid-head22" xml:space="preserve">5 DEFINITIO.</head> <p> <s xml:id="echoid-s91" xml:space="preserve">Gravitatis corporeæ diameter eſt recta infinita per gra-<lb/>vitatis centrum acta: </s> <s xml:id="echoid-s92" xml:space="preserve">Et gravitatis diameter horizonti <lb/>perpendicularis, diameter gravitatis pendula appellatur.</s> <s xml:id="echoid-s93" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div18" type="section" level="1" n="16"> <head xml:id="echoid-head23" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s94" xml:space="preserve">Vtin 4<emph style="sub">æ</emph> definitionis figurâ, quævis recta infinita per gravitatis centrum D <lb/>acta, corporis A B C diameter gravitatis appellatur: </s> <s xml:id="echoid-s95" xml:space="preserve">Verum gravitatis diame-<lb/>ter ad horizontem recta ut A D gravitatis diameter pendula dicatur.</s> <s xml:id="echoid-s96" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div19" type="section" level="1" n="17"> <head xml:id="echoid-head24" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s97" xml:space="preserve">In priore editione gravitatis diameter definita nobis fuit infinita per gravitatis <lb/>ſue cent<unsure/>r@ m pendens, ſufficere enim propoſitæ nobis ſcriptioni videbatur. </s> <s xml:id="echoid-s98" xml:space="preserve">Verum-<lb/>eni@@ver@ in ſequenti additamenio ponderoſorũ genera non paulo diligentius rimantes <pb o="7" file="527.01.007" n="7" rhead="*DE* S*TATICÆ ELEMENTIS.*"/> neceſſarium duximus quamvis rectam infinitam per centrum diametrum gravita-<lb/>tis appellare, distinguere{q́ue} inter pendulam, & </s> <s xml:id="echoid-s99" xml:space="preserve">non pendulam diametrum: </s> <s xml:id="echoid-s100" xml:space="preserve">unde <lb/>etiam diſcrimen inter 5 & </s> <s xml:id="echoid-s101" xml:space="preserve">13 definitionem bujus & </s> <s xml:id="echoid-s102" xml:space="preserve">ſuperior is edition is nature<unsure/> eſt.</s> <s xml:id="echoid-s103" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="1" n="18"> <head xml:id="echoid-head25" xml:space="preserve">6 DEFINITIO.</head> <p> <s xml:id="echoid-s104" xml:space="preserve">Gravitatis planum diametrum eſt quodcunque corpus <lb/>per gravitatis ſuæ centrum ſecat.</s> <s xml:id="echoid-s105" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div21" type="section" level="1" n="19"> <head xml:id="echoid-head26" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s106" xml:space="preserve">Vt quodvis planum quod 4<emph style="sub">tæ</emph> definitionis globum per centrum D ſecat, ejus <lb/>ipſius gravitatis diametrum planum appellatur. </s> <s xml:id="echoid-s107" xml:space="preserve">Idem de aliis corporibus ju-<lb/>dicium eſto. </s> <s xml:id="echoid-s108" xml:space="preserve">Affectio hujus propria eſt, quomodolibet ſecet corpus, in duas <lb/>æqueponderantes partes ſecare.</s> <s xml:id="echoid-s109" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div22" type="section" level="1" n="20"> <head xml:id="echoid-head27" xml:space="preserve">7 DEFINITIO.</head> <p> <s xml:id="echoid-s110" xml:space="preserve">Recta duabus pendulis diametris terminata, jugum <lb/>ſive T*RABS* dicatur.</s> <s xml:id="echoid-s111" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div23" type="section" level="1" n="21"> <head xml:id="echoid-head28" xml:space="preserve">DECLARATIO.</head> <figure> <image file="527.01.007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.007-01"/> </figure> <p> <s xml:id="echoid-s112" xml:space="preserve">A & </s> <s xml:id="echoid-s113" xml:space="preserve">B duo corpora ſunto, & </s> <s xml:id="echoid-s114" xml:space="preserve">pendulæ gravitatis dia-<lb/>metri C D & </s> <s xml:id="echoid-s115" xml:space="preserve">E F, inter quas contingentibus punctis du-<lb/>ctæ rectæ G H, A B, I K aliæq́ue infinitæ pendulis dia-<lb/>metris terminatæ, quas jugum vocamus unde A, B gra-<lb/>vitates dependent, ad Bilancis jugum alludentes.</s> <s xml:id="echoid-s116" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="1" n="22"> <head xml:id="echoid-head29" xml:space="preserve">8 DEFINITIO.</head> <p> <s xml:id="echoid-s117" xml:space="preserve">Iuga@ à pendulâ gravitatis diametro diviſi partes, ex qui-<lb/>bus pondera ſitu æquilibria dependĕt, Radii appellantur.</s> <s xml:id="echoid-s118" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div25" type="section" level="1" n="23"> <head xml:id="echoid-head30" xml:space="preserve">DECLARATIO.</head> <figure> <image file="527.01.007-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.007-02"/> </figure> <p> <s xml:id="echoid-s119" xml:space="preserve">A, B duo corporaſunto, & </s> <s xml:id="echoid-s120" xml:space="preserve">jugum illorum C D partitum <lb/>in E, à pendula diametro F, duo jugi membra ut E C, & </s> <s xml:id="echoid-s121" xml:space="preserve"><lb/>E D, ex quibus<unsure/> iſorropa pondera ſunt ſuſpenſa, radiiappel-<lb/>lantur.</s> <s xml:id="echoid-s122" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div26" type="section" level="1" n="24"> <head xml:id="echoid-head31" xml:space="preserve">9 DEFINITIO.</head> <p> <s xml:id="echoid-s123" xml:space="preserve">Amborum autem ponderum pendula gravitatis dia-<lb/>metrosanſa nobis dicitur.</s> <s xml:id="echoid-s124" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div27" type="section" level="1" n="25"> <head xml:id="echoid-head32" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s125" xml:space="preserve">Vt FE, in 8 definit. </s> <s xml:id="echoid-s126" xml:space="preserve">Anſa eſt.</s> <s xml:id="echoid-s127" xml:space="preserve"/> </p> <pb o="8" file="527.01.008" n="8" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div28" type="section" level="1" n="26"> <head xml:id="echoid-head33" xml:space="preserve">10 DEFINITIO.</head> <p> <s xml:id="echoid-s128" xml:space="preserve">Et Anſæ punctum in jugo fixum. </s> <s xml:id="echoid-s129" xml:space="preserve">Punctum firmum.</s> <s xml:id="echoid-s130" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div29" type="section" level="1" n="27"> <head xml:id="echoid-head34" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s131" xml:space="preserve">Vt E, in 8 definitione, punctum eſt ſtabile.</s> <s xml:id="echoid-s132" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div30" type="section" level="1" n="28"> <head xml:id="echoid-head35" xml:space="preserve">11 DEFINITIO.</head> <p> <s xml:id="echoid-s133" xml:space="preserve">Dicta autem pondera ſitu æquilibria dicimus.</s> <s xml:id="echoid-s134" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div31" type="section" level="1" n="29"> <head xml:id="echoid-head36" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s135" xml:space="preserve">Vt A & </s> <s xml:id="echoid-s136" xml:space="preserve">B, in tertiæ definitionis figurâ, ſive ambo ſuo pondere ſint æqui-<lb/>pondia, ſive inæquipondia, ex ſitu æquilibria vocamus, quòd poſitu æque-<lb/>ponderantia ſunt, A enim ex jugo dependens tantum poteſt ob ſitum, quan-<lb/>tum B, & </s> <s xml:id="echoid-s137" xml:space="preserve">viciſſim B quantum A.</s> <s xml:id="echoid-s138" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s139" xml:space="preserve">Iſta ex ſitu æquilibritas neceſſariò notanda, &</s> <s xml:id="echoid-s140" xml:space="preserve">, quia diverſæ ſunt, ab æqui-<lb/>ponderitate dirimenda. </s> <s xml:id="echoid-s141" xml:space="preserve">Nam, exempli causâ, pondus ex minore radio in ſta-<lb/>tera dependens nonnunquam decuplum eſt ad alterum, æqualem tamen gra-<lb/>vitatem præ ſe ferunt, ſed ſpecie tantum non propriè, & </s> <s xml:id="echoid-s142" xml:space="preserve">tantummodo propter <lb/>ſitum.</s> <s xml:id="echoid-s143" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div32" type="section" level="1" n="30"> <head xml:id="echoid-head37" xml:space="preserve">12 DEFINITIO.</head> <p> <s xml:id="echoid-s144" xml:space="preserve">Pondus elevans dicitur quicquid eſt cauſa ponderis in <lb/>altitudinem elati.</s> <s xml:id="echoid-s145" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div33" type="section" level="1" n="31"> <head xml:id="echoid-head38" xml:space="preserve">DECLARATIO.</head> <figure> <image file="527.01.008-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.008-01"/> </figure> <p> <s xml:id="echoid-s146" xml:space="preserve">Columna A pondus eſto, <lb/>B C linea ex qua ſuſpen-<lb/>ſa tenetur, /<unsure/>o<unsure/>riſmatis pun-<lb/>ctum in quo quieſcat D, E <lb/>vero pondus, corpus A eo-<lb/>dem tenens ſitu. </s> <s xml:id="echoid-s147" xml:space="preserve">Quapro-<lb/>pter E primæ ſecundæq́ue <lb/>figuræ pondus elevans, iſto <lb/>enim pondere corpus ele-<lb/>vatur, aut elevatum eodem <lb/>ſitu tenet. </s> <s xml:id="echoid-s148" xml:space="preserve">E vero tertiæ & </s> <s xml:id="echoid-s149" xml:space="preserve"><lb/>quartæ figuræ pondus de-<lb/>mittens, quod corpus la-<lb/>tere B affixum deſcendere <lb/>faciat, vel illo ſitu demiſſum <lb/>detineat.</s> <s xml:id="echoid-s150" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div34" type="section" level="1" n="32"> <head xml:id="echoid-head39" xml:space="preserve">13 DEFINITIO.</head> <p> <s xml:id="echoid-s151" xml:space="preserve">Lineam à gravitate ſublatâ verſus pondus attollens, quæ <pb o="9" file="527.01.009" n="9" rhead="*DE* S*TATICÆ ELEMENTIS.*"/> eſt inter gravitatis diametrum quæ per firmitudinis pun-<lb/>ctum, ejusq́ue parallelam, elevantem: </s> <s xml:id="echoid-s152" xml:space="preserve">quæ vero à gravita-<lb/>te demiſsâ eſt verſus pondus demittens, ſimiliter inter gra-<lb/>vitatis diametrum, quæ per firmitudinis punctum, ejusq́; <lb/></s> <s xml:id="echoid-s153" xml:space="preserve">parallelam, lineam demittentem dicimus.</s> <s xml:id="echoid-s154" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s155" xml:space="preserve">Vt recta C B in 12 definitione, gravitatis diametro, quæ per firmitudinis <lb/>punctum, ut D B, ejusq́ue parallelâ terminata, in 1 & </s> <s xml:id="echoid-s156" xml:space="preserve">2 figurâ linea attollens, <lb/>in 3 verò & </s> <s xml:id="echoid-s157" xml:space="preserve">4 linea demittens nobis appellabitur.</s> <s xml:id="echoid-s158" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div35" type="section" level="1" n="33"> <head xml:id="echoid-head40" xml:space="preserve">14 DEFINITIO.</head> <p> <s xml:id="echoid-s159" xml:space="preserve">Si linea, & </s> <s xml:id="echoid-s160" xml:space="preserve">attollens, & </s> <s xml:id="echoid-s161" xml:space="preserve">demittens Horizonti perpendi-<lb/>cularis ſit, Recta attollens, & </s> <s xml:id="echoid-s162" xml:space="preserve">Recta demittens, earumq́ue <lb/>pondera, Rectum attollens, Rectum demittens: </s> <s xml:id="echoid-s163" xml:space="preserve">ſin obli-<lb/>qua ſit Horizonti, obliqua attollens, obliqua demittens, <lb/>& </s> <s xml:id="echoid-s164" xml:space="preserve">earum pondera obliquum attollens, obliquum demit-<lb/>tens à ſitu nobis appellabuntur.</s> <s xml:id="echoid-s165" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div36" type="section" level="1" n="34"> <head xml:id="echoid-head41" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s166" xml:space="preserve">Vt in primâ tertiaq́ue duodecimæ definitionis figurâ, attollens, & </s> <s xml:id="echoid-s167" xml:space="preserve">demit-<lb/>tenslineæ, quia ex hypotheſi angulos cum Horizonte rectos faciunt, illa Re-<lb/>cta attollens, hæc Recta demittens, earumq́ue pondera E Rectum attollens, <lb/>Rectum demittens dicantur. </s> <s xml:id="echoid-s168" xml:space="preserve">Sin linea attollens, & </s> <s xml:id="echoid-s169" xml:space="preserve">demittens ut C B in 2 & </s> <s xml:id="echoid-s170" xml:space="preserve">4 <lb/>figurâ horizonti ſit obliqua, obliquæ appellabuntur, & </s> <s xml:id="echoid-s171" xml:space="preserve">obliqua illarum pon-<lb/>dera.</s> <s xml:id="echoid-s172" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div37" type="section" level="1" n="35"> <head xml:id="echoid-head42" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s173" xml:space="preserve">Figura Staticæ & </s> <s xml:id="echoid-s174" xml:space="preserve">Geometricæ columnæeadem eſt, niſi quod hic materia illius æqua-<lb/>bilioris ponderis eſſe ſumatur, operimentum vero & </s> <s xml:id="echoid-s175" xml:space="preserve">baſis quadrangula. </s> <s xml:id="echoid-s176" xml:space="preserve">Artis voca-<lb/>bula ita nobis Belgis uſurpantur.</s> <s xml:id="echoid-s177" xml:space="preserve"/> </p> <note position="right" xml:space="preserve"> <lb/>Materia # # Stof <lb/>Forma # # Form <lb/>Effectus # # Daet <lb/>Subjectum # # Grondt <lb/>Adjunctum # # Aencleving <lb/>Genus # # Gheſlacht <lb/>Species # # Afcomſt <lb/>Definitio # # Bepaling <lb/>Propoſitio # # Voorſtel <lb/>Problema # # Werckſtick <lb/>Theorema # # Vertooch <lb/>Ratio # # Reden <lb/>Proportio # # Everedicheyt <lb/>A Equales # Pro qui- # Even <lb/>Similes # bus uſur- # Ghelijcke <lb/>Exemplum # pavimus # Voorbeelt <pb o="10" file="527.01.010" n="10" rhead="1 L*IBER* S*TATIC Æ*"/> Centrum gravitatis # # Swaerheyts middelpuns<unsure/> <lb/>Axis # # As <lb/>Diameter # # Middellini <lb/>Circumferentia # # Omtreck <lb/>Parallela # # Evewijdighe <lb/>Homologa latera # # Lijckſtandighe ſijden <lb/>Superficies # # Vlack <lb/>Planum # # Plat <lb/>Columna # # Pylaer <lb/>Arithmetica # # Telconſt <lb/>Geometria # # Meetconſt <lb/>Ars Mathematica # # Wiſconſt <lb/>Mathematicus # # Wiſconſtnaer <lb/>Mathematicè # # Wiſconſtlick. <lb/></note> <p style="it"> <s xml:id="echoid-s178" xml:space="preserve">Quæ Latina vocabula, & </s> <s xml:id="echoid-s179" xml:space="preserve">alia nonnulla, major is evidentiæ causâ, in margine non <lb/>nunquam ſuis vernaculis apponimus. </s> <s xml:id="echoid-s180" xml:space="preserve">T res autem literæ P. </s> <s xml:id="echoid-s181" xml:space="preserve">L. </s> <s xml:id="echoid-s182" xml:space="preserve">E. </s> <s xml:id="echoid-s183" xml:space="preserve">in margine aliquan-<lb/>do additæ. </s> <s xml:id="echoid-s184" xml:space="preserve">Propoſitionem, librum, Euclidem notant, ut ex 2 p. </s> <s xml:id="echoid-s185" xml:space="preserve">6. </s> <s xml:id="echoid-s186" xml:space="preserve">l. </s> <s xml:id="echoid-s187" xml:space="preserve">E. </s> <s xml:id="echoid-s188" xml:space="preserve">2 propoſitio-<lb/>nem 6 libri Euclidis intelliges.</s> <s xml:id="echoid-s189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div38" type="section" level="1" n="36"> <head xml:id="echoid-head43" xml:space="preserve">*POSTVLATA.*</head> <p> <s xml:id="echoid-s190" xml:space="preserve">QVandoquidem nonnulla tanquam ſcientiæ principia per communes no-<lb/>tiones ſunt nota, neque demonſtrationis indigent; </s> <s xml:id="echoid-s191" xml:space="preserve">alia vero occultiora <lb/>magisq́ue tecta irridendi materiam cavillatoribus ſuppeditare poſſunt quæ <lb/>irrideriaut culpari minimè debent: </s> <s xml:id="echoid-s192" xml:space="preserve">quapropter Mathematicorum more, ante <lb/>quam ad propoſitiones accedimus, nobis illa concedi poſtulabimus.</s> <s xml:id="echoid-s193" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div39" type="section" level="1" n="37"> <head xml:id="echoid-head44" xml:space="preserve">1 POSTVLATVM.</head> <p> <s xml:id="echoid-s194" xml:space="preserve">Æqualia pondera ex æqualibus radiis ſuſpenſa etiam <lb/>ſitu eſſe æquipondia.</s> <s xml:id="echoid-s195" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div40" type="section" level="1" n="38"> <head xml:id="echoid-head45" xml:space="preserve">2 POSTVLATVM.</head> <p> <s xml:id="echoid-s196" xml:space="preserve">E Mathematicâ lineâ pondus quodvis ſuſpendi, autin <lb/>eâ quieſcere poſſe, ut ne frangatur aut flectatur.</s> <s xml:id="echoid-s197" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div41" type="section" level="1" n="39"> <head xml:id="echoid-head46" xml:space="preserve">3 POSTVLATVM.</head> <figure> <image file="527.01.010-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.010-01"/> </figure> <p> <s xml:id="echoid-s198" xml:space="preserve">Pondus ſive ſublimius, ſive humilius ſit <lb/>ſuſpenſum ejuſdem gravitatis eſſe.</s> <s xml:id="echoid-s199" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div42" type="section" level="1" n="40"> <head xml:id="echoid-head47" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s200" xml:space="preserve">Vt pondus A depreſſum in B ejuſdem gravitatis ma-<lb/>nere, vel tantundem potentiæ in C D obtinere quantum <lb/>in A obtinuit.</s> <s xml:id="echoid-s201" xml:space="preserve"/> </p> <pb o="11" file="527.01.011" n="11" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div43" type="section" level="1" n="41"> <head xml:id="echoid-head48" xml:space="preserve">4 POSTVLATVM.</head> <p> <s xml:id="echoid-s202" xml:space="preserve">Perplanum columnæ, quodillam per axis longitudi-<lb/>nem dividit, datam columnam intelligi.</s> <s xml:id="echoid-s203" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s204" xml:space="preserve">Vt AB columna eſto, & </s> <s xml:id="echoid-s205" xml:space="preserve">ejus axis CD plano <lb/> <anchor type="figure" xlink:label="fig-527.01.011-01a" xlink:href="fig-527.01.011-01"/> aliquo, ut EFGH, per axem ſecta. </s> <s xml:id="echoid-s206" xml:space="preserve">plano EFGH <lb/>nominato, omiſſis cæteris omnibus, totam colum-<lb/>nam intelligi poſtulamus.</s> <s xml:id="echoid-s207" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.011-01" xlink:href="fig-527.01.011-01a"> <image file="527.01.011-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.011-01"/> </figure> </div> </div> <div xml:id="echoid-div45" type="section" level="1" n="42"> <head xml:id="echoid-head49" xml:space="preserve">5 POSTVLATVM.</head> <p> <s xml:id="echoid-s208" xml:space="preserve">Omnes perpendiculares parallelas haberi.</s> <s xml:id="echoid-s209" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div46" type="section" level="1" n="43"> <head xml:id="echoid-head50" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s210" xml:space="preserve">Cauſa ejuſmodi eſt. </s> <s xml:id="echoid-s211" xml:space="preserve">ABCD terræglobus eſto, cujus centrum E, horizon <lb/>AC, & </s> <s xml:id="echoid-s212" xml:space="preserve">FG jugum ad horizontem parallelum, ejusq́; </s> <s xml:id="echoid-s213" xml:space="preserve">æquales radii HF, HG, <lb/>exiisq́; </s> <s xml:id="echoid-s214" xml:space="preserve">æqualia põdera I, K ſuſpenſa; </s> <s xml:id="echoid-s215" xml:space="preserve">unde perpĕdiculares FI & </s> <s xml:id="echoid-s216" xml:space="preserve">GK parallelas <lb/>noneſſe manifeſtũ eſt quod inſeriora illarũ magis <lb/> <anchor type="figure" xlink:label="fig-527.01.011-02a" xlink:href="fig-527.01.011-02"/> annuant, quam ſuperiora, neq; </s> <s xml:id="echoid-s217" xml:space="preserve">diſtent ubiq́ æqualiter. <lb/></s> <s xml:id="echoid-s218" xml:space="preserve">Super ſtabili puncto H, jugũ FG moveatur, <lb/>ut G in locum L, F in M tranſeant, pondusq́ K <lb/>adſcĕdat in N, I vero deſcendat in O: </s> <s xml:id="echoid-s219" xml:space="preserve">& </s> <s xml:id="echoid-s220" xml:space="preserve">angulus <lb/>LME recto propior ſit, quam MLE; </s> <s xml:id="echoid-s221" xml:space="preserve">ex iſto ſitu <lb/>(ut è 24 propoſit. </s> <s xml:id="echoid-s222" xml:space="preserve">patebit) O quam N gravius <lb/>fuerit. </s> <s xml:id="echoid-s223" xml:space="preserve">Ex his conſequens eſft nullum corpus five <lb/>ſolidum in rerum Naturâ eſſe, ut Mathematicè <lb/>loquar, præter Globum, quod ex ſuæ gravitatis <lb/>centro cogitatione ſuſpenſum, quemlibet datum <lb/>ſitum retinet: </s> <s xml:id="echoid-s224" xml:space="preserve">five, per quod planum quodlibet <lb/>ipſum corpus in partes fitu æquipondias dividit, verum propter varios, & </s> <s xml:id="echoid-s225" xml:space="preserve">in-<lb/>finitos ſitus, varia etiam & </s> <s xml:id="echoid-s226" xml:space="preserve">infinita gravitatis centra erunt. </s> <s xml:id="echoid-s227" xml:space="preserve">Neque gravius pon-<lb/>dus (quod 1 propoſitioni repugnat) eam rationem haberet ad levius, quæ <lb/>longioris radii eſt ad breviorem, ſed unum altero ponderoſius eſſe ex ſitu ar-<lb/>gueretur, quod angulus ejus major & </s> <s xml:id="echoid-s228" xml:space="preserve">recto propior eſſet; </s> <s xml:id="echoid-s229" xml:space="preserve">qu i<unsure/>m alterius an-<lb/>gulus. </s> <s xml:id="echoid-s230" xml:space="preserve">Verumenimverò ut exemplo clarius fiat, AB bre-<lb/> <anchor type="figure" xlink:label="fig-527.01.011-03a" xlink:href="fig-527.01.011-03"/> vior radius eſto, ejusq́ue pondus C: </s> <s xml:id="echoid-s231" xml:space="preserve">AD vero longior, ex <lb/>eoque pondus E ſuſpenſum illam rationem habeto ad C; <lb/></s> <s xml:id="echoid-s232" xml:space="preserve">quam AB ad AD, F autem univerſitatis centrum: </s> <s xml:id="echoid-s233" xml:space="preserve">ubi an-<lb/>gulus FBA hebetior, rectoq́ue propior eſſe apparet, quam <lb/>angulus ADE. </s> <s xml:id="echoid-s234" xml:space="preserve">Hinc (ex 24 propoſitionis ſententiâ) C pon-<lb/>deroſius eſſe, quam E, conſequens eſt.</s> <s xml:id="echoid-s235" xml:space="preserve"/> </p> <div xml:id="echoid-div46" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.011-02" xlink:href="fig-527.01.011-02a"> <image file="527.01.011-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.011-02"/> </figure> <figure xlink:label="fig-527.01.011-03" xlink:href="fig-527.01.011-03a"> <image file="527.01.011-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.011-03"/> </figure> </div> <p> <s xml:id="echoid-s236" xml:space="preserve">Hæc inconvenientia inde ſuntnata, quòd in primâ figurâ <lb/>FE & </s> <s xml:id="echoid-s237" xml:space="preserve">GE, in ſecundâ verò BF & </s> <s xml:id="echoid-s238" xml:space="preserve">DF parallelæ non ſint. <lb/></s> <s xml:id="echoid-s239" xml:space="preserve">Verum, quandoquidem diſcrimen illud, in iis quæ ab homi-<lb/>nibus ponderantur, nullum, ſaltem inobſervabile eſt, jugum enim aliquot mi-<lb/>lia longum eſſe deberet, antequam deprehendi poſſet, perpendiculares paral-<lb/>lelas habendas eſſe concedi nobis poſtulamus. </s> <s xml:id="echoid-s240" xml:space="preserve">Verum equidem eſt, ex naturâ <lb/>fuâ illas æſtimantes, perfectè operari poſſeſecundum illarum ſpeciem, ſed quia <lb/>moleſtius illud eſſet, nec tamen ad remipſam, hoc eſt, *STATICES* praxin <lb/>utili<unsure/>us, ſuperſedere conſultius eſt.</s> <s xml:id="echoid-s241" xml:space="preserve"/> </p> <pb o="12" file="527.01.012" n="12" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div48" type="section" level="1" n="44"> <head xml:id="echoid-head51" xml:space="preserve">PARS ALTERA <lb/>DE PROPOSITIONIBVS.</head> <head xml:id="echoid-head52" xml:space="preserve">1 THE OREMA. I PROPOSITIO.</head> <p> <s xml:id="echoid-s242" xml:space="preserve">Duarum gravitatũ ſitu æquilibriũ ponderoſior illam ra-<lb/>tionĕ habet ad leviorĕ, quę lõgioris radii eſt, ad breviorem.</s> <s xml:id="echoid-s243" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div49" type="section" level="1" n="45"> <head xml:id="echoid-head53" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s244" xml:space="preserve">DA*TVM*. </s> <s xml:id="echoid-s245" xml:space="preserve">ABCD 6 ℔ columna eſto in ſex partes æquales à <lb/>planisad baſin AD parallelis partita, ut ſunt EF, GH, IK, LM, <lb/>NO, axem PQ in R, S, T, V, X ſecantibus: </s> <s xml:id="echoid-s246" xml:space="preserve">LMDA gravi-<lb/>tas eſto ponderoſior, ejusq́ue centrum S, LMCB verò levior <lb/>& </s> <s xml:id="echoid-s247" xml:space="preserve">centrum X, partium iſtarum ſecundũ 7 de-<lb/> <anchor type="figure" xlink:label="fig-527.01.012-01a" xlink:href="fig-527.01.012-01"/> finitionem jugum erit SX, T autem columnæ <lb/>totius centrum, TI anſa, ex qua LMDA <lb/>& </s> <s xml:id="echoid-s248" xml:space="preserve">LMCB ſitu æquilibria dependent, & </s> <s xml:id="echoid-s249" xml:space="preserve"><lb/>TX radius longior, TS autem brevior ex 8 <lb/>definit. </s> <s xml:id="echoid-s250" xml:space="preserve">ſententiâ.</s> <s xml:id="echoid-s251" xml:space="preserve"/> </p> <div xml:id="echoid-div49" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.012-01" xlink:href="fig-527.01.012-01a"> <image file="527.01.012-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.012-01"/> </figure> </div> <p> <s xml:id="echoid-s252" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s253" xml:space="preserve">Demonſtrandum nobis <lb/>eſt ſic longiorem radium TX eſſe ad brevio-<lb/>rem TS: </s> <s xml:id="echoid-s254" xml:space="preserve">q@emadmodum ponderoſior gra-<lb/>vitas LMDA eſt ad leviorem LMCB.</s> <s xml:id="echoid-s255" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div51" type="section" level="1" n="46"> <head xml:id="echoid-head54" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s256" xml:space="preserve">LMDA 4 libras pendet, LMCB vero 2, ratio autem longioris radii <lb/>TX ad breviorem TS eſtut 2 ad 1 ex dato: </s> <s xml:id="echoid-s257" xml:space="preserve">Atqui ut 4. </s> <s xml:id="echoid-s258" xml:space="preserve">ad 2: </s> <s xml:id="echoid-s259" xml:space="preserve">ita 2 ad 1, ut <lb/>igitur ponderoſius LMDA ad levius LMCB: </s> <s xml:id="echoid-s260" xml:space="preserve">ita TX longior radius ad <lb/>TS breviorem.</s> <s xml:id="echoid-s261" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s262" xml:space="preserve">VErumenimvero ne caſu potius quam ſolidâ ſcientiâ ita habere ſe iſta vi-<lb/>deantur, Mathematicam demonſtrationem ſubjungemus.</s> <s xml:id="echoid-s263" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div52" type="section" level="1" n="47"> <head xml:id="echoid-head55" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s264" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s265" xml:space="preserve">ABCD iterum eſto columna, ſecta plano EF parallelo ad <lb/>AD, ſecanteaxem GH in puncto contingenti, ut I, ſegmentiq́ue EFDA <lb/>centrum gravitatis K medium in GI, ſegmentiq́ue EFCB centrum L me-<lb/>diumin IH, totiusautem ABCD, M medium in GH, MN verò ſegmen-<lb/>torum EFDA & </s> <s xml:id="echoid-s266" xml:space="preserve">EFCB anſa, unde ſitu æquilibria dependent.</s> <s xml:id="echoid-s267" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s268" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s269" xml:space="preserve">Demonſtrandum eſt, <lb/> <anchor type="figure" xlink:label="fig-527.01.012-02a" xlink:href="fig-527.01.012-02"/> quemadmodum corpus five gravitas (quæ <lb/>hic propter illorum proportionem, unum <lb/>idemq́ue ſunt ut enim corpus EFDA ad <lb/>corpus EFCB: </s> <s xml:id="echoid-s270" xml:space="preserve">ita illius gravitas, ad gra-<lb/>vitatem hujus, columna enim ex poſitu ubiq; <lb/></s> <s xml:id="echoid-s271" xml:space="preserve">æquabilis gravitatis eſt) EFDA ad EFCB: </s> <s xml:id="echoid-s272" xml:space="preserve"><lb/>ita longiorem radium ML eſſe ad brevio-<lb/>rem MK.</s> <s xml:id="echoid-s273" xml:space="preserve"/> </p> <div xml:id="echoid-div52" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.012-02" xlink:href="fig-527.01.012-02a"> <image file="527.01.012-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.012-02"/> </figure> </div> <pb o="13" file="527.01.013" n="13" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div54" type="section" level="1" n="48"> <head xml:id="echoid-head56" xml:space="preserve">DEMONSTRATIO.</head> <head xml:id="echoid-head57" xml:space="preserve">1 MEMBRVM.</head> <p> <s xml:id="echoid-s274" xml:space="preserve">MH ex dato æquatur MG. </s> <s xml:id="echoid-s275" xml:space="preserve">utrique KM addito KH æquabitur MG <lb/>& </s> <s xml:id="echoid-s276" xml:space="preserve">KM. </s> <s xml:id="echoid-s277" xml:space="preserve">ſubductoq́ue deinde hinc GK, inde KI (ex dato autem GK & </s> <s xml:id="echoid-s278" xml:space="preserve">KI <lb/>æqualia ſunt) KM & </s> <s xml:id="echoid-s279" xml:space="preserve">KM reliqua æqualia erunt IH reliquo, eorundemq́ue <lb/>dimidia æqualia fuerint.</s> <s xml:id="echoid-s280" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div55" type="section" level="1" n="49"> <head xml:id="echoid-head58" xml:space="preserve">2 MEMBRVM.</head> <p> <s xml:id="echoid-s281" xml:space="preserve">MI ad KM & </s> <s xml:id="echoid-s282" xml:space="preserve">IL addito tota ML & </s> <s xml:id="echoid-s283" xml:space="preserve">IK æqualia erunt.</s> <s xml:id="echoid-s284" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div56" type="section" level="1" n="50"> <head xml:id="echoid-head59" xml:space="preserve">3 MEMBRVM.</head> <p> <s xml:id="echoid-s285" xml:space="preserve">Vt GI ad ſui dimidium KI: </s> <s xml:id="echoid-s286" xml:space="preserve">ſic IH ad ſui dimidium IL. </s> <s xml:id="echoid-s287" xml:space="preserve">& </s> <s xml:id="echoid-s288" xml:space="preserve">proportio-<lb/>ne alternatâ, ut GI ad IH: </s> <s xml:id="echoid-s289" xml:space="preserve">ita KI ad IL, atqui KI æquatur ML juxta 2 <lb/>membrum, & </s> <s xml:id="echoid-s290" xml:space="preserve">IL ſegmento MK, juxta primum, ideoq́ue ut GI ad IH: </s> <s xml:id="echoid-s291" xml:space="preserve">ita <lb/>ML ad MK. </s> <s xml:id="echoid-s292" xml:space="preserve">Atqui ut GI ad IH: </s> <s xml:id="echoid-s293" xml:space="preserve">ita corpus five gravitas EFDA ad <lb/>EFCB, itaque ut ponderoſior gravitas EFDA, ad leviorem EFCB: </s> <s xml:id="echoid-s294" xml:space="preserve">ita <lb/>longior radius ML, ad breviorem MK.</s> <s xml:id="echoid-s295" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s296" xml:space="preserve">OCcurrerit hic non nemo. </s> <s xml:id="echoid-s297" xml:space="preserve">Propoſitionĕ deſegmentis ejuſdem columnæ, <lb/>& </s> <s xml:id="echoid-s298" xml:space="preserve">quidem æquabilis gravitatis à meeſſe demõſtratam, ignorari tamen an <lb/>veritas in aliis, & </s> <s xml:id="echoid-s299" xml:space="preserve">variis ſegmentis ſigurarum irregularium, & </s> <s xml:id="echoid-s300" xml:space="preserve">materiæ inæqua-<lb/>bilis cadem futura ſit: </s> <s xml:id="echoid-s301" xml:space="preserve">quapropter propoſitionis generalitatem ita dein-<lb/>ceps demonſtrabimus. </s> <s xml:id="echoid-s302" xml:space="preserve">Iugum KL primi modi immotum manere ima-<lb/>ginemur, ſegmentum autem EFDA demitti, lineâ è gravitatis centro edu-<lb/>ctâ ſuſpenſum è puncto K, reliquumq́ue ſe-<lb/> <anchor type="figure" xlink:label="fig-527.01.013-01a" xlink:href="fig-527.01.013-01"/> gmentum EFCB conſimiliter depreſſum <lb/>è gravitatis centro L ſuſpendi, ſegmen-<lb/>tumq́ue EFCB ab EFDA diſtare, & </s> <s xml:id="echoid-s303" xml:space="preserve">ſi-<lb/>tum corũ eſſe qualem diagramma exhibet. <lb/></s> <s xml:id="echoid-s304" xml:space="preserve">Quando corpus primi modi ex anſa MN <lb/>penderet, ſegmenta EFCB & </s> <s xml:id="echoid-s305" xml:space="preserve">EFDA <lb/>ſitu æquipondia erant: </s> <s xml:id="echoid-s306" xml:space="preserve">neque in ſecundo <lb/>pondus EFDA depreſſius altero, plus mi-<lb/>núsve gravitatis quam in primo adſert jugo <lb/>KL, ex 3 poſtulati ſententiâ. </s> <s xml:id="echoid-s307" xml:space="preserve">Neque EFCB pondus ſecundi modi plus gra-<lb/>vitatis jugo adfert quam prius. </s> <s xml:id="echoid-s308" xml:space="preserve">Quapropter gravitates tam primi, quàm ſecun-<lb/>di modi eædem manent, jugiq́ue ſitus idem qui erat prius. </s> <s xml:id="echoid-s309" xml:space="preserve">ideoq́ue EFDA <lb/>& </s> <s xml:id="echoid-s310" xml:space="preserve">EFCB ſitu æquilibria. </s> <s xml:id="echoid-s311" xml:space="preserve">ſegmentaq́ue columnæ tam diviſa, quam conjun-<lb/>cta fitu æquipondia, atque radii eandem rationem, quam habuerunt, reti-<lb/>nent.</s> <s xml:id="echoid-s312" xml:space="preserve"/> </p> <div xml:id="echoid-div56" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.013-01" xlink:href="fig-527.01.013-01a"> <image file="527.01.013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.013-01"/> </figure> </div> <p> <s xml:id="echoid-s313" xml:space="preserve">Hoc probato, corpora EFDA & </s> <s xml:id="echoid-s314" xml:space="preserve">EFCB ſe-<lb/> <anchor type="figure" xlink:label="fig-527.01.013-02a" xlink:href="fig-527.01.013-02"/> cundi modi aliter premendo ſingendoq́ue figure-<lb/>mus (materiam ceream, argillaceam, aliamve tra-<lb/>ctabilem eſſe ponamus) ut EFDA & </s> <s xml:id="echoid-s315" xml:space="preserve">EFCB <lb/>modi ſecundi, EFDA & </s> <s xml:id="echoid-s316" xml:space="preserve">EFCB fiant tertii; <lb/></s> <s xml:id="echoid-s317" xml:space="preserve">KL jugum eundem poſitum ſervare, radiosq́ue <lb/>ML, & </s> <s xml:id="echoid-s318" xml:space="preserve">KM eandem rationem manifeſtum eſt, <lb/>ideoq́ue EFDA & </s> <s xml:id="echoid-s319" xml:space="preserve">EFCB ſitu æquilibria ma-<lb/>nere conſequens eſt, quia manente materiâ, muta-<lb/>tio ſormæ mutationem gravitatis non adfert.</s> <s xml:id="echoid-s320" xml:space="preserve"/> </p> <div xml:id="echoid-div57" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.013-02" xlink:href="fig-527.01.013-02a"> <image file="527.01.013-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.013-02"/> </figure> </div> <pb o="14" file="527.01.014" n="14" rhead="1 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s321" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s322" xml:space="preserve">EFDA & </s> <s xml:id="echoid-s323" xml:space="preserve">EFBC tertiimodi mutatis, pro illo corpus plumbeum, <lb/>pro hoc ligneum ejuſdem ponderis ſuſpĕduntor, <lb/> <anchor type="figure" xlink:label="fig-527.01.014-01a" xlink:href="fig-527.01.014-01"/> eritq́ue modus quartus, uthîcvidere eſt. </s> <s xml:id="echoid-s324" xml:space="preserve">Iugum <lb/>in eodem ſitu manere, ideoq́; </s> <s xml:id="echoid-s325" xml:space="preserve">EFDA & </s> <s xml:id="echoid-s326" xml:space="preserve">EFCB <lb/>ſitu æquipondia, radiosq́ue ejuſdem rationis eſſe <lb/>nemini dubium eſſe poteſt.</s> <s xml:id="echoid-s327" xml:space="preserve"/> </p> <div xml:id="echoid-div58" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.014-01" xlink:href="fig-527.01.014-01a"> <image file="527.01.014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.014-01"/> </figure> </div> </div> <div xml:id="echoid-div60" type="section" level="1" n="51"> <head xml:id="echoid-head60" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s328" xml:space="preserve">Ponderibus corporeo etiam jugo ſuſpenſis <lb/>idem demonſtrari poteſt, & </s> <s xml:id="echoid-s329" xml:space="preserve">quidem iſto pacto. <lb/></s> <s xml:id="echoid-s330" xml:space="preserve">Columna ABCD plano per axem EF eſto bi-<lb/>ſecta, axisq́ue inferioris biſegmĕti EC, eſto GH. </s> <s xml:id="echoid-s331" xml:space="preserve"><lb/>EC porro ſecta plano IK, ad baſin ED <lb/> <anchor type="figure" xlink:label="fig-527.01.014-02a" xlink:href="fig-527.01.014-02"/> parallelo, axem GH ſecante in L, cen-<lb/>trúmque gravitatis ſegmĕti IKDE eſto <lb/>M, in medio GL, ſegmĕti verò IKCF <lb/>ſit N, in medio LH, O denique totius <lb/>ABCD, in medio EF. </s> <s xml:id="echoid-s332" xml:space="preserve">OP gravitatis <lb/>diameter, totius corporis ABCD, MQ <lb/>ſegmĕti IKDE, NR ſegmĕti IKCF. <lb/></s> <s xml:id="echoid-s333" xml:space="preserve">His poſitis, obſcurum eſſe non poteſt, <lb/>quin columnæ ſegmentum dextrum ſi-<lb/>niſtro æquilibre ſit ſitu.</s> <s xml:id="echoid-s334" xml:space="preserve"/> </p> <div xml:id="echoid-div60" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.014-02" xlink:href="fig-527.01.014-02a"> <image file="527.01.014-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.014-02"/> </figure> </div> <p> <s xml:id="echoid-s335" xml:space="preserve">Inferiore ſegmento EFCD deorſum detracto, & </s> <s xml:id="echoid-s336" xml:space="preserve">ex MQ & </s> <s xml:id="echoid-s337" xml:space="preserve">NR lineis, <lb/>ut in paradigmate exhibetur, ſuſpenſo, ABFE corporeum jugum antiquum <lb/>nihilo minus obtinere ſitum dubium non eſt. </s> <s xml:id="echoid-s338" xml:space="preserve">Segmentum IKDE ab IKCF <lb/>diviſum ſingamus, & </s> <s xml:id="echoid-s339" xml:space="preserve">utrumque ab altero ita <lb/> <anchor type="figure" xlink:label="fig-527.01.014-03a" xlink:href="fig-527.01.014-03"/> ſeparatum, ut quò Natura ſert cadere poſſit, <lb/>verum cum utrumq; </s> <s xml:id="echoid-s340" xml:space="preserve">è ſuo gravitatis centro <lb/>M, N dependeat, primum obtinebit ſitum <lb/>per 4 deſinitionem, & </s> <s xml:id="echoid-s341" xml:space="preserve">propterea neque in <lb/>ABFE quicquam mutabitur. </s> <s xml:id="echoid-s342" xml:space="preserve">Sed IKDE <lb/>eandem rationem ad IKCF habere, quam <lb/>radium OR ad OQ, jam antea experien-<lb/>tia docuit, adeo quidem ut quod prius in <lb/>S*TATICO* jugo (hoc eſt lineâ) exhibui-<lb/>mus, id jam nunc in corporeo exhibeamus. </s> <s xml:id="echoid-s343" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s344" xml:space="preserve">Quapropter <lb/>duarum gravitatum ſitu æquilibrium ponderoſior, illam rationem habet ad <lb/>leviorem (cujuſcunque vel materiæ, vel formæ ſint corpora) quæ longioris <lb/>radii eſt ad breviorem, quod nobis demonſtrandum fuit.</s> <s xml:id="echoid-s345" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.014-03" xlink:href="fig-527.01.014-03a"> <image file="527.01.014-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.014-03"/> </figure> </div> </div> <div xml:id="echoid-div63" type="section" level="1" n="52"> <head xml:id="echoid-head61" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s346" xml:space="preserve">E`<unsure/> primæ propoſitionis converſo conſequens eſt; </s> <s xml:id="echoid-s347" xml:space="preserve">Si ponderis gravio-<lb/>ris ea ratio eſt ad levius, quæ longioris radii ad breviorem, pondera eſſe ſitu <lb/>æquipondia.</s> <s xml:id="echoid-s348" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div64" type="section" level="1" n="53"> <head xml:id="echoid-head62" xml:space="preserve">1 PROBLEMA. 2 PROPOSITIO.</head> <p> <s xml:id="echoid-s349" xml:space="preserve">Dati & </s> <s xml:id="echoid-s350" xml:space="preserve">cogniti ponderis anſam invenire.</s> <s xml:id="echoid-s351" xml:space="preserve"/> </p> <pb o="15" file="527.01.015" n="15" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div65" type="section" level="1" n="54"> <head xml:id="echoid-head63" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s352" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s353" xml:space="preserve">Pondus A ſuſpenſum è C libras tres, pondusalterum B ſuſpen-<lb/>ſum è D unam pendeat, denique CD jugum ſit.</s> <s xml:id="echoid-s354" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s355" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s356" xml:space="preserve">Anſa invenienda nobis eſt.</s> <s xml:id="echoid-s357" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div66" type="section" level="1" n="55"> <head xml:id="echoid-head64" xml:space="preserve">PRAGMATIA.</head> <figure> <image file="527.01.015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.015-01"/> </figure> <p> <s xml:id="echoid-s358" xml:space="preserve">CD ita dividitor, ut majus ſegmentum quod minoris <lb/>ponderis eſt pendulam diametrum verſus, illam rationem <lb/>habeat ad minus, quam majus pondus ad minus. </s> <s xml:id="echoid-s359" xml:space="preserve">Sitq́ue <lb/>in E, ut quemadmodum 3 ℔ A, ad 1 ℔ B: </s> <s xml:id="echoid-s360" xml:space="preserve">ſic ſit ED, <lb/>ad EC. </s> <s xml:id="echoid-s361" xml:space="preserve">Dico lineam per E pendulam, eſſe anſam.</s> <s xml:id="echoid-s362" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div67" type="section" level="1" n="56"> <head xml:id="echoid-head65" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s363" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s364" xml:space="preserve">Datum pondus eſto columna ABCD 6 ℔ pendens, ſimiliter <lb/>columnæ primæ propoſitionis diviſa. </s> <s xml:id="echoid-s365" xml:space="preserve">& </s> <s xml:id="echoid-s366" xml:space="preserve">ex Q ſuſpenſum pondus Y ℔ 12.</s> <s xml:id="echoid-s367" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s368" xml:space="preserve">Q*VAE SITVM*. </s> <s xml:id="echoid-s369" xml:space="preserve">Anſa invenienda eſt.</s> <s xml:id="echoid-s370" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div68" type="section" level="1" n="57"> <head xml:id="echoid-head66" xml:space="preserve">*PRAGMATIA*.</head> <figure> <image file="527.01.015-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.015-02"/> </figure> <p> <s xml:id="echoid-s371" xml:space="preserve">Columnæ pendula gravitatis diametrus <lb/>eſt IT, ponderis verò Y eſt BQ, TQ ju-<lb/>gum ita dirimédum eſt, ut ejus ſegmenta ra-<lb/>tionem habeant, quæ eſt inter 12 ℔ Y, & </s> <s xml:id="echoid-s372" xml:space="preserve"><lb/>6 ℔ columnæ, breviusq́; </s> <s xml:id="echoid-s373" xml:space="preserve">ſegmentum pon-<lb/>deroſioris Y diametrum verſus ſit, quæ eſt <lb/>in X. </s> <s xml:id="echoid-s374" xml:space="preserve">ut NX anſa ſit quæſita.</s> <s xml:id="echoid-s375" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div69" type="section" level="1" n="58"> <head xml:id="echoid-head67" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s376" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s377" xml:space="preserve">ABCD iterum columna eſto. </s> <s xml:id="echoid-s378" xml:space="preserve">ſecta ut paulò ante, & </s> <s xml:id="echoid-s379" xml:space="preserve">Y 6 ℔ <lb/>ex X pendeto.</s> <s xml:id="echoid-s380" xml:space="preserve"/> </p> <figure> <image file="527.01.015-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.015-03"/> </figure> <p> <s xml:id="echoid-s381" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s382" xml:space="preserve">Anſa eſt nobis quærĕda.</s> <s xml:id="echoid-s383" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div70" type="section" level="1" n="59"> <head xml:id="echoid-head68" xml:space="preserve">*PRAGMATIA.*</head> <p> <s xml:id="echoid-s384" xml:space="preserve">Columnę pendula diameter eſt IT pon-<lb/>deris verò Y, NX, TX jugum ita parti-<lb/>tum, ut ſegmenta rationem habeant 6 ℔ Y, <lb/>ad 6 ℔ columnæ, quod in V incidit, eritq́ue <lb/>VL anſa quæſita.</s> <s xml:id="echoid-s385" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div71" type="section" level="1" n="60"> <head xml:id="echoid-head69" xml:space="preserve">*PRAGMATIA ALIVSMODI.*</head> <p> <s xml:id="echoid-s386" xml:space="preserve">PEndula diametros ponderis MLBCY eſt NX, MLAD verò eſt SG, <lb/>& </s> <s xml:id="echoid-s387" xml:space="preserve">jugum XS, quod ita ſecandum eſt, ut ſegmentorũ ratio ſit eadem quæ <lb/>8 ℔ pondens MLBCY ad 4 ℔ MLAD, & </s> <s xml:id="echoid-s388" xml:space="preserve">illorum brevius à diametro pen-<lb/>dulâ quæ eſt V gravius pondus verſus ſit, atque hoc modo VL quæſita erit <lb/>anſa, ut prius.</s> <s xml:id="echoid-s389" xml:space="preserve"/> </p> <pb o="16" file="527.01.016" n="16" rhead="*I* L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div72" type="section" level="1" n="61"> <head xml:id="echoid-head70" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s390" xml:space="preserve">*Datvm.</s> <s xml:id="echoid-s391" xml:space="preserve">* ABCD columna eſto, partita, ut prius, pendeatq́ue Y 6 ℔ ex <lb/>X, Z vero 24 ℔ ex R. </s> <s xml:id="echoid-s392" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s393" xml:space="preserve">* Anſa quærenda eſt.</s> <s xml:id="echoid-s394" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div73" type="section" level="1" n="62"> <head xml:id="echoid-head71" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s395" xml:space="preserve">Diametros pendula põderis ABCDY <lb/> <anchor type="figure" xlink:label="fig-527.01.016-01a" xlink:href="fig-527.01.016-01"/> eſt L V, ex 3 exemplo, ponderis autem <lb/>Z, R E, R V itaque jugum in duo ſe-<lb/>gmenta ſecandum, ut ratio illorum ſit 12 <lb/>A B C D Y ad 24 Z. </s> <s xml:id="echoid-s396" xml:space="preserve">& </s> <s xml:id="echoid-s397" xml:space="preserve">à pendulâ dia-<lb/>metro quæ incidet in S, brevius ſegmen-<lb/>tum gravius pondus verſus ſit, eritq́; </s> <s xml:id="echoid-s398" xml:space="preserve">S G <lb/>quæſita anſa.</s> <s xml:id="echoid-s399" xml:space="preserve"/> </p> <div xml:id="echoid-div73" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.016-01" xlink:href="fig-527.01.016-01a"> <image file="527.01.016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.016-01"/> </figure> </div> </div> <div xml:id="echoid-div75" type="section" level="1" n="63"> <head xml:id="echoid-head72" xml:space="preserve">PRAGMATIA ALIVSMODI.</head> <p> <s xml:id="echoid-s400" xml:space="preserve">PEndula gravitatis diametros ponderis A B C D Z eſto Æ W ex 3 propo-<lb/>ſitione, ut S Æ valeat {2/3} S R, pendulaq́ue diametros Y, X N eſto, jugum <lb/>vero Æ X ita partitum ut ſegmentorum ratio ſit 30 ℔ A B C D Z ad 6 ℔ Y, <lb/>& </s> <s xml:id="echoid-s401" xml:space="preserve">illorum brevius ponderum gravius verſus ſità pendula diametro, quæ eſt S, <lb/>atque iſto pacto S G quæſita erit anſa.</s> <s xml:id="echoid-s402" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div76" type="section" level="1" n="64"> <head xml:id="echoid-head73" xml:space="preserve">PRAGMATIA ALIVSMODI.</head> <p> <s xml:id="echoid-s403" xml:space="preserve">PEndula gravitatis diametros Y Z (per primum exemplum) eſt Φ Δ, ut S Φ <lb/>ſit {1/5} S R, & </s> <s xml:id="echoid-s404" xml:space="preserve">columnæ diametros pendula T I, & </s> <s xml:id="echoid-s405" xml:space="preserve">T Φ jugum ita partitum, <lb/>ut ratio ſegmentorum ſit 30 ℔ Y cum Z, ad 6 ℔ columnæ, & </s> <s xml:id="echoid-s406" xml:space="preserve">S G hoc modo, <lb/>ut prius, erit anſa quæſita.</s> <s xml:id="echoid-s407" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="1" n="65"> <head xml:id="echoid-head74" style="it" xml:space="preserve">5 Exemplum.</head> <p> <s xml:id="echoid-s408" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s409" xml:space="preserve">* A B C D columna eſto partita ut prius, & </s> <s xml:id="echoid-s410" xml:space="preserve">Y 6 ℔ ex X, Z vero <lb/>24 ℔ ex R pendeat, & </s> <s xml:id="echoid-s411" xml:space="preserve">Æ 12 ℔ è Q. </s> <s xml:id="echoid-s412" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s413" xml:space="preserve">* Anſanobis quæ-<lb/>renda eſt.</s> <s xml:id="echoid-s414" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div78" type="section" level="1" n="66"> <head xml:id="echoid-head75" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s415" xml:space="preserve">Diametros pendula A B C D Y Z eſt S G, ex 4 exempliſententiâ, & </s> <s xml:id="echoid-s416" xml:space="preserve">Æ, <lb/>Q B, S Q eſt jugum in duo ſegmenta partiendum ut illorum ratio ſit, quæ eſt <lb/>36 ℔ columnæ cum Y & </s> <s xml:id="echoid-s417" xml:space="preserve">Z, ad 12 ℔ Æ minus ſegmentum pendulam diame-<lb/>trum verſus gravioris ſegmenti, quæ incidit in T, ut T I anſa ſit quæſita.</s> <s xml:id="echoid-s418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s419" xml:space="preserve">Si ex P præterea 24 ℔ eſſent ſuſpenſæ, S G eſſet anſa, & </s> <s xml:id="echoid-s420" xml:space="preserve">ita deinceps cum <lb/>quovis alio pondere, quod ex jugo ſuſpendi poteſt.</s> <s xml:id="echoid-s421" xml:space="preserve"/> </p> <pb o="17" file="527.01.017" n="17" rhead="*DE STATICÆ* ELEMENTIS."/> </div> <div xml:id="echoid-div79" type="section" level="1" n="67"> <head xml:id="echoid-head76" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s422" xml:space="preserve">Gravioris põderis A, in primo paradigma-<lb/> <anchor type="figure" xlink:label="fig-527.01.017-01a" xlink:href="fig-527.01.017-01"/> te, ea eſt ratio ad levius B, quę longioris radii <lb/>E D ad breviorem E C. </s> <s xml:id="echoid-s423" xml:space="preserve">E F itaque per 9 <lb/>definitionem anſa erit. </s> <s xml:id="echoid-s424" xml:space="preserve">Reliquorum exem-<lb/>plorum eadem demonſtratio fuerit, quibus <lb/>brevitatis cauſa ſuperſedemus.</s> <s xml:id="echoid-s425" xml:space="preserve"/> </p> <div xml:id="echoid-div79" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.017-01" xlink:href="fig-527.01.017-01a"> <image file="527.01.017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.017-01"/> </figure> </div> <p> <s xml:id="echoid-s426" xml:space="preserve">*CONCLVSIO.</s> <s xml:id="echoid-s427" xml:space="preserve">* Cognitis igitur ponde-<lb/>ribus datis anſam illorum invenerimus.</s> <s xml:id="echoid-s428" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div81" type="section" level="1" n="68"> <head xml:id="echoid-head77" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s429" xml:space="preserve">Si ad γ, 2 paradigmatis pondus, 1 ℔ adderetur, & </s> <s xml:id="echoid-s430" xml:space="preserve">ex V 1 ℔ ſuſpenderetur, at <lb/>hîc infra ponitur, ex antecedentibus manifeſtum <lb/> <anchor type="figure" xlink:label="fig-527.01.017-02a" xlink:href="fig-527.01.017-02"/> eſt X N anſam nibilo minus manere, & </s> <s xml:id="echoid-s431" xml:space="preserve">quæcung <lb/>ex ea dependĕt ſitu æquilibria eſſe. </s> <s xml:id="echoid-s432" xml:space="preserve">Idem N X ma-<lb/>nebit ſi Z 1 ℔ pendeat ex T, & </s> <s xml:id="echoid-s433" xml:space="preserve">γ 14 ℔ valeat; <lb/></s> <s xml:id="echoid-s434" xml:space="preserve">aut Z 1 ℔ ex S, & </s> <s xml:id="echoid-s435" xml:space="preserve">γ ſit ℔ 15: </s> <s xml:id="echoid-s436" xml:space="preserve">itidem Z 1 ℔ ex <lb/>R, & </s> <s xml:id="echoid-s437" xml:space="preserve">γ ſit 16 ℔, aut ex P, & </s> <s xml:id="echoid-s438" xml:space="preserve">γ ſit 17 ℔. </s> <s xml:id="echoid-s439" xml:space="preserve">& </s> <s xml:id="echoid-s440" xml:space="preserve"><lb/>ita deinceps ſi jugum longius fuerit, perpetuo 1 ℔ <lb/>ad γ addendo, pro longitudine cujus{q́ue} partis <lb/>æquantis X V, quò Z promovetur. </s> <s xml:id="echoid-s441" xml:space="preserve">Vnde quali-<lb/>tates & </s> <s xml:id="echoid-s442" xml:space="preserve">affectiones Stater æ cognoſcuntur, ut ple-<lb/>nius in Statices praxi tr actabitur.</s> <s xml:id="echoid-s443" xml:space="preserve"/> </p> <div xml:id="echoid-div81" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.017-02" xlink:href="fig-527.01.017-02a"> <image file="527.01.017-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.017-02"/> </figure> </div> </div> <div xml:id="echoid-div83" type="section" level="1" n="69"> <head xml:id="echoid-head78" xml:space="preserve">2 PROBLEMA. 3 PROPOSITIO.</head> <p> <s xml:id="echoid-s444" xml:space="preserve">Datis ponderibus ſitu æquipondiis, altero cognito, al-<lb/>tero incognito, unà cum ansâ: </s> <s xml:id="echoid-s445" xml:space="preserve">incognitum cognitum <lb/>reddere.</s> <s xml:id="echoid-s446" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div84" type="section" level="1" n="70"> <head xml:id="echoid-head79" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s447" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s448" xml:space="preserve">* A & </s> <s xml:id="echoid-s449" xml:space="preserve">B duo pondera ſitu æquipondia ſunto <lb/> <anchor type="figure" xlink:label="fig-527.01.017-03a" xlink:href="fig-527.01.017-03"/> quorum A ex C ſuſpenſum 3 ℔ pendeat, B verò ſuſpenſum <lb/>ex D ignorator, E F anſa. </s> <s xml:id="echoid-s450" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s451" xml:space="preserve">* Pondus B <lb/>innoteſcat.</s> <s xml:id="echoid-s452" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.017-03" xlink:href="fig-527.01.017-03a"> <image file="527.01.017-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.017-03"/> </figure> </div> </div> <div xml:id="echoid-div86" type="section" level="1" n="71"> <head xml:id="echoid-head80" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s453" xml:space="preserve">In rationem radii E D ad radium E C inquirendum eſt, <lb/>ſit autem ex hypotheſi, ut 3 ad 1, dico igitur quemadmodum <lb/>E D 3, ad E C 1: </s> <s xml:id="echoid-s454" xml:space="preserve">ita & </s> <s xml:id="echoid-s455" xml:space="preserve">3 ℔ ad quem? </s> <s xml:id="echoid-s456" xml:space="preserve">proportione conclu-<lb/>ditur 1 ℔.</s> <s xml:id="echoid-s457" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div87" type="section" level="1" n="72"> <head xml:id="echoid-head81" style="it" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s458" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s459" xml:space="preserve">* Quemadmodum 2 propoſitionis 2 exemplo, columna A B C D <lb/>pro altero pondere 6 ℔ pendeat, reliquum pondus incognitũ, & </s> <s xml:id="echoid-s460" xml:space="preserve">inde ſuſpen-<lb/>ſum Y ſit, anſa autĕ X N. </s> <s xml:id="echoid-s461" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s462" xml:space="preserve">* In põdus X inquirendũ nobis eſt.</s> <s xml:id="echoid-s463" xml:space="preserve"/> </p> <pb o="18" file="527.01.018" n="18" rhead="*I* L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div88" type="section" level="1" n="73"> <head xml:id="echoid-head82" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s464" xml:space="preserve">Quandoquidem TI columnę diameter pendula eſt, Q B autĕ ponderis Y, <lb/>T Q jugum erit, ejusq́ue radius brevior X Q, X T vero longior. </s> <s xml:id="echoid-s465" xml:space="preserve">Inquiren-<lb/>dum igitur quæ ſit ratio X Q radii ad radium X T: </s> <s xml:id="echoid-s466" xml:space="preserve">eſto ex hypotheſi 1 ad 2. <lb/></s> <s xml:id="echoid-s467" xml:space="preserve">Dico igitur ut X Q 1 ad X T 2: </s> <s xml:id="echoid-s468" xml:space="preserve">ita columna 6 ℔ ad quem? </s> <s xml:id="echoid-s469" xml:space="preserve">pro Y conclu-<lb/>ditur 12 ℔. </s> <s xml:id="echoid-s470" xml:space="preserve">Hujuſmodi plura exempla 2 propoſitionis exemplorum conſimi-<lb/>lia proponi poſſent, niſi jam ex antecedentibus innotuiſſent.</s> <s xml:id="echoid-s471" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div89" type="section" level="1" n="74"> <head xml:id="echoid-head83" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s472" xml:space="preserve">B primi exempli, ſi poſſit fieri, 1 ℔ ponderoſius ſit, non erit gravioris pon-<lb/>deris ea ratio adlevius, quæ longioris radii eſt ad breviorem, quod 1 propoſi-<lb/>tioni repugnat. </s> <s xml:id="echoid-s473" xml:space="preserve">B igitur 1 ℔ ponderoſius non eſt. </s> <s xml:id="echoid-s474" xml:space="preserve">eodemq́ue pacto nequele-<lb/>vius eſſe demonſtrabitur. </s> <s xml:id="echoid-s475" xml:space="preserve">Ideoq́ue unam tantum ℔ pendebit, quod demon-<lb/>ſtrandum erat. </s> <s xml:id="echoid-s476" xml:space="preserve">*CONCLVSIO.</s> <s xml:id="echoid-s477" xml:space="preserve">* Datis igitur duobus ponderibus ſitu æ qui-<lb/>pondiis cognito ſcilicet, & </s> <s xml:id="echoid-s478" xml:space="preserve">incognito, datâ item anſâ. </s> <s xml:id="echoid-s479" xml:space="preserve">Incognitum pondus <lb/>cognitum fecimus, quod fuit quæſitum.</s> <s xml:id="echoid-s480" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div90" type="section" level="1" n="75"> <head xml:id="echoid-head84" xml:space="preserve">3 PROBLEMA. 4 PROPOSITIO.</head> <p> <s xml:id="echoid-s481" xml:space="preserve">Datis ponderibus cognitis ſitu æquipondiis, unàcum <lb/>lõgitudine radii alterius: </s> <s xml:id="echoid-s482" xml:space="preserve">reliqui radii lõgitudinĕ invenire.</s> <s xml:id="echoid-s483" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s484" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s485" xml:space="preserve">* A & </s> <s xml:id="echoid-s486" xml:space="preserve">B pondera ſitu æquipõdia ſunto, A quidem ex C ſuſpen-<lb/>ſum 3 ℔, B vero ex D 1 ℔ pendeat, & </s> <s xml:id="echoid-s487" xml:space="preserve">radius D E 6 pedes ſit longus.</s> <s xml:id="echoid-s488" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s489" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s490" xml:space="preserve">* Reliqui radii longitudo nobis invenienda eſt.</s> <s xml:id="echoid-s491" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div91" type="section" level="1" n="76"> <head xml:id="echoid-head85" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s492" xml:space="preserve">Proportio ſic erit, ut A 3 ℔ ad B 1 ℔. </s> <s xml:id="echoid-s493" xml:space="preserve">ita D E 6 pedes ad <lb/> <anchor type="figure" xlink:label="fig-527.01.018-01a" xlink:href="fig-527.01.018-01"/> E C 2. </s> <s xml:id="echoid-s494" xml:space="preserve">Plura exempla 2 propoſitionis exemplis conformia <lb/>hucadducere poſſemus, niſi ex antecedente doctrinâ ſatis no-<lb/>ta eſſent.</s> <s xml:id="echoid-s495" xml:space="preserve"/> </p> <div xml:id="echoid-div91" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.018-01" xlink:href="fig-527.01.018-01a"> <image file="527.01.018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.018-01"/> </figure> </div> </div> <div xml:id="echoid-div93" type="section" level="1" n="77"> <head xml:id="echoid-head86" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s496" xml:space="preserve">Si E C duobus pedibus longior eſſe fingatur, longioris ra-<lb/>dii minor ratio fuerit ad breviorem, quam gravioris ponderis <lb/>ad levius, quod contra primam propoſitionem eſt. </s> <s xml:id="echoid-s497" xml:space="preserve">E C igi-<lb/>tur 2 pedibus nequaquam longior eſt. </s> <s xml:id="echoid-s498" xml:space="preserve">Similiter neq; </s> <s xml:id="echoid-s499" xml:space="preserve">brevior <lb/>eſſe demonſtrabitur, ut duos tantum pedes longum eſſe conſequens ſit, quod <lb/>erat demonſtrandum.</s> <s xml:id="echoid-s500" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s501" xml:space="preserve">*CONCLVSIO.</s> <s xml:id="echoid-s502" xml:space="preserve">* Datis igitur duobus ponderibus ſitu æquipondiis, & </s> <s xml:id="echoid-s503" xml:space="preserve">al-<lb/>terius radiorum longitudine: </s> <s xml:id="echoid-s504" xml:space="preserve">etiam reliqui longitudinem invenerimus, ut <lb/>petitum erat.</s> <s xml:id="echoid-s505" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div94" type="section" level="1" n="78"> <head xml:id="echoid-head87" xml:space="preserve">4 PROBLEMA. 5 PROPOSITIO.</head> <p> <s xml:id="echoid-s506" xml:space="preserve">Datâ columnâ pondus invenire, quod ad columnam <lb/>habeat datam rationem.</s> <s xml:id="echoid-s507" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s508" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s509" xml:space="preserve">* A B C D columna eſto, cujus axis E F, centrum G ſit, data au-<lb/>tem ratio 2 ad 3. </s> <s xml:id="echoid-s510" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s511" xml:space="preserve">* Pondus ejus rationis erit ad datam co-<lb/>lumnam: </s> <s xml:id="echoid-s512" xml:space="preserve">quæ eſt 2 ad 3, hoc eſt columnæ {2/3}.</s> <s xml:id="echoid-s513" xml:space="preserve"/> </p> <pb o="19" file="527.01.019" n="19" rhead="*DE STATICÆ ELEMENTIS.*"/> </div> <div xml:id="echoid-div95" type="section" level="1" n="79"> <head xml:id="echoid-head88" xml:space="preserve">NOTATO.</head> <p> <s xml:id="echoid-s514" xml:space="preserve">Quemadmodum Arithmeticæ & </s> <s xml:id="echoid-s515" xml:space="preserve">Geometricæ propoſitiones diverſas pra-<lb/>gmatias habent, ita etiam *STATICAE*, de columna enim ſegmentum deſe-<lb/>cari poſſet, cujus ratio ad totam eſſepoſſet, <lb/> <anchor type="figure" xlink:label="fig-527.01.019-01a" xlink:href="fig-527.01.019-01"/> quæ eſt 2 ad 3. </s> <s xml:id="echoid-s516" xml:space="preserve">Aut etiam columnâ integrâ <lb/>manente quævis alia materia contra illam <lb/>ponderari poſſet, indeq́ue {1/3} auferri, verum <lb/>Staticè illud ipſum efficere volumus, hoc <lb/>pacto.</s> <s xml:id="echoid-s517" xml:space="preserve"/> </p> <div xml:id="echoid-div95" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.019-01" xlink:href="fig-527.01.019-01a"> <image file="527.01.019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.019-01"/> </figure> </div> </div> <div xml:id="echoid-div97" type="section" level="1" n="80"> <head xml:id="echoid-head89" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s518" xml:space="preserve">A centro G, F verſus 5 puncta (5 ſcilicet <lb/>pro toto datorum terminorum 2 & </s> <s xml:id="echoid-s519" xml:space="preserve">3) ut <lb/>H, I, K, L, M æqualiter ſpacio inter ſe diſſi-<lb/>ta, ſignanda erunt, & </s> <s xml:id="echoid-s520" xml:space="preserve">in ſecundo puncto I (à ſecundo puncto inquam, quia <lb/>2 datorum numerorum alter eſt) columna è pendulâ gravitatis diametro I N <lb/>ſuſpendenda, necnon ex quinto puncto M aliquod pondus demittendum, <lb/>ut O tantæ gravitatis, ut omnia in ſitus æquilibritate pendeant. </s> <s xml:id="echoid-s521" xml:space="preserve">Dico pondus <lb/>O eâ eſſe in rationead columnæ pondus, in qua eſt 2 ad 3, aut O æquare {2/3} co-<lb/>lumnæ.</s> <s xml:id="echoid-s522" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div98" type="section" level="1" n="81"> <head xml:id="echoid-head90" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s523" xml:space="preserve">G gravitatis centrum eſt columnæ A B C D, M P vero pendula gravi-<lb/>tatis diametros ipſius O, propterea ut radius I G ad radium I M: </s> <s xml:id="echoid-s524" xml:space="preserve">ita O ad co-<lb/>lumnam per primam propoſitionem. </s> <s xml:id="echoid-s525" xml:space="preserve">Atqui I G rationem habet ad I M, <lb/>quam 2 ad 3, ergo & </s> <s xml:id="echoid-s526" xml:space="preserve">O ad columnam habet eandem rationem 2 ad 3, quod <lb/>nobis fuit demonſtrandum.</s> <s xml:id="echoid-s527" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div99" type="section" level="1" n="82"> <head xml:id="echoid-head91" xml:space="preserve">NOTATO.</head> <p> <s xml:id="echoid-s528" xml:space="preserve">Etiam <anchor type="note" xlink:href="" symbol="*"/> incommenſurabilium terminorum exempla in medium adducere <anchor type="note" xlink:label="note-527.01.019-01a" xlink:href="note-527.01.019-01"/> poſſemus, niſi ex antecedentibus manifeſta eſſent, etiam ex iis, quæ de incom-<lb/>menſurabilibus magnitudinibus alibi præcepimus.</s> <s xml:id="echoid-s529" xml:space="preserve"/> </p> <div xml:id="echoid-div99" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-527.01.019-01" xlink:href="note-527.01.019-01a" xml:space="preserve"> aſymmetre-<lb/>rum.</note> </div> </div> <div xml:id="echoid-div101" type="section" level="1" n="83"> <head xml:id="echoid-head92" xml:space="preserve">2 THEOREMA. 6 PROPOSITIO.</head> <p> <s xml:id="echoid-s530" xml:space="preserve">Pendulâ columnâ per gravitatis centrum à plano ad <lb/>baſin parallelo ſectâ, firmitudinis autem puncto ſupra <lb/>gravitatis centrum fixo: </s> <s xml:id="echoid-s531" xml:space="preserve">Axis horizonti eſt parallelus.</s> <s xml:id="echoid-s532" xml:space="preserve"/> </p> <figure> <image file="527.01.019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.019-02"/> </figure> <p> <s xml:id="echoid-s533" xml:space="preserve">*DATVM.</s> <s xml:id="echoid-s534" xml:space="preserve">* A B C D columna eſto <lb/>per gravitatis centrum à plano F G ad <lb/>baſin A D parallelo ſecta, H autem <lb/>firmitudinis punctum in pendulâ gra-<lb/>vitatis diametro I G fixum, ſupra gra-<lb/>vitatis centrum E. </s> <s xml:id="echoid-s535" xml:space="preserve">& </s> <s xml:id="echoid-s536" xml:space="preserve">K L axis, M N <lb/>denique horizon.</s> <s xml:id="echoid-s537" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s538" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s539" xml:space="preserve">* K L axem ad ho-<lb/>rizontem M N parallelum eſſe demon-<lb/>ſtrari oportet.</s> <s xml:id="echoid-s540" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div102" type="section" level="1" n="84"> <head xml:id="echoid-head93" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s541" xml:space="preserve">Axis K L, ſi fieri quidem poteſt ab horizonte M N inæqualiter diſtans <pb o="20" file="527.01.020" n="20" rhead="*I LIBER STATICÆ*"/> eſto, ut ſecunda figura exhibet, & </s> <s xml:id="echoid-s542" xml:space="preserve">I H producatur in O, A B ſecans in P, ſe-<lb/>gmentumq́ue columnæ P O C B contra P O D A æquilibre pendeat, atqui <lb/>illud iſto & </s> <s xml:id="echoid-s543" xml:space="preserve">majus & </s> <s xml:id="echoid-s544" xml:space="preserve">ponderoſius eſt (C F G D A enim æquatur F G C B, <lb/>triangulum autem FHI deſectum de F G C B minus eſt triangulo O H G <lb/>de F G C B deſecto, ideo & </s> <s xml:id="echoid-s545" xml:space="preserve">c.) </s> <s xml:id="echoid-s546" xml:space="preserve">ponderoſius itaque ſeleviori æquilibre erit, <lb/>quod planè abſurdum eſt. </s> <s xml:id="echoid-s547" xml:space="preserve">Quapropter K L <lb/> <anchor type="figure" xlink:label="fig-527.01.020-01a" xlink:href="fig-527.01.020-01"/> ad horizontem M N parallelus eſt, ut in <lb/>primo diagrammate.</s> <s xml:id="echoid-s548" xml:space="preserve"/> </p> <div xml:id="echoid-div102" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.020-01" xlink:href="fig-527.01.020-01a"> <image file="527.01.020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.020-01"/> </figure> </div> <p> <s xml:id="echoid-s549" xml:space="preserve">Illud quoque tanquam Statices generale <lb/>theorema habendum eſt.</s> <s xml:id="echoid-s550" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s551" xml:space="preserve">Gravitatis centrum pendentis corporis in pen-<lb/>dulâ gravitatis diametro eſſe.</s> <s xml:id="echoid-s552" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s553" xml:space="preserve">Atqui gravitatis centrum E ſecundi dia-<lb/>grammatis non eſt in I O pendulâ gravitatis <lb/>diametro. </s> <s xml:id="echoid-s554" xml:space="preserve">Impoſſibile igitur.</s> <s xml:id="echoid-s555" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s556" xml:space="preserve">*CONCLVSIO.</s> <s xml:id="echoid-s557" xml:space="preserve">* Columnâigitur ſecta, & </s> <s xml:id="echoid-s558" xml:space="preserve">c.</s> <s xml:id="echoid-s559" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div104" type="section" level="1" n="85"> <head xml:id="echoid-head94" xml:space="preserve">3 THEOREMA. 7 PROPOSITIO.</head> <p> <s xml:id="echoid-s560" xml:space="preserve">Si punctum firmitudinis centrum gravitatis ſit penden-<lb/>tis columnæ, quemcunque ei ſitum dederis, ſervat.</s> <s xml:id="echoid-s561" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s562" xml:space="preserve">*DATVM*. </s> <s xml:id="echoid-s563" xml:space="preserve">A B C D columna eſto, ejusq́; <lb/></s> <s xml:id="echoid-s564" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.020-02a" xlink:href="fig-527.01.020-02"/> centrum E firmum fixumq́ue, quo de linea <lb/>E F ſit ſuſpenſa, axis G H ad horizontem I K <lb/>parallelus. </s> <s xml:id="echoid-s565" xml:space="preserve">*QVAESITVM.</s> <s xml:id="echoid-s566" xml:space="preserve">* Columnam <lb/>A B C D, quemcunqueſitum dederis, reti-<lb/>nere demonſtrandum eſt.</s> <s xml:id="echoid-s567" xml:space="preserve"/> </p> <div xml:id="echoid-div104" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.020-02" xlink:href="fig-527.01.020-02a"> <image file="527.01.020-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.020-02"/> </figure> </div> </div> <div xml:id="echoid-div106" type="section" level="1" n="86"> <head xml:id="echoid-head95" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s568" xml:space="preserve">Datæ columnæ (E puncto immoto) alium <lb/>affingamus ſitum, quam prius, ut ſecundâ <lb/>hæc figurâ exhibetur, producaturq́ue F E, ut in L uſque, ſecans A B in M, <lb/>eq́ue ſuo ſitu, ſi quidem poſſit, emoveatur, & </s> <s xml:id="echoid-s569" xml:space="preserve">ſegmentum M L D A, vel <lb/>M L C B nutet deſcendatq́ue. </s> <s xml:id="echoid-s570" xml:space="preserve">Atqui duo iſta ſegmenta magnitudine æqua-<lb/>lia ſunt, ideoq́ue æquilibria, æquilibrium <lb/> <anchor type="figure" xlink:label="fig-527.01.020-03a" xlink:href="fig-527.01.020-03"/> igitur alterum ponderoſius eſſe altero conſe-<lb/>quens erit, quod prorſus abſurdum eſt. </s> <s xml:id="echoid-s571" xml:space="preserve">Co-<lb/>lumna igitur ſitum ſuum obtinet, aut alium <lb/>quemvis, quicunque ei tributus fuerit.</s> <s xml:id="echoid-s572" xml:space="preserve"/> </p> <div xml:id="echoid-div106" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.020-03" xlink:href="fig-527.01.020-03a"> <image file="527.01.020-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.020-03"/> </figure> </div> <p> <s xml:id="echoid-s573" xml:space="preserve">*CONCLVSIO.</s> <s xml:id="echoid-s574" xml:space="preserve">* Si itaque firmitudinis <lb/> punctum columnæ centrum fuerit, quemli-<lb/>bet datum ſitum ſervabit.</s> <s xml:id="echoid-s575" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div108" type="section" level="1" n="87"> <head xml:id="echoid-head96" xml:space="preserve">4 THEOREMA. 8 PROPOSITIO.</head> <p> <s xml:id="echoid-s576" xml:space="preserve">Sicolumna per gravitatis punctum ſit ſecta à plano ad <pb o="21" file="527.01.021" n="21" rhead="*DE* S*TATICÆ ELEMENTIS*."/> bafin parallelo, fueritq́ue firmitudinis punctum in ſecan-<lb/>te plano, infra gravitatis centrum: </s> <s xml:id="echoid-s577" xml:space="preserve">Columna (naturæ du-<lb/>ctu) ſeſe invertit donec gravitatis centrum in pendulâ gra-<lb/>vitatis diametro ſit.</s> <s xml:id="echoid-s578" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s579" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s580" xml:space="preserve">A B C D columna eſto, per gravitatis centrum E ſecta, plano <lb/>FG ad baſin A D parallelo, G autem fir-<lb/>mitudinis punctum infra gravitatis cen-<lb/> <anchor type="figure" xlink:label="fig-527.01.021-01a" xlink:href="fig-527.01.021-01"/> trum E, quo faſtigio H incumbit, & </s> <s xml:id="echoid-s581" xml:space="preserve"><lb/>quieſcit, I K porro axis horizonti L M pa-<lb/>rallelus. </s> <s xml:id="echoid-s582" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s583" xml:space="preserve">Columnam in-<lb/>vertere ſe probandum eſt, uſque dum gra-<lb/>vitatis centrum in pendulâ gravitatis dia-<lb/>metro fuerit. </s> <s xml:id="echoid-s584" xml:space="preserve">Sed naturæ, ut dixi, ductu <lb/>nutuq́;</s> <s xml:id="echoid-s585" xml:space="preserve">, Mathematicè enim in eo quieſce-<lb/>repoteſt.</s> <s xml:id="echoid-s586" xml:space="preserve"/> </p> <div xml:id="echoid-div108" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.021-01" xlink:href="fig-527.01.021-01a"> <image file="527.01.021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.021-01"/> </figure> </div> </div> <div xml:id="echoid-div110" type="section" level="1" n="88"> <head xml:id="echoid-head97" xml:space="preserve">DEMONSTRATIO.</head> <p style="it"> <s xml:id="echoid-s587" xml:space="preserve">A. </s> <s xml:id="echoid-s588" xml:space="preserve">Luicquid jacet ſolum aliquod neceſſe eſt habeat ubi quieſcat.</s> <s xml:id="echoid-s589" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s590" xml:space="preserve">E. </s> <s xml:id="echoid-s591" xml:space="preserve">Atqui columnæ nihil eſt ubi quieſcat.</s> <s xml:id="echoid-s592" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s593" xml:space="preserve">F. </s> <s xml:id="echoid-s594" xml:space="preserve">Columnia igitur jacere nequit.</s> <s xml:id="echoid-s595" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s596" xml:space="preserve">Syllogiſmi hujus aſſumptio inde manifeſta eſt, quòd punctum non ſit ma-<lb/>gnitudo, ideo neque ſolum. </s> <s xml:id="echoid-s597" xml:space="preserve">Verum quidem eſt, fingere nos ali quando corpus <lb/>ex hypotheſi ita quieſcere, ſed re verâ effectum illud reddere hominum nemo <lb/>poteſt. </s> <s xml:id="echoid-s598" xml:space="preserve">Adeo ut quantumvis axis I K horizonti L M parallelus ſtatuatur; <lb/></s> <s xml:id="echoid-s599" xml:space="preserve">columnatamen ( puncto G immoto) in illud latus ſeſe invertet à quo motus <lb/>principium exſtiterit. </s> <s xml:id="echoid-s600" xml:space="preserve">Quod autem eo uſque ſeſe motabit atq; </s> <s xml:id="echoid-s601" xml:space="preserve">invertet, donec <lb/>gravitatis centrum in pendulam gravitatis diametrum incidat, è 6 propoſitio-<lb/>ne manifeſtum eſt. </s> <s xml:id="echoid-s602" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s603" xml:space="preserve">* Sectâ igitur columnâ, & </s> <s xml:id="echoid-s604" xml:space="preserve">c.</s> <s xml:id="echoid-s605" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div111" type="section" level="1" n="89"> <head xml:id="echoid-head98" xml:space="preserve">1 NOTA.</head> <p style="it"> <s xml:id="echoid-s606" xml:space="preserve">Hic ſi aliquis diſcriminationem inter jacere, & </s> <s xml:id="echoid-s607" xml:space="preserve">pendère declar ari ſibi postulaverit, <lb/>hoc reſponſum à nobis ferat. </s> <s xml:id="echoid-s608" xml:space="preserve">Pendere corpus, quando gravitatis centrum deorſum eſt, <lb/>vel in propinquo puncti in quo quieſcit; </s> <s xml:id="echoid-s609" xml:space="preserve">ſin gravitatis centrum ſurſum eſt jacere, ſtare, <lb/>ſedere existimamus. </s> <s xml:id="echoid-s610" xml:space="preserve">Iacere quidem quando longius latus ſecundum horizontem ſeſe <lb/>exporrigit, ſtare quando horizonti perpendiculare eſt, hinc illud eſt quod cubum (quia <lb/>latera habet æqualia) tam ſtare quam jacere, tam{q́ue} jacere quam ſtare propriè dici-<lb/>mus. </s> <s xml:id="echoid-s611" xml:space="preserve">Sedere inter utrumque medium eſſe.</s> <s xml:id="echoid-s612" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div112" type="section" level="1" n="90"> <head xml:id="echoid-head99" xml:space="preserve">2 NOTA.</head> <p style="it"> <s xml:id="echoid-s613" xml:space="preserve">Si quis argumentum trium propoſitionum experientiâ edoceri cupidus eſt, regulam <lb/>e<unsure/>x ligno, aliáve quâlibet materiâ æquabiliter & </s> <s xml:id="echoid-s614" xml:space="preserve">craßâ & </s> <s xml:id="echoid-s615" xml:space="preserve">põderosâ ſumat, ut A B C D. <lb/></s> <s xml:id="echoid-s616" xml:space="preserve">in ea{q́ue} punctis E, F, G, H in mediis lineis A B, B C, C D, D A ſignatis, rectas E G, <lb/>& </s> <s xml:id="echoid-s617" xml:space="preserve">H F ſeſe in I ſecantes ducat. </s> <s xml:id="echoid-s618" xml:space="preserve">Deinde exiguum <lb/>foramen, & </s> <s xml:id="echoid-s619" xml:space="preserve">tanquam acu punctum, terebretur in <lb/>I, item ſupra I in K, denique infra in L. </s> <s xml:id="echoid-s620" xml:space="preserve">Tum <lb/> <anchor type="figure" xlink:label="fig-527.01.021-02a" xlink:href="fig-527.01.021-02"/> acu in K inſertâ, ut liberè moveri poßit, diame-<lb/>trum H F, nunquã non horizonti parallelam mu-<lb/>nere; </s> <s xml:id="echoid-s621" xml:space="preserve">in I verò tranſlatâ, regulam quemcunque <lb/>ſitum dederis ſervare, denique in L inditâ, omnia <lb/>@@ illud latus inverti, unde primum motus incipit, donec I in gravitatis ſuæ centre <pb o="22" file="527.01.022" n="22" rhead="*I* L*IBER* S*TATICÆ*"/> fuerit, experientia testabitur, cujus rei cauſa è 6, 7, 8 propoſitionibus mani-<lb/>festa eſt.</s> <s xml:id="echoid-s622" xml:space="preserve"/> </p> <div xml:id="echoid-div112" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.021-02" xlink:href="fig-527.01.021-02a"> <image file="527.01.021-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.021-02"/> </figure> </div> </div> <div xml:id="echoid-div114" type="section" level="1" n="91"> <head xml:id="echoid-head100" xml:space="preserve">5 THEOREMA. 9 PROPOSITIO.</head> <p> <s xml:id="echoid-s623" xml:space="preserve">Anſa infinitum cõtinuata binorum ponderum jugum <lb/>quodvis in ſuos radios ſecat.</s> <s xml:id="echoid-s624" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s625" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s626" xml:space="preserve">* A, B duo pondera ſunto, C D & </s> <s xml:id="echoid-s627" xml:space="preserve">E F eorum diametri. </s> <s xml:id="echoid-s628" xml:space="preserve">& </s> <s xml:id="echoid-s629" xml:space="preserve">ju-<lb/>gum CE, anſa denique G H, ita ut C G ſit ad G E, ut pondus B ad A. <lb/></s> <s xml:id="echoid-s630" xml:space="preserve">Eſto & </s> <s xml:id="echoid-s631" xml:space="preserve">I K jugum inæqualiter à C E diſtans, & </s> <s xml:id="echoid-s632" xml:space="preserve">G H infinitum continuator <lb/>L verſus ſecans jugum I K in M. </s> <s xml:id="echoid-s633" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s634" xml:space="preserve">* Demonſtrandum nobis <lb/>eſt I M & </s> <s xml:id="echoid-s635" xml:space="preserve">M K etiam radios eſſe ponderum A, B. </s> <s xml:id="echoid-s636" xml:space="preserve">id eſt, ut B ad A: </s> <s xml:id="echoid-s637" xml:space="preserve">ſic etiam <lb/>M I eſſe ad M K.</s> <s xml:id="echoid-s638" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s639" xml:space="preserve">P*RAEPARATIO.</s> <s xml:id="echoid-s640" xml:space="preserve">* C N ducaturad I K parallela, ſe-<lb/>cans H L in O.</s> <s xml:id="echoid-s641" xml:space="preserve"/> </p> <figure> <image file="527.01.022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.022-01"/> </figure> </div> <div xml:id="echoid-div115" type="section" level="1" n="92"> <head xml:id="echoid-head101" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s642" xml:space="preserve">Quemadmodum C G ad G E: </s> <s xml:id="echoid-s643" xml:space="preserve">ita C O ad O N. </s> <s xml:id="echoid-s644" xml:space="preserve">Atqui <lb/>C O æquatur I M, & </s> <s xml:id="echoid-s645" xml:space="preserve">N O ipſi M K, quapropter ut C G <lb/>ad G E: </s> <s xml:id="echoid-s646" xml:space="preserve">ita I M ad M K. </s> <s xml:id="echoid-s647" xml:space="preserve">Atqui ut B ad A: </s> <s xml:id="echoid-s648" xml:space="preserve">ita ex con-<lb/>ceſſo C G ad G E, ideoq́ue ut B ad A: </s> <s xml:id="echoid-s649" xml:space="preserve">ita M I ad M K: <lb/></s> <s xml:id="echoid-s650" xml:space="preserve">eadem cujuſvis jugi demonſtratio eſt lineis C D & </s> <s xml:id="echoid-s651" xml:space="preserve">E F ter-<lb/>minati, ut P Q ſecti in R, & </s> <s xml:id="echoid-s652" xml:space="preserve">quæcunque alia lineari poſſuntinter dictos ter-<lb/>minos. </s> <s xml:id="echoid-s653" xml:space="preserve">C*ONCLUSIO.</s> <s xml:id="echoid-s654" xml:space="preserve">* Anſa in infinitum cõtinuata ſecat quodvis jugum <lb/>in ſuos radios, quod nobis demonſtrandum fuit.</s> <s xml:id="echoid-s655" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div116" type="section" level="1" n="93"> <head xml:id="echoid-head102" xml:space="preserve">1 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s656" xml:space="preserve">Hinc conſequens eſt, ut duorum ponderum pendula gravitatis diametros <lb/>inveniatur, non neceſſe eſſe ut jugum horizonti ſit parallelum. </s> <s xml:id="echoid-s657" xml:space="preserve">Verum quoli-<lb/>bet modo ſitum iſti uſui ſufficere.</s> <s xml:id="echoid-s658" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div117" type="section" level="1" n="94"> <head xml:id="echoid-head103" xml:space="preserve">2 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s659" xml:space="preserve">Quandoquidé centrum gravitatis in pendulâ gravitatis diametro eſt, quam-<lb/>libet rectam inter duo gravitatis centra terminatam, etiam ponderum jugum <lb/>eſſe cõſequens eſt, & </s> <s xml:id="echoid-s660" xml:space="preserve">radiorum jugi diſcriminationem gravitatis centrum eſſe <lb/>amborum ponderum.</s> <s xml:id="echoid-s661" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div118" type="section" level="1" n="95"> <head xml:id="echoid-head104" xml:space="preserve">5 PROBLEMA. 10 PROPOSITIO.</head> <p> <s xml:id="echoid-s662" xml:space="preserve">Datis, firmitudinis puncto notæ columnæ, notisque <lb/>ponderibus ſitu æquipondiis inde dependentibus: </s> <s xml:id="echoid-s663" xml:space="preserve">inveni-<lb/>rian axis parallelus futurus eſt horizonti; </s> <s xml:id="echoid-s664" xml:space="preserve">an quem dede-<lb/>ris ſitum fervaturus: </s> <s xml:id="echoid-s665" xml:space="preserve">an verò ſe inverſurus, donec gravita-<lb/>tis centrum in pendul à gravitatis diametro ſit.</s> <s xml:id="echoid-s666" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s667" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s668" xml:space="preserve">* A B C D columna eſto 4 ℔, fecta per gravitatis centrum E. </s> <s xml:id="echoid-s669" xml:space="preserve">pla-<lb/>no F G ad baſin A D parallelo, H firmitudinis punctum inſra centrum E, <pb o="23" file="527.01.023" n="23" rhead="*DE* S*TATICÆ ELEMENTIS.*"/> medio inter E, Gloco, è columnâ <lb/> <anchor type="figure" xlink:label="fig-527.01.023-01a" xlink:href="fig-527.01.023-01"/> autem duo pondera I, K, depen-<lb/>dento, ſingula 4 ℔, & </s> <s xml:id="echoid-s670" xml:space="preserve">diametro-<lb/>rum firmitudinis puncta C, D, <lb/>axis L M, horizon N O.</s> <s xml:id="echoid-s671" xml:space="preserve"/> </p> <div xml:id="echoid-div118" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.023-01" xlink:href="fig-527.01.023-01a"> <image file="527.01.023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.023-01"/> </figure> </div> <p> <s xml:id="echoid-s672" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s673" xml:space="preserve">* Inveniendum <lb/>nobis, an axis L M ad horizon-<lb/>tem N O parallelus futurus ſit; <lb/></s> <s xml:id="echoid-s674" xml:space="preserve">an quicunque datus fuerit ſitus, <lb/>retinebit; </s> <s xml:id="echoid-s675" xml:space="preserve">an denique invertet ſe <lb/>donec, E centrum gravitatis in <lb/>pendulâ gravitatis diametro ſit, <lb/>quæ eſt per H, quæ diverſitates <lb/>evenire poſſunt, pro variâ ratio-<lb/>ne gravitatis columnæ ad ponde-<lb/>ra, quæ inde dependent.</s> <s xml:id="echoid-s676" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div120" type="section" level="1" n="96"> <head xml:id="echoid-head105" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s677" xml:space="preserve">Ducatur P Q pendula gravi-<lb/>tatis diameter columnæ per E, <lb/>hinc per G pĕdula diameter R S <lb/>ponderum I, K, jugum erit E G, deinde ſecundæ propoſit. </s> <s xml:id="echoid-s678" xml:space="preserve">adjumento, quo <lb/>firmum anſae; </s> <s xml:id="echoid-s679" xml:space="preserve">punctum incidat, cognoſcetur. </s> <s xml:id="echoid-s680" xml:space="preserve">Si enim infra H ſit locus ejus, <lb/>movet ſe L M donec ad horizontem N O parallelus ſit; </s> <s xml:id="echoid-s681" xml:space="preserve">ſi in H, quicunque <lb/>datus fuerit ſitus retinetur; </s> <s xml:id="echoid-s682" xml:space="preserve">ſin ſupra H, omnia invertuntur. </s> <s xml:id="echoid-s683" xml:space="preserve">Atqui columna <lb/>4 ℔ pendet, ponderum I, K, item ſingula 4 ℔, ut amborum totus 8 ℔ ſit ex <lb/>conceſſo, E G itaque ſecta in T, ut E T ad T G illam rationem habeat quæ <lb/>eſt 8 ad 4. </s> <s xml:id="echoid-s684" xml:space="preserve">Dico L M ſeſe moturum (quod T infra H ſit) donec ad horizon-<lb/>tem parallelus frierit. </s> <s xml:id="echoid-s685" xml:space="preserve">Columna 4 ℔ pendeat, pondera vero I, K binas, ſumma <lb/>utriuſque 4 ℔ fuerit: </s> <s xml:id="echoid-s686" xml:space="preserve">ſectâ igitur E Gín H (eſt autem H ex cõceſſo inter E G <lb/>medium) utratio E H ad H G eaſit: </s> <s xml:id="echoid-s687" xml:space="preserve">quæ eſt 4 ad 4; </s> <s xml:id="echoid-s688" xml:space="preserve">dico L M (quòd in H <lb/>incidit) quemcunque ſitum dederis ſervare. </s> <s xml:id="echoid-s689" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s690" xml:space="preserve">columna 4 ℔ eſto, pon-<lb/>dera I, K, vero ſingula 1 ℔, ut ſimul utrumque ſit 2 ℔. </s> <s xml:id="echoid-s691" xml:space="preserve">quapropter E G ſecta <lb/>in V, ut E V eam rationem habeat ad V G: </s> <s xml:id="echoid-s692" xml:space="preserve">quæ eſt 2 ad 4: </s> <s xml:id="echoid-s693" xml:space="preserve">inquio columnam <lb/>& </s> <s xml:id="echoid-s694" xml:space="preserve">ſe, & </s> <s xml:id="echoid-s695" xml:space="preserve">omnia reliqua inverſuram (quòd V ſupra H ſit) uſque dum H in ſua <lb/>gravitatis diametro fuerit.</s> <s xml:id="echoid-s696" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div121" type="section" level="1" n="97"> <head xml:id="echoid-head106" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s697" xml:space="preserve">Primum L M movere ſe donec horizonti ſit parallela, I & </s> <s xml:id="echoid-s698" xml:space="preserve">K 4 ℔ penden-<lb/>tibus, ita liquet. </s> <s xml:id="echoid-s699" xml:space="preserve">Perpendicularis per T, ut T X, eſt pendula gravitatis diame-<lb/>ter totius, eâ igitur omiſsâ, totoq́ue ex perpendiculari per H, ut H Y, ſuſpen-<lb/>ſo (H autem firmitudinis punctum eſt) ſegmentum B A, K, verſus ponde-<lb/>roſius erit, quam quod A D, I, verſus eſt, ideòque B A, K, deorſum verget, <lb/>donec H in pendulâ gravitatis diametro totius fuerit, atque tunc L M ad ho-<lb/>rizontem N O parallela fuerit.</s> <s xml:id="echoid-s700" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s701" xml:space="preserve">Secundò, I & </s> <s xml:id="echoid-s702" xml:space="preserve">K binas ℔ pendentibus, L M quemvis datum ſitum ſerva-<lb/>re, iſto pacto arguitur. </s> <s xml:id="echoid-s703" xml:space="preserve">I & </s> <s xml:id="echoid-s704" xml:space="preserve">K in altitudinem elata eſſe fingamus, ut pro I & </s> <s xml:id="echoid-s705" xml:space="preserve">K, <lb/>D & </s> <s xml:id="echoid-s706" xml:space="preserve">C centra gravitatis ſint, nulla, ex 3 poſtulato, gravitatis mutatio columnæ <pb o="24" file="527.01.024" n="24" rhead="*I* L*IBER* S*TATICÆ*"/> adferetur: </s> <s xml:id="echoid-s707" xml:space="preserve">Eoq́uepoſito & </s> <s xml:id="echoid-s708" xml:space="preserve">conceſſo, H gravitatis centrum eſt corporis è co-<lb/>lumnâ, & </s> <s xml:id="echoid-s709" xml:space="preserve">duobus ponderibus I & </s> <s xml:id="echoid-s710" xml:space="preserve">K compoſiti, & </s> <s xml:id="echoid-s711" xml:space="preserve">ſuper eo centro quem de-<lb/>deris ſitum ſervabit per 4 definitionem, quod quovis in ſitu, in quo L M col-<lb/>locabitur, demonſtrari poterit.</s> <s xml:id="echoid-s712" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s713" xml:space="preserve">Denique I & </s> <s xml:id="echoid-s714" xml:space="preserve">K 1 ℔ pendentibus ſingulis, omnia inverti ita demonſtratur. <lb/></s> <s xml:id="echoid-s715" xml:space="preserve">Perpendicularis per V, ut V Z, pendula gravitatis diameter eſt totius, eâ igi-<lb/>tur omiſsâ, totoq́ue è perpendiculari H Y, per H datum firmitudinis pun-<lb/>ctum, ſuſpenſo, ſegmentum A D I verſus ponderoſius erit illo, quod B C K <lb/>verſus eſt, ideoq́ue idem illud deorſum verget, donec H in pendulâ diame-<lb/>tro totius fuerit, & </s> <s xml:id="echoid-s716" xml:space="preserve">tametſi maximè L M (toto in H firmitudinis puncto mo-<lb/>to) ad horizontem N O parallela collocetur, iſtum tamen ſitum per 8 propo-<lb/>ſitionem non retinebit, ſed totum invertetur, quod nobis probandum fuit.</s> <s xml:id="echoid-s717" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s718" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s719" xml:space="preserve">* Dato firmitudinis puncto notæ columnæ, &</s> <s xml:id="echoid-s720" xml:space="preserve">c.</s> <s xml:id="echoid-s721" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s722" xml:space="preserve">Satis ex antecedentibus liquet, quomodo in aliis procedendum operan-<lb/>dumq́ue ſit, ut in columnis quarum firmitudinis punctum extra G F lineam <lb/>eſt, ponderum puncta firmitudinis alio loco quam in C & </s> <s xml:id="echoid-s723" xml:space="preserve">D. </s> <s xml:id="echoid-s724" xml:space="preserve">Verumenim-<lb/>vero, quia cauſas qualitatum & </s> <s xml:id="echoid-s725" xml:space="preserve">affectionum libræ è fundamĕtis eruere & </s> <s xml:id="echoid-s726" xml:space="preserve">ape-<lb/>rire potiſſimum hic contendimus (qua de re in Statices exercitatione & </s> <s xml:id="echoid-s727" xml:space="preserve">praxi <lb/>fuſius dicetur) nulla admodum irregularium formarum exempla damus.</s> <s xml:id="echoid-s728" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div122" type="section" level="1" n="98"> <head xml:id="echoid-head107" xml:space="preserve">6 PROBLEMA. 11 PROPOSITIO.</head> <p> <s xml:id="echoid-s729" xml:space="preserve">Datis, notâ columnâ, notisq́; </s> <s xml:id="echoid-s730" xml:space="preserve">ponderibus inde ſuſpen-<lb/>ſis: </s> <s xml:id="echoid-s731" xml:space="preserve">invenire ſirmitudinis punctum, in quo quemlibet <lb/>datum ſitum ſervabit.</s> <s xml:id="echoid-s732" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div123" type="section" level="1" n="99"> <head xml:id="echoid-head108" xml:space="preserve">1 NOTA.</head> <p style="it"> <s xml:id="echoid-s733" xml:space="preserve">Si duorum æqualium ponderum, ut A, B, firmitudinis <lb/>puncta in columnæ axe ſint à centro E æquidi stantia, ut in <lb/> <anchor type="figure" xlink:label="fig-527.01.024-01a" xlink:href="fig-527.01.024-01"/> istâ figurâ; </s> <s xml:id="echoid-s734" xml:space="preserve">dubium non eſt, quin E per ſecundam demen-<lb/>ſtrationis partem 10 propoſitionis, quæſitum ſit punctum. <lb/></s> <s xml:id="echoid-s735" xml:space="preserve">Sed exemplum irregularis formæ esto.</s> <s xml:id="echoid-s736" xml:space="preserve"/> </p> <div xml:id="echoid-div123" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.024-01" xlink:href="fig-527.01.024-01a"> <image file="527.01.024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.024-01"/> </figure> </div> </div> <div xml:id="echoid-div125" type="section" level="1" n="100"> <head xml:id="echoid-head109" xml:space="preserve">2 NOTA.</head> <p style="it"> <s xml:id="echoid-s737" xml:space="preserve">Firmitudinis punctis ut C, D, duorum ponderum, item <lb/>anſæ ut E, in unâ rectâ lineâ conſtitutis, ut paulo ſupra, e{q́ue} <lb/>C, D ponderibus æquipondiis ſuſpenſis contingentis, ſive <lb/>cujuſlibet magnitudinis, E perpetuò firmitudinis punctum manere manifestum eſt, in <lb/>quo quemcunque dederis ſitum ſervabunt: </s> <s xml:id="echoid-s738" xml:space="preserve">& </s> <s xml:id="echoid-s739" xml:space="preserve">tribus istis punctis C, E, D in eadem <lb/>recta conſtitutis, C vero & </s> <s xml:id="echoid-s740" xml:space="preserve">D non æqualiter ab E diſtantibus, & </s> <s xml:id="echoid-s741" xml:space="preserve">deillis ſuſpenſis <lb/>ponderibus ad radios proportionalibus, E firmitudinis punctum nihilo minus manere <lb/>certum eſt, in quo datum ſitum ſervabunt.</s> <s xml:id="echoid-s742" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s743" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s744" xml:space="preserve">* A B C D columna 10 ℔ pendeat, cujus centrum gravitatis E, <lb/>pondera inde ſuſpenſa F 1 ℔, H 4 ℔ pendeant, firmitudinis punctum illius <lb/>G, hujus autem H.</s> <s xml:id="echoid-s745" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s746" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s747" xml:space="preserve">* Inveniendum nobis firmitudinis punctum, in quo quem-<lb/>vis datum ſitum ſervant.</s> <s xml:id="echoid-s748" xml:space="preserve"/> </p> <pb o="25" file="527.01.025" n="25" rhead="*DE* S*TATICÆ ELEMENTIS.*"/> </div> <div xml:id="echoid-div126" type="section" level="1" n="101"> <head xml:id="echoid-head110" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s749" xml:space="preserve">Iugum G I ponderum F, H <lb/> <anchor type="figure" xlink:label="fig-527.01.025-01a" xlink:href="fig-527.01.025-01"/> lineetur, hinc radii 2 propoſi-<lb/>tionis adjumento inveniantur, <lb/>quorum K I ad K G ratio ſit, <lb/>quæ eſt F 1 ℔ ad H 4 ℔, tum <lb/>F K, hinc columnæ, inde pon-<lb/>derum F & </s> <s xml:id="echoid-s750" xml:space="preserve">H, jugum ducatur, <lb/>ita fectum in L, ut radius E L <lb/>ſit ad L K: </s> <s xml:id="echoid-s751" xml:space="preserve">quemadmodũ 5 ℔ <lb/>F, & </s> <s xml:id="echoid-s752" xml:space="preserve">H ad 10 ℔ columnæ. </s> <s xml:id="echoid-s753" xml:space="preserve">L <lb/>optatum erit punctum, in quo <lb/>quemlibet datum ſitum ſerva-<lb/>bunt, cujus demonſtratio ex 7 <lb/>propoſitione manifeſta eſt.</s> <s xml:id="echoid-s754" xml:space="preserve"/> </p> <div xml:id="echoid-div126" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.025-01" xlink:href="fig-527.01.025-01a"> <image file="527.01.025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.025-01"/> </figure> </div> </div> <div xml:id="echoid-div128" type="section" level="1" n="102"> <head xml:id="echoid-head111" xml:space="preserve">7 PROBLEMA. 12 PROPOSITIO.</head> <p> <s xml:id="echoid-s755" xml:space="preserve">Datis, notâ columnâ cum firmitudinis puncto, notis <lb/>item ponderibusinde ſuſpenſis, quę axem horizonti paral-<lb/>lelum ſervant: </s> <s xml:id="echoid-s756" xml:space="preserve">pondus invenire, quod optato columnæ <lb/>loco ſuſpenſum axem in dato ſitu ſervabit.</s> <s xml:id="echoid-s757" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div129" type="section" level="1" n="103"> <head xml:id="echoid-head112" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s758" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s759" xml:space="preserve">* A B C D columna 6 ℔ <lb/>pendeat, cujus firmitudinis punctum E, <lb/> <anchor type="figure" xlink:label="fig-527.01.025-02a" xlink:href="fig-527.01.025-02"/> anſa verò E F, duoq́ue pondera G, H <lb/>quorum utrumque 3 ℔ ſit, I K axis ad <lb/>horizontem L M parallelus, D optati lo-<lb/>ci punctus. </s> <s xml:id="echoid-s760" xml:space="preserve">Axis denique I K (univer-<lb/>ſa in puncto E mobilia intellige) ſuſtol-<lb/>latur ut in ſecundâ formulâ.</s> <s xml:id="echoid-s761" xml:space="preserve"/> </p> <div xml:id="echoid-div129" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.025-02" xlink:href="fig-527.01.025-02a"> <image file="527.01.025-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.025-02"/> </figure> </div> <p> <s xml:id="echoid-s762" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s763" xml:space="preserve">* Inveniendum nobis <lb/>pondus eſt, quod ſuſpenſum è D, axem <lb/>in dato ſitu teneat.</s> <s xml:id="echoid-s764" xml:space="preserve"/> </p> <pb o="26" file="527.01.026" n="26" rhead="I L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div131" type="section" level="1" n="104"> <head xml:id="echoid-head113" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s765" xml:space="preserve">Inveniendum eſt, ex 11 propoſitio-<lb/> <anchor type="figure" xlink:label="fig-527.01.026-01a" xlink:href="fig-527.01.026-01"/> nis doctrinâ, firmitudinis punctum, ſu-<lb/>per quo quemvis datum ſitum axis reti-<lb/>net, ſitq́ue N, hinc recta D N ducen-<lb/>da, perpendicularisq́; </s> <s xml:id="echoid-s766" xml:space="preserve">E O, ſecans N D <lb/>in O, poſtea ratio N O ad O D expen-<lb/>denda, ſitq́ue 1 ad 2, ex D igitur pon-<lb/>dere P 6 ℔ ſuſpenſo, quod ad columnã <lb/>unà cum G, H ponderibus, quæ ſimul <lb/>12 ℔ pendent, eam rationem habeat, <lb/>quæ eſt 1 ad 2, dico P 6 ℔ quæſitum <lb/>pondus eſſe.</s> <s xml:id="echoid-s767" xml:space="preserve"/> </p> <div xml:id="echoid-div131" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.026-01" xlink:href="fig-527.01.026-01a"> <image file="527.01.026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.026-01"/> </figure> </div> </div> <div xml:id="echoid-div133" type="section" level="1" n="105"> <head xml:id="echoid-head114" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s768" xml:space="preserve">Gravius pondus 12 ℔ radii O N, il-<lb/>lam rationem habet ad levius 6 ℔ radii <lb/>O D, quam longior radius O D ad bre-<lb/>viorem O N. </s> <s xml:id="echoid-s769" xml:space="preserve">Igitur ex anſa E F ſitu <lb/>æquipondia dependent per 1 propoſit. <lb/></s> <s xml:id="echoid-s770" xml:space="preserve">& </s> <s xml:id="echoid-s771" xml:space="preserve">per conſequens axis I K datum ſitum <lb/>ſervat.</s> <s xml:id="echoid-s772" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div134" type="section" level="1" n="106"> <head xml:id="echoid-head115" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s773" xml:space="preserve">Columna A B C D 6 ℔ pendeat, cujus firmitudinis punctum E, anſa E F <lb/>G pondus 2 ℔, I pondus 1 ℔, firmitu-<lb/>dinis punctum illius H, hujus verò K, <lb/> <anchor type="figure" xlink:label="fig-527.01.026-02a" xlink:href="fig-527.01.026-02"/> axis L M, parallelus horizonti N O. <lb/></s> <s xml:id="echoid-s774" xml:space="preserve">Punctum autem P in columnâ quæſi-<lb/>tus locus eſto. </s> <s xml:id="echoid-s775" xml:space="preserve">Hinc axis L M (univer-<lb/>ſis in puncto E mobilibus) ut in ſecun-<lb/>dâ formulâ ſuſtollitur.</s> <s xml:id="echoid-s776" xml:space="preserve"/> </p> <div xml:id="echoid-div134" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.026-02" xlink:href="fig-527.01.026-02a"> <image file="527.01.026-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.026-02"/> </figure> </div> <p> <s xml:id="echoid-s777" xml:space="preserve">Q*VAE SITVM*. </s> <s xml:id="echoid-s778" xml:space="preserve">è P ſuſpendendum <lb/>pondus, quod axem L M in dato ſitu <lb/>ſervet.</s> <s xml:id="echoid-s779" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div136" type="section" level="1" n="107"> <head xml:id="echoid-head116" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s780" xml:space="preserve">Firmitudinis punctum, ex 11 propo-<lb/>ſitionis doctrinâ, invenitor, in quo ſi <lb/>vertitur, datus ſitus quilibet retinetur. <lb/></s> <s xml:id="echoid-s781" xml:space="preserve">eſto autĕ Q. </s> <s xml:id="echoid-s782" xml:space="preserve">Hinc P Q ducitur, perpen-<lb/>dicularisq́ue E R, ſecans P Q in R, <lb/>inventaq́ue ratione R Q ad R P, eſto <lb/>autem 1 ad 2, de P põdus S 4 {1/2} ℔, ſuſpen-<lb/>ditur, eâ ſcilicet ratione ad columnam <pb o="27" file="527.01.027" n="27" rhead="*DE* S*TATIGÆ ELEMENTIS*."/> unà cum ponderibus G & </s> <s xml:id="echoid-s783" xml:space="preserve">I, quorum <lb/> <anchor type="figure" xlink:label="fig-527.01.027-01a" xlink:href="fig-527.01.027-01"/> omnium totus eſt 9 ℔, quæ eſt 1 ad <lb/>2. </s> <s xml:id="echoid-s784" xml:space="preserve">S 4 {1/2} quæſitum pondus eſſe dico.</s> <s xml:id="echoid-s785" xml:space="preserve"/> </p> <div xml:id="echoid-div136" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.027-01" xlink:href="fig-527.01.027-01a"> <image file="527.01.027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.027-01"/> </figure> </div> </div> <div xml:id="echoid-div138" type="section" level="1" n="108"> <head xml:id="echoid-head117" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s786" xml:space="preserve">Gravius pódus 9 ℔ radii R Q eam <lb/>habet rationem ad levius 4 {1/2} ℔ radii <lb/>R P, quæ longioris radii eſt R P ad <lb/>breviorem R Q, ſitu igitur æquipon-<lb/>dia ſunt ex ansâ E F per 1 propoſitio-<lb/>nem, &</s> <s xml:id="echoid-s787" xml:space="preserve">, quod inde conſequitur, axis <lb/>L M in dato ſitu manet, quod demon-<lb/>ſtrandum fuit.</s> <s xml:id="echoid-s788" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s789" xml:space="preserve">C*ONCLU<unsure/>SIO*. </s> <s xml:id="echoid-s790" xml:space="preserve">Datâ igitur & </s> <s xml:id="echoid-s791" xml:space="preserve"><lb/>cognitâ columnâ unà cum pun-<lb/>cto, &</s> <s xml:id="echoid-s792" xml:space="preserve">c.</s> <s xml:id="echoid-s793" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div139" type="section" level="1" n="109"> <head xml:id="echoid-head118" xml:space="preserve">6 THEOREMA. 13 PROPOSITIO.</head> <p> <s xml:id="echoid-s794" xml:space="preserve">Æqualia pondera, unumelevans, alterum demittens <lb/>æqualibus & </s> <s xml:id="echoid-s795" xml:space="preserve">angulis, & </s> <s xml:id="echoid-s796" xml:space="preserve">radiis, æquales potentias habent.</s> <s xml:id="echoid-s797" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div140" type="section" level="1" n="110"> <head xml:id="echoid-head119" xml:space="preserve">I Exemplum rectorum ponderum.</head> <p> <s xml:id="echoid-s798" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s799" xml:space="preserve">A punctum eſto, in jugo ſive trabe B C firmum, A B & </s> <s xml:id="echoid-s800" xml:space="preserve">A C <lb/>æquales radii, pendeatq́ue de B rectum pondus demittens ſive deſcendens, <lb/>de C vero adſcendens ſive attollens, hujusq́ue jugum F G, firmumq́ue ejus <lb/>punctum H, æquales autem radii H F, H G, angulusq́ue A B I æquetur an-<lb/>gulo A C F. </s> <s xml:id="echoid-s801" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s802" xml:space="preserve">Rectum pondus deſcendens D, rectumq́ue <lb/>adſcendens E, ex æqualibus radiis A B, A C æquales potentias habere de-<lb/>monſtrandum nobis eſt. </s> <s xml:id="echoid-s803" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s804" xml:space="preserve">DeC pondus K, æquale pon-<lb/>deri D, pendeto.</s> <s xml:id="echoid-s805" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div141" type="section" level="1" n="111"> <head xml:id="echoid-head120" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s806" xml:space="preserve">Amoto E, potentiam D eſſe radios A B, <lb/> <anchor type="figure" xlink:label="fig-527.01.027-02a" xlink:href="fig-527.01.027-02"/> A C in dato ſitu retinere, manifeſtum <lb/>eſt, pondera enim D & </s> <s xml:id="echoid-s807" xml:space="preserve">K, item radii A B <lb/>& </s> <s xml:id="echoid-s808" xml:space="preserve">A C æqualia ſunt. </s> <s xml:id="echoid-s809" xml:space="preserve">Amoto viciſſim D, <lb/>appenditor E, & </s> <s xml:id="echoid-s810" xml:space="preserve">hujus potentia eſt, radios <lb/>A B & </s> <s xml:id="echoid-s811" xml:space="preserve">A C in dato ſitu retinere, pondera <lb/>enim K & </s> <s xml:id="echoid-s812" xml:space="preserve">E, radiiq́ue H F & </s> <s xml:id="echoid-s813" xml:space="preserve">H G æquan-<lb/>tur, Eigitur & </s> <s xml:id="echoid-s814" xml:space="preserve">D pari potentiâ & </s> <s xml:id="echoid-s815" xml:space="preserve">vi in ra-<lb/>dios A B & </s> <s xml:id="echoid-s816" xml:space="preserve">A C agunt.</s> <s xml:id="echoid-s817" xml:space="preserve"/> </p> <div xml:id="echoid-div141" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.027-02" xlink:href="fig-527.01.027-02a"> <image file="527.01.027-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.027-02"/> </figure> </div> <pb o="28" file="527.01.028" n="28" rhead="I L*IBER* S*TATIGÆ*"/> </div> <div xml:id="echoid-div143" type="section" level="1" n="112"> <head xml:id="echoid-head121" xml:space="preserve">2 Exemplum obliquorum ponderum.</head> <p> <s xml:id="echoid-s818" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s819" xml:space="preserve">A firmum anſæ punctum eſto, A B verò & </s> <s xml:id="echoid-s820" xml:space="preserve">A C radii, D pon-<lb/>dus deſcendĕs obliquũ, ex B ſuſpenſum, cujus <lb/> <anchor type="figure" xlink:label="fig-527.01.028-01a" xlink:href="fig-527.01.028-01"/> linea deſcendens obliqua B E, è Cautem obli-<lb/>quum pondus attollens F, ponderi Dæquale, <lb/>cujus linea attollens obliqua C G, anguliq́ue <lb/>A B E & </s> <s xml:id="echoid-s821" xml:space="preserve">A C G æquentur.</s> <s xml:id="echoid-s822" xml:space="preserve"/> </p> <div xml:id="echoid-div143" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.028-01" xlink:href="fig-527.01.028-01a"> <image file="527.01.028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.028-01"/> </figure> </div> <p> <s xml:id="echoid-s823" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s824" xml:space="preserve">Demõſtrandum nobis eſt, <lb/>D pondus deſcendens obliquum, & </s> <s xml:id="echoid-s825" xml:space="preserve">F obli-<lb/>quum attollens, ex paribus radiis A B, A G æ-<lb/>quales habere potentias. </s> <s xml:id="echoid-s826" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s827" xml:space="preserve">Ex C pondus H deſcendens <lb/>obliquum ſuſpenditor, æquale ponderi D, & </s> <s xml:id="echoid-s828" xml:space="preserve">illius linea C I obliqua deſcen-<lb/>dens parallela ſit ad B E, & </s> <s xml:id="echoid-s829" xml:space="preserve">C B in K continuata.</s> <s xml:id="echoid-s830" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div145" type="section" level="1" n="113"> <head xml:id="echoid-head122" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s831" xml:space="preserve">Amoto F, dubium non eſt, quin potentia D contra H ſit radios A B, A C <lb/>in dato ſitu retinere, æquatur enim ipſi H, radii A B, & </s> <s xml:id="echoid-s832" xml:space="preserve">A C, itemq́ue angu-<lb/>li A C I & </s> <s xml:id="echoid-s833" xml:space="preserve">K B E æquales ſunt. </s> <s xml:id="echoid-s834" xml:space="preserve">Amoto viciſſim D, appenditor F, cujus iti-<lb/>dem potentia eſt radios A B, A C in dato ſitu retinere quod pondus H æque-<lb/>tur ponderi F.</s> <s xml:id="echoid-s835" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div146" type="section" level="1" n="114"> <head xml:id="echoid-head123" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s836" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s837" xml:space="preserve">A firmum anſæ punctum eſto, A B verò & </s> <s xml:id="echoid-s838" xml:space="preserve">A C æquales radii, <lb/>& </s> <s xml:id="echoid-s839" xml:space="preserve">de B pondus D obliquè deſcendens, cujus obliqua linea B E, de C verò <lb/>pondus F obliquè attollens dependeat, cujus linea obliquè attollens ſit C G, <lb/>angulusq́ue K C G angulo K B E æqualis.</s> <s xml:id="echoid-s840" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s841" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s842" xml:space="preserve">Demonſtrandum nobis pondus D obliquè deſcendens, <lb/>pondúſque F obliquè attollens, in æquales radios A B, A C æqualem po-<lb/>tentiam obtinere. </s> <s xml:id="echoid-s843" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s844" xml:space="preserve">De C pondus H obliquè deſcen-<lb/>dens, ponderi D æquale, dependeto, cujus obliquè deſcendens linea C I, ut <lb/>angulus A C I æquet angulum A B E.</s> <s xml:id="echoid-s845" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div147" type="section" level="1" n="115"> <head xml:id="echoid-head124" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s846" xml:space="preserve">Amoto F, potentiam ponderis D eſſe, ra-<lb/> <anchor type="figure" xlink:label="fig-527.01.028-02a" xlink:href="fig-527.01.028-02"/> dios A B, A C in dato ſitu ſervare non eſt inco-<lb/>gnitum, quod D æquale ſit H, & </s> <s xml:id="echoid-s847" xml:space="preserve">A B radius, <lb/>A C radio, angulúſque A C I angulo A B E. <lb/></s> <s xml:id="echoid-s848" xml:space="preserve">Amoto viciſſim D, ponderi F appenſo eadem <lb/>potentia erit, A B & </s> <s xml:id="echoid-s849" xml:space="preserve">A C radios in dato ſitu <lb/>ſervare, quod pondus H ponderi F ſit æquale.</s> <s xml:id="echoid-s850" xml:space="preserve"/> </p> <div xml:id="echoid-div147" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.028-02" xlink:href="fig-527.01.028-02a"> <image file="527.01.028-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.028-02"/> </figure> </div> <p> <s xml:id="echoid-s851" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s852" xml:space="preserve">Pondus igitur deſcendens, <lb/>& </s> <s xml:id="echoid-s853" xml:space="preserve">attollens illi æquale, æqualibus angulis in <lb/>æquales radios æqualem potentiam exercent.</s> <s xml:id="echoid-s854" xml:space="preserve"/> </p> <pb o="29" file="527.01.029" n="29" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div149" type="section" level="1" n="116"> <head xml:id="echoid-head125" xml:space="preserve">8 PROBLEMA. 14 PROPOSITIO.</head> <p> <s xml:id="echoid-s855" xml:space="preserve">Datis, columnâ, inque ejus axe duobus punctis, uno <lb/>fixo, altero in longiore ſegmento mobili: </s> <s xml:id="echoid-s856" xml:space="preserve">invenire pondus <lb/>rectè attollens ex puncto mobili, quod datam columnam <lb/>in dato ſitu conſervet, retineatque.</s> <s xml:id="echoid-s857" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s858" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s859" xml:space="preserve">A B C D columna, ſecta utin 1 propoſit. </s> <s xml:id="echoid-s860" xml:space="preserve">initio, 6 ℔ pendeat, <lb/>punctum ejus firmum R, mobile autem in longiore axis R Q ſegmento, V <lb/>eſto, in breviore enim ſieri neutiquam poteſt, ut ullum pondus rectè attollens <lb/>axem in dato ſitu detineat.</s> <s xml:id="echoid-s861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s862" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s863" xml:space="preserve">Pondus recte attollens nobis eſt inveniendum, quod in <lb/>ſuo ſitu columnam ſervet.</s> <s xml:id="echoid-s864" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div150" type="section" level="1" n="117"> <head xml:id="echoid-head126" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s865" xml:space="preserve">Linea Q R continuanda <lb/> <anchor type="figure" xlink:label="fig-527.01.029-01a" xlink:href="fig-527.01.029-01"/> eſt in Y, ut R Y æquet R V: <lb/></s> <s xml:id="echoid-s866" xml:space="preserve">hinc pondus Z invenien-<lb/>dum, quod de Y ſuſpen-<lb/>ſum columnæ ſit ſitu æqui-<lb/>libre: </s> <s xml:id="echoid-s867" xml:space="preserve">illud ipſum, quia R <lb/>eſt punctum firmum, per 3 <lb/>propoſit 4 ℔ pendebit. </s> <s xml:id="echoid-s868" xml:space="preserve">Dico <lb/>itaque Æ quæſitum pondus <lb/>rectè attollens 4 ℔ eſſe.</s> <s xml:id="echoid-s869" xml:space="preserve"/> </p> <div xml:id="echoid-div150" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.029-01" xlink:href="fig-527.01.029-01a"> <image file="527.01.029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.029-01"/> </figure> </div> </div> <div xml:id="echoid-div152" type="section" level="1" n="118"> <head xml:id="echoid-head127" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s870" xml:space="preserve">Quandoquidem radius R V ponderis Æ rectè attollentis æquatur radio <lb/>R Y ponderis Z. </s> <s xml:id="echoid-s871" xml:space="preserve">ipſaq́ue pondera Æ & </s> <s xml:id="echoid-s872" xml:space="preserve">Z æqualia, iſtorum quoque poten-<lb/>tias, ex 13 propoſit. </s> <s xml:id="echoid-s873" xml:space="preserve">æquari conſequens eſt. </s> <s xml:id="echoid-s874" xml:space="preserve">Atqui potentia Z eſt (amoto Æ) <lb/>columnam in ſuo ſitu retinere: </s> <s xml:id="echoid-s875" xml:space="preserve">itaque & </s> <s xml:id="echoid-s876" xml:space="preserve">potentia Æ (amoto Z) eadem eſt, <lb/>quod nobis fuit demonſtrandum.</s> <s xml:id="echoid-s877" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s878" xml:space="preserve">C*ONCLUSIO*. </s> <s xml:id="echoid-s879" xml:space="preserve">Datis igitur, columnâ, & </s> <s xml:id="echoid-s880" xml:space="preserve">in axe duobus punctis, altero <lb/>fixo, reliquo ſegmenti longioris mobili: </s> <s xml:id="echoid-s881" xml:space="preserve">rectum pondus attollens invenimus <lb/>quod in puncto mobili columnam in dato ſitu conſervat, quod fuit quæ-<lb/>ſitum.</s> <s xml:id="echoid-s882" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div153" type="section" level="1" n="119"> <head xml:id="echoid-head128" xml:space="preserve">NOTATO</head> <p style="it"> <s xml:id="echoid-s883" xml:space="preserve">Compendio concludi poſſe quemadmodum V R 3, ad R T 2: </s> <s xml:id="echoid-s884" xml:space="preserve">ita columna 6 ℔ ad <lb/>quem terminum quartum? </s> <s xml:id="echoid-s885" xml:space="preserve">concluditur pro Æ 4 ℔, utpaulò ante, cujus cauſa 15 <lb/>propoſit. </s> <s xml:id="echoid-s886" xml:space="preserve">patebit.</s> <s xml:id="echoid-s887" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div154" type="section" level="1" n="120"> <head xml:id="echoid-head129" xml:space="preserve">1 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s888" xml:space="preserve">QVandoquidem univerſa columna ex conceſſo 6 ℔ pendet, quarum 4 ℔ <lb/>Æ attollit, neceſſariò ſequitur in puncto R, hoc eſt, faſtigio coni, vel py-<lb/>ramidis OE reliquas 2 ℔ quieſcere.</s> <s xml:id="echoid-s889" xml:space="preserve"/> </p> <pb o="30" file="527.01.030" n="30" rhead="I L*IBER* S*TATICÆ*"/> <figure> <image file="527.01.030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.030-01"/> </figure> <p> <s xml:id="echoid-s890" xml:space="preserve">VEl, ſiad R, pro cono OE, <lb/>pondus rectè attollĕs ad-<lb/>datur, ut hic videre eſt, II 2 ℔ <lb/>pendebit.</s> <s xml:id="echoid-s891" xml:space="preserve"/> </p> <figure> <image file="527.01.030-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.030-02"/> </figure> <p> <s xml:id="echoid-s892" xml:space="preserve">VEl, ſi ad V, loco ponderis OE rectè <lb/>attollentis, conus Φ adjungatur, ut <lb/>videre hic eſt, quod ſuper OE quieſcit <lb/>2 ℔, quod verò ſuper Φ 4 ℔ fuerit.</s> <s xml:id="echoid-s893" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s894" xml:space="preserve">VEl, ſi è duabus parallelis OE R, & </s> <s xml:id="echoid-s895" xml:space="preserve">Φ V <lb/> <anchor type="figure" xlink:label="fig-527.01.030-03a" xlink:href="fig-527.01.030-03"/> columna ſuſpenſa ſit: </s> <s xml:id="echoid-s896" xml:space="preserve">quod quidem <lb/>de OE R dependet 2 ℔, quod verò de Φ V <lb/>4 ℔ eſt.</s> <s xml:id="echoid-s897" xml:space="preserve"/> </p> <div xml:id="echoid-div154" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.030-03" xlink:href="fig-527.01.030-03a"> <image file="527.01.030-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.030-03"/> </figure> </div> </div> <div xml:id="echoid-div156" type="section" level="1" n="121"> <head xml:id="echoid-head130" xml:space="preserve">2 C*ONSECTARIUM*.</head> <figure> <image file="527.01.030-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.030-04"/> </figure> <p> <s xml:id="echoid-s898" xml:space="preserve">SI de columnâ (R puncto, <lb/>utſupra, fixo) pondus, vel <lb/>pondera ſuſpenſa ſint, etiam <lb/>pondus rectè attollens inno-<lb/>teſcet. </s> <s xml:id="echoid-s899" xml:space="preserve">Exempli gratiâ, ſi de X <lb/>6 ℔ dependent, Z 12 ℔ erit, <lb/>per 3 propoſit. </s> <s xml:id="echoid-s900" xml:space="preserve">ideoq́ue & </s> <s xml:id="echoid-s901" xml:space="preserve">Æ <lb/>totidem.</s> <s xml:id="echoid-s902" xml:space="preserve"/> </p> <pb o="31" file="527.01.031" n="31" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div157" type="section" level="1" n="122"> <head xml:id="echoid-head131" xml:space="preserve">7 THEOREMA. 15 PROPOSITIO.</head> <p> <s xml:id="echoid-s903" xml:space="preserve">Duorum punctorum in axe columnæ altero fixo, alte-<lb/>ro mobili: </s> <s xml:id="echoid-s904" xml:space="preserve">Pondus rectèattollĕs ex mobili cum columnâ <lb/>ſitu æquipondium, illam habet rationem ad columnam, <lb/>quæ eſt ſegmenti axis quod inter centrum gravitatis & </s> <s xml:id="echoid-s905" xml:space="preserve"><lb/>punctum fixum eſt, ad ſegmentum ejuſdem quod inter <lb/>fixum & </s> <s xml:id="echoid-s906" xml:space="preserve">mobile intercipitur.</s> <s xml:id="echoid-s907" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div158" type="section" level="1" n="123"> <head xml:id="echoid-head132" xml:space="preserve">DECLARATIO.</head> <p> <s xml:id="echoid-s908" xml:space="preserve">Figuras è 14 propoſit. </s> <s xml:id="echoid-s909" xml:space="preserve">repetamus, è quibus patet ita eſſe T R ad R V: <lb/></s> <s xml:id="echoid-s910" xml:space="preserve">quemadmodum Æ 4 ℔ ad columnam 6 ℔. </s> <s xml:id="echoid-s911" xml:space="preserve">Sed ut cauſam hujus Mathema-<lb/>iicèaperiamus, ſciendum eſt, ita eſſe R T ad R Y: </s> <s xml:id="echoid-s912" xml:space="preserve">ut pondus Z ad pondus <lb/>columnæ, per 1 propoſit. </s> <s xml:id="echoid-s913" xml:space="preserve">Atqui Æ æquale eſt Z, & </s> <s xml:id="echoid-s914" xml:space="preserve">R V ex conceſſo æquale <lb/>eſt R Y, itaque ut Æ ad columnam: </s> <s xml:id="echoid-s915" xml:space="preserve">ita T R ad R V. </s> <s xml:id="echoid-s916" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s917" xml:space="preserve">Duo-<lb/>rum igitur punctorum altero fixo, altero mobili, &</s> <s xml:id="echoid-s918" xml:space="preserve">c.</s> <s xml:id="echoid-s919" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div159" type="section" level="1" n="124"> <head xml:id="echoid-head133" xml:space="preserve">8 THEOREMA. 16 PROPOSITIO.</head> <p> <s xml:id="echoid-s920" xml:space="preserve">Duorum punctorum in axe columnæ, altero fixo, al-<lb/>tero mobili: </s> <s xml:id="echoid-s921" xml:space="preserve">Pondus rectèattollens è puncto mobili ſer-<lb/>vans columnam in uno aliquo ſitu, in quovis alio ſervare <lb/>poterit.</s> <s xml:id="echoid-s922" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s923" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s924" xml:space="preserve">Columnam 14 propoſitionis cum ſuis póderibus, in fixo pun-<lb/>cto R non nihil vertamus mutemusq́ue, maneatq́ue Æ pondus rectum ex-<lb/>tollens, cæteráque ſint bujuſmodi, quemadmodum hîc exhibentur.</s> <s xml:id="echoid-s925" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s926" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s927" xml:space="preserve">Demonſtrandum nobis eſt, Æ pondus rectè attollens <lb/>columnam in dato ſitu ſervare.</s> <s xml:id="echoid-s928" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div160" type="section" level="1" n="125"> <head xml:id="echoid-head134" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s929" xml:space="preserve">Amoto Æ, Z vero 4 ℔ ap-<lb/> <anchor type="figure" xlink:label="fig-527.01.031-01a" xlink:href="fig-527.01.031-01"/> penſo, columna ex 10 propoſ. <lb/></s> <s xml:id="echoid-s930" xml:space="preserve">in dato manebit ſitu. </s> <s xml:id="echoid-s931" xml:space="preserve">Atqui <lb/>Æ in puncto V, & </s> <s xml:id="echoid-s932" xml:space="preserve">Z in pun-<lb/>cto Y vim potentiamq́ue pa-<lb/>rilem columnæ adferunt ex <lb/>13 propoſit. </s> <s xml:id="echoid-s933" xml:space="preserve">Amoto itaque Z, <lb/>Æ appenſum eodem in ſitu <lb/>columnam tenebit.</s> <s xml:id="echoid-s934" xml:space="preserve"/> </p> <div xml:id="echoid-div160" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.031-01" xlink:href="fig-527.01.031-01a"> <image file="527.01.031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.031-01"/> </figure> </div> <p> <s xml:id="echoid-s935" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s936" xml:space="preserve">Duorum <lb/>igitur punctorum in axe, alte-<lb/>ro fixo, altero mobili, rectè attollens pondus mobili appenſum in uno aliquo <lb/>ſitu columnam ſervans, in quovis alio ſervare poterit.</s> <s xml:id="echoid-s937" xml:space="preserve"/> </p> <pb o="32" file="527.01.032" n="32" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div162" type="section" level="1" n="126"> <head xml:id="echoid-head135" xml:space="preserve">9 THEOREMA. 17 PROPOSITIO.</head> <p> <s xml:id="echoid-s938" xml:space="preserve">Columnâ ſuper duobus in axe punctis quieſcente: <lb/></s> <s xml:id="echoid-s939" xml:space="preserve">quemadmodum axis ſegmentum inter gravitatis cen-<lb/>trum punctumq́ue ſiniſtrum, ad ejuſdem ſegmentum in-<lb/>ter gravitatis centrum punctumq́ue dextrum: </s> <s xml:id="echoid-s940" xml:space="preserve">ita co-<lb/>lumnæ pondus ſuper puncto dextro quieſcens, ad reli-<lb/>quum ponderis ſuper ſiniſtro quieſcentis.</s> <s xml:id="echoid-s941" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s942" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s943" xml:space="preserve">ABCD columna 6 ℔ pendeat, ſecta quemadmodum in 1 pro-<lb/>poſitione, duobus punctis R, V, ſuper OE, Æ quieſcens.</s> <s xml:id="echoid-s944" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s945" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s946" xml:space="preserve">Demonſtrandũ nobis eſt, quemadmodum axis ſegmen-<lb/>tum T R ad ejuſdem T V: </s> <s xml:id="echoid-s947" xml:space="preserve">ita eſſe pondus puncto V quieſcens in Æ, ad re-<lb/>liquum ponderis puncto R, ſuper OE quieſcentis.</s> <s xml:id="echoid-s948" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div163" type="section" level="1" n="127"> <head xml:id="echoid-head136" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s949" xml:space="preserve">T R duplum eſtad T V extheſi, & </s> <s xml:id="echoid-s950" xml:space="preserve">ſuper <lb/> <anchor type="figure" xlink:label="fig-527.01.032-01a" xlink:href="fig-527.01.032-01"/> Æ 4 ℔ ſuper OE verò 2 ℔ quieſcunt, ex <lb/>1 conſect. </s> <s xml:id="echoid-s951" xml:space="preserve">14 propoſitionis, atqui 4 ℔ ad <lb/>2 ℔ etiam dupla eſtratio; </s> <s xml:id="echoid-s952" xml:space="preserve">quemadmodum <lb/>T R ad T V: </s> <s xml:id="echoid-s953" xml:space="preserve">ita & </s> <s xml:id="echoid-s954" xml:space="preserve">pondus quod ſuper pun-<lb/>cto Æ eſt, ad reliquum ponderis quieſcen-<lb/>tis ſuper OE.</s> <s xml:id="echoid-s955" xml:space="preserve"/> </p> <div xml:id="echoid-div163" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.032-01" xlink:href="fig-527.01.032-01a"> <image file="527.01.032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.032-01"/> </figure> </div> <p> <s xml:id="echoid-s956" xml:space="preserve">Verumenimvero generalis conſectarii <lb/>neceſſitas demonſtretur; </s> <s xml:id="echoid-s957" xml:space="preserve">V R in Z cõtinua-<lb/>tor, ut R Z æquetur R V, ſumptoq́ue R <lb/>pro puncto fixo, ex Z pondus <lb/> <anchor type="figure" xlink:label="fig-527.01.032-02a" xlink:href="fig-527.01.032-02"/> 114 ℔ ſuſpendi neceſſe eſt, ut <lb/>columna ſuo in ſitu cõſervetur, <lb/>ex 3 propoſit. </s> <s xml:id="echoid-s958" xml:space="preserve">quod verò ex V, <lb/>columnam eodĕ in ſitu, quo Æ, <lb/>ſervat, parem cum 11 potentiam <lb/>habere ex 13 propoſitione ne-<lb/>ceſſe eſt. </s> <s xml:id="echoid-s959" xml:space="preserve">In Æ igitur pondus par <lb/>ipſi 11 quieſcit. </s> <s xml:id="echoid-s960" xml:space="preserve">Cõſimiliter R V <lb/>in Φ continuator, ut V Φ æque-<lb/>tur V R, ſumptoq́ue V pro pun-<lb/>cto firmo, de Φ ſuſpendi Δ 2 ℔ <lb/>neceſſe eſt, ut columna eodem in ſitu ſuſtineatur, per 3 exemplum. </s> <s xml:id="echoid-s961" xml:space="preserve">quod <lb/>verò ex R columnam ſive vectem eodem in ſitu ſuſtinet, quo OE, r<unsure/>antun-<lb/>dem potentiæ habet, quantum Δ, per 13 propoſit. </s> <s xml:id="echoid-s962" xml:space="preserve">pondus igitur in OE quieſ-<lb/>cens æquatur ponderi Δ. </s> <s xml:id="echoid-s963" xml:space="preserve">Quandoquidem autem 11, ex R communi fulci-<lb/>menti puncto, contra columnam ſitu æquilibre eſt, ratio radii T R eſt ad ra-<lb/>dium R Z, quæ eſt 11 ad columnam, per 1 propoſitionem. </s> <s xml:id="echoid-s964" xml:space="preserve">Cõſimiliter V pro <lb/>firmo puncto uſurpato, ratio radii T V ad radium V Φ eadem eſt cum ra-<lb/>z<unsure/>ione Δ, ad columnam, atque R Z æquatur V Φ Duæ igitur proportiones <lb/>nobishic ſunt quaternûm terminorum, quorum ſecundi quartique æquales <pb o="33" file="527.01.033" n="33" rhead="*DE* S*TATICÆ ELEMENTIS*."/> funt. </s> <s xml:id="echoid-s965" xml:space="preserve">Verum quæcunque binæ proportiones quaternûm terminorum, ſecun-<lb/>dos quartosq́ue terminos equales habent, reliquos æquè rationales, id eſt pro-<lb/>portionales habebunt. </s> <s xml:id="echoid-s966" xml:space="preserve">Vt T R igitur ad T V: </s> <s xml:id="echoid-s967" xml:space="preserve">ita 11 ad Δ. </s> <s xml:id="echoid-s968" xml:space="preserve">Atqui pondus 11 <lb/>æquatur columnæ ponderi, quod puncto V, ſuper puncto Æ quieſcit; <lb/></s> <s xml:id="echoid-s969" xml:space="preserve">pondusq́ue Δ ponderi, quod R puncto quieſcit ſuper OE. </s> <s xml:id="echoid-s970" xml:space="preserve">Ideoq́ue ut T R <lb/>ad T V: </s> <s xml:id="echoid-s971" xml:space="preserve">ita pondus puncto Æ innitens, ad pondus OE innixum.</s> <s xml:id="echoid-s972" xml:space="preserve"/> </p> <div xml:id="echoid-div164" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.032-02" xlink:href="fig-527.01.032-02a"> <image file="527.01.032-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.032-02"/> </figure> </div> <p> <s xml:id="echoid-s973" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s974" xml:space="preserve">Columnâigitur duobus punctis axis quieſcente, &</s> <s xml:id="echoid-s975" xml:space="preserve">c.</s> <s xml:id="echoid-s976" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div166" type="section" level="1" n="128"> <head xml:id="echoid-head137" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s977" xml:space="preserve">Si puncta, in quibus columna quieſcit, in perpendicularibus ſint per R & </s> <s xml:id="echoid-s978" xml:space="preserve"><lb/>V ductis, pondera quæ antea ſuper quieſcent@bus punctis erant, etiam nunc <lb/>eſſe poſſunt. </s> <s xml:id="echoid-s979" xml:space="preserve">Per puncta R & </s> <s xml:id="echoid-s980" xml:space="preserve">V perpendiculares, exempli cauſa, ducantur, <lb/>in iiſque puncta ut Y, & </s> <s xml:id="echoid-s981" xml:space="preserve">λ ſignentur. </s> <s xml:id="echoid-s982" xml:space="preserve">Si columna in Y & </s> <s xml:id="echoid-s983" xml:space="preserve">λ quieſcit, in <lb/>Y 2 ℔, in λ 4 ℔ quieſcere manifeſtũ eſt, unde theorematis veritas manifeſta eſt.</s> <s xml:id="echoid-s984" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div167" type="section" level="1" n="129"> <head xml:id="echoid-head138" xml:space="preserve">10 THEOREMA. 18 PROPOSITIO.</head> <p> <s xml:id="echoid-s985" xml:space="preserve">Columna duobus in punctis quieſcĕte:</s> <s xml:id="echoid-s986" xml:space="preserve">erit ut ſegmen-<lb/>tum axis inter gravitatis centrum & </s> <s xml:id="echoid-s987" xml:space="preserve">perpendicularem per <lb/>punctum ſiniſtrum, ad eju ſdem ſegmentum inter gravi-<lb/>tatis centrum & </s> <s xml:id="echoid-s988" xml:space="preserve">perpĕdicularem per punctum dextrum: <lb/></s> <s xml:id="echoid-s989" xml:space="preserve">ita ſuſtentatum pondus columnæ dextro puncto, ad pon-<lb/>dus quod ſuſtinetur ſiniſtro.</s> <s xml:id="echoid-s990" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s991" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s992" xml:space="preserve">A B C D columna eſto, <lb/> <anchor type="figure" xlink:label="fig-527.01.033-01a" xlink:href="fig-527.01.033-01"/> ejusq́ue axis E F gravitatis centrum G, <lb/>puncta quibus columna ſuſtinetur H, I, quà <lb/>perpendiculares K L, M N ductæ axem <lb/>in O, P ſecant. </s> <s xml:id="echoid-s993" xml:space="preserve">Dico quemadmodũ G O ad <lb/>G P: </s> <s xml:id="echoid-s994" xml:space="preserve">ita pondus puncto I ſuſtentatum, ad <lb/>pondus reliquum quod H ſuſtinet: </s> <s xml:id="echoid-s995" xml:space="preserve">cujus <lb/>demonſtratio ex conſectario 17 propoſit. <lb/></s> <s xml:id="echoid-s996" xml:space="preserve">manifeſta eſt. </s> <s xml:id="echoid-s997" xml:space="preserve">Verumenimverò, ut paulo <lb/>fuſius de neceſſaria hujus veritateagatur, ſi <lb/>Hloco O eſſe fingamus, ratio põderis pun-<lb/> <anchor type="figure" xlink:label="fig-527.01.033-02a" xlink:href="fig-527.01.033-02"/> cto H ſuſtentati ad pondus P ſuſtentatum <lb/>erit, quæ eſt G P ad G O, per 17 propoſit. <lb/></s> <s xml:id="echoid-s998" xml:space="preserve">Puncto H ſixo, columnam in dato ſitu <lb/>deſcĕdere ponamus intervallo ab H uſque <lb/>in O, pondus H puncto ſuſtentatũ per 3 po-<lb/>ſtulatum, idem manèt. </s> <s xml:id="echoid-s999" xml:space="preserve">Cõſimiliter pondus <lb/>quod in puncto P quieſcit, etiam puncto I <lb/>quieſcere oſtĕdetur, ut igitur G O ad G P: </s> <s xml:id="echoid-s1000" xml:space="preserve"><lb/>ita põdus quod I ſuſtinet ad pondus quod <lb/>ſuſtinetur in H. </s> <s xml:id="echoid-s1001" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1002" xml:space="preserve">Quieſcente igitur columnâ in duobus <lb/>punctis, &</s> <s xml:id="echoid-s1003" xml:space="preserve">c.</s> <s xml:id="echoid-s1004" xml:space="preserve"/> </p> <div xml:id="echoid-div167" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.033-01" xlink:href="fig-527.01.033-01a"> <image file="527.01.033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.033-01"/> </figure> <figure xlink:label="fig-527.01.033-02" xlink:href="fig-527.01.033-02a"> <image file="527.01.033-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.033-02"/> </figure> </div> <pb o="34" file="527.01.034" n="34" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div169" type="section" level="1" n="130"> <head xml:id="echoid-head139" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1005" xml:space="preserve">Vnde conſequitur. </s> <s xml:id="echoid-s1006" xml:space="preserve">Si ratio ponderis in I quieſcentis, ad pondus H quære-<lb/>retur, ductis perpendicularibus K L, M N, ſecantibus E F axem in O, P, ra-<lb/>tionem G O ad G P fore quæſitam. </s> <s xml:id="echoid-s1007" xml:space="preserve">Vnde & </s> <s xml:id="echoid-s1008" xml:space="preserve">iſtud deducitur, columnæ gra-<lb/>vitate cognitâ: </s> <s xml:id="echoid-s1009" xml:space="preserve">pondera quoque cognoſci quæ cuique puncto, ut H & </s> <s xml:id="echoid-s1010" xml:space="preserve">I, in-<lb/>nituntur.</s> <s xml:id="echoid-s1011" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div170" type="section" level="1" n="131"> <head xml:id="echoid-head140" xml:space="preserve">HACTENVS RECTORVM</head> <head xml:id="echoid-head141" xml:space="preserve">PONDERVM GENERA DICTA SVNT; OBLI-<lb/>QVORVM PROPRIETATES DEINCEPS</head> <head xml:id="echoid-head142" style="it" xml:space="preserve">deſcribendæ ſunt, quarum omnium genera-<lb/>lem veritatem tanquam fundamentum istud <lb/>theoremata complectitur.</head> <head xml:id="echoid-head143" xml:space="preserve">11 THEOREMA. 19 PROPOSITIO.</head> <p> <s xml:id="echoid-s1012" xml:space="preserve">Si triangulum planum horizonti eſt perpendiculare, ba-<lb/>ſis parallela, reliquis autem lateribus globi ſinguli addan-<lb/> <anchor type="note" xlink:label="note-527.01.034-01a" xlink:href="note-527.01.034-01"/> trum ad ſiniſtrum: </s> <s xml:id="echoid-s1013" xml:space="preserve">ita ſacoma globi ſiniſtri ad antiſacoma <lb/>globi dextri.</s> <s xml:id="echoid-s1014" xml:space="preserve"/> </p> <div xml:id="echoid-div170" type="float" level="2" n="1"> <note position="left" xlink:label="note-527.01.034-01" xlink:href="note-527.01.034-01a" xml:space="preserve">Intellige ſaco-<lb/>ma põdus eſſe <lb/>quod additur <lb/>ad æquipon-<lb/>dium faci@n-<lb/>dum. Cui an-<lb/>tiſac<unsure/>oma op-<lb/>poſuimus.</note> </div> <p> <s xml:id="echoid-s1015" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1016" xml:space="preserve">A B C triangulum eſto cujus planũ ad horizontem ſit rectum <lb/>baſis vero parallela. </s> <s xml:id="echoid-s1017" xml:space="preserve">additorq́ue lateri A B, quod ad B C eſt duplum, globus <lb/>D. </s> <s xml:id="echoid-s1018" xml:space="preserve">lateri vero B C globus E & </s> <s xml:id="echoid-s1019" xml:space="preserve">ponde-<lb/> <anchor type="figure" xlink:label="fig-527.01.034-01a" xlink:href="fig-527.01.034-01"/> re & </s> <s xml:id="echoid-s1020" xml:space="preserve">magnitudine æqualis cum D.</s> <s xml:id="echoid-s1021" xml:space="preserve"/> </p> <div xml:id="echoid-div171" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.034-01" xlink:href="fig-527.01.034-01a"> <image file="527.01.034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.034-01"/> </figure> </div> <p> <s xml:id="echoid-s1022" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1023" xml:space="preserve">Demonſtrandũ no-<lb/>bis eſt, quemadmodum latus A B 2, ad <lb/>latus B C 1: </s> <s xml:id="echoid-s1024" xml:space="preserve">ita ſacoma globi E, ad an-<lb/>tiſacoma globi D.</s> <s xml:id="echoid-s1025" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1026" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1027" xml:space="preserve">Triangulũ A B C <lb/>quatuordecim globorum pondere & </s> <s xml:id="echoid-s1028" xml:space="preserve"><lb/>magnitudine æqualium, quaſi coronâ <lb/>ut E, F, G, H, I, K, L, M, N, O, P, Q, R, D, <lb/>cunctum ſingamus, qui omnes lineâ per <lb/>cĕtro ipſorum, ut in illis moveri poſſint, <lb/>tranſeunte, colligati æquali inter ſeſpa-<lb/>cio diſtent, ut illorũ bini lateri B C, qua-<lb/>terni vero B A accommodentur, hoc eſt, quemadmodum linea ad lineam; </s> <s xml:id="echoid-s1029" xml:space="preserve">ita <lb/>globi ſint ad globos. </s> <s xml:id="echoid-s1030" xml:space="preserve">Inſuper in S, T, V tria ſint puncta immota ac ſixa, quæà <lb/>lineâ ſive globorum funiculo, cum movetur, raduntur, ac ſtringuntur: </s> <s xml:id="echoid-s1031" xml:space="preserve">duæq́ <lb/>funiculi partes, quæ ſupra trianguli baſin, lateribus A B, B C ſint parallelæ, <lb/>ut, quando connexio illa ſeriesq́; </s> <s xml:id="echoid-s1032" xml:space="preserve">globorum adſcendit, deſcenditve, globi pes <lb/>crura A B, B C volui poſſint.</s> <s xml:id="echoid-s1033" xml:space="preserve"/> </p> <pb o="35" file="527.01.035" n="35" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div173" type="section" level="1" n="132"> <head xml:id="echoid-head144" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1034" xml:space="preserve">Si ſacoma quaternûm globorum D, R, Q,-P, non æquaretur antiſacoma-<lb/>ti binorum, E F. </s> <s xml:id="echoid-s1035" xml:space="preserve">alterutri graviores erunt: </s> <s xml:id="echoid-s1036" xml:space="preserve">ſunto autem (ſi fieri poteſt) quatuor <lb/>iſti D, R, Q, P; </s> <s xml:id="echoid-s1037" xml:space="preserve">Atqui O, N, M, L, æquiponderant quatuor G, H, I, K. </s> <s xml:id="echoid-s1038" xml:space="preserve">Latus <lb/>igitur octo globorum D, R, Q, P, O, N, M, L, ponderoſius eſt latere ſex glo-<lb/>borum E, F, G, H, I, K. </s> <s xml:id="echoid-s1039" xml:space="preserve">Quia vero gravius præpõderat leviori, octo deorſum <lb/>volventur, ſex vero reliqui ſurſum. </s> <s xml:id="echoid-s1040" xml:space="preserve">Deſcenderit D, in O, & </s> <s xml:id="echoid-s1041" xml:space="preserve">E, F, G, H ſint, <lb/>loco P, Q, R, D, denique I, K, loco E, F. </s> <s xml:id="echoid-s1042" xml:space="preserve">Atqui hoc ſi ſit, globorum ſeries <lb/>ſive corona eundem ſitum cum priore habebit, eadem q́ue de cauſa octo glo-<lb/>bi ſiniſtri ponderoſiores erunt ſex dextris, ideoq́ue rurſus octo illi deſcen-<lb/>dent, ſex iſti adſcendent, ipſiq́ue globi ex ſeſe continuum & </s> <s xml:id="echoid-s1043" xml:space="preserve">æternum motum <lb/>efficient, quod eſt falſum. </s> <s xml:id="echoid-s1044" xml:space="preserve">Pars igitur coronæ D, R, Q, P, O, N, M, L, parti <lb/>E, F, G, H, I, K, ſitu æquilibris eſt. </s> <s xml:id="echoid-s1045" xml:space="preserve">Si verò ab æquilibribus æquilibria tollantur <lb/>reliqua manent æquilibria. </s> <s xml:id="echoid-s1046" xml:space="preserve">illinc igitur O, N, M, L, hinc vero G, H, I, K, qui <lb/>æquantur O, N, M, L ſublatis, reliqui D, R, P, Q, reliquis E, F ſitu æquilibres <lb/>erunt. </s> <s xml:id="echoid-s1047" xml:space="preserve">Atqui duobus iſtis quatuor illis æquilibribus, E duplo ponderoſior erit <lb/>ſitu, quam D. </s> <s xml:id="echoid-s1048" xml:space="preserve">Quemadmodum igitur latus B A 2 ad latus B C 1, ita ſaco-<lb/>ma globi E ad antiſacoma globi D. </s> <s xml:id="echoid-s1049" xml:space="preserve">C*ONCLUSIO*. </s> <s xml:id="echoid-s1050" xml:space="preserve">Si igitur trianguli <lb/>planum horizonti ſit perpendiculare &</s> <s xml:id="echoid-s1051" xml:space="preserve">c.</s> <s xml:id="echoid-s1052" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div174" type="section" level="1" n="133"> <head xml:id="echoid-head145" xml:space="preserve">1 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1053" xml:space="preserve">Si A B C triangulum ſit, ut ante, ejusq́ue latus A B duplum lateris B C, <lb/>inq́ue A B jaceat globus D, in B C verò globus E, ſubduplus ponderi D, & </s> <s xml:id="echoid-s1054" xml:space="preserve"><lb/>in F fixus ſit punctus quâ linea ſive funiculus DFE <lb/> <anchor type="figure" xlink:label="fig-527.01.035-01a" xlink:href="fig-527.01.035-01"/> (è centro ſcilicet D, per F, in centrum E uſque) <lb/>motus radit F fixum punctum, ut D F ab A B, <lb/>& </s> <s xml:id="echoid-s1055" xml:space="preserve">F E à B C æquidiſtans ſit: </s> <s xml:id="echoid-s1056" xml:space="preserve">quia quatuor globi <lb/>P, Q, R, D, ſitu æquilibres fuerunt duobus E, F, <lb/>etiam globus D ſitu æquilibris erit globo E. </s> <s xml:id="echoid-s1057" xml:space="preserve">Vt <lb/>enim P, Q, R, D ad E, F: </s> <s xml:id="echoid-s1058" xml:space="preserve">ita D ad E. </s> <s xml:id="echoid-s1059" xml:space="preserve">Igitur quem-<lb/>admodum latus A B, ad B C: </s> <s xml:id="echoid-s1060" xml:space="preserve">ita globus D ad globum E.</s> <s xml:id="echoid-s1061" xml:space="preserve"/> </p> <div xml:id="echoid-div174" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.035-01" xlink:href="fig-527.01.035-01a"> <image file="527.01.035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.035-01"/> </figure> </div> </div> <div xml:id="echoid-div176" type="section" level="1" n="134"> <head xml:id="echoid-head146" xml:space="preserve">2 C*ONSECTARIUM*.</head> <figure> <image file="527.01.035-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.035-02"/> </figure> <p> <s xml:id="echoid-s1062" xml:space="preserve">SI latus trianguli B C, cui A B duplum eſt, re-<lb/>ctum ad A C collocetur, ut expreſſum hîcvi-<lb/>des; </s> <s xml:id="echoid-s1063" xml:space="preserve">globus D duplus ad globum E, cum E ſitu <lb/>æquilibris erit, utenim A B ad B C: </s> <s xml:id="echoid-s1064" xml:space="preserve">ita globus D <lb/>ad globum E.</s> <s xml:id="echoid-s1065" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div177" type="section" level="1" n="135"> <head xml:id="echoid-head147" xml:space="preserve">3 C*ONSECTARIUM*.</head> <figure> <image file="527.01.035-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.035-03"/> </figure> <p> <s xml:id="echoid-s1066" xml:space="preserve">SI loco puncti F trochlea ita collocetur, <lb/>ut linea D F, obliquè extollens, ad A B <lb/>ſit parallela, & </s> <s xml:id="echoid-s1067" xml:space="preserve">proglobo E pondus ſit con-<lb/>tingĕti quidem figura, ſed eodem cum illo <lb/>po dere, erit cum D ſitu æquilibre. </s> <s xml:id="echoid-s1068" xml:space="preserve">Ideoq́; <lb/></s> <s xml:id="echoid-s1069" xml:space="preserve">quemadmodum A B ad B C: </s> <s xml:id="echoid-s1070" xml:space="preserve">ita globus <lb/>D ad pondus E.</s> <s xml:id="echoid-s1071" xml:space="preserve"/> </p> <pb o="36" file="527.01.036" n="36" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div178" type="section" level="1" n="136"> <head xml:id="echoid-head148" xml:space="preserve">4 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1072" xml:space="preserve">QVandoquidem 3 confectarii globus li-<lb/> <anchor type="figure" xlink:label="fig-527.01.036-01a" xlink:href="fig-527.01.036-01"/> neam A B tangit in puncto G, tanquam <lb/>firmo, globi axis G H perpendicularis eſt ad <lb/>A B: </s> <s xml:id="echoid-s1073" xml:space="preserve">quapropter amoto globo, loco ejus co-<lb/>lumna D ejuſdem ponderis cum globo po-<lb/>natur, ut axis G H (cujus punctum firmũ G) <lb/>perpendicularis ſit lateri A B, & </s> <s xml:id="echoid-s1074" xml:space="preserve">D F linea <lb/>obliquè extollens ad A B parallela ſecans la-<lb/>tus columnæ in I; </s> <s xml:id="echoid-s1075" xml:space="preserve">erit etiam nunc, quemad-<lb/>modum A B ad B C (ratio autem eſt dupla, <lb/>utante) ita columna D ad pondus E.</s> <s xml:id="echoid-s1076" xml:space="preserve"/> </p> <div xml:id="echoid-div178" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.036-01" xlink:href="fig-527.01.036-01a"> <image file="527.01.036-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.036-01"/> </figure> </div> </div> <div xml:id="echoid-div180" type="section" level="1" n="137"> <head xml:id="echoid-head149" xml:space="preserve">5 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1077" xml:space="preserve">PErpendiculari D K è columnæ centro D productâ, ut ejus latus in L ſe-<lb/>cet, triangulum LDI ſimile erit triangulo A B C, anguli enim A C B & </s> <s xml:id="echoid-s1078" xml:space="preserve"><lb/>L I D recti ſunt, & </s> <s xml:id="echoid-s1079" xml:space="preserve">rectæ L D, D I, pa-<lb/> <anchor type="figure" xlink:label="fig-527.01.036-02a" xlink:href="fig-527.01.036-02"/> rallelæ ſunt ad BC, AB. </s> <s xml:id="echoid-s1080" xml:space="preserve">Vtigitur AB <lb/>ad B C: </s> <s xml:id="echoid-s1081" xml:space="preserve">ita L D ad D I. </s> <s xml:id="echoid-s1082" xml:space="preserve">Atquiut A B <lb/>ad B C: </s> <s xml:id="echoid-s1083" xml:space="preserve">ita columna ad pondus E, per <lb/>4 confectarium, & </s> <s xml:id="echoid-s1084" xml:space="preserve">propterea ut L D <lb/>ad D I: </s> <s xml:id="echoid-s1085" xml:space="preserve">ita columna eſt ad E. </s> <s xml:id="echoid-s1086" xml:space="preserve">Ad li-<lb/>neam K D ſi pondus M, quod rectè <lb/>extollat, & </s> <s xml:id="echoid-s1087" xml:space="preserve">columnæ ſitu æquilibre eſt, <lb/>adjungatur, ad columnam etiam æqui-<lb/>pondium erit, per 14 propoſit. </s> <s xml:id="echoid-s1088" xml:space="preserve">utigitur <lb/>L D ad D I: </s> <s xml:id="echoid-s1089" xml:space="preserve">ita M ad E.</s> <s xml:id="echoid-s1090" xml:space="preserve"/> </p> <div xml:id="echoid-div180" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.036-02" xlink:href="fig-527.01.036-02a"> <image file="527.01.036-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.036-02"/> </figure> </div> </div> <div xml:id="echoid-div182" type="section" level="1" n="138"> <head xml:id="echoid-head150" xml:space="preserve">6 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1091" xml:space="preserve">B N ducatur, ſecans A C continuatam in N, conſimiliter D O ſecans <lb/>continuatam L I, hoc eſt, latus columnæ in O, ut angulus I D O æqualis ſit <lb/>angulo C B N. </s> <s xml:id="echoid-s1092" xml:space="preserve">Appendatur quoque ad D O pondus P obliquè attollens, <lb/>quod (amotis M, E ponderibus) columnam in ſuo ſitu conſervet. </s> <s xml:id="echoid-s1093" xml:space="preserve">Quia vero <lb/>D L & </s> <s xml:id="echoid-s1094" xml:space="preserve">B A, item D I & </s> <s xml:id="echoid-s1095" xml:space="preserve">B C latera trian-<lb/> <anchor type="figure" xlink:label="fig-527.01.036-03a" xlink:href="fig-527.01.036-03"/> gulorum D L I & </s> <s xml:id="echoid-s1096" xml:space="preserve">B A C homologa <lb/>ſunt, hujuſmodi concluſio inde dedu-<lb/>citur. </s> <s xml:id="echoid-s1097" xml:space="preserve">Quemadmodum B A ad B C: <lb/></s> <s xml:id="echoid-s1098" xml:space="preserve">ita ſacoma lateris B A ad antiſacoma <lb/>lateris B C (per 2 conſectarium.) </s> <s xml:id="echoid-s1099" xml:space="preserve">item <lb/>quemadmodum D L ad D I: </s> <s xml:id="echoid-s1100" xml:space="preserve">ita ſaco-<lb/>ma lateris D L ad antiſacoma lateris <lb/>D I, hoc eſtita M ad E. </s> <s xml:id="echoid-s1101" xml:space="preserve">Sed homolo-<lb/>ga latera triãgulorum ſimilium A B N, <lb/>L D O ſunt A B & </s> <s xml:id="echoid-s1102" xml:space="preserve">D L, item B N, & </s> <s xml:id="echoid-s1103" xml:space="preserve"><lb/>D O. </s> <s xml:id="echoid-s1104" xml:space="preserve">Itaque ut ſupra, quemadmodum B A ad B N: </s> <s xml:id="echoid-s1105" xml:space="preserve">ir<unsure/>a ſacoma B A ad an-<lb/>tiſacoma B N (per 1 confectarium) Et quemadmodum D L ad D O: </s> <s xml:id="echoid-s1106" xml:space="preserve">ita il-<lb/>lius ſacoma ad hujus antiſacoma, id eſt, M ad P. </s> <s xml:id="echoid-s1107" xml:space="preserve">Si linea B N à puncto B <lb/>aliovorſum; </s> <s xml:id="echoid-s1108" xml:space="preserve">A ſcilicet verſus, ultra B C fuiſſet ducta, etiam recta D O à D <pb o="37" file="527.01.037" n="37" rhead="*DE* S*TATICÆ ELEMENTIS*."/> u@@a D I cecidiſſet, hoc eſt, ut nunc citra: </s> <s xml:id="echoid-s1109" xml:space="preserve">ita tunc ultra cecidiſſet, & </s> <s xml:id="echoid-s1110" xml:space="preserve">præce-<lb/>dens demonſtratio etiam iſtiſitui accommoda fuiſſet, hoc eſt, quemadmodum <lb/>B A ad B N ita ſacoma lateris B A, ad antiſacoma lateris B N eſſet: </s> <s xml:id="echoid-s1111" xml:space="preserve">& </s> <s xml:id="echoid-s1112" xml:space="preserve">quem-<lb/>admodum D L ad D O: </s> <s xml:id="echoid-s1113" xml:space="preserve">ita ſacoma lateris D L, ad antiſacoma lateris D O. <lb/></s> <s xml:id="echoid-s1114" xml:space="preserve">hoc eſt M ad P. </s> <s xml:id="echoid-s1115" xml:space="preserve">Vtiſta proportio non tantum in exemplis valeat, in quibus <lb/>linea attollens, ut D I, perpendicularis eſt axi, ſed etiam in aliis cujuſmodi-<lb/>cunque ſint anguli.</s> <s xml:id="echoid-s1116" xml:space="preserve"/> </p> <div xml:id="echoid-div182" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.036-03" xlink:href="fig-527.01.036-03a"> <image file="527.01.036-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.036-03"/> </figure> </div> <p> <s xml:id="echoid-s1117" xml:space="preserve">ISta etiam deglobo in lineâ, ut A B, jacente intelligi poſſunt, nam & </s> <s xml:id="echoid-s1118" xml:space="preserve">hic, ut <lb/>L D ad D O: </s> <s xml:id="echoid-s1119" xml:space="preserve">ita M ad P (modo C L ad A B perpendicularis ſit, hoc eſt, <lb/>patallela ad axem G H globi D) atqui pon-<lb/> <anchor type="figure" xlink:label="fig-527.01.037-01a" xlink:href="fig-527.01.037-01"/> dus Mglobo D æquatur, ideo etiam ut L D <lb/>ad D O: </s> <s xml:id="echoid-s1120" xml:space="preserve">ita pondus globi ad pondus P. </s> <s xml:id="echoid-s1121" xml:space="preserve">Ve-<lb/>rumenimvero, quia L D & </s> <s xml:id="echoid-s1122" xml:space="preserve">D O intra glo-<lb/>biſoliditatem re ipſa delineari cõmodè non <lb/>poſſunt, perpendiculari C E ductâ, extra <lb/>globi ſolidum comprehĕdetur C E O trian-<lb/>gulum L D O triangulo ſimile, cujus latera <lb/>L D & </s> <s xml:id="echoid-s1123" xml:space="preserve">C E, item D O & </s> <s xml:id="echoid-s1124" xml:space="preserve">E O homologa <lb/>erunt. </s> <s xml:id="echoid-s1125" xml:space="preserve">Quemadmodum igitur L D ad D O: <lb/></s> <s xml:id="echoid-s1126" xml:space="preserve">ita C E ad E O, & </s> <s xml:id="echoid-s1127" xml:space="preserve">per conſequens ut C E ad E O: </s> <s xml:id="echoid-s1128" xml:space="preserve">itaglobi pondus ad P.</s> <s xml:id="echoid-s1129" xml:space="preserve"/> </p> <div xml:id="echoid-div183" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.037-01" xlink:href="fig-527.01.037-01a"> <image file="527.01.037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.037-01"/> </figure> </div> <p> <s xml:id="echoid-s1130" xml:space="preserve">VT major claritudo hujus ſit, ſublatis aliis li-<lb/> <anchor type="figure" xlink:label="fig-527.01.037-02a" xlink:href="fig-527.01.037-02"/> neis omnibus dicatur ut C E ad C O: </s> <s xml:id="echoid-s1131" xml:space="preserve">ita <lb/>pondus globi D ad pondus P.</s> <s xml:id="echoid-s1132" xml:space="preserve"/> </p> <div xml:id="echoid-div184" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.037-02" xlink:href="fig-527.01.037-02a"> <image file="527.01.037-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.037-02"/> </figure> </div> <p> <s xml:id="echoid-s1133" xml:space="preserve">NEque illud de globis tantum verum eſt, ſed <lb/>etiam de quibuſvis corporibus, puncta vel li-<lb/>neas ſtringentibus, aut etiam per illa volutis, ut in-<lb/>fra videre eſt. </s> <s xml:id="echoid-s1134" xml:space="preserve">Sed de his in S*TATICES* praxi <lb/> <anchor type="figure" xlink:label="fig-527.01.037-03a" xlink:href="fig-527.01.037-03"/> preſſius dicemus. </s> <s xml:id="echoid-s1135" xml:space="preserve">Nam & </s> <s xml:id="echoid-s1136" xml:space="preserve">hîc dicimus quemadmodum C E ad E O: </s> <s xml:id="echoid-s1137" xml:space="preserve">ita pon-<lb/>dus corporis D, ad pondus P.</s> <s xml:id="echoid-s1138" xml:space="preserve"/> </p> <div xml:id="echoid-div185" type="float" level="2" n="4"> <figure xlink:label="fig-527.01.037-03" xlink:href="fig-527.01.037-03a"> <image file="527.01.037-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.037-03"/> </figure> </div> <p> <s xml:id="echoid-s1139" xml:space="preserve">VNde etiam hoc manifeſtum: </s> <s xml:id="echoid-s1140" xml:space="preserve">Si recta A B horizonti eſt parallela, qua-<lb/>lem figuram hic juxta poſitam videre eſt, rectas C E & </s> <s xml:id="echoid-s1141" xml:space="preserve">C O in unam & </s> <s xml:id="echoid-s1142" xml:space="preserve"><lb/>candem lineam coïre, ideoq́ue inter E & </s> <s xml:id="echoid-s1143" xml:space="preserve">O nullam longitudinĕ & </s> <s xml:id="echoid-s1144" xml:space="preserve">propterea <lb/>rectæ C E ad rectam E O nullam rationĕ fore. </s> <s xml:id="echoid-s1145" xml:space="preserve">Hinc intelligere in proclivi eſt, <pb o="38" file="527.01.038" n="38" rhead="1 L*IBER* S*TATICÆ*"/> nullum pondus, quantulumcunque fuerit in P, ſitu æquipondium eſſe poſſe <lb/>cõtra corpus D, verùm (mathematicè intel-<lb/> <anchor type="figure" xlink:label="fig-527.01.038-01a" xlink:href="fig-527.01.038-01"/> ligitor) loco promovere poſſe quantumvis <lb/>póderoſium fuerit. </s> <s xml:id="echoid-s1146" xml:space="preserve">Vnde cõſequitur, quæ-<lb/>vis põdera ſecundum horizontem promo-<lb/>ta, cujuſmodi ſunt naves in aquis, plauſtra <lb/>in camporũ æquoribus, &</s> <s xml:id="echoid-s1147" xml:space="preserve">c. </s> <s xml:id="echoid-s1148" xml:space="preserve">ne muſcæ qui-<lb/>dem potentiam ad motum ſui requirere, <lb/>niſi quantum circumſtantia obſtacula offi-<lb/>ciunt, motumq́; </s> <s xml:id="echoid-s1149" xml:space="preserve">impediunt. </s> <s xml:id="echoid-s1150" xml:space="preserve">ut Aqua, Aër, <lb/>axium in modiolos ſuos, rotarumq́; </s> <s xml:id="echoid-s1151" xml:space="preserve">in via-<lb/>rum ſtrata offenſationes, & </s> <s xml:id="echoid-s1152" xml:space="preserve">impactiones, & </s> <s xml:id="echoid-s1153" xml:space="preserve">alia hujuſcemodi.</s> <s xml:id="echoid-s1154" xml:space="preserve"/> </p> <div xml:id="echoid-div186" type="float" level="2" n="5"> <figure xlink:label="fig-527.01.038-01" xlink:href="fig-527.01.038-01a"> <image file="527.01.038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.038-01"/> </figure> </div> <p> <s xml:id="echoid-s1155" xml:space="preserve">SEd triangulũ A B N 6 conſectar<unsure/>ii, quan-<lb/> <anchor type="figure" xlink:label="fig-527.01.038-02a" xlink:href="fig-527.01.038-02"/> doquidem ad proportionem hanc nihil <lb/>adjumenti, neque adfert, nequeinde aufert, <lb/>ſublatum fingamus, & </s> <s xml:id="echoid-s1156" xml:space="preserve">G firmum columnæ <lb/>punctum coni, aut pyramidis faſtigio inniti, <lb/>ut vides; </s> <s xml:id="echoid-s1157" xml:space="preserve">nihilo tamen minus erit, quemad-<lb/>modum L D ad D O: </s> <s xml:id="echoid-s1158" xml:space="preserve">ita M ad P.</s> <s xml:id="echoid-s1159" xml:space="preserve"/> </p> <div xml:id="echoid-div187" type="float" level="2" n="6"> <figure xlink:label="fig-527.01.038-02" xlink:href="fig-527.01.038-02a"> <image file="527.01.038-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.038-02"/> </figure> </div> </div> <div xml:id="echoid-div189" type="section" level="1" n="139"> <head xml:id="echoid-head151" xml:space="preserve">7 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1160" xml:space="preserve">VErum ne iſta proportio ſolius columnæ eſſe videatur, ubilinea rectè ex-<lb/>tollens, ut D L, è centro eſt ducta, & </s> <s xml:id="echoid-s1161" xml:space="preserve">firmum punctum axis extremitas <lb/>eſt, ut G 6 conſectarii: </s> <s xml:id="echoid-s1162" xml:space="preserve">Triangulum A B C eſto, cujus latus A B duplum <lb/>ſit lateris B C, & </s> <s xml:id="echoid-s1163" xml:space="preserve">hoc ipſum B C perpendiculare reliquo A C: </s> <s xml:id="echoid-s1164" xml:space="preserve">D E co-<lb/>lumna eſto, ejusq́ue axis F G, perpendicularis in latus A B, illudq́ue ſecans in <lb/>H; </s> <s xml:id="echoid-s1165" xml:space="preserve">I verò contingens in eodĕ axe punctum. <lb/></s> <s xml:id="echoid-s1166" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.038-03a" xlink:href="fig-527.01.038-03"/> Eſto & </s> <s xml:id="echoid-s1167" xml:space="preserve">altera K L æqualis & </s> <s xml:id="echoid-s1168" xml:space="preserve">ſimilis D E <lb/>columnæ, ejusq́; </s> <s xml:id="echoid-s1169" xml:space="preserve">axis M N, & </s> <s xml:id="echoid-s1170" xml:space="preserve">O punctum <lb/>tangens B C, eodem ſitu in ſua columna, <lb/>quo H in D E. </s> <s xml:id="echoid-s1171" xml:space="preserve">P quoque punctum alterum <lb/>eo ſitu eſto in K L, quo I in D E. </s> <s xml:id="echoid-s1172" xml:space="preserve">Q fir-<lb/>mum punctum ſtatuitor, quod linea I Q P <lb/>ducta ac reducta ſtringat, radatq́ue, ut I Q <lb/>ad A B, Q P vero ad B C parallela ſit. </s> <s xml:id="echoid-s1173" xml:space="preserve">Pro-<lb/>pter cauſas, 19 propoſitione de quatuorde-<lb/>cim globis enucleatas, (quæ etſi hic quoque de columnis ſimiliter motis, & </s> <s xml:id="echoid-s1174" xml:space="preserve"><lb/>puncta firma ſtring entibus demonſtrari poſſunt, præterire tamen animus eſt, <lb/>quia iſtinc repetita facile intelligi poſſunt) ſacoma columnæ K L duplum eſt, <lb/>ad antiſacoma columnæ D E.</s> <s xml:id="echoid-s1175" xml:space="preserve"/> </p> <div xml:id="echoid-div189" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.038-03" xlink:href="fig-527.01.038-03a"> <image file="527.01.038-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.038-03"/> </figure> </div> </div> <div xml:id="echoid-div191" type="section" level="1" n="140"> <head xml:id="echoid-head152" xml:space="preserve">8 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1176" xml:space="preserve">A D I, 7 confectarii pondus R rectè extollens, & </s> <s xml:id="echoid-s1177" xml:space="preserve">columnæ æquilibreap-<lb/>penditor, cujus linea rectè attollens I S ſit, ſecans latus columnæ in T, & </s> <s xml:id="echoid-s1178" xml:space="preserve"><lb/>I Q idem latus ſecet in V, & </s> <s xml:id="echoid-s1179" xml:space="preserve">de P Q pondus X, pro columna K L, depen- <pb o="39" file="527.01.039" n="39" rhead="*DE* S*TATICÆ ELEMENTIS*."/> deto, ſub duplum æquilibris ponderis ejuſdĕ <lb/> <anchor type="figure" xlink:label="fig-527.01.039-01a" xlink:href="fig-527.01.039-01"/> columnæ, ſublatoq́ue triangulo A B C, co-<lb/>lumna D E quieſcat in H, ut hîc vides. </s> <s xml:id="echoid-s1180" xml:space="preserve">Ob <lb/>cauſas jam nunc cõmem oratas, quemadmo-<lb/>dum T I ad IV: </s> <s xml:id="echoid-s1181" xml:space="preserve">ita R eritad X. </s> <s xml:id="echoid-s1182" xml:space="preserve">neque hoc <lb/>tantũ quando I V perpĕdicularis eſt & </s> <s xml:id="echoid-s1183" xml:space="preserve">recta <lb/>ad axem F G, verum etiam quando contin-<lb/>gĕter obliqua. </s> <s xml:id="echoid-s1184" xml:space="preserve">Cujus rei argumĕta documĕ-<lb/>taq; </s> <s xml:id="echoid-s1185" xml:space="preserve">ſpeciatim dari poſſent, niſi hoc è 6 con-<lb/>fectario clarum ſatis ac manifeſtum eſſet.</s> <s xml:id="echoid-s1186" xml:space="preserve"/> </p> <div xml:id="echoid-div191" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.039-01" xlink:href="fig-527.01.039-01a"> <image file="527.01.039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.039-01"/> </figure> </div> </div> <div xml:id="echoid-div193" type="section" level="1" n="141"> <head xml:id="echoid-head153" xml:space="preserve">9 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1187" xml:space="preserve">8 Confectario proportio declarata fuit, ubi I mobile punctum ſupra H fuir <lb/>punctum fixum, & </s> <s xml:id="echoid-s1188" xml:space="preserve">linea IV obliquè extollens H firmum punctum verſus <lb/>inclinata: </s> <s xml:id="echoid-s1189" xml:space="preserve">eadem proportio in alio quovis ſitu demonſtranda eſt, & </s> <s xml:id="echoid-s1190" xml:space="preserve">primum <lb/>quidem in illis, ubi mobile punctum infra fixum eſt, lineaq́ue obliquè extol-<lb/>lens à firmo inclinata eſt. </s> <s xml:id="echoid-s1191" xml:space="preserve">& </s> <s xml:id="echoid-s1192" xml:space="preserve">quidem iſto pacto.</s> <s xml:id="echoid-s1193" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1194" xml:space="preserve">A B columna eſto, ejusq́ue axis C D, punctum firmum E, mobile vero F, <lb/> <anchor type="figure" xlink:label="fig-527.01.039-02a" xlink:href="fig-527.01.039-02"/> pondus obliquè extollens G, cujus <lb/>obliqua linea FH, FI verò linea re-<lb/>ctè attollens, cujus rectum pondus K. <lb/></s> <s xml:id="echoid-s1195" xml:space="preserve">Etiam L M columna æqualis & </s> <s xml:id="echoid-s1196" xml:space="preserve">ſimi-<lb/>lis eſto A B columnæ, ejusq́ue axis <lb/>N O, punctum firmum E, mobile F, <lb/>ut E N æquetur E D, E F verò E P, <lb/>pondus obliquè extollens Q æquale <lb/>G, cujus linea obliqua ſit parallela ad <lb/>F H: </s> <s xml:id="echoid-s1197" xml:space="preserve">pondus rectè extollens S æqua-<lb/>le ponderi K, & </s> <s xml:id="echoid-s1198" xml:space="preserve">linea illius recta P T. </s> <s xml:id="echoid-s1199" xml:space="preserve">His ita poſitis & </s> <s xml:id="echoid-s1200" xml:space="preserve">conceſſis A B & </s> <s xml:id="echoid-s1201" xml:space="preserve">L M <lb/>addantur, fiantq́ue una columna AM, cujus centrum gravitatis erit E, ex theſi-<lb/>Ponderibus K, G, S, Q, amotis, columna A M quemvis datũ ſitum ſervabit <lb/>in E puncto, per 7 propoſit. </s> <s xml:id="echoid-s1202" xml:space="preserve">eritq́ columna A B cõtra L M columnam æquili-<lb/>bris. </s> <s xml:id="echoid-s1203" xml:space="preserve">Rurſus pondera Q, G æquiponderantia æquipõderantibus & </s> <s xml:id="echoid-s1204" xml:space="preserve">quidem <lb/>ſimili ſitu appendamus, Q & </s> <s xml:id="echoid-s1205" xml:space="preserve">G, per 13 propoſitionem, in A M columnam <lb/>cjuſdem potentiæ ſunt, ideoq́ue quantum potentiæ eſt ponderi Q in L M <lb/>columnam, tantundem quoque & </s> <s xml:id="echoid-s1206" xml:space="preserve">G fueritin ſuam A B. </s> <s xml:id="echoid-s1207" xml:space="preserve">Atqui potentia G <lb/>eſt, in ſitu ſuo retinere A B, per 6 confect. </s> <s xml:id="echoid-s1208" xml:space="preserve">eadem igitur & </s> <s xml:id="echoid-s1209" xml:space="preserve">Q erit in L M. </s> <s xml:id="echoid-s1210" xml:space="preserve"><lb/>Conſimiliter eadem potentia K eſt in A B, eadem igitur S fuerit in L M. </s> <s xml:id="echoid-s1211" xml:space="preserve"><lb/>Quemadmodum itaque IF ad FH ita K ad G per 8 conſectar. </s> <s xml:id="echoid-s1212" xml:space="preserve">atqui TP <lb/>æquatur IF, & </s> <s xml:id="echoid-s1213" xml:space="preserve">PR, ipſi FH, item pondus S ponderi, K, pondusq́ue Q <lb/>ponderi G: </s> <s xml:id="echoid-s1214" xml:space="preserve">ut igitur TP ad PR ita S ad Q. </s> <s xml:id="echoid-s1215" xml:space="preserve">Quapropter iſta proportio, <lb/>ut diximus, non minus conſtans eſt in exemplis, ubi mobile punctum P in-<lb/>fra E firmum eſt, quam ubi ſupra, ubiq́ue linea P R rectè extollens à latere <lb/>firmi puncti E declinat, quam ubiſupra eſt, & </s> <s xml:id="echoid-s1216" xml:space="preserve">obliquè extollens linea idem <lb/>firmum punctum verſus inclinat.</s> <s xml:id="echoid-s1217" xml:space="preserve"/> </p> <div xml:id="echoid-div193" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.039-02" xlink:href="fig-527.01.039-02a"> <image file="527.01.039-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.039-02"/> </figure> </div> <pb o="40" file="527.01.040" n="40" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div195" type="section" level="1" n="142"> <head xml:id="echoid-head154" xml:space="preserve">10 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1218" xml:space="preserve">Eſto exemplat 9 conſectarii fimile, eo <lb/> <anchor type="figure" xlink:label="fig-527.01.040-01a" xlink:href="fig-527.01.040-01"/> tantũ diſſimile, quod F H ultra F I, ver-<lb/>ſus C declinet, quodq́; </s> <s xml:id="echoid-s1219" xml:space="preserve">H F C angulus <lb/>R P O angulo ſit æqualis, quapropter <lb/>põderi G in columnam A M tantumdĕ <lb/>potentiæ eſt, quantũ ponderi Q: </s> <s xml:id="echoid-s1220" xml:space="preserve">& </s> <s xml:id="echoid-s1221" xml:space="preserve">pro-<lb/>pter cauſas 9 cõſect. </s> <s xml:id="echoid-s1222" xml:space="preserve">cõmemoratas (quas <lb/>brevitatis causâ omittimus) G tantam <lb/>vim columnæ A B adfert: </s> <s xml:id="echoid-s1223" xml:space="preserve">quantam Q <lb/>columnæ L M. </s> <s xml:id="echoid-s1224" xml:space="preserve">Itaque ut T P ad P R: </s> <s xml:id="echoid-s1225" xml:space="preserve">ita <lb/>S ad Q, per 9 conſectarium: </s> <s xml:id="echoid-s1226" xml:space="preserve">at qui I F <lb/>æquatur T P, & </s> <s xml:id="echoid-s1227" xml:space="preserve">F H ipſi P R, pondusq́; <lb/></s> <s xml:id="echoid-s1228" xml:space="preserve">K ponderi S, & </s> <s xml:id="echoid-s1229" xml:space="preserve">G ipſi Q. </s> <s xml:id="echoid-s1230" xml:space="preserve">Quemadmo-<lb/>dum igitur I F ad F H: </s> <s xml:id="echoid-s1231" xml:space="preserve">ita K ad G.</s> <s xml:id="echoid-s1232" xml:space="preserve"/> </p> <div xml:id="echoid-div195" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.040-01" xlink:href="fig-527.01.040-01a"> <image file="527.01.040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.040-01"/> </figure> </div> </div> <div xml:id="echoid-div197" type="section" level="1" n="143"> <head xml:id="echoid-head155" xml:space="preserve">11 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1233" xml:space="preserve">ESto & </s> <s xml:id="echoid-s1234" xml:space="preserve">10 conſectarii ſimile exemplũ, <lb/> <anchor type="figure" xlink:label="fig-527.01.040-02a" xlink:href="fig-527.01.040-02"/> eo tantũ diſſimile quod P R in alterũ <lb/>latus à P T declinet & </s> <s xml:id="echoid-s1235" xml:space="preserve">P R ad F H pa-<lb/>rallela ſit, ut Q & </s> <s xml:id="echoid-s1236" xml:space="preserve">G columnæ A M æ-<lb/>qualem vim inferant, &</s> <s xml:id="echoid-s1237" xml:space="preserve">, propter cauſas 9 <lb/>conſectario dictas, quantum potentię eſt <lb/>ponderi G in columnam A B, tantundĕ <lb/>& </s> <s xml:id="echoid-s1238" xml:space="preserve">põdus Q obtinet in columnam L M. <lb/></s> <s xml:id="echoid-s1239" xml:space="preserve">Vtigitur I F ad F H: </s> <s xml:id="echoid-s1240" xml:space="preserve">ita, per 6 conſecta-<lb/>rium, K ad G. </s> <s xml:id="echoid-s1241" xml:space="preserve">Atqui T P æquatur I F, & </s> <s xml:id="echoid-s1242" xml:space="preserve"><lb/>P R ipſi F H, & </s> <s xml:id="echoid-s1243" xml:space="preserve">pondus S ponderi K, & </s> <s xml:id="echoid-s1244" xml:space="preserve"><lb/>Q ipſi G. </s> <s xml:id="echoid-s1245" xml:space="preserve">Quemadmodum igitur TP ad <lb/>P R: </s> <s xml:id="echoid-s1246" xml:space="preserve">ita S ad Q. </s> <s xml:id="echoid-s1247" xml:space="preserve">Similiter aliorum <lb/>omnium ſituum proportio ex contrario demonſtrabitur.</s> <s xml:id="echoid-s1248" xml:space="preserve"/> </p> <div xml:id="echoid-div197" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.040-02" xlink:href="fig-527.01.040-02a"> <image file="527.01.040-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.040-02"/> </figure> </div> </div> <div xml:id="echoid-div199" type="section" level="1" n="144"> <head xml:id="echoid-head156" xml:space="preserve">12 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1249" xml:space="preserve">HAncautem proportionem etiam in illis <lb/> <anchor type="figure" xlink:label="fig-527.01.040-03a" xlink:href="fig-527.01.040-03"/> locum obtinere ubi axis horizõti paral-<lb/>lelus eſt, ita demõſtratur. </s> <s xml:id="echoid-s1250" xml:space="preserve">A B columna eſto, <lb/>ejusq́; </s> <s xml:id="echoid-s1251" xml:space="preserve">axis C D ad horizontem parallelus, E <lb/>punctũ firmum, F mobile, G põdus obliquè <lb/>extollens, quod columnam eo in ſitu ſervat, <lb/>cujus obliqua linea F H, I vero pondus rectè <lb/>extollens, columnam itidem eodem ſervans <lb/>in ſitu, ejusq́; </s> <s xml:id="echoid-s1252" xml:space="preserve">linea recta F K. </s> <s xml:id="echoid-s1253" xml:space="preserve">His poſitis, diſ-<lb/>ſimilis ratio eſto (ſi fieri poteſt) K F ad F H, <lb/>atq; </s> <s xml:id="echoid-s1254" xml:space="preserve">I ad G: </s> <s xml:id="echoid-s1255" xml:space="preserve">exempli gratià, ratio K F ad F H <lb/>ſit 1 ad 2: </s> <s xml:id="echoid-s1256" xml:space="preserve">ratio verò I ad G 3 ad 7. </s> <s xml:id="echoid-s1257" xml:space="preserve">Et hoc ita <lb/>poſito, prioris exempli columna demittitor, <lb/>aut poſterioris extollitor, uſq; </s> <s xml:id="echoid-s1258" xml:space="preserve">dum ratio K F <lb/>ad F H, ſit 3 ad 7: </s> <s xml:id="echoid-s1259" xml:space="preserve">tunc G contra columnam <lb/>æquilibre erit, per antecedentia conſectaria, <lb/>ut columna ſive altius elata, ſive humilius de-<lb/>preſſa contra G æquilibris manſura ſit: </s> <s xml:id="echoid-s1260" xml:space="preserve">At-<lb/>qui illud @@ ἀδύναπις eſſe manifeſtum eſt, <pb o="41" file="527.01.041" n="41" rhead="*DE* S*TATICÆ ELEMENTIS*."/> quod Mathematicè quoque ex 22 propoſit. </s> <s xml:id="echoid-s1261" xml:space="preserve">patebit. </s> <s xml:id="echoid-s1262" xml:space="preserve">Quapropter ratio KF <lb/>ad FH non eſt alia, quam I ad G.</s> <s xml:id="echoid-s1263" xml:space="preserve"/> </p> <div xml:id="echoid-div199" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.040-03" xlink:href="fig-527.01.040-03a"> <image file="527.01.040-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.040-03"/> </figure> </div> <p> <s xml:id="echoid-s1264" xml:space="preserve">Ex hifce jam dictis hujuſmodi theoremata deducimus.</s> <s xml:id="echoid-s1265" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div201" type="section" level="1" n="145"> <head xml:id="echoid-head157" xml:space="preserve">12 THE OREMA. 20 PROPOSITIO.</head> <p> <s xml:id="echoid-s1266" xml:space="preserve">Siaxis columnæ puncta habeat, firmũ, & </s> <s xml:id="echoid-s1267" xml:space="preserve">mobile, & </s> <s xml:id="echoid-s1268" xml:space="preserve">ex <lb/>iſto dependentia pondera, unum rectè, alterum obliquè <lb/>extollens, in dato ſitu columnam cõſervant: </s> <s xml:id="echoid-s1269" xml:space="preserve">erit quemad-<lb/>modum linea rectè extollens ad lineam obliquè extollen-<lb/>tem; </s> <s xml:id="echoid-s1270" xml:space="preserve">ita illius pondus, ad pondus hujus.</s> <s xml:id="echoid-s1271" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1272" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1273" xml:space="preserve">AB columna eſto, cujus axis ſit C D, in eoq́ue E punctum fir-<lb/>mum, F mobile, cui G pondus rectè extollens appenſum columnam in dato <lb/>ſitu ſervat; </s> <s xml:id="echoid-s1274" xml:space="preserve">indidĕ etiam obliquum pondus H dependens (coërcito vel amo-<lb/>to G) in ſuo ſitu eandem detinet. </s> <s xml:id="echoid-s1275" xml:space="preserve">Latus columnæ à lineâ r<unsure/>ectè extollente in I, <lb/>ab obliquè extollente in K ſecatur. </s> <s xml:id="echoid-s1276" xml:space="preserve">Dico igitur quemadmodũ rectè extollens <lb/>I F ad obliquè tollentem FK: </s> <s xml:id="echoid-s1277" xml:space="preserve">ita rectum pondus G, ad pondus obliquum <lb/>H. </s> <s xml:id="echoid-s1278" xml:space="preserve">Proportionis iſtius demonſtratio, ex doctrinâ antecedĕtemanifeſta eſt.</s> <s xml:id="echoid-s1279" xml:space="preserve"/> </p> <figure> <image file="527.01.041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.041-01"/> </figure> <pb o="42" file="527.01.042" n="42" rhead="1 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s1280" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1281" xml:space="preserve">Si igitur axis columnæ puncta habeat firmum & </s> <s xml:id="echoid-s1282" xml:space="preserve">mo-<lb/>bile, &</s> <s xml:id="echoid-s1283" xml:space="preserve">c.</s> <s xml:id="echoid-s1284" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div202" type="section" level="1" n="146"> <head xml:id="echoid-head158" xml:space="preserve">NOTATO</head> <p> <s xml:id="echoid-s1285" xml:space="preserve">Si linearum alteralatus columnæ non ſecet, eò uſque continuandum eſſe <lb/>latus donec ſecetur, ut in proximè antecedentium figurarum noviſſimâ vi-<lb/>dere eſt.</s> <s xml:id="echoid-s1286" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div203" type="section" level="1" n="147"> <head xml:id="echoid-head159" xml:space="preserve">13 THEOREMA. 21 PROPOSITIO.</head> <p> <s xml:id="echoid-s1287" xml:space="preserve">Si axis columnæ puncta habeat firmum, & </s> <s xml:id="echoid-s1288" xml:space="preserve">mobile, ex <lb/>quo dependentia pondera, unum rectè, alterum obliquè <lb/>demittens in dato ſitu columnam conſervant: </s> <s xml:id="echoid-s1289" xml:space="preserve">erit quem-<lb/>admodum linea rectè demittens ad lineam obliquè de-<lb/>mitten tem: </s> <s xml:id="echoid-s1290" xml:space="preserve">ita pondus rectum ad pondus obliquum.</s> <s xml:id="echoid-s1291" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1292" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1293" xml:space="preserve">AB eſto columna, cujus axis ſit CD, in eoq́ue E punctum fir-<lb/>mum, F mobile, cui G pondus rectè demittens appenſum columnam in da-<lb/>to ſitu ſervat: </s> <s xml:id="echoid-s1294" xml:space="preserve">indidem quoque dependĕs pondus H obliquè demittens (coër-<lb/>cito velamoto G) in ſitu ſuo eandem ſervat: </s> <s xml:id="echoid-s1295" xml:space="preserve">linea rectè demittens ſecat la-<lb/>tus illius in I, obliquè autem demittens in K.</s> <s xml:id="echoid-s1296" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1297" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1298" xml:space="preserve">Demonſtrandum nobis eſt, quemadmodum I F rectè de-<lb/>mittens, ad FK obliquè demittentem: </s> <s xml:id="echoid-s1299" xml:space="preserve">ita eſſe pondus G rectè demittens, ad <lb/>pondus H demittens obliquè.</s> <s xml:id="echoid-s1300" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1301" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1302" xml:space="preserve">Punctum L ſignator, ut EL æquetur EF, hinc pun-<lb/>cto L accommodetur pondus M rectè attollens, columnamq́ue in ſitu ſer-<lb/>vans, cujus recta attollens L N: </s> <s xml:id="echoid-s1303" xml:space="preserve">itidem pondus O obliquè attollens, colu-<lb/>mnamq́ue in ſitu ſervans, cujus linea obliquè attollens LP parallela ſit ad F K.</s> <s xml:id="echoid-s1304" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div204" type="section" level="1" n="148"> <head xml:id="echoid-head160" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1305" xml:space="preserve">Quemadmodum N L ad L P: </s> <s xml:id="echoid-s1306" xml:space="preserve">ita M <lb/> <anchor type="figure" xlink:label="fig-527.01.042-01a" xlink:href="fig-527.01.042-01"/> ad O per 20 propoſitionem, atqui po-<lb/>tentia G in columnam æquatur poten-<lb/>tiæ M, & </s> <s xml:id="echoid-s1307" xml:space="preserve">potentia H potentiæ O per <lb/>13 propoſitionĕ, & </s> <s xml:id="echoid-s1308" xml:space="preserve">recta I F rectæ L N, <lb/>atque F K æqualis eſt LP. </s> <s xml:id="echoid-s1309" xml:space="preserve">Quemad-<lb/>modum igitur IF rectè demittens ad <lb/>F K obliquè demittentem : </s> <s xml:id="echoid-s1310" xml:space="preserve">ita pondus <lb/>G rectè demittens, ad H demittens <lb/>obliquè. </s> <s xml:id="echoid-s1311" xml:space="preserve">eadem demonſtratio erit alio-<lb/>rum omnium, ut videre eſt in ſubje-<lb/>ctis exemplis.</s> <s xml:id="echoid-s1312" xml:space="preserve"/> </p> <div xml:id="echoid-div204" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.042-01" xlink:href="fig-527.01.042-01a"> <image file="527.01.042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.042-01"/> </figure> </div> <pb o="43" file="527.01.043" n="43" rhead="*DE* S*TATICÆ ELEMENTIS*."/> <figure> <image file="527.01.043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.043-01"/> </figure> <p> <s xml:id="echoid-s1313" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1314" xml:space="preserve">Si igitur axis columnæ puncta habeat.</s> <s xml:id="echoid-s1315" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div206" type="section" level="1" n="149"> <head xml:id="echoid-head161" xml:space="preserve">9 PROBLEMA. 22 PROPOSITIO.</head> <p> <s xml:id="echoid-s1316" xml:space="preserve">Datis, notâ columnâ, punctisq́ue in axe, altero firmo, <lb/>altero mobili, ex quo ignotum pondus ſuſpenſum in dato <lb/>fitu columnam conſervat: </s> <s xml:id="echoid-s1317" xml:space="preserve">pondus notum facere.</s> <s xml:id="echoid-s1318" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1319" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1320" xml:space="preserve">Columna ABCD 6 ℔ pendeat, ſecta ut in 1 propoſit. </s> <s xml:id="echoid-s1321" xml:space="preserve">cujus <lb/>punctum X ſit firmum, S mobile, ex quo ſuſpenſum ignotum pondus Y, <lb/>obliquè extollens, columnæ ſit ſitu æquilibre, cujus linea obliquè extollens <lb/>ſecat latus columnæ A B in OE.</s> <s xml:id="echoid-s1322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1323" xml:space="preserve">Q*VAESITVM* Ignotum pondus Y obliquè extollens notum facien-<lb/>dum eſt.</s> <s xml:id="echoid-s1324" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div207" type="section" level="1" n="150"> <head xml:id="echoid-head162" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1325" xml:space="preserve">Primum omnium quodnam pondus rectè extollens de S ſuſpenſum co-<lb/>lumnam in dato ſitu fervet, videndum eſt: </s> <s xml:id="echoid-s1326" xml:space="preserve">invenitur autem, per 14 propoſitio-<lb/>nem, 4 ℔ eſſe: </s> <s xml:id="echoid-s1327" xml:space="preserve">deinde quæ ratio ſit perpendicularis lineæ, ut Z Æ, ad Z OE, <lb/>eſto autem 2 ad 1. </s> <s xml:id="echoid-s1328" xml:space="preserve">quare dico, quia 2 dant 1: </s> <s xml:id="echoid-s1329" xml:space="preserve">pondus rectè extollens 4 ℔, da-<lb/>bit Y 2 ℔. </s> <s xml:id="echoid-s1330" xml:space="preserve">illudq́ue quæſitum pondus eſſe affirmo.</s> <s xml:id="echoid-s1331" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1332" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1333" xml:space="preserve">Perpendicularis per S, nimirum AS, ducatur.</s> <s xml:id="echoid-s1334" xml:space="preserve"/> </p> <pb o="44" file="527.01.044" n="44" rhead="1 L*IBER* S*TATIC Æ*"/> </div> <div xml:id="echoid-div208" type="section" level="1" n="151"> <head xml:id="echoid-head163" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1335" xml:space="preserve">Quemadmodũ AS ad S OE: </s> <s xml:id="echoid-s1336" xml:space="preserve">ita rectè extol-<lb/> <anchor type="figure" xlink:label="fig-527.01.044-01a" xlink:href="fig-527.01.044-01"/> lens põdus ad Y extollĕs obliquè, per 20 propoſ. <lb/></s> <s xml:id="echoid-s1337" xml:space="preserve">Atqui triangulũ Æ OE Z triangulo OE S A ſimi-<lb/>le eſt, quorum homologa latera ſunt, OE Z cum <lb/>OE S, & </s> <s xml:id="echoid-s1338" xml:space="preserve">Z Æ, cum S A. </s> <s xml:id="echoid-s1339" xml:space="preserve">erit igitur quemadmo-<lb/>dum A S ad S OE: </s> <s xml:id="echoid-s1340" xml:space="preserve">ita Æ Z ad Z OE, & </s> <s xml:id="echoid-s1341" xml:space="preserve">con-<lb/>ſequenter quemadmodum Æ Z 2 ad Z OE 1: </s> <s xml:id="echoid-s1342" xml:space="preserve">ita <lb/>pondus rectè extollens 4 lib. </s> <s xml:id="echoid-s1343" xml:space="preserve">ad Y. </s> <s xml:id="echoid-s1344" xml:space="preserve">Innotuit igi-<lb/>tur Y 2 ℔ pendere, quod probandum ſuit. </s> <s xml:id="echoid-s1345" xml:space="preserve">Si-<lb/>militer in quibuſvis aliis exemplis proceditur.</s> <s xml:id="echoid-s1346" xml:space="preserve"/> </p> <div xml:id="echoid-div208" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.044-01" xlink:href="fig-527.01.044-01a"> <image file="527.01.044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.044-01"/> </figure> </div> <p> <s xml:id="echoid-s1347" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1348" xml:space="preserve">Datis igitur, notâ columnâ <lb/>punctis q́ue in axe ſirmo, &</s> <s xml:id="echoid-s1349" xml:space="preserve">c.</s> <s xml:id="echoid-s1350" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div210" type="section" level="1" n="152"> <head xml:id="echoid-head164" xml:space="preserve">1 NOTA.</head> <p style="it"> <s xml:id="echoid-s1351" xml:space="preserve">Etiamiſto pacto concludere licuiſſet: </s> <s xml:id="echoid-s1352" xml:space="preserve">AS 2 dat S OE 1: </s> <s xml:id="echoid-s1353" xml:space="preserve">ego pondus 4 ℔ rectè<unsure/> ex-<lb/>t<unsure/>ollens dabit γ 2 ℔. </s> <s xml:id="echoid-s1354" xml:space="preserve">Verum ut operatio ipſirei & </s> <s xml:id="echoid-s1355" xml:space="preserve">natur æ magis conformis ſit (intra <lb/>ſolidx<unsure/>m corpus enim AS & </s> <s xml:id="echoid-s1356" xml:space="preserve">S OE delineari nequeunt ) externam perpendicularen<unsure/>s <lb/>in exemplo pro internâ aſſumere placuit.</s> <s xml:id="echoid-s1357" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div211" type="section" level="1" n="153"> <head xml:id="echoid-head165" xml:space="preserve">2 NOTA.</head> <p style="it"> <s xml:id="echoid-s1358" xml:space="preserve">Quomodo autem terminus ignotus, ut pondus rectè extollens, linea tam rectè. </s> <s xml:id="echoid-s1359" xml:space="preserve">quàns <lb/>obliquè extollens, columna, datis tribus inversâ & </s> <s xml:id="echoid-s1360" xml:space="preserve">alternâ proportione innoteſcat, igno-<lb/>tum eſſe non poteſt, quapropter brevitati ſtudentes, omittemus.</s> <s xml:id="echoid-s1361" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div212" type="section" level="1" n="154"> <head xml:id="echoid-head166" xml:space="preserve">14 THE OREMA. 23 PROPOSITIO.</head> <p> <s xml:id="echoid-s1362" xml:space="preserve">Æqualia pondera ſuſpenſa de ductariis lineis, quæ ex <lb/>eodem axis puncto in contrarias partes ductę æquales cum <lb/>axe angulos faciunt: </s> <s xml:id="echoid-s1363" xml:space="preserve">in columnam æqualem vim poten-<lb/>tia mq́ue exercent.</s> <s xml:id="echoid-s1364" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1365" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1366" xml:space="preserve">A B columna, C D axis, E firmum, F mobile punctum eſto, <lb/>unde G obliquè extollens pondus dependeat, in ſuo ſitu columnam ſervans, <lb/>cujus obliqua linea F H. </s> <s xml:id="echoid-s1367" xml:space="preserve">Indi<unsure/>dem à puncto <lb/> <anchor type="figure" xlink:label="fig-527.01.044-02a" xlink:href="fig-527.01.044-02"/> ſcilicet F, & </s> <s xml:id="echoid-s1368" xml:space="preserve">pondus I itidem obliquum, <lb/>aliovorſum depĕdeat<unsure/>, ejuſdem cum G pon-<lb/>deris, cujus obliqua linea F K, æquans K F D <lb/>angulum H F C angulo.</s> <s xml:id="echoid-s1369" xml:space="preserve"/> </p> <div xml:id="echoid-div212" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.044-02" xlink:href="fig-527.01.044-02a"> <image file="527.01.044-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.044-02"/> </figure> </div> <p> <s xml:id="echoid-s1370" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1371" xml:space="preserve">Demonſtrandũ eſt pon-<lb/>deris I tantundĕ potentiæ eſſe in columnam <lb/>A B, quantum eſt ponderis G, id eſt, & </s> <s xml:id="echoid-s1372" xml:space="preserve">I <lb/>pondus (coërcito eſt amoto G) columnam <lb/>eodem in ſitu tenere.</s> <s xml:id="echoid-s1373" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1374" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1375" xml:space="preserve">Adidem punctum F, pondus L rectè extollens adda-<lb/>tur, quod non minus, in illo ſitu columnam ſuſpendat, cujus recta extol-<lb/>lens eſt F M.</s> <s xml:id="echoid-s1376" xml:space="preserve"/> </p> <pb o="45" file="527.01.045" n="45" rhead="*DE* S*TATIC Æ ELEMENTIS*."/> </div> <div xml:id="echoid-div214" type="section" level="1" n="155"> <head xml:id="echoid-head167" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1377" xml:space="preserve">Quoniam rectæ F H, F K inter eaſdem ſunt parallelas angulusq́ue HFC, <lb/>ex conceſſo, æqualis eſt angulo K F D, etiam F H & </s> <s xml:id="echoid-s1378" xml:space="preserve">F K æquales ſunt; </s> <s xml:id="echoid-s1379" xml:space="preserve">un-<lb/>de conſequens eſt, ita eſſe M F ad F K: </s> <s xml:id="echoid-s1380" xml:space="preserve">quemadmodum eſt M F ad F H. <lb/></s> <s xml:id="echoid-s1381" xml:space="preserve">Atqui quemadmodum eſt M F ad F H: </s> <s xml:id="echoid-s1382" xml:space="preserve">ita eſt L ad G: </s> <s xml:id="echoid-s1383" xml:space="preserve">ideoq́ue ut M F ad <lb/>F K: </s> <s xml:id="echoid-s1384" xml:space="preserve">ita L ad G. </s> <s xml:id="echoid-s1385" xml:space="preserve">I autĕ æquatur G extheſi. </s> <s xml:id="echoid-s1386" xml:space="preserve">itaque ut M Fad F K: </s> <s xml:id="echoid-s1387" xml:space="preserve">ita L ad I: </s> <s xml:id="echoid-s1388" xml:space="preserve"><lb/>quo poſito, etiam I columnamin eodem ſitu ſuſtinet. </s> <s xml:id="echoid-s1389" xml:space="preserve">per 20 propoſitionem. </s> <s xml:id="echoid-s1390" xml:space="preserve"><lb/>Conſimilis planè in quibuſvis aliis exemplis demonſtratio fuerit.</s> <s xml:id="echoid-s1391" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1392" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1393" xml:space="preserve">Æqualia pondera ſuſpenſa de ductariis lineis, quæ ex co-<lb/>dem axis puncto in contrarias partes ductæ æquales cum axe angulos faciunt: <lb/></s> <s xml:id="echoid-s1394" xml:space="preserve">in columnam æqualem vim potentiamq́ue exercent; </s> <s xml:id="echoid-s1395" xml:space="preserve">quod nobis erat demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s1396" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div215" type="section" level="1" n="156"> <head xml:id="echoid-head168" xml:space="preserve">15 THEOREMA. 24 PROPOSITIO.</head> <p> <s xml:id="echoid-s1397" xml:space="preserve">Potentia ponderis, cujus ductaria linea axi perpendi-<lb/>cularis eſt, in columnam dati ſitus omnium eſt maxima.</s> <s xml:id="echoid-s1398" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1399" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1400" xml:space="preserve">AB columna, CD axis, E firmum, F mobile punctum eſto, <lb/>eique G pondus obliquè extollens affigatur in ſitu columnam conſervans, <lb/>ut linea extollens H F horizonti obliqua axi C D ſit recta. </s> <s xml:id="echoid-s1401" xml:space="preserve">eidemq́; </s> <s xml:id="echoid-s1402" xml:space="preserve">F pondus <lb/>I obliquè extollens, æquali cum G pondere, obliquâ linea K F affigatur.</s> <s xml:id="echoid-s1403" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1404" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1405" xml:space="preserve">Demonſtrandum eſt ponderis G in columnam majo-<lb/>rem eſſe potentiam, quam ponderis I, eamq́ue potentiam omnium eſſe maxi-<lb/>mam. </s> <s xml:id="echoid-s1406" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1407" xml:space="preserve">A D punctum F pondus L rectè extollens ad-<lb/>figatur, quod columnam in ſitu ſuo retineat, cujus rectè extollens ſit F M.</s> <s xml:id="echoid-s1408" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div216" type="section" level="1" n="157"> <head xml:id="echoid-head169" xml:space="preserve">DEMONSTRATIO.</head> <p style="it"> <s xml:id="echoid-s1409" xml:space="preserve">A. </s> <s xml:id="echoid-s1410" xml:space="preserve">Quodcunque pondus extollens minorem rationem habet ad L, quam ſua linea <lb/>extollens ad F M, levius eſt quam ut columnam in ſuo ſitu detineat. </s> <s xml:id="echoid-s1411" xml:space="preserve">per <lb/>20 propoſit.</s> <s xml:id="echoid-s1412" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1413" xml:space="preserve">I. </s> <s xml:id="echoid-s1414" xml:space="preserve">Atqui I pondus extollens minorem rationem habet ad L, quam ſua linea K F <lb/>extollens ad F M.</s> <s xml:id="echoid-s1415" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1416" xml:space="preserve">I. </s> <s xml:id="echoid-s1417" xml:space="preserve">I pondus extollens igitur levius eſt, quam ut columnam in ſuo ſitu detineat.</s> <s xml:id="echoid-s1418" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1419" xml:space="preserve">Syllogiſmi aſſumptio ita approbatur. </s> <s xml:id="echoid-s1420" xml:space="preserve">Pondus G (quod columnam in ſuo <lb/>ſitu ſuſtinet) eam habet rationem ad L; </s> <s xml:id="echoid-s1421" xml:space="preserve">quam H F ad F M, atqui I æquatur <lb/>G, K F vero major eſt quam F H. </s> <s xml:id="echoid-s1422" xml:space="preserve">I igitur <lb/> <anchor type="figure" xlink:label="fig-527.01.045-01a" xlink:href="fig-527.01.045-01"/> minorem rationem habet, ad L: </s> <s xml:id="echoid-s1423" xml:space="preserve">quam K F <lb/>ad F M. </s> <s xml:id="echoid-s1424" xml:space="preserve">& </s> <s xml:id="echoid-s1425" xml:space="preserve">propter ea,<unsure/> ut paulo ante monui-<lb/>mus, I levius eſt, quàm ut columnam eo ſitu <lb/>ſuſtineat: </s> <s xml:id="echoid-s1426" xml:space="preserve">at G ſuſtinere poteſt, potentia igi-<lb/>t<unsure/>ur G major eſt, quam potentia I. </s> <s xml:id="echoid-s1427" xml:space="preserve">Poten-<lb/>@iam vero ponderis G majorem effe nõ poſſe <lb/>@n<unsure/>de cõſtat, quod ab F, ea quidem columnæ <lb/>parte, brevior linea quam F H ducinon poſ-<lb/>ſit, quandoquidem perpendicularis eſt.</s> <s xml:id="echoid-s1428" xml:space="preserve"/> </p> <div xml:id="echoid-div216" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.045-01" xlink:href="fig-527.01.045-01a"> <image file="527.01.045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.045-01"/> </figure> </div> <p> <s xml:id="echoid-s1429" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1430" xml:space="preserve">Si igitur ductaria linea axi perpendicularis eſt, maximam <lb/>potentiam in columnam dati ſitus habet, quod demonſtrandum fuit.</s> <s xml:id="echoid-s1431" xml:space="preserve"/> </p> <pb o="46" file="527.01.046" n="46" rhead="1 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div218" type="section" level="1" n="158"> <head xml:id="echoid-head170" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1432" xml:space="preserve">Hinc facile iſtud deducitur. </s> <s xml:id="echoid-s1433" xml:space="preserve">Quò anguli ductariarum linearum unde pon-<lb/>dera ſuſpenſa ſunt, recto angulo proximiores ſunt: </s> <s xml:id="echoid-s1434" xml:space="preserve">eò ponderum por<unsure/>étias eſſe <lb/>majores. </s> <s xml:id="echoid-s1435" xml:space="preserve">Et contra, quo longius indidem, id eſt ab angulo recto diffident, po-<lb/>tentias eorundem eò quoque minores eſſe.</s> <s xml:id="echoid-s1436" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div219" type="section" level="1" n="159"> <head xml:id="echoid-head171" xml:space="preserve">16 THEOREMA. 25 PROPOSITIO.</head> <p> <s xml:id="echoid-s1437" xml:space="preserve">Duæ<anchor type="note" xlink:href="" symbol="*"/> annuentes lineæ, unde columna dependet, infi- <anchor type="note" xlink:label="note-527.01.046-01a" xlink:href="note-527.01.046-01"/> nitùm continuatæ, in columnæ pendula gravitatis dia-<lb/>metro ſeſe interſecant.</s> <s xml:id="echoid-s1438" xml:space="preserve"/> </p> <div xml:id="echoid-div219" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.046-01" xlink:href="note-527.01.046-01a" xml:space="preserve">Inaæquali-<lb/>ter diftantes, <lb/>n<unsure/>on paral<unsure/>-<lb/>le@e<unsure/>.</note> </div> </div> <div xml:id="echoid-div221" type="section" level="1" n="160"> <head xml:id="echoid-head172" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s1439" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1440" xml:space="preserve">AB columna, ex duabus lineis annuentibus C D, E F, depen-<lb/>deto, quæ in G & </s> <s xml:id="echoid-s1441" xml:space="preserve">H continuatæ, ſeſe inutuò in I ſecant.</s> <s xml:id="echoid-s1442" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1443" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1444" xml:space="preserve">Punctum I in columnæ A B pendulâ gravitatis diame-<lb/>tro eſſe, demonſtrandum eſt.</s> <s xml:id="echoid-s1445" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div222" type="section" level="1" n="161"> <head xml:id="echoid-head173" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1446" xml:space="preserve">Anguli FEC, IEC, HEC unus idemq́; <lb/></s> <s xml:id="echoid-s1447" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.046-01a" xlink:href="fig-527.01.046-01"/> eſt angulus; </s> <s xml:id="echoid-s1448" xml:space="preserve">idem de DCE, ICE, GCE <lb/>judicum eſto, ut quicunq; </s> <s xml:id="echoid-s1449" xml:space="preserve">punctus in rectis <lb/>HE, GF pro extremo ſumptus fuerit, co-<lb/>lumna, ex eo datum ſitum ſervet. </s> <s xml:id="echoid-s1450" xml:space="preserve">Eſto autern<unsure/> <lb/>I extremus punctus, utriuſque lineæ com-<lb/>munis, ex illo igitur columna ſitum ſuum <lb/>retinebit. </s> <s xml:id="echoid-s1451" xml:space="preserve">Atqui columnâ ex I dependente, <lb/>perpendicularis per I columnæ pendula <lb/>gravitatis diametris fuerit, in qua eſt I.</s> <s xml:id="echoid-s1452" xml:space="preserve"/> </p> <div xml:id="echoid-div222" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.046-01" xlink:href="fig-527.01.046-01a"> <image file="527.01.046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.046-01"/> </figure> </div> </div> <div xml:id="echoid-div224" type="section" level="1" n="162"> <head xml:id="echoid-head174" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s1453" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1454" xml:space="preserve">AB columna ex duabus lineis annuentibus CD, EF depen-<lb/>deto, continuatis in G, H uſque, mutuò ſe in I ſecantibus.</s> <s xml:id="echoid-s1455" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1456" xml:space="preserve">Q*VAESITVM* Punctum I in columnæ AB pendulâ gravitatis diame-<lb/>tro eſſe, demonſtrandum eſt.</s> <s xml:id="echoid-s1457" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div225" type="section" level="1" n="163"> <head xml:id="echoid-head175" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1458" xml:space="preserve">DG, & </s> <s xml:id="echoid-s1459" xml:space="preserve">FH tibicines & </s> <s xml:id="echoid-s1460" xml:space="preserve">fulcra ſunto, vel <lb/> <anchor type="figure" xlink:label="fig-527.01.046-02a" xlink:href="fig-527.01.046-02"/> rigidæ & </s> <s xml:id="echoid-s1461" xml:space="preserve">inflexibiles lineæ, per 2 poſtulat. <lb/></s> <s xml:id="echoid-s1462" xml:space="preserve">quibus columna ſuffulcitur; </s> <s xml:id="echoid-s1463" xml:space="preserve">quarũ potentiæ <lb/>æquales ſunt potétiis CD. </s> <s xml:id="echoid-s1464" xml:space="preserve">DF, ut enim iftę, <lb/>ita etiam illæ columnam in ſuo ſitu ſuſtinent. </s> <s xml:id="echoid-s1465" xml:space="preserve"><lb/>Et quodcun que punctum in illis extremum <lb/>nobis fuerit, illud columnam in ſuo ſitu ſer-<lb/>vaverit. </s> <s xml:id="echoid-s1466" xml:space="preserve">Eſto autem I extremus & </s> <s xml:id="echoid-s1467" xml:space="preserve">utriuſque <lb/>lineæ communis punctus; </s> <s xml:id="echoid-s1468" xml:space="preserve">ex iſto igitur co-<lb/>lumna (Mathematicè intelligas) datum ſi-<lb/>rum retinet, & </s> <s xml:id="echoid-s1469" xml:space="preserve">pendula gravitatis diametrus <lb/>per I fuerit, & </s> <s xml:id="echoid-s1470" xml:space="preserve">in eâ punctum I.</s> <s xml:id="echoid-s1471" xml:space="preserve"/> </p> <div xml:id="echoid-div225" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.046-02" xlink:href="fig-527.01.046-02a"> <image file="527.01.046-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.046-02"/> </figure> </div> <pb o="47" file="527.01.047" n="47" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div227" type="section" level="1" n="164"> <head xml:id="echoid-head176" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s1472" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1473" xml:space="preserve">AB columna, lineis CD, & </s> <s xml:id="echoid-s1474" xml:space="preserve">EF, obliquè illâ demittente, hae <lb/>extollente in dato ſitu conſervetur, iſtæ autem continuatæ in I ſe interſecent.</s> <s xml:id="echoid-s1475" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1476" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1477" xml:space="preserve">Punctum I in columnæ AB pendulâ gravitatis diame-<lb/>tro eſſe, nobis demonſtrandum eſt.</s> <s xml:id="echoid-s1478" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div228" type="section" level="1" n="165"> <head xml:id="echoid-head177" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1479" xml:space="preserve">Primum GC tibicen, & </s> <s xml:id="echoid-s1480" xml:space="preserve">inflexibilis li-<lb/> <anchor type="figure" xlink:label="fig-527.01.047-01a" xlink:href="fig-527.01.047-01"/> nea, deinde potétia ducendi deorſum, quæ <lb/>erat in D, eſto deorium, contingenti inter <lb/>C & </s> <s xml:id="echoid-s1481" xml:space="preserve">G puncto, quoquo loco ſumatur. </s> <s xml:id="echoid-s1482" xml:space="preserve">Co-<lb/>lumna itaq; </s> <s xml:id="echoid-s1483" xml:space="preserve">AB quovis puncto inter GC <lb/>& </s> <s xml:id="echoid-s1484" xml:space="preserve">EH medio, quod pro extremo ſumitur, <lb/>datum ſitum ſervabit. </s> <s xml:id="echoid-s1485" xml:space="preserve">I autem extremus, <lb/>& </s> <s xml:id="echoid-s1486" xml:space="preserve">communis utriuſque lineę terminus fue-<lb/>rit; </s> <s xml:id="echoid-s1487" xml:space="preserve">columna itaq; </s> <s xml:id="echoid-s1488" xml:space="preserve">inde ſuſpenſa datũ ſitum <lb/>retinebit, & </s> <s xml:id="echoid-s1489" xml:space="preserve">propterea ipſius pendula gra-<lb/>vitatis linea per I erit: </s> <s xml:id="echoid-s1490" xml:space="preserve">& </s> <s xml:id="echoid-s1491" xml:space="preserve">I in ipsâ.</s> <s xml:id="echoid-s1492" xml:space="preserve"/> </p> <div xml:id="echoid-div228" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.047-01" xlink:href="fig-527.01.047-01a"> <image file="527.01.047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.047-01"/> </figure> </div> </div> <div xml:id="echoid-div230" type="section" level="1" n="166"> <head xml:id="echoid-head178" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s1493" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1494" xml:space="preserve">Columnam AB in fitu ſuo ſervent, hinc quidem CD obli-<lb/>què demittens, illinc EF obliquè extollens, quæ continuatæ interſecant ſe <lb/>in I. </s> <s xml:id="echoid-s1495" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1496" xml:space="preserve">I in columnæ pendulâ gravitatis diametro eſſe nobis <lb/>demonſtrandum eſt.</s> <s xml:id="echoid-s1497" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div231" type="section" level="1" n="167"> <head xml:id="echoid-head179" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1498" xml:space="preserve">HE pro tibicine nobis eſto, & </s> <s xml:id="echoid-s1499" xml:space="preserve">linea inflexibili, potentia quoque ſurſum <lb/>ducendi, quæ fuerat in E, eſto ſurſum contingenti inter C & </s> <s xml:id="echoid-s1500" xml:space="preserve">G puncto, <lb/>quoquo loco ſumatur. </s> <s xml:id="echoid-s1501" xml:space="preserve">Columna AB quo-<lb/>vis puncto inter CG, & </s> <s xml:id="echoid-s1502" xml:space="preserve">EH medio, quod <lb/> <anchor type="figure" xlink:label="fig-527.01.047-02a" xlink:href="fig-527.01.047-02"/> pro extremo ſumitur, datum fitũ retinebit. <lb/></s> <s xml:id="echoid-s1503" xml:space="preserve">I autem extremus, & </s> <s xml:id="echoid-s1504" xml:space="preserve">utriuſque lineæ com-<lb/>munis terminus fuerit, in quo datum ſitum <lb/>columna retinebit, atqui in eo quieſcĕte co-<lb/>lumnâ, pĕdula gravitatis diametrus eſt per I, <lb/>& </s> <s xml:id="echoid-s1505" xml:space="preserve">I in ipsâ diametro. </s> <s xml:id="echoid-s1506" xml:space="preserve">C*ONCLUSIO*. </s> <s xml:id="echoid-s1507" xml:space="preserve">Dua-<lb/>bus igitur lineis annuĕtibus, unde columna <lb/>dependet, in infinitum continuatis, in co-<lb/>lumnæ pendulâ gravitatis diametro ſeſe in-<lb/>terſecant, quod nobis demóſtrandum erat.</s> <s xml:id="echoid-s1508" xml:space="preserve"/> </p> <div xml:id="echoid-div231" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.047-02" xlink:href="fig-527.01.047-02a"> <image file="527.01.047-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.047-02"/> </figure> </div> </div> <div xml:id="echoid-div233" type="section" level="1" n="168"> <head xml:id="echoid-head180" xml:space="preserve">17 THEOREMA. 26 PROPOSITIO.</head> <p> <s xml:id="echoid-s1509" xml:space="preserve">Si duarum linearum unde columna dependet, alt<unsure/>era <lb/>horizontieſt perpendicularis, erit & </s> <s xml:id="echoid-s1510" xml:space="preserve">reliqua: </s> <s xml:id="echoid-s1511" xml:space="preserve">ſin obliqua, <pb o="48" file="527.01.048" n="48" rhead="1 L*IBER* S*TATICÆ*"/> obliqua. </s> <s xml:id="echoid-s1512" xml:space="preserve">Siilla huic <anchor type="note" xlink:href="" symbol="*"/> annuit; </s> <s xml:id="echoid-s1513" xml:space="preserve">annuet & </s> <s xml:id="echoid-s1514" xml:space="preserve">hæcilli: </s> <s xml:id="echoid-s1515" xml:space="preserve">ſin abnuit, <anchor type="note" xlink:label="note-527.01.048-01a" xlink:href="note-527.01.048-01"/> abnuet.</s> <s xml:id="echoid-s1516" xml:space="preserve"/> </p> <div xml:id="echoid-div233" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.048-01" xlink:href="note-527.01.048-01a" xml:space="preserve">Ann@@ere & <lb/>abnudere pa-<lb/>ralleliſm<unsure/>o op-<lb/>ponuntur.</note> </div> <p> <s xml:id="echoid-s1517" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1518" xml:space="preserve">AB columna duabus lineis eſto ſuſtentata, CD quidem ad <lb/>horizontem perpendiculari, EF vero (ſi poſſit) ad eandem obliquâ, & </s> <s xml:id="echoid-s1519" xml:space="preserve">GH <lb/>columnæ pendula gravitatis diametrus. </s> <s xml:id="echoid-s1520" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1521" xml:space="preserve">Veritas propofitio-<lb/>nis à nobis demonſtrari debet. </s> <s xml:id="echoid-s1522" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1523" xml:space="preserve">CD, & </s> <s xml:id="echoid-s1524" xml:space="preserve">EF infinitum <lb/>continuantor mutuo ſe in I interſecantes.</s> <s xml:id="echoid-s1525" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div235" type="section" level="1" n="169"> <head xml:id="echoid-head181" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1526" xml:space="preserve">Quocunque in ſitu columna de lineis CD, EF fuerit ſuſpenſa, eundem<unsure/> <lb/>in quovis puncto firmo continuatarũ linearum ſervabit, quod anguli I CE & </s> <s xml:id="echoid-s1527" xml:space="preserve"><lb/>IEC non mutentur. </s> <s xml:id="echoid-s1528" xml:space="preserve">Poſito igitur, I firmum pun-<lb/> <anchor type="figure" xlink:label="fig-527.01.048-01a" xlink:href="fig-527.01.048-01"/> ctum eſſe duarum linearum commune, columna <lb/>ex eo ſuſpenſa datum ſitum retinebit, & </s> <s xml:id="echoid-s1529" xml:space="preserve">IC pen-<lb/>dula gravitatis diametrus erit. </s> <s xml:id="echoid-s1530" xml:space="preserve">Atqui illud neuti-<lb/>quam fieri poteſt, quod GH iſti parallela, ea ipſa <lb/>ſit. </s> <s xml:id="echoid-s1531" xml:space="preserve">Eadem demonſtratio fuerit, rectà EF in oppo-<lb/>ſitam partem inclinatâ. </s> <s xml:id="echoid-s1532" xml:space="preserve">Si igitur IC horizonti eſt <lb/>perpendicularis reliqua EF eidem nó poteſt eſſe <lb/>obliqua, neceſſariò igitur perpendicularis etiam <lb/>fuerit: </s> <s xml:id="echoid-s1533" xml:space="preserve">Et per conſequens ſi EF horizonti eft <lb/>obliqua, etiam reliqua obliqua fuerit.</s> <s xml:id="echoid-s1534" xml:space="preserve"/> </p> <div xml:id="echoid-div235" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.048-01" xlink:href="fig-527.01.048-01a"> <image file="527.01.048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.048-01"/> </figure> </div> <p> <s xml:id="echoid-s1535" xml:space="preserve">Porro, quia EF, A verſus annuit, etiam reliquam, quæ in dato ſitu co-<lb/>lumnam ſervat, EF verſus annuere neceſſe eſt. </s> <s xml:id="echoid-s1536" xml:space="preserve">Ab ea enim (ſi poteſt) decli-<lb/>net, ut CK, ſecans EI continuatam in K, perpendicularis per K, ob dictam <lb/>cauſam, columnæ pendula gravitatis diametrus erit, quod magis abſurdũ eft, <lb/>quam eandem illam per I cadere. </s> <s xml:id="echoid-s1537" xml:space="preserve">Neque hic igitur reliqua linea, columnam <lb/>in dato ſitu ſervans, ab EF declinat, neque parallela eſt, quod antea demon-<lb/>ftratum eſt, in latum autem decedere manifeſtè abſurdum eſt. </s> <s xml:id="echoid-s1538" xml:space="preserve">Quapropter ne-<lb/>ceſſariò EF verſus annuit. </s> <s xml:id="echoid-s1539" xml:space="preserve">Si vero EF in oppoſitam partem abiret, conſimi-<lb/>liter etiam reliquam ab illa abire, & </s> <s xml:id="echoid-s1540" xml:space="preserve">abnuere demonſtrabitur.</s> <s xml:id="echoid-s1541" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1542" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1543" xml:space="preserve">Siigitur, &</s> <s xml:id="echoid-s1544" xml:space="preserve">c.</s> <s xml:id="echoid-s1545" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div237" type="section" level="1" n="170"> <head xml:id="echoid-head182" xml:space="preserve">18 PROBLEMA. 27 PROPOSITIO.</head> <p> <s xml:id="echoid-s1546" xml:space="preserve">Si columna, & </s> <s xml:id="echoid-s1547" xml:space="preserve">duo pondera obliquè extollentia ſitu <lb/>æquilibria ſunt, erit quemadmodum linea obliquè extol-<lb/> <anchor type="note" xlink:label="note-527.01.048-02a" xlink:href="note-527.01.048-02"/> lans, ad lineam rectè extollĕtem: </s> <s xml:id="echoid-s1548" xml:space="preserve">ita <anchor type="note" xlink:href="" symbol="*"/> ponderum quodq́ue obliquum ad ſuum pondus rectum.</s> <s xml:id="echoid-s1549" xml:space="preserve"/> </p> <div xml:id="echoid-div237" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.048-02" xlink:href="note-527.01.048-02a" xml:space="preserve">Pondus o<unsure/>bls-<lb/>quium & re-<lb/>ctum intelli-<lb/>ge rectè & <lb/>obliquè extol-<lb/>lens.</note> </div> <p> <s xml:id="echoid-s1550" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1551" xml:space="preserve">AB columna, CD ejus axis eſto, & </s> <s xml:id="echoid-s1552" xml:space="preserve">in iſto E, F puncta, quo-<lb/>rum pondera, columnam in dato ſitu detinentia, ſunt G, H, quidem obliquè, <lb/>I, K verò rectè extollentia, & </s> <s xml:id="echoid-s1553" xml:space="preserve">lineæ EL, FM obliquè, EN & </s> <s xml:id="echoid-s1554" xml:space="preserve">FO rectè ex-<lb/>tollentes. </s> <s xml:id="echoid-s1555" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1556" xml:space="preserve">Quemadmodum LE ad EN: </s> <s xml:id="echoid-s1557" xml:space="preserve">ita eſſe G ad I, <lb/>& </s> <s xml:id="echoid-s1558" xml:space="preserve">quemadmodum MF ad FO: </s> <s xml:id="echoid-s1559" xml:space="preserve">ita eſſe H ad K, demonſtrabimus.</s> <s xml:id="echoid-s1560" xml:space="preserve"/> </p> <pb o="49" file="527.01.049" n="49" rhead="*DE* S*TATICÆ ELEMENTIS*."/> </div> <div xml:id="echoid-div239" type="section" level="1" n="171"> <head xml:id="echoid-head183" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1561" xml:space="preserve">F firmum nobis eſto punctum, Eautem <lb/> <anchor type="figure" xlink:label="fig-527.01.049-01a" xlink:href="fig-527.01.049-01"/> mobile. </s> <s xml:id="echoid-s1562" xml:space="preserve">Itaque (per 20 propoſit.) </s> <s xml:id="echoid-s1563" xml:space="preserve">Quem-<lb/>admodum LE ad EN: </s> <s xml:id="echoid-s1564" xml:space="preserve">ita G ad I. </s> <s xml:id="echoid-s1565" xml:space="preserve">E <lb/>nunc firmum nobis eſt punctum, F mobile. <lb/></s> <s xml:id="echoid-s1566" xml:space="preserve">itaque (per dictam 20 propoſit.) </s> <s xml:id="echoid-s1567" xml:space="preserve">ut MF <lb/>ad FO: </s> <s xml:id="echoid-s1568" xml:space="preserve">ita H ad K.</s> <s xml:id="echoid-s1569" xml:space="preserve"/> </p> <div xml:id="echoid-div239" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.049-01" xlink:href="fig-527.01.049-01a"> <image file="527.01.049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.049-01"/> </figure> </div> <p> <s xml:id="echoid-s1570" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1571" xml:space="preserve">Igitur, ſi columna, & </s> <s xml:id="echoid-s1572" xml:space="preserve"><lb/>duo pondera obliquè extollentia ſitu æqui-<lb/>libria ſunt, erit quemadmodũ linea obliquè <lb/>extollens ad lineam rectè extollentem: </s> <s xml:id="echoid-s1573" xml:space="preserve">ita <lb/>quodque pondus obliquum ad ſuum pon-<lb/>dus rectum, quod nobis demonſtrandum <lb/>erat.</s> <s xml:id="echoid-s1574" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div241" type="section" level="1" n="172"> <head xml:id="echoid-head184" xml:space="preserve">C*ONSECTARIUM*.</head> <figure> <image file="527.01.049-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.049-02"/> </figure> <p> <s xml:id="echoid-s1575" xml:space="preserve">Cognitâ columnâ è duabus lineis non <lb/>parallelis ſuſpensâ, ut hîc juxtà vides, quan-<lb/>tum ponderis de quaque lineâ pendeat, <lb/>quantumve potentiæ quæque linea obti-<lb/>neat, innotel<unsure/>cere poſſe manifeſtum eſt.</s> <s xml:id="echoid-s1576" xml:space="preserve"/> </p> <figure> <image file="527.01.049-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.049-03"/> </figure> </div> <div xml:id="echoid-div242" type="section" level="1" n="173"> <head xml:id="echoid-head185" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s1577" xml:space="preserve">Pleriſque omnibus in propoſitionibus hujus libri, columnam nos loco exempli uſurpa-<lb/>viſſe, tanquam figuram ad propoſiti no ſtri declar ationem accommodatißimam, in axe <lb/>prœterea fixum & </s> <s xml:id="echoid-s1578" xml:space="preserve">mobile punctum effinxiſſe: </s> <s xml:id="echoid-s1579" xml:space="preserve">hac novißimâ tandem propoſitione ge-<lb/>neralem illarum veritatem, omnium{q́ue} figurarum, qualiacunque corpora, & </s> <s xml:id="echoid-s1580" xml:space="preserve">quocung <lb/>loco puncta, fixum & </s> <s xml:id="echoid-s1581" xml:space="preserve">mobile fuerint, communem eſſe oſtendere.</s> <s xml:id="echoid-s1582" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div243" type="section" level="1" n="174"> <head xml:id="echoid-head186" xml:space="preserve">19 THEOREMA. 28 PROPOSITIO.</head> <p> <s xml:id="echoid-s1583" xml:space="preserve">Quæcunque proportiones ſunt columnæ ad pondera <lb/>inde ſuſpenſa, ponderumq́; </s> <s xml:id="echoid-s1584" xml:space="preserve">lineas: </s> <s xml:id="echoid-s1585" xml:space="preserve">eaſdem cujuſvis etiam <lb/>corporis eſſe ad ſua pondera, conſimiliter inde pendentia, <lb/>ponderumq́ue lineas.</s> <s xml:id="echoid-s1586" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1587" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1588" xml:space="preserve">Exempli loco proportionem 20 propofitionis repetamus, iſto <lb/>pacto. </s> <s xml:id="echoid-s1589" xml:space="preserve">Columnæ AB, axis eſto CD, gravitatis centrum E, punctum fixum F, <pb o="50" file="527.01.050" n="50" rhead="1 L*IBER* S*TATICÆ*"/> mobile G, cuipondus H obliquè extollens <lb/> <anchor type="figure" xlink:label="fig-527.01.050-01a" xlink:href="fig-527.01.050-01"/> affixum ſuo in ſitu ſervat columnam, & </s> <s xml:id="echoid-s1590" xml:space="preserve">linea <lb/>obliquè extollens GI. </s> <s xml:id="echoid-s1591" xml:space="preserve">Pondus autem rectè <lb/>extollens K, quo columna ſuo in ſitu conſi-<lb/>militer retinetur, ejusq́ue rectè extollens li-<lb/>nea GL. </s> <s xml:id="echoid-s1592" xml:space="preserve">Dico igitur quemadmodum IG <lb/>ad GL: </s> <s xml:id="echoid-s1593" xml:space="preserve">ita H ad K.</s> <s xml:id="echoid-s1594" xml:space="preserve"/> </p> <div xml:id="echoid-div243" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.050-01" xlink:href="fig-527.01.050-01a"> <image file="527.01.050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.050-01"/> </figure> </div> <p> <s xml:id="echoid-s1595" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1596" xml:space="preserve">Demonſtrandum eſt, <lb/>proportionem iſtam, non ſolum in corpore <lb/>AB, quæ eſt columna: </s> <s xml:id="echoid-s1597" xml:space="preserve">ſed etiam in quoli-<lb/>bet corpore contingentis figuræ veram & </s> <s xml:id="echoid-s1598" xml:space="preserve">conſtantem eſſe.</s> <s xml:id="echoid-s1599" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div245" type="section" level="1" n="175"> <head xml:id="echoid-head187" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1600" xml:space="preserve">Lineis FG & </s> <s xml:id="echoid-s1601" xml:space="preserve">IL loco immotis, co-<lb/> <anchor type="figure" xlink:label="fig-527.01.050-02a" xlink:href="fig-527.01.050-02"/> lumna AB deorfum deducitor è ſuo gra-<lb/>vitatis centro E ſuſpenſa, quemadmodum, <lb/>hîc vides. </s> <s xml:id="echoid-s1602" xml:space="preserve">Iſta loci mutatio, ex 3 poſtulato, <lb/>aliam gravitatis & </s> <s xml:id="echoid-s1603" xml:space="preserve">ponderis contentionem <lb/>punctis FG non adfert, omniaq́ue ſitu <lb/>æquilibria manent, atque etiam nunc ut <lb/>GI ad GL: </s> <s xml:id="echoid-s1604" xml:space="preserve">ita H ad K.</s> <s xml:id="echoid-s1605" xml:space="preserve"/> </p> <div xml:id="echoid-div245" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.050-02" xlink:href="fig-527.01.050-02a"> <image file="527.01.050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.050-02"/> </figure> </div> <p> <s xml:id="echoid-s1606" xml:space="preserve">Figura columnæ, manente materiâ, in <lb/> <anchor type="figure" xlink:label="fig-527.01.050-03a" xlink:href="fig-527.01.050-03"/> aliam & </s> <s xml:id="echoid-s1607" xml:space="preserve">quidem irregularem transforme-<lb/>tur, qualem hîc juxta vides in AB, cujus <lb/>centrum gravitatis E, & </s> <s xml:id="echoid-s1608" xml:space="preserve">recta per illud <lb/>CD (quorum inventio in S*TATICES* <lb/>praxi Mechanicè non Mathematicè doce-<lb/>bitur) omnia ſitu æquilibria manent, <lb/>atque ut GI ad GL: </s> <s xml:id="echoid-s1609" xml:space="preserve">ita A ad K etiam <lb/>nunc eſt.</s> <s xml:id="echoid-s1610" xml:space="preserve"/> </p> <div xml:id="echoid-div246" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.050-03" xlink:href="fig-527.01.050-03a"> <image file="527.01.050-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.050-03"/> </figure> </div> <p> <s xml:id="echoid-s1611" xml:space="preserve">Corpus AB ſurſum reducatur, donec <lb/> <anchor type="figure" xlink:label="fig-527.01.050-04a" xlink:href="fig-527.01.050-04"/> FG in rectam CD incidat, ut ſitus ejus ſit, <lb/>quem vides, omnia ſitu manent ęquilibria. <lb/></s> <s xml:id="echoid-s1612" xml:space="preserve">Solidum enim AB five altius, ſive humi-<lb/>lius pendeat, per 3 poſtulatum, nihilo mi-<lb/>nus ejuſdem ponderis eſt, & </s> <s xml:id="echoid-s1613" xml:space="preserve">per cõſequens <lb/>etiam hic, ut IG ad GL ita H ad K. </s> <s xml:id="echoid-s1614" xml:space="preserve">Pro-<lb/>portio enim 20 propoſitionis non tantum <lb/>columnæ eſt, ſed cujuſlibet etiam corpo- <pb o="51" file="527.01.051" n="51" rhead="*DE* S*TATICÆ ELEMENTIS*."/> ris. </s> <s xml:id="echoid-s1615" xml:space="preserve">Atque conſimiliter cætera omnia, quæ de columnâ aliæ propoſitiones <lb/>præceperunt, demonſtrabuntur.</s> <s xml:id="echoid-s1616" xml:space="preserve"/> </p> <div xml:id="echoid-div247" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.050-04" xlink:href="fig-527.01.050-04a"> <image file="527.01.050-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.050-04"/> </figure> </div> <p> <s xml:id="echoid-s1617" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1618" xml:space="preserve">Quæcunq; </s> <s xml:id="echoid-s1619" xml:space="preserve">proportiones ſunt columnæ ad pondera inde <lb/>ſuſpenſa, ponderumq́ue lineas: </s> <s xml:id="echoid-s1620" xml:space="preserve">eædem ſunt corporis cujuſvis ad ſua pondera, <lb/>ponderumq́ue lineas conſimiliter dependentia.</s> <s xml:id="echoid-s1621" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div249" type="section" level="1" n="176"> <head xml:id="echoid-head188" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s1622" xml:space="preserve">Dato puncta F, G neceſſariò in C D non eſſe, ſed contingenti etiam loco <lb/>eſſe poſſe manifeſtum eſt, ut in extremo etiam, exempli gratia, corpore M N. <lb/></s> <s xml:id="echoid-s1623" xml:space="preserve">Lineâ enim I N continuata in rectam uſque C D, incidat autem in G: </s> <s xml:id="echoid-s1624" xml:space="preserve">per-<lb/>pendiculari etiam per M in rectam C D ductâ, cadat autem in F; </s> <s xml:id="echoid-s1625" xml:space="preserve">dicta pro-<lb/>portio (quemamodum I G ad G L: </s> <s xml:id="echoid-s1626" xml:space="preserve">ita H ad K) firma & </s> <s xml:id="echoid-s1627" xml:space="preserve">conſtans manet.</s> <s xml:id="echoid-s1628" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div250" type="section" level="1" n="177"> <head xml:id="echoid-head189" xml:space="preserve">FINIS</head> <head xml:id="echoid-head190" xml:space="preserve">LIBRI PRIMI.</head> <pb file="527.01.052" n="52"/> <pb file="527.01.053" n="53"/> </div> <div xml:id="echoid-div251" type="section" level="1" n="178"> <head xml:id="echoid-head191" xml:space="preserve">STATICES <lb/>LIBER SECVNDVS <lb/>QVI EST <lb/>DE INVENIENDO <lb/>GRAVITATIS CENTRO.</head> <pb file="527.01.054" n="54"/> <pb o="55" file="527.01.055" n="55"/> <p style="it"> <s xml:id="echoid-s1629" xml:space="preserve">PRimo quidem libro in ponderum affe-<lb/>ctionibus deſcribendis, ut institutæ do-<lb/>ctrinæ fidem faceremus, pro omnibus <lb/>unam columnam uſurpavimus, cujus <lb/>gravitatis centrum, vel gener ali noti-<lb/>tiâ, notum eſſe poteſt, in mult is tamen <lb/>aliis co poribus multò aliares eſt. </s> <s xml:id="echoid-s1630" xml:space="preserve">Brevi <lb/>& </s> <s xml:id="echoid-s1631" xml:space="preserve">gener alipræcepto, in omnibus mecha-<lb/>nicè reperiri poſſe verum equidem eſt, ut prima propoſit. <lb/></s> <s xml:id="echoid-s1632" xml:space="preserve">πζάξεως patebit, ſed Mathematicæ inventionis diſpar ratio <lb/>eſt, quam rem in planis Archimedes, in ſolidis verò Frede-<lb/>ricus Comandinus monumentis ſuis nobis prodiderunt. </s> <s xml:id="echoid-s1633" xml:space="preserve">Ad <lb/>utrunque (quia utriuſque ſpeciei idem principium, anteceden-<lb/>ti doctrinæ non inutile, conſequenti verò, tam H*YDROSTA-<lb/>TICÆ* quam S*TATIC Æ PRAXI* valde neceſſarium) noſtra <lb/>inventa adjunximus, omnia{q́ue} nostro more, & </s> <s xml:id="echoid-s1634" xml:space="preserve">methodo diſto-<lb/>nentes fecundum element orum librum conſcripſimus.</s> <s xml:id="echoid-s1635" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1636" xml:space="preserve">Definitiones Geometricarum figurarum, filector fortè de-<lb/>ſideras, it a habeto: </s> <s xml:id="echoid-s1637" xml:space="preserve">illas ipſas ex Geometrià, tanquam ex by-<lb/>potbeſinotas, à nobis aſſumi; </s> <s xml:id="echoid-s1638" xml:space="preserve">illud tantum monendus. </s> <s xml:id="echoid-s1639" xml:space="preserve">Para-<lb/>bolam, ſive Rectam coni ſectionem, *Bi<unsure/>anti<unsure/>nee*/ Conoi-<lb/>dale Rectangulum *Bi<unsure/>anber* vocabulo nobis vernaculo nos <lb/>appellaſſe, nominum autem etymologiam ab effectis eſſe, vis <lb/>enim istarum figur arum in accendendo, urendo{q́ue} potiſsimum <lb/>conſiſtit.</s> <s xml:id="echoid-s1640" xml:space="preserve"/> </p> <pb o="56" file="527.01.056" n="56"/> </div> <div xml:id="echoid-div252" type="section" level="1" n="179"> <head xml:id="echoid-head192" xml:space="preserve">DE INVENIENDO <lb/>GRAVITATIS CENTRO <lb/>IN PLANIS, PARS PRIOR.</head> <p> <s xml:id="echoid-s1641" xml:space="preserve">SI planis vel minimum pondusineſſet, illudq́ue ratio-<lb/>nem adipſorum magnitudinem habere cõcederetur, <lb/>de planorum ponderibus, ponderũ centris, diametris, & </s> <s xml:id="echoid-s1642" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s1643" xml:space="preserve">accuratè præcipi poſſet. </s> <s xml:id="echoid-s1644" xml:space="preserve">Quia vero nullum pondus plano <lb/>ineſt, neque gravitas igitur, neque gravitatis centrum, aut <lb/>diameter, propriè & </s> <s xml:id="echoid-s1645" xml:space="preserve">ſecundum naturam conſiderata. </s> <s xml:id="echoid-s1646" xml:space="preserve">Mo-<lb/>dificatè igitur, & </s> <s xml:id="echoid-s1647" xml:space="preserve">quidem metaphoricè, intelligenda ſint, <lb/>quaſiex theſi gravitas planis, pro ipſorum magnitudine, <lb/>ineſſet. </s> <s xml:id="echoid-s1648" xml:space="preserve">Falſum enim conceditur, ut verum inde adstruatur.</s> <s xml:id="echoid-s1649" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div253" type="section" level="1" n="180"> <head xml:id="echoid-head193" xml:space="preserve">1 THEOREMA. 1 PROPOSITIO.</head> <p> <s xml:id="echoid-s1650" xml:space="preserve">In omni plano figuræ centrum, gravitatis quoque cen-<lb/>trum eſt.</s> <s xml:id="echoid-s1651" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div254" type="section" level="1" n="181"> <head xml:id="echoid-head194" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s1652" xml:space="preserve">DATVM. </s> <s xml:id="echoid-s1653" xml:space="preserve">ABC triangulum æquilaterum eſto, & </s> <s xml:id="echoid-s1654" xml:space="preserve">figuræ centrum D.</s> <s xml:id="echoid-s1655" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1656" xml:space="preserve">QVAESITVM. </s> <s xml:id="echoid-s1657" xml:space="preserve">Idem D gravitatis quoque centrum eſſe trianguli A B C <lb/>demonſtrandum eſt. </s> <s xml:id="echoid-s1658" xml:space="preserve">PRAEPARATIO. </s> <s xml:id="echoid-s1659" xml:space="preserve">Ab angulo A recta A E in me-<lb/>dium latus B C, conſimiliter ab angulo C recta C F in medium latus A B <lb/>ducatur.</s> <s xml:id="echoid-s1660" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div255" type="section" level="1" n="182"> <head xml:id="echoid-head195" xml:space="preserve">DEMONSTRATIO.</head> <figure> <image file="527.01.056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.056-01"/> </figure> <p> <s xml:id="echoid-s1661" xml:space="preserve">Triangulo A B C lineâ A E ſuſpenſo, ſegmentum A E C <lb/>ſegmento A E B æquilibre erit, ſunt enim æqualia, ſimilia, <lb/>& </s> <s xml:id="echoid-s1662" xml:space="preserve">ſimiliter ſita: </s> <s xml:id="echoid-s1663" xml:space="preserve">quapropter A E gravitatis diameter eſt trian-<lb/>guli A B C. </s> <s xml:id="echoid-s1664" xml:space="preserve">Eademq́ue de causâ F C quoque ejuſdem trian-<lb/>guli gravitatis diameter fuerit. </s> <s xml:id="echoid-s1665" xml:space="preserve">Atqui iſtæ in figuræ centro <lb/>D ſeſe interſecant, quarum quæque gravitatis centrum in ſe <lb/>habet, illud ipſum igitur D fuerit.</s> <s xml:id="echoid-s1666" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div256" type="section" level="1" n="183"> <head xml:id="echoid-head196" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s1667" xml:space="preserve">A B C D Quadrangulum parallelogrammum eſto, & </s> <s xml:id="echoid-s1668" xml:space="preserve">figuræ centrum E.</s> <s xml:id="echoid-s1669" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1670" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1671" xml:space="preserve">E etiam gravitatis centrum eſſe demonſtrandum eſt.</s> <s xml:id="echoid-s1672" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1673" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1674" xml:space="preserve">Rectæ F G & </s> <s xml:id="echoid-s1675" xml:space="preserve">HI inter laterum A D & </s> <s xml:id="echoid-s1676" xml:space="preserve">B C, item <lb/>A B & </s> <s xml:id="echoid-s1677" xml:space="preserve">D C puncta media ducuntor.</s> <s xml:id="echoid-s1678" xml:space="preserve"/> </p> <pb o="57" file="527.01.057" n="57" rhead="DE INVENIENDO GRAVITATIS CENTRO."/> </div> <div xml:id="echoid-div257" type="section" level="1" n="184"> <head xml:id="echoid-head197" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1679" xml:space="preserve">Quadrangulo delineâ HI ſuſpenſo, ſe-<lb/> <anchor type="figure" xlink:label="fig-527.01.057-01a" xlink:href="fig-527.01.057-01"/> gmentum H I D A ſegmĕto H I C B æqui-<lb/>libre pendebit, quia æqualia ſunt, ſimilia, & </s> <s xml:id="echoid-s1680" xml:space="preserve"><lb/>ſimiliter ſita. </s> <s xml:id="echoid-s1681" xml:space="preserve">H I igitur in parallelogrammo <lb/>A B C D gravitatis diameter eſt, eandemq́; <lb/></s> <s xml:id="echoid-s1682" xml:space="preserve">ob cauſam & </s> <s xml:id="echoid-s1683" xml:space="preserve">F G. </s> <s xml:id="echoid-s1684" xml:space="preserve">Atqui iſtæ in E mutuo <lb/>ſe interſecantes gravitatis centrum in ſeſe <lb/>habent. </s> <s xml:id="echoid-s1685" xml:space="preserve">Quapropter E illud eſſe conclu-<lb/>ditur.</s> <s xml:id="echoid-s1686" xml:space="preserve"/> </p> <div xml:id="echoid-div257" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.057-01" xlink:href="fig-527.01.057-01a"> <image file="527.01.057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.057-01"/> </figure> </div> </div> <div xml:id="echoid-div259" type="section" level="1" n="185"> <head xml:id="echoid-head198" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s1687" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1688" xml:space="preserve">A B C D E ordinatum ſive circulo inſcriptũ quinquangulum <lb/>eſto, & </s> <s xml:id="echoid-s1689" xml:space="preserve">figuræ centrum F. </s> <s xml:id="echoid-s1690" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1691" xml:space="preserve">F gravitatis centrum quoq; </s> <s xml:id="echoid-s1692" xml:space="preserve">eſſe <lb/>demonſtrandum eſt. </s> <s xml:id="echoid-s1693" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1694" xml:space="preserve">Ab A in medium latus D C recta <lb/>A G; </s> <s xml:id="echoid-s1695" xml:space="preserve">conſimiliter à B in medium latus E D recta B H ducatur.</s> <s xml:id="echoid-s1696" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div260" type="section" level="1" n="186"> <head xml:id="echoid-head199" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1697" xml:space="preserve">Quinquangulo de A G ſuſpenſo, ſegmentũ A G D E ſegmento A G C B <lb/>æquilibre erit. </s> <s xml:id="echoid-s1698" xml:space="preserve">ſunt enim æqualia, ſimilia, & </s> <s xml:id="echoid-s1699" xml:space="preserve">ſimiliter ſi-<lb/> <anchor type="figure" xlink:label="fig-527.01.057-02a" xlink:href="fig-527.01.057-02"/> ta. </s> <s xml:id="echoid-s1700" xml:space="preserve">A G igitur nec non B H in codem quinquangulo <lb/>gravitatis diametereſt. </s> <s xml:id="echoid-s1701" xml:space="preserve">Atqui mutuò ſe in F figuræ cen-<lb/>tro interſecant, & </s> <s xml:id="echoid-s1702" xml:space="preserve">illarum quæq́ue gravitatis centrum in <lb/>ſe habet. </s> <s xml:id="echoid-s1703" xml:space="preserve">F igitur illud ipſum eſt. </s> <s xml:id="echoid-s1704" xml:space="preserve">Eadem demonſtratio <lb/>aliarum omnium fuerit, quæcunque figuræ, centrum <lb/>habebunt, cujuſmodi ſunt ſexangulum, Circulus, & </s> <s xml:id="echoid-s1705" xml:space="preserve">c.</s> <s xml:id="echoid-s1706" xml:space="preserve"/> </p> <div xml:id="echoid-div260" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.057-02" xlink:href="fig-527.01.057-02a"> <image file="527.01.057-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.057-02"/> </figure> </div> <p> <s xml:id="echoid-s1707" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1708" xml:space="preserve">In omni igitur plano figuræ cen-<lb/>trum, gravitatis quoque centrum eſt, quod nobis de-<lb/>monſtrandum fuit.</s> <s xml:id="echoid-s1709" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div262" type="section" level="1" n="187"> <head xml:id="echoid-head200" xml:space="preserve">2 THEOREMA. 2 PROPOSITIO.</head> <p> <s xml:id="echoid-s1710" xml:space="preserve">Trianguli cujusq́ue gravitatis centrum eſt in rectâ ab <lb/>angulo in oppoſitum latus medium ductâ.</s> <s xml:id="echoid-s1711" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1712" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1713" xml:space="preserve">A B C contingentis figuræ triangulum eſto, ab ejusq́ue angu-<lb/>lo, A in D medium oppoſiti lateris B C punctum, recta A D ducta.</s> <s xml:id="echoid-s1714" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1715" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1716" xml:space="preserve">Gravitatis centrum dati trianguli in rectâ A D eſſe, de-<lb/>monſtrandum eſt. </s> <s xml:id="echoid-s1717" xml:space="preserve">PRAEPARATIO. </s> <s xml:id="echoid-s1718" xml:space="preserve">Rectæ E F, G H, I K ad B C paral-<lb/>lelæ ducuntor, ſecantes A D in L, M, N. </s> <s xml:id="echoid-s1719" xml:space="preserve">ducuntor conſimiliter E O, G P, <lb/>I Q, K R, H S, F T ad A D parallelæ.</s> <s xml:id="echoid-s1720" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div263" type="section" level="1" n="188"> <head xml:id="echoid-head201" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1721" xml:space="preserve">Quandoquidem E F ad B C parallela eſt, idemq́ue E O & </s> <s xml:id="echoid-s1722" xml:space="preserve">F T ad L D, <lb/>quadrang ulum E F T O parallelogrammum erit, in quo E L, L F, O D & </s> <s xml:id="echoid-s1723" xml:space="preserve"><lb/>D T æqualia ſunt, ideoq́ue gravitatis centrum in D L per 1 hujus propoſit. <lb/></s> <s xml:id="echoid-s1724" xml:space="preserve">eandemq́ue ob cauſam parallelogrammi G H S P gravitatis centrum in L M.</s> <s xml:id="echoid-s1725" xml:space="preserve"> <pb o="58" file="527.01.058" n="58" rhead="2 LIBER STATICÆ"/> & </s> <s xml:id="echoid-s1726" xml:space="preserve">I K R Q in N M, & </s> <s xml:id="echoid-s1727" xml:space="preserve">per conſequens idem centrum figuræ I K R H S F T <lb/>O E P G Q, è tribus parallelogrammis compoſitæ, erit in recta N D vel A D. <lb/></s> <s xml:id="echoid-s1728" xml:space="preserve">Quemadmodum vero in dato triangulo tria quadrangula in-<lb/> <anchor type="figure" xlink:label="fig-527.01.058-01a" xlink:href="fig-527.01.058-01"/> ſcripta ſunt, ita infinita inſcribi poſlunt, & </s> <s xml:id="echoid-s1729" xml:space="preserve">inſcriptæ figuræ <lb/>gravitatis centrum nihilo minus, ob cauſas jam commemo-<lb/>ratas, in A D rectâ erit. </s> <s xml:id="echoid-s1730" xml:space="preserve">Verumenimvero quò plura quadran-<lb/>gula inſcribuntur, eo minor trianguli A B C ab inſcriptis <lb/>differentia fuerit. </s> <s xml:id="echoid-s1731" xml:space="preserve">Parallelis enim à latere A B per media ſe-<lb/>gmenta A N, N M, M L, L D. </s> <s xml:id="echoid-s1732" xml:space="preserve">ductis, differentia poſterio-<lb/>ris ſitus erit dimidium differentiæ prioris. </s> <s xml:id="echoid-s1733" xml:space="preserve">Quapropter infinita hujuſmodi <lb/>progreſſione, & </s> <s xml:id="echoid-s1734" xml:space="preserve">appropinquatione figura tandem invenietur, ut differentia in-<lb/>ter ipſam & </s> <s xml:id="echoid-s1735" xml:space="preserve">triangulum quovis plano, quantumvis minimo, minorſit. </s> <s xml:id="echoid-s1736" xml:space="preserve">Vnde <lb/>ſequitur, Si A D gravitatis diameter eſt, differentiã põderis ſegmenti A D C <lb/>à pondere ſegmenti A D B quovis plano, quantumvis minimo, minorem <lb/>eſſe. </s> <s xml:id="echoid-s1737" xml:space="preserve">Quare ſic argumentor.</s> <s xml:id="echoid-s1738" xml:space="preserve"/> </p> <div xml:id="echoid-div263" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.058-01" xlink:href="fig-527.01.058-01a"> <image file="527.01.058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.058-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s1739" xml:space="preserve">A. </s> <s xml:id="echoid-s1740" xml:space="preserve">Inæqualibus ponderibus aliquod pondus inveniri poteſt, quod ipſorum diffe-<lb/>rentiâ ſit minus.</s> <s xml:id="echoid-s1741" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1742" xml:space="preserve">O. </s> <s xml:id="echoid-s1743" xml:space="preserve">Atqui hiſce ponderibus A D C, A D B nullum pondus inveniri poteſt, <lb/>quod differentia ipſorum ſit minus.</s> <s xml:id="echoid-s1744" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1745" xml:space="preserve">O. </s> <s xml:id="echoid-s1746" xml:space="preserve">Ponder a igitur A D C, A D B non differunt.</s> <s xml:id="echoid-s1747" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1748" xml:space="preserve">Ideoq́ue A D gravitatis diameter eſt, in eaq́ue propterea etiam gravitatis <lb/>centrum trianguli A B C. </s> <s xml:id="echoid-s1749" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1750" xml:space="preserve">Cujusq́ue trianguli gravitatis <lb/>centrum eſt in rectâ, ab angulo in medium oppoſiti lateris punctum ductâ, <lb/>quod demonſtrari oportuit.</s> <s xml:id="echoid-s1751" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div265" type="section" level="1" n="189"> <head xml:id="echoid-head202" xml:space="preserve">1 PROBLEMA. 3 PROPOSITIO.</head> <p> <s xml:id="echoid-s1752" xml:space="preserve">Dato triangulo, gravitatis centrum invenire.</s> <s xml:id="echoid-s1753" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1754" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1755" xml:space="preserve">A B C triangulum eſto.</s> <s xml:id="echoid-s1756" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1757" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1758" xml:space="preserve">Centrum gravitatis inveniendum eſt.</s> <s xml:id="echoid-s1759" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div266" type="section" level="1" n="190"> <head xml:id="echoid-head203" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1760" xml:space="preserve">Ab A in medium B C recta A D ducatur, conſimiliter à C in medium <lb/>A B recta C E: </s> <s xml:id="echoid-s1761" xml:space="preserve">Gravitatis centrum F eſſe dico.</s> <s xml:id="echoid-s1762" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div267" type="section" level="1" n="191"> <head xml:id="echoid-head204" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1763" xml:space="preserve">Gravitatis centrum trianguli A B C eſt in re-<lb/> <anchor type="figure" xlink:label="fig-527.01.058-02a" xlink:href="fig-527.01.058-02"/> ctis A D & </s> <s xml:id="echoid-s1764" xml:space="preserve">C E per 2 propoſ. </s> <s xml:id="echoid-s1765" xml:space="preserve">quod demonſtran-<lb/>dum fuit.</s> <s xml:id="echoid-s1766" xml:space="preserve"/> </p> <div xml:id="echoid-div267" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.058-02" xlink:href="fig-527.01.058-02a"> <image file="527.01.058-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.058-02"/> </figure> </div> <p> <s xml:id="echoid-s1767" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1768" xml:space="preserve">Dato igitur triangulo, gravi-<lb/>tatis centrum invenimus, quod quærebatur.</s> <s xml:id="echoid-s1769" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div269" type="section" level="1" n="192"> <head xml:id="echoid-head205" xml:space="preserve">3 THEOREMA. 4 PROPOSITIO.</head> <p> <s xml:id="echoid-s1770" xml:space="preserve">Centrum gravitatis cujusq́ue trianguli, rectam ab an-<lb/>gulo in oppoſitum latus medium ita ſecat: </s> <s xml:id="echoid-s1771" xml:space="preserve">ut ſegmentum <lb/>interipſum & </s> <s xml:id="echoid-s1772" xml:space="preserve">angulum, duplum ſit reliqui.</s> <s xml:id="echoid-s1773" xml:space="preserve"/> </p> <pb o="59" file="527.01.059" n="59" rhead="DE INVENIENDO GRAVITATIS CENTRO."/> <p> <s xml:id="echoid-s1774" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1775" xml:space="preserve">Ab angulo B, trianguli A B C recta ducatur in D, medium <lb/>punctum oppoſiti lateris, conſimiliter & </s> <s xml:id="echoid-s1776" xml:space="preserve">à C recta in E punctum medium la-<lb/>teris A B, ſecans B D in F, gravitatis centro trianguli A B C.</s> <s xml:id="echoid-s1777" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1778" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1779" xml:space="preserve">C F ad F E duplum eſſe demonſtrandum eſt.</s> <s xml:id="echoid-s1780" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div270" type="section" level="1" n="193"> <head xml:id="echoid-head206" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1781" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> Subductâ ratione E B 1 ad B A 2, de ratione <anchor type="figure" xlink:label="fig-527.01.059-01a" xlink:href="fig-527.01.059-01"/> <anchor type="note" xlink:label="note-527.01.059-01a" xlink:href="note-527.01.059-01"/> D C 1 ad D A 1 (id eſt ratione {1/2} de ratione {1/1}) <lb/>C F ad FE reliqua eſt. </s> <s xml:id="echoid-s1782" xml:space="preserve">Atqui ratione {1/2} ſubductâ <lb/>de ratione {1/1} relinquitur ratio {2/1}. </s> <s xml:id="echoid-s1783" xml:space="preserve">C Figitur ad F E <lb/>eſt, ut 2 ad 1. </s> <s xml:id="echoid-s1784" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1785" xml:space="preserve">Gravitatis igitur <lb/>centrum in triangulo ita ſecat rectam ab angulo in <lb/>medium oppoſiti lateris, ut ſegmentũ inter ipſum <lb/>& </s> <s xml:id="echoid-s1786" xml:space="preserve">angulum ad reliquum duplum ſit, quod fuit demonſtrandum.</s> <s xml:id="echoid-s1787" xml:space="preserve"/> </p> <div xml:id="echoid-div270" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.059-01" xlink:href="fig-527.01.059-01a"> <image file="527.01.059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.059-01"/> </figure> <note symbol="*" position="right" xlink:label="note-527.01.059-01" xlink:href="note-527.01.059-01a" xml:space="preserve"> Per inver-<lb/>ſionĕ cap. 12. <lb/>Almag. Pto-<lb/>lem.</note> </div> </div> <div xml:id="echoid-div272" type="section" level="1" n="194"> <head xml:id="echoid-head207" xml:space="preserve">4 THEOREMA. 5 PROPOSITIO.</head> <p> <s xml:id="echoid-s1788" xml:space="preserve">Trianguli duorum laterum unumquoq; </s> <s xml:id="echoid-s1789" xml:space="preserve">in tria æqua-<lb/>lia ſegmenta partito: </s> <s xml:id="echoid-s1790" xml:space="preserve">recta per ſectionum puncta tertio la-<lb/>teri proxima, pergravitatis centrum eſt ducta.</s> <s xml:id="echoid-s1791" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1792" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1793" xml:space="preserve">A B C trianguli duo latera A B & </s> <s xml:id="echoid-s1794" xml:space="preserve">A C utrumq; </s> <s xml:id="echoid-s1795" xml:space="preserve">in tria æqua-<lb/>lia ſegmenta ſecta ſunto, illud punctis D, E, iſtud vero F, G. </s> <s xml:id="echoid-s1796" xml:space="preserve">perq́ue E, G, ter-<lb/>tio lateri B C proxima, recta E G ſit ducta.</s> <s xml:id="echoid-s1797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1798" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s1799" xml:space="preserve">E G per trianguli A B C gravitatis centrum eſſe, demon-<lb/>ſtrandum eſt. </s> <s xml:id="echoid-s1800" xml:space="preserve">P*ARASCEVE*. </s> <s xml:id="echoid-s1801" xml:space="preserve">Ab A in medium B C recta A H ducatur, <lb/>ſecans E G in I.</s> <s xml:id="echoid-s1802" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div273" type="section" level="1" n="195"> <head xml:id="echoid-head208" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1803" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> Quandoquidem ratio A E ad E B, eſt ratio A G ad G C recta E G ad <anchor type="figure" xlink:label="fig-527.01.059-02a" xlink:href="fig-527.01.059-02"/> <anchor type="note" xlink:label="note-527.01.059-02a" xlink:href="note-527.01.059-02"/> rectam B C parallela erit, item E I ad B H. </s> <s xml:id="echoid-s1804" xml:space="preserve">Quemadmo-<lb/>dum igitur A E ad E B: </s> <s xml:id="echoid-s1805" xml:space="preserve">ita A I ad I H, atqui A E ad E B <lb/>ex conceſſo, eſt dupla; </s> <s xml:id="echoid-s1806" xml:space="preserve">dupla igitur erit & </s> <s xml:id="echoid-s1807" xml:space="preserve">A I ad H I. </s> <s xml:id="echoid-s1808" xml:space="preserve">Quia <lb/>vero A I dupla eſt ad I H, I gravitatis centrum eſt triangu-<lb/>li A BC per 4 propoſit. </s> <s xml:id="echoid-s1809" xml:space="preserve">E G igitur per gravitatis centrum <lb/>eſt ducta. </s> <s xml:id="echoid-s1810" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1811" xml:space="preserve">Trianguli igitur duorum la-<lb/>terum unoquoque in tria æqualia ſegmenta partito: </s> <s xml:id="echoid-s1812" xml:space="preserve">recta <lb/>perſectionum puncta tertio lateri proxima, per gravitatis <lb/>centrum eſt ducta.</s> <s xml:id="echoid-s1813" xml:space="preserve"/> </p> <div xml:id="echoid-div273" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.059-02" xlink:href="fig-527.01.059-02a"> <image file="527.01.059-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.059-02"/> </figure> <note symbol="*" position="right" xlink:label="note-527.01.059-02" xlink:href="note-527.01.059-02a" xml:space="preserve">1. prop. 62 <lb/>lib. Eucl.</note> </div> </div> <div xml:id="echoid-div275" type="section" level="1" n="196"> <head xml:id="echoid-head209" xml:space="preserve">2 PROBLEMA. 6 PROPOSITIO.</head> <p> <s xml:id="echoid-s1814" xml:space="preserve">Dato plano rectilineo; </s> <s xml:id="echoid-s1815" xml:space="preserve">gravitatis centrum invenire.</s> <s xml:id="echoid-s1816" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div276" type="section" level="1" n="197"> <head xml:id="echoid-head210" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s1817" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1818" xml:space="preserve">A B C D inordinatum quadrangulum eſto.</s> <s xml:id="echoid-s1819" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1820" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s1821" xml:space="preserve">Gravitatis centrum inveniendum nobis eſt.</s> <s xml:id="echoid-s1822" xml:space="preserve"/> </p> <pb o="60" file="527.01.060" n="60" rhead="2 LIBER STATICÆ"/> </div> <div xml:id="echoid-div277" type="section" level="1" n="198"> <head xml:id="echoid-head211" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1823" xml:space="preserve">Quadrangulum rectâ A C in duo triangula dividun-<lb/>dum eſt, & </s> <s xml:id="echoid-s1824" xml:space="preserve">cujusq́ue gravitatis centrum, per 3 propoſit. </s> <s xml:id="echoid-s1825" xml:space="preserve">in-<lb/> <anchor type="figure" xlink:label="fig-527.01.060-01a" xlink:href="fig-527.01.060-01"/> veniendum: </s> <s xml:id="echoid-s1826" xml:space="preserve">trianguli A B C eſto E, A C D vero F, re-<lb/>ctaq́ue E F jugum <anchor type="note" xlink:href="" symbol="*"/>, hinc duo quadrãgula parallelogram- <anchor type="note" xlink:label="note-527.01.060-01a" xlink:href="note-527.01.060-01"/> ma æquæalta cõſtituenda, ut G H I K & </s> <s xml:id="echoid-s1827" xml:space="preserve">G H L M, æqua-<lb/>lia triangulis, illud quidem A C D hoc vero A C B, de-<lb/> <anchor type="note" xlink:label="note-527.01.060-02a" xlink:href="note-527.01.060-02"/> nique jugo F E in N ita ſecto, utradius N E ſit ad radium <lb/>N F: </s> <s xml:id="echoid-s1828" xml:space="preserve">quemadmodum HI, ad HL: </s> <s xml:id="echoid-s1829" xml:space="preserve">N quæſitum gravi-<lb/>tatis centrum eſſe dico.</s> <s xml:id="echoid-s1830" xml:space="preserve"/> </p> <div xml:id="echoid-div277" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.060-01" xlink:href="fig-527.01.060-01a"> <image file="527.01.060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.060-01"/> </figure> <note symbol="*" position="left" xlink:label="note-527.01.060-01" xlink:href="note-527.01.060-01a" xml:space="preserve">45 propoſ. <lb/>z. lib. Euclid.</note> <note position="left" xlink:label="note-527.01.060-02" xlink:href="note-527.01.060-02a" xml:space="preserve">10. propoſ. <lb/>G. lib. Euclid.</note> </div> </div> <div xml:id="echoid-div279" type="section" level="1" n="199"> <head xml:id="echoid-head212" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s1831" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1832" xml:space="preserve">A B C D F inordinatum quinquangulum eſto.</s> <s xml:id="echoid-s1833" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1834" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s1835" xml:space="preserve">Gravitatis centrum quærendum eſt.</s> <s xml:id="echoid-s1836" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div280" type="section" level="1" n="200"> <head xml:id="echoid-head213" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1837" xml:space="preserve">Ductâ A C, centrum gravitatis trianguli A C B, <lb/>per 1 propoſitionem, inveniendũ ſit vero F: </s> <s xml:id="echoid-s1838" xml:space="preserve">itemq́ue <lb/> <anchor type="figure" xlink:label="fig-527.01.060-02a" xlink:href="fig-527.01.060-02"/> quadranguli A C D E, per 1 hujus exemplum, ſitq́ue <lb/>G, & </s> <s xml:id="echoid-s1839" xml:space="preserve">recta F G jugum, hinc duo quadrangula paral-<lb/>lelogramma æquealta conſtituenda, ut H I K L & </s> <s xml:id="echoid-s1840" xml:space="preserve"><lb/>HIM N, quorum illud quadrangulo A C D E iſtud <lb/>vero triangulo A C B æquetur. </s> <s xml:id="echoid-s1841" xml:space="preserve">Denique jugo G F <lb/>diviſo in O, ut radius O F ſitad radium O G; </s> <s xml:id="echoid-s1842" xml:space="preserve">quem-<lb/>admodum I K ad IM: </s> <s xml:id="echoid-s1843" xml:space="preserve">O quæſitum gravitatis cen-<lb/>trum eſſe ajo.</s> <s xml:id="echoid-s1844" xml:space="preserve"/> </p> <div xml:id="echoid-div280" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.060-02" xlink:href="fig-527.01.060-02a"> <image file="527.01.060-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.060-02"/> </figure> </div> </div> <div xml:id="echoid-div282" type="section" level="1" n="201"> <head xml:id="echoid-head214" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s1845" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1846" xml:space="preserve">A B C D E F inordinatum ſexangulum eſto.</s> <s xml:id="echoid-s1847" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1848" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s1849" xml:space="preserve">Gravitatis centrum nobis quærendum eſt.</s> <s xml:id="echoid-s1850" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div283" type="section" level="1" n="202"> <head xml:id="echoid-head215" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1851" xml:space="preserve">Ductâ A C gravitatis centrum trianguli A C B, <lb/> <anchor type="figure" xlink:label="fig-527.01.060-03a" xlink:href="fig-527.01.060-03"/> per 3 propoſit. </s> <s xml:id="echoid-s1852" xml:space="preserve">quærendum, eſtq́ue G, nec non & </s> <s xml:id="echoid-s1853" xml:space="preserve"><lb/>quinquanguli A C D E F, per 2 exemplum hujus, <lb/>eſtq́ue H. </s> <s xml:id="echoid-s1854" xml:space="preserve">& </s> <s xml:id="echoid-s1855" xml:space="preserve">recta G H jugum. </s> <s xml:id="echoid-s1856" xml:space="preserve">Deinde duo quadran-<lb/>gula parallelogramma æquealta fabricanda ſunt, ut <lb/>I K L M æquale quinquãgulo A C D E F, & </s> <s xml:id="echoid-s1857" xml:space="preserve">I K N O <lb/>æquale triangulo A C B: </s> <s xml:id="echoid-s1858" xml:space="preserve">diviſoq́ue jugo H G in P, <lb/>ut P G radius illam rationem habe<unsure/> at ad P H radium; <lb/></s> <s xml:id="echoid-s1859" xml:space="preserve">quæ eſt K L ad K N: </s> <s xml:id="echoid-s1860" xml:space="preserve">P quæſitum gravitatis centrum <lb/>eſſe dico. </s> <s xml:id="echoid-s1861" xml:space="preserve">Atque iſta pragmatiæ ratio in reliquis mul-<lb/>tilateris planis planiſſime eadem eſt.</s> <s xml:id="echoid-s1862" xml:space="preserve"/> </p> <div xml:id="echoid-div283" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.060-03" xlink:href="fig-527.01.060-03a"> <image file="527.01.060-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.060-03"/> </figure> </div> </div> <div xml:id="echoid-div285" type="section" level="1" n="203"> <head xml:id="echoid-head216" xml:space="preserve">NOTATO</head> <p> <s xml:id="echoid-s1863" xml:space="preserve">Etſi hactenus exempla fuere, in quibus datum planum in quadrangula pa-<lb/>rallelogramma æquealta mutatur, poſſe tamen abſque transfiguratione hujuſ-<lb/>modirem expediri: </s> <s xml:id="echoid-s1864" xml:space="preserve">ejusq́ue rei nos varia exempla ſubjungere.</s> <s xml:id="echoid-s1865" xml:space="preserve"/> </p> <pb o="61" file="527.01.061" n="61" rhead="*DE* S*TATICÆ PRINCIPIIS*."/> </div> <div xml:id="echoid-div286" type="section" level="1" n="204"> <head xml:id="echoid-head217" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s1866" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1867" xml:space="preserve">ABCD irregulare & </s> <s xml:id="echoid-s1868" xml:space="preserve">inordinatum quadran-<lb/>gulum eſto. </s> <s xml:id="echoid-s1869" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1870" xml:space="preserve">Gravitatis centrum nobis l<unsure/>in-<lb/>veniendum eſt.</s> <s xml:id="echoid-s1871" xml:space="preserve"/> </p> <figure> <image file="527.01.061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.061-01"/> </figure> </div> <div xml:id="echoid-div287" type="section" level="1" n="205"> <head xml:id="echoid-head218" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1872" xml:space="preserve">Quadrangulum rectâ A C in duo triangulaſecto ipſorum <lb/>gravitatis centra, 3 propoſ. </s> <s xml:id="echoid-s1873" xml:space="preserve">adjumento, inveniuntor. </s> <s xml:id="echoid-s1874" xml:space="preserve">Trian-<lb/>guli A B C, eſto E; </s> <s xml:id="echoid-s1875" xml:space="preserve">A C D vero F; </s> <s xml:id="echoid-s1876" xml:space="preserve">recta denique E F jugum. </s> <s xml:id="echoid-s1877" xml:space="preserve">quo facto D G, <lb/>B H perpendiculares ducuntor in A C. </s> <s xml:id="echoid-s1878" xml:space="preserve">jugo F E, ſecto in I, ut radius I E, ſit <lb/>ad radium I F, quemadmodum D G ad B H; </s> <s xml:id="echoid-s1879" xml:space="preserve">I gravitatis centrum eſſe dico.</s> <s xml:id="echoid-s1880" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div288" type="section" level="1" n="206"> <head xml:id="echoid-head219" xml:space="preserve">5 Exemplum.</head> <p> <s xml:id="echoid-s1881" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1882" xml:space="preserve">ABCDE quinquangulum inordinatum eſto.</s> <s xml:id="echoid-s1883" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1884" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1885" xml:space="preserve">Gravitatis centrum inveniendum eſt.</s> <s xml:id="echoid-s1886" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div289" type="section" level="1" n="207"> <head xml:id="echoid-head220" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1887" xml:space="preserve">Quinquangulo duabus diagoniis AC, AD in tria triangula reſoluto, qua-<lb/>dranguli A C D E gravitatis centrum F per 4 propoſ. </s> <s xml:id="echoid-s1888" xml:space="preserve">& </s> <s xml:id="echoid-s1889" xml:space="preserve">trian-<lb/> <anchor type="figure" xlink:label="fig-527.01.061-02a" xlink:href="fig-527.01.061-02"/> guli A C B, G per 3 propoſ. </s> <s xml:id="echoid-s1890" xml:space="preserve">inveniuntor, quæ connectat ju-<lb/>gum F G; </s> <s xml:id="echoid-s1891" xml:space="preserve">tum B G in A C, CI & </s> <s xml:id="echoid-s1892" xml:space="preserve">E K in A D perpendicu-<lb/>lares ſunto, & </s> <s xml:id="echoid-s1893" xml:space="preserve">tribus rectis A D, A C, H B in eadem analogia <lb/>quarta inveniatur LM, deniqueſecato jugum F G in N ut <lb/>ratio ſegmentorum GN, NF eadem ſit quæ C I & </s> <s xml:id="echoid-s1894" xml:space="preserve">E K ad <lb/>ipſam L M. </s> <s xml:id="echoid-s1895" xml:space="preserve">N optatum gravitatis centrum eſſe dico.</s> <s xml:id="echoid-s1896" xml:space="preserve"/> </p> <div xml:id="echoid-div289" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.061-02" xlink:href="fig-527.01.061-02a"> <image file="527.01.061-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.061-02"/> </figure> </div> </div> <div xml:id="echoid-div291" type="section" level="1" n="208"> <head xml:id="echoid-head221" xml:space="preserve">6 Exemplum.</head> <p> <s xml:id="echoid-s1897" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1898" xml:space="preserve">ABCDEF inordinatum ſexangulum eſto.</s> <s xml:id="echoid-s1899" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1900" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1901" xml:space="preserve">Gravitatis centrum inveniendum eſt.</s> <s xml:id="echoid-s1902" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div292" type="section" level="1" n="209"> <head xml:id="echoid-head222" xml:space="preserve">PRAGMATIA.</head> <p> <s xml:id="echoid-s1903" xml:space="preserve">Sexangulum tribus diagoniis in quatuor triangula dirimito, & </s> <s xml:id="echoid-s1904" xml:space="preserve">quadrangu-<lb/>lorum ADCB, ADEF gravitatis centra G, H per 4 propoſ. </s> <s xml:id="echoid-s1905" xml:space="preserve">invenito, quæ <lb/>connectat jugum GH. </s> <s xml:id="echoid-s1906" xml:space="preserve">deinde in AC perpendiculares <lb/>demittuntor BI, DK. </s> <s xml:id="echoid-s1907" xml:space="preserve">ſimiliter AL, EM in FD, jam <lb/> <anchor type="figure" xlink:label="fig-527.01.061-03a" xlink:href="fig-527.01.061-03"/> tribus rectis quarum prima F D ſecunda AC, tertia <lb/>compoſita ex BI & </s> <s xml:id="echoid-s1908" xml:space="preserve">K D, in eâdem analogiâ invenito <lb/>quartam N O, tumq́ue jugum H G ſecato in P ut ratio <lb/>ſegmentorũ G P, P H eadem ſit quæ compoſitæ ex A L <lb/>& </s> <s xml:id="echoid-s1909" xml:space="preserve">E M ad ipſam N O. </s> <s xml:id="echoid-s1910" xml:space="preserve">Ajo P quæſitum eſſe gravi-<lb/>tatis centrum. </s> <s xml:id="echoid-s1911" xml:space="preserve">Atque ita deinceps in cæteris multangulis.</s> <s xml:id="echoid-s1912" xml:space="preserve"/> </p> <div xml:id="echoid-div292" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.061-03" xlink:href="fig-527.01.061-03a"> <image file="527.01.061-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.061-03"/> </figure> </div> </div> <div xml:id="echoid-div294" type="section" level="1" n="210"> <head xml:id="echoid-head223" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1913" xml:space="preserve">In primo exemplo eſtradius N E ad radium N F, ſicut H I ad H L, at ſic <lb/>quoque eſt parallelogrammum ut G H I K ad parallelogrammum G H L M; <lb/></s> <s xml:id="echoid-s1914" xml:space="preserve">& </s> <s xml:id="echoid-s1915" xml:space="preserve">æqueordinatè ut G H I K ad G H L M, hoc eſt per conſtructionem triangu-<lb/>lum A C D ad triangulum A C B ſicut N E ad N F. </s> <s xml:id="echoid-s1916" xml:space="preserve">Punctum igitur N <lb/>(per primam 1 lib. </s> <s xml:id="echoid-s1917" xml:space="preserve">propoſitionem) eſt expoſiti quadranguli gravitatis cen-<lb/>trum. </s> <s xml:id="echoid-s1918" xml:space="preserve">Simillima eritſecundi tertiiq́ue exempli demonſtratio.</s> <s xml:id="echoid-s1919" xml:space="preserve"/> </p> <pb o="62" file="527.01.062" n="62" rhead="2 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s1920" xml:space="preserve">Atquarti exempli demonſtratio pendet è proportione rectarum D G, H B <lb/>& </s> <s xml:id="echoid-s1921" xml:space="preserve">triangulorum A C D, A C B; </s> <s xml:id="echoid-s1922" xml:space="preserve">Et enim ut D G ad H B ſic erit, ſumpta com-<lb/>muni altitudine A C, rectangulum ſub D G in A C ad rectangulum ſub H B <lb/>in A C, hoc eſt (quia iſtorum dimidia ſunt) triangulum A C D ad triangu-<lb/>lum A C B.</s> <s xml:id="echoid-s1923" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1924" xml:space="preserve">Pari ratione quinti exempli demonſtratio, pendet ab analogia rectæ EK <lb/>cum I C ad L M & </s> <s xml:id="echoid-s1925" xml:space="preserve">quadranguli A C D E ad triangulum A C B. </s> <s xml:id="echoid-s1926" xml:space="preserve">Enimverò <lb/>cum L M ſit quarta in analogia rectarum A D, A C, H B rectangulum extre. <lb/></s> <s xml:id="echoid-s1927" xml:space="preserve">marum A D in L M æquatur mediarum rectangulo A C in H B. </s> <s xml:id="echoid-s1928" xml:space="preserve">Hinc tres <lb/>rectæ E K, I C, L M pro baſibus parallelogrammorum nobis ſunto, quarum <lb/>altitudo ſit eadem A D, ideoque ut E K & </s> <s xml:id="echoid-s1929" xml:space="preserve">C I ad L M ſic rectangula duo <lb/>E K in A D & </s> <s xml:id="echoid-s1930" xml:space="preserve">CI in A D ad L M in A D ſed duo illa rectangula ſunt dupli-<lb/>cia ſuorum triangulorũ hoc eſt quadranguli A E D C; </s> <s xml:id="echoid-s1931" xml:space="preserve">& </s> <s xml:id="echoid-s1932" xml:space="preserve">rectangulum L M in <lb/>A D duplum eſt trianguli A B C quia æquale eſt rectangulo A C in H D ut <lb/>ſupra jam patuit; </s> <s xml:id="echoid-s1933" xml:space="preserve">quamobrem erit quadrangulum A E D C ad triangulum <lb/>A B C ſicut E K & </s> <s xml:id="echoid-s1934" xml:space="preserve">I C ad B H ſed ſic quoque eſt propter conſtructionem, <lb/>G N ad N F quare N eſt optatum gravitatis centrum.</s> <s xml:id="echoid-s1935" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1936" xml:space="preserve">Sexti exempli demonſtratio huic affinis eſt. </s> <s xml:id="echoid-s1937" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1938" xml:space="preserve">Itaque dati <lb/>rectilinei cujuſcunque gravitatis centrum invenimus. </s> <s xml:id="echoid-s1939" xml:space="preserve">Quod oportuit.</s> <s xml:id="echoid-s1940" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div295" type="section" level="1" n="211"> <head xml:id="echoid-head224" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s1941" xml:space="preserve">Interim dum hæc pralo ſubjiciuntur nactus ſuns Federici Commandini Com-<lb/>mentarium in Archimedis quadraturam paraboles, ubi ad 6 propoſitionem recti-<lb/>lineorum gravitatis centrum invenire docet, modo ab horum utroque diverſo. </s> <s xml:id="echoid-s1942" xml:space="preserve">Siquis <lb/>cognoſcendi ſit cupidus ipſum conſulat.</s> <s xml:id="echoid-s1943" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div296" type="section" level="1" n="212"> <head xml:id="echoid-head225" xml:space="preserve">5 THEOREMA. 7 PROPOSITIO.</head> <p> <s xml:id="echoid-s1944" xml:space="preserve">Securiculæ gravitatis centrum eſt in recta laterum paral-<lb/>lelorum biſectionem connectente.</s> <s xml:id="echoid-s1945" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1946" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1947" xml:space="preserve">A B C D ſecuricula eſt qualem in Geometricis definivimus, <lb/>duobus lateribus A B, D C parallela, recta ab E biſegmento A B, connexa cum <lb/>F biſegmento ipſius D E. </s> <s xml:id="echoid-s1948" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s1949" xml:space="preserve">Quadrilateri A B C D gravitatis <lb/>centrum in jungente E F conſiſtere demonſtretur.</s> <s xml:id="echoid-s1950" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1951" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1952" xml:space="preserve">Tres rectæ D A, E F, C B, propter homologiam ſeg-<lb/>mentorum A E, E B, D F, F C eodem coïbunt in G.</s> <s xml:id="echoid-s1953" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div297" type="section" level="1" n="213"> <head xml:id="echoid-head226" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1954" xml:space="preserve">Triangulum G D C ſuſpenſum ex rectâ G F faciet ſegmen-<lb/> <anchor type="figure" xlink:label="fig-527.01.062-01a" xlink:href="fig-527.01.062-01"/> ta GFC, G F D per 2 propoſ. </s> <s xml:id="echoid-s1955" xml:space="preserve">ſitu ęquilibria; </s> <s xml:id="echoid-s1956" xml:space="preserve">ideoq́ue triangu-<lb/>li G D C gravitatis centrum in recta G F conſiſtet. </s> <s xml:id="echoid-s1957" xml:space="preserve">Sed G E B <lb/>triangulum triangulo G E A itidem ſitu æquilibre eſt, æqualia <lb/>igitur & </s> <s xml:id="echoid-s1958" xml:space="preserve">ſitu ęquilibria utrimque deducta relinquent quadran-<lb/>gula A E F D, E F B C quoque ſitu æquilibria, & </s> <s xml:id="echoid-s1959" xml:space="preserve">gravitatis <lb/>centrum in ipſa G E, neque tamen in ſegmento exteriore <lb/>E G, quamobrem in ipſâ E F. </s> <s xml:id="echoid-s1960" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1961" xml:space="preserve">Itaque ſecu-<lb/>riculæ gravitatis centrum eſtin rectâ parallelorum laterum bi-<lb/>ſectrice.</s> <s xml:id="echoid-s1962" xml:space="preserve"/> </p> <div xml:id="echoid-div297" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.062-01" xlink:href="fig-527.01.062-01a"> <image file="527.01.062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.062-01"/> </figure> </div> <pb o="63" file="527.01.063" n="63" rhead="*DE* S*TATICÆ PRINCIPIIS*."/> </div> <div xml:id="echoid-div299" type="section" level="1" n="214"> <head xml:id="echoid-head227" xml:space="preserve">6 THEOREMA. 8 PROPOSITIO.</head> <p> <s xml:id="echoid-s1963" xml:space="preserve">Securiculæ gravitatis centrum rectam parallelorum la-<lb/>terum biſectricen<unsure/> ita ſecat, ut ſegmentum biſectricis mi-<lb/>nori latericon terminum ad reliquum ſit, ut majoris pa-<lb/>ralleli lateris duplum minore auctum, ad duplum mino-<lb/>ris cum majore.</s> <s xml:id="echoid-s1964" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1965" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s1966" xml:space="preserve">Latera A B, D C, ſecuriculæ A B C D parallela ſunto, biſectrix <lb/>E F, & </s> <s xml:id="echoid-s1967" xml:space="preserve">gravitatis centrum G, Q*VAESITVM*. </s> <s xml:id="echoid-s1968" xml:space="preserve">Duplam D C auctam ipſa <lb/>A B, & </s> <s xml:id="echoid-s1969" xml:space="preserve">Duplam A B cum D C, ſegmentis G E, G F proportionales eſſe de-<lb/>monſtrandum eſto. </s> <s xml:id="echoid-s1970" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s1971" xml:space="preserve">Diagonia D B tripartito dividatur <lb/>in punctis H, I, parallelæ ab his terminis K L, M N, contra latus D C inter-<lb/>ſecent E F in O & </s> <s xml:id="echoid-s1972" xml:space="preserve">P. </s> <s xml:id="echoid-s1973" xml:space="preserve">Deniqueacta E D interſecet M I in Q; </s> <s xml:id="echoid-s1974" xml:space="preserve">F B verò ipſam <lb/>K L in puncto R, atque harum interſectionum puncta connectat Q R.</s> <s xml:id="echoid-s1975" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div300" type="section" level="1" n="215"> <head xml:id="echoid-head228" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s1976" xml:space="preserve">Quandoquidem centrum gravitatis trianguli B D C per 2 propoſ. </s> <s xml:id="echoid-s1977" xml:space="preserve">eſt in <lb/>recta B F, & </s> <s xml:id="echoid-s1978" xml:space="preserve">per 5 propoſ. </s> <s xml:id="echoid-s1979" xml:space="preserve">etiam in recta <lb/>K L, centrum erit in concurſu R, eâdem ra-<lb/>tione Q erit centrum gravitatis trianguli <lb/> <anchor type="figure" xlink:label="fig-527.01.063-01a" xlink:href="fig-527.01.063-01"/> A B D. </s> <s xml:id="echoid-s1980" xml:space="preserve">Quamobrem Q R horum triangu-<lb/>lorum jugum crit, in quo utriuſq; </s> <s xml:id="echoid-s1981" xml:space="preserve">ſeu quod <lb/>idem eſt ſecuriculæ A B C D gravitatis cen-<lb/>trum conſiſtit, ſed idem per propoſ. </s> <s xml:id="echoid-s1982" xml:space="preserve">7 quoq; <lb/></s> <s xml:id="echoid-s1983" xml:space="preserve">eſtin F E; </s> <s xml:id="echoid-s1984" xml:space="preserve">Itaque G centrum gravitatis erit <lb/>quadranguli A B C D. </s> <s xml:id="echoid-s1985" xml:space="preserve">Triangula autem <lb/>C D B, A B D, intra eaſdem parallelas ex <lb/>hypotheſi cõſiſtentia, erunt ut baſes, hoc eſt, D C ad A B, ut C D B ad A B D, <lb/>ſed ſic per 1 propoſ. </s> <s xml:id="echoid-s1986" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s1987" xml:space="preserve">radius G Q ad G R, atque ita P G ad G O (quia <lb/>clauduntur parallelis M N, K L) omiſſis itaque mediis, ut D C ad A B ſic <lb/>G P ad G O. </s> <s xml:id="echoid-s1988" xml:space="preserve">Ideoq́ue (per 15, 16 & </s> <s xml:id="echoid-s1989" xml:space="preserve">24 propoſ. </s> <s xml:id="echoid-s1990" xml:space="preserve">5. </s> <s xml:id="echoid-s1991" xml:space="preserve">lib. </s> <s xml:id="echoid-s1992" xml:space="preserve">Euclid.) </s> <s xml:id="echoid-s1993" xml:space="preserve">ut dupla D C <lb/>cum A B, ad duplam A B auctam ipſa D C, ſic dupla G O, aucta G P, ad du-<lb/>plam G P plus ipſa G O. </s> <s xml:id="echoid-s1994" xml:space="preserve">Verum G E æquatur duplici G P cum G O; </s> <s xml:id="echoid-s1995" xml:space="preserve">Et <lb/>G F item duplici G O plus G P. </s> <s xml:id="echoid-s1996" xml:space="preserve">Quamobrem ut D C bis plus A B, ad A B <lb/>bis plus D C, ſic G E ad G F. </s> <s xml:id="echoid-s1997" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s1998" xml:space="preserve">Itaque ſecuriculæ gravi-<lb/>tatis centrum, &</s> <s xml:id="echoid-s1999" xml:space="preserve">c.</s> <s xml:id="echoid-s2000" xml:space="preserve"/> </p> <div xml:id="echoid-div300" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.063-01" xlink:href="fig-527.01.063-01a"> <image file="527.01.063-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.063-01"/> </figure> </div> </div> <div xml:id="echoid-div302" type="section" level="1" n="216"> <head xml:id="echoid-head229" xml:space="preserve">3 PROBLEMA. 9 PROPOSITIO.</head> <p> <s xml:id="echoid-s2001" xml:space="preserve">Dato cum totius plani, tum ſegmenti cujus ad reliquum <lb/>ratio ſit nota, gravitatis centro; </s> <s xml:id="echoid-s2002" xml:space="preserve">ejuſdem reliqui centrum <lb/>invenire.</s> <s xml:id="echoid-s2003" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div303" type="section" level="1" n="217"> <head xml:id="echoid-head230" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s2004" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2005" xml:space="preserve">Rectilinei plani A B C D gravitatis centrum E, ſegmenti verò <lb/>B D A, F centrum eſto. </s> <s xml:id="echoid-s2006" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2007" xml:space="preserve">Reliqui ſegmenti B D C gravita-<lb/>tis centrum invenire.</s> <s xml:id="echoid-s2008" xml:space="preserve"/> </p> <pb o="64" file="527.01.064" n="64" rhead="2 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div304" type="section" level="1" n="218"> <head xml:id="echoid-head231" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2009" xml:space="preserve">Continuator F E in G, ita ut ratio F E ad E G, ſit eadem rationi ſegmen-<lb/>ti B D C ad ſegmentum B D A; </s> <s xml:id="echoid-s2010" xml:space="preserve">ajo G reliqui ſegmenti B D C optatum eſſe <lb/>gravitatis centrum.</s> <s xml:id="echoid-s2011" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div305" type="section" level="1" n="219"> <head xml:id="echoid-head232" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2012" xml:space="preserve">Cum F centrum ſit B D A, & </s> <s xml:id="echoid-s2013" xml:space="preserve">E totius A B C D, reliqui ſegmenti centrum <lb/>erit in F E infinitum continuata. </s> <s xml:id="echoid-s2014" xml:space="preserve">(Secus enim, ſi fieri poſsit, cadat extra in H, <lb/>totius igitur rectilinei gravitatis centrum conſiſteret <lb/>in recta F H, quod tamen theſi repugnat, nam ſta-<lb/> <anchor type="figure" xlink:label="fig-527.01.064-01a" xlink:href="fig-527.01.064-01"/> tuitur in E) quamobrem inquam cum ſit in ipſa F E <lb/>infinitum continuata autultra aut citra G, E verſum <lb/>cadet, ſi citra ceciderit ut in I ratio lõgioris radii E F, <lb/>ad breviorem E I, major fuerit, quam gravitatis pon-<lb/>deroſioris B C D ad leviorem B A D contra 1 pro-<lb/>poſ. </s> <s xml:id="echoid-s2015" xml:space="preserve">lib. </s> <s xml:id="echoid-s2016" xml:space="preserve">1 quamobrem citra G, E verſus non cadet: </s> <s xml:id="echoid-s2017" xml:space="preserve">neque ultra G quod ſi-<lb/>millima ratione evincetur. </s> <s xml:id="echoid-s2018" xml:space="preserve">Neceſſariò itaque in puncto G. </s> <s xml:id="echoid-s2019" xml:space="preserve">Quod demon-<lb/>ſtrari oportuit.</s> <s xml:id="echoid-s2020" xml:space="preserve"/> </p> <div xml:id="echoid-div305" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.064-01" xlink:href="fig-527.01.064-01a"> <image file="527.01.064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.064-01"/> </figure> </div> </div> <div xml:id="echoid-div307" type="section" level="1" n="220"> <head xml:id="echoid-head233" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s2021" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2022" xml:space="preserve">Circuli A B C D ſemidiameter eſt E A, E centrum gravitatis, <lb/>circelli A F G H eidem inſcripti gravitatis centrum I, diameter A G.</s> <s xml:id="echoid-s2023" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2024" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2025" xml:space="preserve">Reliqui ſegmenti A B C D H G F gravitatis centrum in-<lb/>venire.</s> <s xml:id="echoid-s2026" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div308" type="section" level="1" n="221"> <head xml:id="echoid-head234" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2027" xml:space="preserve">Continuator I E in K, ut I E ad continuationem E K habeat rationem <lb/>quam ſpatium A B C D H G F ad circulum A F G H; </s> <s xml:id="echoid-s2028" xml:space="preserve">ajo K eſſe optatum gra-<lb/>vitatis centrum, cujus demon-<lb/>ſtratio ſimillima ſuperiori. </s> <s xml:id="echoid-s2029" xml:space="preserve">Ve-<lb/> <anchor type="figure" xlink:label="fig-527.01.064-02a" xlink:href="fig-527.01.064-02"/> rùm quô arbeli hujus ad reli-<lb/>quum circulum ratio ad rectas li-<lb/>neas revocetur, ſic ages; </s> <s xml:id="echoid-s2030" xml:space="preserve">Si inſcri-<lb/>ptæ C L diametro A G æqualis <lb/>terminum L cum reliquo dia-<lb/>metri termino C connectat adia-<lb/>metrum A L, & </s> <s xml:id="echoid-s2031" xml:space="preserve">rectis A L, L C <lb/>diametro & </s> <s xml:id="echoid-s2032" xml:space="preserve">inter ſe conterminis <lb/>tertia proportionalis ſit M, ratio <lb/>ſpatii ad circulum AFGH (cùm <lb/>A L C angulus in ſemicirculo ſit <lb/>rectus) erit eadem quæ primæ re-<lb/>ctæ A L ad tertiam M, & </s> <s xml:id="echoid-s2033" xml:space="preserve">circulus <lb/>diametri A L, ſpatio dicto æqua-<lb/>lis, nam A L ad M ratio eſt dupli-<lb/>cata A L ad L C, hoc eſtad A G.</s> <s xml:id="echoid-s2034" xml:space="preserve"/> </p> <div xml:id="echoid-div308" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.064-02" xlink:href="fig-527.01.064-02a"> <image file="527.01.064-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.064-02"/> </figure> </div> <p> <s xml:id="echoid-s2035" xml:space="preserve">Eadem planè ratio fuerit ſi plures circelli ex integro A B C D forent exem-<lb/>pti; </s> <s xml:id="echoid-s2036" xml:space="preserve">dicis gatia, deſit præterea circulus N O, cujus centrum erat P. </s> <s xml:id="echoid-s2037" xml:space="preserve">Continue-<lb/>tur P K centra connectens ad Q uſque ut P K ad K Q ſit quemadmodum re-<lb/>liquum ad circulum N O. </s> <s xml:id="echoid-s2038" xml:space="preserve">Quare erit optatum gravitatis centrum, atque ita <pb o="65" file="527.01.065" n="65" rhead="DE INVENTIONE GRAVITATIS CENTRO."/> deinceps in cæteris ſimili machinatione, quorum ſegmentorum ratio per ar-<lb/>tem cognoſci poſsit. </s> <s xml:id="echoid-s2039" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2040" xml:space="preserve">Quamobrem datis planę ſuperficiæ <lb/>& </s> <s xml:id="echoid-s2041" xml:space="preserve">ſegmenti ejuſdem gravitatis centris & </s> <s xml:id="echoid-s2042" xml:space="preserve">C.</s> <s xml:id="echoid-s2043" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div310" type="section" level="1" n="222"> <head xml:id="echoid-head235" xml:space="preserve">7 THEOREMA. 10 PROPOSITIO.</head> <p> <s xml:id="echoid-s2044" xml:space="preserve">Parabo.</s> <s xml:id="echoid-s2045" xml:space="preserve">æ gravitatis centrum eſt in diametro.</s> <s xml:id="echoid-s2046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2047" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2048" xml:space="preserve">Parabola A B C, axis A D. </s> <s xml:id="echoid-s2049" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s2050" xml:space="preserve">Centrum gravita-<lb/>tis in A D conſiltere demonſtrato. </s> <s xml:id="echoid-s2051" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2052" xml:space="preserve">E F, G H, I K baſi <lb/>B C parallelæ interſecent diametrum A D in punctis L, M, N, & </s> <s xml:id="echoid-s2053" xml:space="preserve">eædem in-<lb/>tercipiant rectas E O, G P, I Q, K R, H S, F T axi A D parallelas.</s> <s xml:id="echoid-s2054" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div311" type="section" level="1" n="223"> <head xml:id="echoid-head236" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2055" xml:space="preserve">Cum enim parallelæ E F, B C, claudantur E O, F T, parallelis, E F T O <lb/>parallelogrammum erit, cujus oppoſita latera E F, O T in L & </s> <s xml:id="echoid-s2056" xml:space="preserve">D bifariam <lb/>dividuntur, quare centrum gravitatis per 1 propoſ. </s> <s xml:id="echoid-s2057" xml:space="preserve">in L D conſiſtet. </s> <s xml:id="echoid-s2058" xml:space="preserve">Eadem ra-<lb/>tione centrũ gravitatis quadranguli G H S P erit in L M, itemq́; </s> <s xml:id="echoid-s2059" xml:space="preserve">ipſius IKR Q <lb/>in M N. </s> <s xml:id="echoid-s2060" xml:space="preserve">Quamobrem gravitatis centrum rectilinei I K R H S F T O E P G Q <lb/>è tribus iſtis parallelogrammis cõflati in DN, <lb/> <anchor type="figure" xlink:label="fig-527.01.065-01a" xlink:href="fig-527.01.065-01"/> ſeu D A conſiſtet. </s> <s xml:id="echoid-s2061" xml:space="preserve">Sed quò frequentiora hu-<lb/>juſmodi parallelogramma in parabolam in-<lb/>ſcribuntur, eò minor erit inſcriptæ figuræ à <lb/>parabola defectus. </s> <s xml:id="echoid-s2062" xml:space="preserve">Quamobrem infinita hac <lb/>parallelogrammorum inſcriptione eo adſcen-<lb/>ditur ut ejus à parabola defectus quacunque <lb/>minima propoſita ſuperficie minor ſit, conſe-<lb/>quens igitur eſt, ſumpta A D gravitatis dia-<lb/>metro, æquilibritatem ſitus ſtgmenti A D C <lb/>ab æquilibritate ſitus ſegmenti A D B, mi-<lb/>nori intervallo abeſſe quam vel minimæ quæ <lb/>dari poſſit ſuperficiei planę differentia: </s> <s xml:id="echoid-s2063" xml:space="preserve">unde <lb/>concludo.</s> <s xml:id="echoid-s2064" xml:space="preserve"/> </p> <div xml:id="echoid-div311" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.065-01" xlink:href="fig-527.01.065-01a"> <image file="527.01.065-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.065-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s2065" xml:space="preserve">Ponderum inaqualium ſitu gravium differentiâ minus pondus exhiberi poteſt: <lb/></s> <s xml:id="echoid-s2066" xml:space="preserve">Atqui ponderum horum A D C, A B D ſitu gravium differentiâ pondus minus <lb/>exhiberinullum poteſt.</s> <s xml:id="echoid-s2067" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2068" xml:space="preserve">Ponderum igitur A D C, A B D ſitu gravium differentia nulla eſt.</s> <s xml:id="echoid-s2069" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2070" xml:space="preserve">A D igitur crit diameter gravitatis, & </s> <s xml:id="echoid-s2071" xml:space="preserve">propterea parabolæ A B C gravitatis <lb/>centrum in ipſa. </s> <s xml:id="echoid-s2072" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2073" xml:space="preserve">Itaque paraboles gravitatis centrum eſt in <lb/>diametro. </s> <s xml:id="echoid-s2074" xml:space="preserve">Quod demonſtraſſe oportuit.</s> <s xml:id="echoid-s2075" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div313" type="section" level="1" n="224"> <head xml:id="echoid-head237" xml:space="preserve">8 THE OREMA. 11 PROPOSITIO.</head> <p> <s xml:id="echoid-s2076" xml:space="preserve">Parabolarum diametri à gravitatis centro in homologa <lb/>fegmenta dirimuntur.</s> <s xml:id="echoid-s2077" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2078" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2079" xml:space="preserve">Sunto A B C D & </s> <s xml:id="echoid-s2080" xml:space="preserve">a b c d, diſſimiles parabolæ, harum diame-<lb/>@ri A D, & </s> <s xml:id="echoid-s2081" xml:space="preserve">a d, denique gravitatis centra E & </s> <s xml:id="echoid-s2082" xml:space="preserve">e.</s> <s xml:id="echoid-s2083" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2084" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2085" xml:space="preserve">Segmenta A E, E D, ſegmentis a e, e d proportionalia <lb/>@ſſe demonſtrator. </s> <s xml:id="echoid-s2086" xml:space="preserve">P*RÆPARATIO*. </s> <s xml:id="echoid-s2087" xml:space="preserve">Rectas A B, A C, à vertice parabolæ <pb o="66" file="527.01.066" n="66" rhead="L*IBER* S*TATICÆ*"/> ad baſis terminos eductas biſecet F G, in G & </s> <s xml:id="echoid-s2088" xml:space="preserve">F, & </s> <s xml:id="echoid-s2089" xml:space="preserve">diametrum A D in H, & </s> <s xml:id="echoid-s2090" xml:space="preserve"><lb/>ab ipſis biſectionum punctis ſint F I, G K parallelæ contra A D, quarum ver-<lb/>tices cum verticeſectionis & </s> <s xml:id="echoid-s2091" xml:space="preserve">termino baſis proximo connectantur rectis I A, <lb/>I B, K A, K C; </s> <s xml:id="echoid-s2092" xml:space="preserve">deinde eædem illæ F I, G K æquales (parallelæ diameter enim <lb/>A D, parallelarum I F, K G ſi ad B C baſin educantur ſeſquitertia eſſet per <lb/>19 propoſ. </s> <s xml:id="echoid-s2093" xml:space="preserve">Archimed. </s> <s xml:id="echoid-s2094" xml:space="preserve">de quad. </s> <s xml:id="echoid-s2095" xml:space="preserve">parab. </s> <s xml:id="echoid-s2096" xml:space="preserve">& </s> <s xml:id="echoid-s2097" xml:space="preserve">ſublatis æqualibus, reliquæ I F, K G <lb/>æquales erunt) fecentur ratione dupla in L & </s> <s xml:id="echoid-s2098" xml:space="preserve">M, tum recta L M connexa, in<unsure/>-<lb/> <anchor type="figure" xlink:label="fig-527.01.066-01a" xlink:href="fig-527.01.066-01"/> terſecet diametrum A D in N, & </s> <s xml:id="echoid-s2099" xml:space="preserve">I K eandem in O. </s> <s xml:id="echoid-s2100" xml:space="preserve">Præterea tota diameter <lb/>A D ſecetur dupla ratione in P, parallela autem I F continuata occurrat baſi <lb/>B C in Q. </s> <s xml:id="echoid-s2101" xml:space="preserve">Quandoquidem igitur A P dupla eſt ipſius P D, P erit trianguli <lb/>A B C gravitatis centrum, eadem ratione L, M, erunt centra gravitatis trian-<lb/>gulorum A B I, A C K, Ideoq́ue N (ſunt enim triangula æqualia) utriuſque <lb/>commune centrum, quare N P jugum erit, quod ſecetur in R, ut ratio N R <lb/>ad R P ſit eadem quæ trianguli A B C ad duo triangula A B I, A C K, hoc <lb/>eſt ut 4 ad 1 (parabola enim trianguli æquealti in eadem baſi ſeſquitertia eſt, <lb/>demonſtrante Archimede propoſ. </s> <s xml:id="echoid-s2102" xml:space="preserve">24. </s> <s xml:id="echoid-s2103" xml:space="preserve">de quadratura paraboles. </s> <s xml:id="echoid-s2104" xml:space="preserve">Simili planè viâ <lb/>fecetur parabola a b c.</s> <s xml:id="echoid-s2105" xml:space="preserve"/> </p> <div xml:id="echoid-div313" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.066-01" xlink:href="fig-527.01.066-01a"> <image file="527.01.066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.066-01"/> </figure> </div> </div> <div xml:id="echoid-div315" type="section" level="1" n="225"> <head xml:id="echoid-head238" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2106" xml:space="preserve">Vt A D ad A O, ſic per 20 prop. </s> <s xml:id="echoid-s2107" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2108" xml:space="preserve">Apoll. </s> <s xml:id="echoid-s2109" xml:space="preserve">quadratum D B ad quadra-<lb/>tum O I, hoc eſtad Q D, ſed Q D dimidia eſt ipſius B D, nam F Q parallela <lb/>contra A D biſecat inſcriptam A B, quadratum itaque D Q hoc eſt O I ſub-<lb/>quadruplum erit quadrati B D, & </s> <s xml:id="echoid-s2110" xml:space="preserve">ſegmentum igitur A O {1/4} erit totius A D, cui <lb/>O H æqualis eſt, nam integra A D biſecatur in H, & </s> <s xml:id="echoid-s2111" xml:space="preserve">N H {1/12} ejuſdem, quæ <lb/>ad H D {1/2} addita exhibet N D {7/12} de qua deducta P D {1/3} relinquet P N {1/4}, fed <lb/>R P ſubquadrupla eſt ipſius N R, & </s> <s xml:id="echoid-s2112" xml:space="preserve">totius igitur A D ſubvigecupla, quæ ad-<lb/>dita ad P D {1/3} dabit D R {23/60} & </s> <s xml:id="echoid-s2113" xml:space="preserve">reliquam R A {37/60}. </s> <s xml:id="echoid-s2114" xml:space="preserve">Quamobrem ut 37 ad 23 ſic <lb/>A R ad R D. </s> <s xml:id="echoid-s2115" xml:space="preserve">eodem modo evincetur ſegmenta alterius parabolæ a r, r d, eſſe <lb/>ut 37 ad 23. </s> <s xml:id="echoid-s2116" xml:space="preserve">Itaque rectilinea ſimili ratione in diſſimilibus parabolis inſcripta <lb/>centrum gravitatis habent in diametris, à quibus ipſæ diametri in homologa <lb/>ſegmenta dividuntur. </s> <s xml:id="echoid-s2117" xml:space="preserve">Ac denique ſi in parabolæ ſegmentis B I, I A, A K, K C <lb/>triangula itidem ut in ſegmentis B I A, A K C inſcribantur, & </s> <s xml:id="echoid-s2118" xml:space="preserve">rectilineorum <lb/>gravitatis centra S & </s> <s xml:id="echoid-s2119" xml:space="preserve">ſ inveniantur, tandem ſimiliter concludes A S, S R, ſeg-<lb/>mentis aſ, ſr proportionalia eſſe, verum infinita hujuſmodi inſcriptione con-<lb/>tinuô ad E & </s> <s xml:id="echoid-s2120" xml:space="preserve">e propius acceditur. </s> <s xml:id="echoid-s2121" xml:space="preserve">Itaque hujuſmodi rectilineorum γνωζί-<lb/> <anchor type="note" xlink:label="note-527.01.066-01a" xlink:href="note-527.01.066-01"/> μως ſcitè (ut cum Archimeàe loquar) in parabolas inſcriptorum gravitatis cen-<lb/>tra, diametros A D & </s> <s xml:id="echoid-s2122" xml:space="preserve">a d in ſegmenta homologa perpetuò tribuent; </s> <s xml:id="echoid-s2123" xml:space="preserve">atque <lb/>adeò ipſæ quibus inſcribuntur parabolæ A B C, a b c, ſegmenta diametri pro- <pb o="67" file="527.01.067" n="67" rhead="DE INVENTIONE GRAVITATIS CENTRO."/> portionalia habebunt. </s> <s xml:id="echoid-s2124" xml:space="preserve">Etenim ſi T ſumatur centrũ gravitatis parabolæ A B C, <lb/>hinc t ita quidem ftatuatur in a d, ut E T, T S, ipſis et, & </s> <s xml:id="echoid-s2125" xml:space="preserve">t s, proportionales <lb/>ſint, cùm multilaterarum figurarum inſcriptione in hac ad t deventum erit, <lb/>in illa itidem ad T devenietur, Quamobrem T centrum erit inſcripti multan-<lb/>guli, & </s> <s xml:id="echoid-s2126" xml:space="preserve">ipſius quoque parabolæ A B C, quod abſurdum eſt.</s> <s xml:id="echoid-s2127" xml:space="preserve"/> </p> <div xml:id="echoid-div315" type="float" level="2" n="1"> <note position="left" xlink:label="note-527.01.066-01" xlink:href="note-527.01.066-01a" xml:space="preserve">Deſinit <lb/>Archimed. <lb/>prop. 1. lib. 2. <lb/>iſerrhopiewr<unsure/>.</note> </div> <p> <s xml:id="echoid-s2128" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2129" xml:space="preserve">Itaque omnium parabolarum diametri à gravitatis centro <lb/>in homologa ſegmenta dividuntur. </s> <s xml:id="echoid-s2130" xml:space="preserve">Quod demonſtraſſe oportuit.</s> <s xml:id="echoid-s2131" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div317" type="section" level="1" n="226"> <head xml:id="echoid-head239" xml:space="preserve">4 PROBLEMA. 12 PROPOSITIO.</head> <p> <s xml:id="echoid-s2132" xml:space="preserve">Datæ parabolæ gravitatis centrum invenire.</s> <s xml:id="echoid-s2133" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2134" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2135" xml:space="preserve">A B C parabola, ejus diameter A D.</s> <s xml:id="echoid-s2136" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2137" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2138" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2139" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div318" type="section" level="1" n="227"> <head xml:id="echoid-head240" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2140" xml:space="preserve">Fiat ut 3 ad 2 ſic diametri ſegmentum A E ad E D.</s> <s xml:id="echoid-s2141" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2142" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2143" xml:space="preserve">Biſectrix rectarum A B, A C, interſecet A D in H, hinc <lb/>F I, G K diametro parallelæ quasq́ue in antecedentis theorematis conſtructio-<lb/>ne æquales oſten dimus in L & </s> <s xml:id="echoid-s2144" xml:space="preserve">M, ita ſecentur ut I L, L F, item K M, M G, <lb/>diametri ſegmentis A E, E D proportionales ſint; </s> <s xml:id="echoid-s2145" xml:space="preserve">hinc I F continuata occur-<lb/>rat baſi B C in Q, & </s> <s xml:id="echoid-s2146" xml:space="preserve">fiat A P ſegmentum duplum P D, P erit trianguli A B C <lb/>gravitatis centrum: </s> <s xml:id="echoid-s2147" xml:space="preserve">ſiquidem M, L centra gravitatis ſint portionum A C K, <lb/>A B I, igitur N (nam per 4 propoſ. </s> <s xml:id="echoid-s2148" xml:space="preserve">Arch. </s> <s xml:id="echoid-s2149" xml:space="preserve">de Conoid. </s> <s xml:id="echoid-s2150" xml:space="preserve">& </s> <s xml:id="echoid-s2151" xml:space="preserve">Sphæroïd. </s> <s xml:id="echoid-s2152" xml:space="preserve">portiones <lb/>parabolicæ iſtæ inter ſe æquantur) harum commune gravitatis centrum eſt. <lb/></s> <s xml:id="echoid-s2153" xml:space="preserve">Quamobrem Iugo P N, ſecundum rationem trianguli A B C ad duas para-<lb/>bolicas portiones, diviſo, habebimus optatum: </s> <s xml:id="echoid-s2154" xml:space="preserve">ſed integra parabola A B C eſt, <lb/>per 24 propoſ. </s> <s xml:id="echoid-s2155" xml:space="preserve">Archimed. </s> <s xml:id="echoid-s2156" xml:space="preserve">de quadr. </s> <s xml:id="echoid-s2157" xml:space="preserve">parab. </s> <s xml:id="echoid-s2158" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-527.01.067-01a" xlink:href="fig-527.01.067-01"/> ſeſquitertia trianguli A B C, quamobrem <lb/>A B C triangulum triplum erit duarum pa-<lb/>raboles portionum, ſecetur igitur P N in E <lb/>ratione tripla, hoc eſt ut ſegmentum N E <lb/>vertici vicinius triplum ſit reliqui E P. </s> <s xml:id="echoid-s2159" xml:space="preserve">Di-<lb/>co E optatum eſſe parabolæ centrum: </s> <s xml:id="echoid-s2160" xml:space="preserve">& </s> <s xml:id="echoid-s2161" xml:space="preserve"><lb/>& </s> <s xml:id="echoid-s2162" xml:space="preserve">ſegmenti A E ad E D rationem eſſe ſeſ-<lb/>quialteram, quod ex opere & </s> <s xml:id="echoid-s2163" xml:space="preserve">ſectionis ra-<lb/>tione patet.</s> <s xml:id="echoid-s2164" xml:space="preserve"/> </p> <div xml:id="echoid-div318" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.067-01" xlink:href="fig-527.01.067-01a"> <image file="527.01.067-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.067-01"/> </figure> </div> </div> <div xml:id="echoid-div320" type="section" level="1" n="228"> <head xml:id="echoid-head241" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2165" xml:space="preserve">A O, O H ſunt quartæ partes totius A D, quod 11 prop. </s> <s xml:id="echoid-s2166" xml:space="preserve">oſtendimus; </s> <s xml:id="echoid-s2167" xml:space="preserve">Ve-<lb/>rum ut 3 ad 2 ſic A E ad E D, ſic item I L ad L F, ſic quoque O N ad N H, <lb/>quamobrem N H erit {1/4}, hoc eſt ſubdecupla totius A D, hinc N H {1/10} addita <lb/>ad A D {1/2} exhibet N D {1/3} quæ multata P D {1/3} relinquit N P {4/23}. </s> <s xml:id="echoid-s2168" xml:space="preserve">Verum hæc <lb/>ex fabrica in E ita diviſa eſt ut N E tripla ſit ipſius E P. </s> <s xml:id="echoid-s2169" xml:space="preserve">Itaque E P valet {1/15} hæc <lb/>addita ad P D {1/3} dabit E D {2/3<unsure/>} diametri A D. </s> <s xml:id="echoid-s2170" xml:space="preserve">Et E A valebit ejuſdem {3/5}. </s> <s xml:id="echoid-s2171" xml:space="preserve">Quam-<lb/>obrem A E ad E D eſt ut 3 ad 2, & </s> <s xml:id="echoid-s2172" xml:space="preserve">conſequenter E gravitatis eſt centrum pa-<lb/>rabolæ A B C. </s> <s xml:id="echoid-s2173" xml:space="preserve">quod fuit propoſitum. </s> <s xml:id="echoid-s2174" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2175" xml:space="preserve">Itaque. </s> <s xml:id="echoid-s2176" xml:space="preserve">Data ellipſi <lb/>centrum gravitatis invenimus.</s> <s xml:id="echoid-s2177" xml:space="preserve"/> </p> <pb o="68" file="527.01.068" n="68" rhead="L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div321" type="section" level="1" n="229"> <head xml:id="echoid-head242" xml:space="preserve">NOTA.</head> <p style="it"> <s xml:id="echoid-s2178" xml:space="preserve">Videtur Archimedes, @altero horum modorum problematis hujus inventionens <lb/>aſſecutus, ut dum aut parabolici ſui ſpeculi exemplar fabricatur, aut alterius gratia pa-<lb/>rabolam ſolidam, boc eſt conoïdale rectangulum efformat, reapſe edoctus ſit, ſegmentums <lb/>vertici co nterminum reliqui eſſe ſeſquialterum, in cujus cauſam hâc viâ inquiſierit & </s> <s xml:id="echoid-s2179" xml:space="preserve"><lb/>quaſi inſpexerit: </s> <s xml:id="echoid-s2180" xml:space="preserve">Cum ambæ B A I, B A C parabolæ ſint, diametros I F, A D àgra-<lb/>vitatis centris in homologa ſegmenta per 11 propoſ. </s> <s xml:id="echoid-s2181" xml:space="preserve">ſecari neceſſe erit, ideo{q́ue} I L, L F, <lb/>hoc eſt O N, N H ipſis æquales, rectis A E, E D proportionales erunt: </s> <s xml:id="echoid-s2182" xml:space="preserve">ſed ſi N com-<lb/>mune utriuſque par abolicæ portionis gravitatis centrum foret, P verò centrum trian-<lb/>guli A B C, quia triangulum ſimulutriuſque portionis eſt triplum, etiam jugum N E <lb/>jugi E P quoque triplum erit. </s> <s xml:id="echoid-s2183" xml:space="preserve">Vnde propoſitio iſtiuſmodi exiſtit. </s> <s xml:id="echoid-s2184" xml:space="preserve">Invenire duo puncta <lb/>N, E, quæ ſegmentorum O N, N H rationem faciant eandem quam A E habet <lb/>ad E D. </s> <s xml:id="echoid-s2185" xml:space="preserve">aſſumpta deinde A E {3/5} totius A D, & </s> <s xml:id="echoid-s2186" xml:space="preserve">E D {2/5}, facto{q́ue} periculo, quid ex his dedu-<lb/>catur; </s> <s xml:id="echoid-s2187" xml:space="preserve">tandem iſtudipſum veritati congruere comperit. </s> <s xml:id="echoid-s2188" xml:space="preserve">Aut ſi coniectur â huius ſeſ-<lb/>quialteræ rationis id ipſum aſſecutus non ſit, verùm arte duce in hæc penetralia pene-<lb/>traverit, videtur numeris hæc primùm expertus: </s> <s xml:id="echoid-s2189" xml:space="preserve">Dati duo numeri O H {1/4}, H P {1/6} am-<lb/>bo ita dividuntor, ut minus ſegmentum rectæ O H cum majore ipſius H P, tri-<lb/>plum ſit ſegmenti minoris rectæ H P cum majore ipſius H O, ea lege ut majus <lb/>ſegmentum rectæ O H ad minus habeat rationem, quam majus ſegmentum <lb/>H P + {1/3} habet ad minus ſegmentum H P + {1/3}.</s> <s xml:id="echoid-s2190" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div322" type="section" level="1" n="230"> <head xml:id="echoid-head243" xml:space="preserve">5 PROBLEMA. 15 PROPOSITIO.</head> <p> <s xml:id="echoid-s2191" xml:space="preserve">Datâ parabolâ curtâ, gravitatis centrum invenire.</s> <s xml:id="echoid-s2192" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2193" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2194" xml:space="preserve">A B C D parabola curta, oppoſitas rectas habeat parallelas, quas <lb/>biſecat diameter E F. </s> <s xml:id="echoid-s2195" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2196" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2197" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div323" type="section" level="1" n="231"> <head xml:id="echoid-head244" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2198" xml:space="preserve">Parabolam curtam abſolvito, defectu A B G addito, hinc G E ſecetur in <lb/>H ut ſegmentum G H vertici vicinum reliqui H E ſit ſeſquialterum, itemq́uc <lb/>G I ipſius I F; </s> <s xml:id="echoid-s2199" xml:space="preserve">denique fiat ut A B C D ad A B G ſic H I ad I K: </s> <s xml:id="echoid-s2200" xml:space="preserve">Ajo K <lb/>optatu m gravitatis centrum eſſe.</s> <s xml:id="echoid-s2201" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div324" type="section" level="1" n="232"> <head xml:id="echoid-head245" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2202" xml:space="preserve">Integræ parabolæ gravitatis cen-<lb/>trum eſt I, & </s> <s xml:id="echoid-s2203" xml:space="preserve">H portionis, quia verò <lb/> <anchor type="figure" xlink:label="fig-527.01.068-01a" xlink:href="fig-527.01.068-01"/> eſt H I ad I K ut parabola curta ad <lb/>dictam portionem, K curtæ parabolæ <lb/>centrum erit.</s> <s xml:id="echoid-s2204" xml:space="preserve"/> </p> <div xml:id="echoid-div324" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.068-01" xlink:href="fig-527.01.068-01a"> <image file="527.01.068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.068-01"/> </figure> </div> <p> <s xml:id="echoid-s2205" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2206" xml:space="preserve">Itaque, ut opor-<lb/>tuit, curtæ parabolæ centrum gravita-<lb/>tis invenimus, Generaliter autem ſive <lb/>A B parallela ſit contra D C, ſive an-<lb/>nuat ita efficies. </s> <s xml:id="echoid-s2207" xml:space="preserve">inveniatur H centrum <lb/>gravitatis parabolæ A G B & </s> <s xml:id="echoid-s2208" xml:space="preserve">I centrum totius D G C quæ connectantur ju-<lb/>go H I & </s> <s xml:id="echoid-s2209" xml:space="preserve">fiat H I ad continuationem I K ſicut parabola curta A B C D ad <lb/>complementum ſui A G B. </s> <s xml:id="echoid-s2210" xml:space="preserve">utriuſque autem ratio ad rectilineas figuras revo-<lb/>cari poteſt, cum utraque D G C, A G B trianguli quæ ipſis & </s> <s xml:id="echoid-s2211" xml:space="preserve">baſin & </s> <s xml:id="echoid-s2212" xml:space="preserve">altitu-<lb/>dinem habet æqualem ſeſquitertia ſit; </s> <s xml:id="echoid-s2213" xml:space="preserve">Et demonſtratio antecedens huic omni-<lb/>no congruet.</s> <s xml:id="echoid-s2214" xml:space="preserve"/> </p> <pb o="69" file="527.01.069" n="69" rhead="*DE* S*TATICÆ PRINCIPITS.*"/> </div> <div xml:id="echoid-div326" type="section" level="1" n="233"> <head xml:id="echoid-head246" xml:space="preserve">CENTROBARICA SOLIDORVM <lb/>DEINCEPS SVCCEDVNT.</head> <head xml:id="echoid-head247" xml:space="preserve">9 THE OREMA. 14 PROPOSITIO.</head> <p> <s xml:id="echoid-s2215" xml:space="preserve">Solidi cujuſlibet, & </s> <s xml:id="echoid-s2216" xml:space="preserve">figuræ & </s> <s xml:id="echoid-s2217" xml:space="preserve">gravitatis idem eſt cen-<lb/>trum.</s> <s xml:id="echoid-s2218" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2219" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2220" xml:space="preserve">Tetraë dri A B C D centrum ſit E, axis autem ab A per E cen-<lb/>trum occurrens baſi B C D in F, ſit A F.</s> <s xml:id="echoid-s2221" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2222" xml:space="preserve">Q*VALSITVM*. </s> <s xml:id="echoid-s2223" xml:space="preserve">Ipſum E gravitatis quoque centrum eſſe oſtenditor.</s> <s xml:id="echoid-s2224" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div327" type="section" level="1" n="234"> <head xml:id="echoid-head248" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2225" xml:space="preserve">Solidum hoc ſuſpendito ex A E; </s> <s xml:id="echoid-s2226" xml:space="preserve">cum igitur tetraëdrum componatur è qua-<lb/>tuor pyramidibus ſimilibus & </s> <s xml:id="echoid-s2227" xml:space="preserve">inter ſe æqualibus quorum vertex communis ſit <lb/>E, ipſa A F erit ejus gravitatis diameter; </s> <s xml:id="echoid-s2228" xml:space="preserve">eadem ratio erit rectæ <lb/> <anchor type="figure" xlink:label="fig-527.01.069-01a" xlink:href="fig-527.01.069-01"/> C E; </s> <s xml:id="echoid-s2229" xml:space="preserve">quare E centrum erit in concurſu diametrorum. </s> <s xml:id="echoid-s2230" xml:space="preserve">Similis <lb/>demonſtratio erit in reliquis corporibus cum auctis & </s> <s xml:id="echoid-s2231" xml:space="preserve">immi-<lb/>nutis tum etiam abſolutè ordinatis quæ centrum ſoliditatis <lb/>habebunt, nam ipſa ex diametris vel per angulum ſolidum, vel <lb/>per hedrarũ centra eductis ſuſpenſa, ratio ſitus componentum <lb/>pyramidum (quarum quidem vertices eodem coëunt, & </s> <s xml:id="echoid-s2232" xml:space="preserve">baſes <lb/>ſolidi ipſius ſint hedræ) ad latera omnia par erit, quamobrem <lb/>ex communi notitia, & </s> <s xml:id="echoid-s2233" xml:space="preserve">1 poſtulatum 1 lib. </s> <s xml:id="echoid-s2234" xml:space="preserve">univerſa ab hac recta <lb/>æquilibria dependebunt; </s> <s xml:id="echoid-s2235" xml:space="preserve">& </s> <s xml:id="echoid-s2236" xml:space="preserve">conſequenter mutua in centro <lb/>figuræ iſtiuſmodi diametrorum interſectio, ceutrum quoque gravitatis erit.</s> <s xml:id="echoid-s2237" xml:space="preserve"/> </p> <div xml:id="echoid-div327" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.069-01" xlink:href="fig-527.01.069-01a"> <image file="527.01.069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.069-01"/> </figure> </div> <p> <s xml:id="echoid-s2238" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2239" xml:space="preserve">Itaque in ſolido & </s> <s xml:id="echoid-s2240" xml:space="preserve">figuræ & </s> <s xml:id="echoid-s2241" xml:space="preserve">gravitatis idem eſt centrum.</s> <s xml:id="echoid-s2242" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div329" type="section" level="1" n="235"> <head xml:id="echoid-head249" xml:space="preserve">10 THE OREMA. 15 PROPOSITIO.</head> <p> <s xml:id="echoid-s2243" xml:space="preserve">Priſmatis gravitatis centrum eſt in axis medio.</s> <s xml:id="echoid-s2244" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div330" type="section" level="1" n="236"> <head xml:id="echoid-head250" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s2245" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2246" xml:space="preserve">Eſto A B priſma baſis triangulæ A C D.</s> <s xml:id="echoid-s2247" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2248" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2249" xml:space="preserve">Axem à gravitatis ſuæ centro biſecari demonſtrator.</s> <s xml:id="echoid-s2250" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2251" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2252" xml:space="preserve">D E biſecet latus A C, parallelæ autem H I, F G, ipſam <lb/>biſectricem interſecent in punctis K, L, & </s> <s xml:id="echoid-s2253" xml:space="preserve">parallelæ ſint F M, H N, I O, G P <lb/>ad ipſam D E; </s> <s xml:id="echoid-s2254" xml:space="preserve">dein de A Q oppoſitam D C bifariam ſecet in Q Ac denique <lb/>reliquæ he@ræ parallelogrammæ biſecentur plano R S baſi A D C parallelo, <lb/>à quò C B bifariam dividatur in puncto S.</s> <s xml:id="echoid-s2255" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div331" type="section" level="1" n="237"> <head xml:id="echoid-head251" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2256" xml:space="preserve">Planum actum per D E & </s> <s xml:id="echoid-s2257" xml:space="preserve">rectam ſibi parallelam in plano R S reliquas he-<lb/>dras biſecante, tribuit priſma H N F M P G I in duo priſmata æqualia & </s> <s xml:id="echoid-s2258" xml:space="preserve">ſimi-<lb/>lia; </s> <s xml:id="echoid-s2259" xml:space="preserve">tranſit igitur per hujus inſcriptæ priſmatis gravitatis centrum, quo autem <pb o="70" file="527.01.070" n="70" rhead="2 L*IBER* S*TATICÆ*"/> plura priſmata baſis quadrangulæ in datum <lb/> <anchor type="figure" xlink:label="fig-527.01.070-01a" xlink:href="fig-527.01.070-01"/> inſcribuntur eo minus ab eodem differunt; <lb/></s> <s xml:id="echoid-s2260" xml:space="preserve">quamobrem infinita iſta inſcriptione eô <lb/>tandem adſcenditur ut inſcripti & </s> <s xml:id="echoid-s2261" xml:space="preserve">circum-<lb/>ſcripti differentia quamcunque minimo ſo-<lb/>lido minor adhuc ſit. </s> <s xml:id="echoid-s2262" xml:space="preserve">Vnde efficitur gravi-<lb/>tatem ſitus unius ſegmenti D F C B, a gra-<lb/>vitate ſitus reliqui ſegmenti abeſſe etiam mi-<lb/>nori differentia quam cujuſcunque minimi <lb/>corporis quod quidem exhiberi poſſit. </s> <s xml:id="echoid-s2263" xml:space="preserve"><lb/>quamobrem ſic ediſſero.</s> <s xml:id="echoid-s2264" xml:space="preserve"/> </p> <div xml:id="echoid-div331" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.070-01" xlink:href="fig-527.01.070-01a"> <image file="527.01.070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.070-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s2265" xml:space="preserve">Inæqualium & </s> <s xml:id="echoid-s2266" xml:space="preserve">ſitu @ravium ponderum differentiâ pondus minus exhiberi poteſt. <lb/></s> <s xml:id="echoid-s2267" xml:space="preserve">Sed horum ponderũ ſitu gravium differentiâ pondus minus exhiberi nullum poteſt. </s> <s xml:id="echoid-s2268" xml:space="preserve"><lb/>Itaque horum ponderum ſitu gravium differentia nulla eſt.</s> <s xml:id="echoid-s2269" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2270" xml:space="preserve">Ideoq́; </s> <s xml:id="echoid-s2271" xml:space="preserve">planum actum per D E & </s> <s xml:id="echoid-s2272" xml:space="preserve">rectam in plano R S ſibi homologam, dati <lb/>priſmatis gravitatis centrum tranſit, ſimillimo argumento planum A Q per in-<lb/>clinationem laterum A D, A C, & </s> <s xml:id="echoid-s2273" xml:space="preserve">biſectionem rectæ D C eductum, idem <lb/>gravitatis centrum induere evinces; </s> <s xml:id="echoid-s2274" xml:space="preserve">ſed horum planorum communis ſectio, eſt <lb/>recta cõnectens centra gravitatis oppoſitarum baſium, qui axis eſt d@ti priſmatis <lb/>itaq; </s> <s xml:id="echoid-s2275" xml:space="preserve">centrum gravitatis conſiſtit in axe, eſt item in plano per R S oppoſitis <lb/>baſibus parallelo, hoc enim & </s> <s xml:id="echoid-s2276" xml:space="preserve">priſma & </s> <s xml:id="echoid-s2277" xml:space="preserve">axem bipartitò & </s> <s xml:id="echoid-s2278" xml:space="preserve">ſimili partium ſitu <lb/>diſpeſcit; </s> <s xml:id="echoid-s2279" xml:space="preserve">Quare centrum gravitatis ex in axis medio.</s> <s xml:id="echoid-s2280" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div333" type="section" level="1" n="238"> <head xml:id="echoid-head252" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s2281" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2282" xml:space="preserve">Priſma A B eſto quadrangulæ baſis A C D E.</s> <s xml:id="echoid-s2283" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2284" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s2285" xml:space="preserve">Gravitatis centrum in axe conſiſtere.</s> <s xml:id="echoid-s2286" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2287" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2288" xml:space="preserve">Solidum datum plano A D B in priſmata triangulę ba-<lb/>ſis ipſum componentia dirimatur. </s> <s xml:id="echoid-s2289" xml:space="preserve">Singulorum igitur gravitatis centrum per <lb/>1 exempl. </s> <s xml:id="echoid-s2290" xml:space="preserve">axem ſuum biſecat, Quare jugum cen-<lb/>tra connectens, pro ratione ponderum reciproce <lb/> <anchor type="figure" xlink:label="fig-527.01.070-02a" xlink:href="fig-527.01.070-02"/> tributum, centrum quæſitum exhibebit, punctum <lb/>autem ipſum incidet in centro gravitatis plani priſ-<lb/>ma biſecantis & </s> <s xml:id="echoid-s2291" xml:space="preserve">baſibus paralleli, hoc eſt in ipſum <lb/>ſolidi axem quem medium ſecat.</s> <s xml:id="echoid-s2292" xml:space="preserve"/> </p> <div xml:id="echoid-div333" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.070-02" xlink:href="fig-527.01.070-02a"> <image file="527.01.070-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.070-02"/> </figure> </div> <p> <s xml:id="echoid-s2293" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2294" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s2295" xml:space="preserve">priſmatis gravitatis cen-<lb/>trum axem medium incîdit.</s> <s xml:id="echoid-s2296" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div335" type="section" level="1" n="239"> <head xml:id="echoid-head253" xml:space="preserve">11 THEOREMA. 16 PROPOSITIO.</head> <p> <s xml:id="echoid-s2297" xml:space="preserve">Pyramidis gravitatis centrum eſt in axe.</s> <s xml:id="echoid-s2298" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2299" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2300" xml:space="preserve">Pyramidis A B C D baſis triangula B C D, gravitatis centrum <lb/>E, axis eſto A E. </s> <s xml:id="echoid-s2301" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s2302" xml:space="preserve">Centrum gravitatis in ipſa A E conſiſtere <lb/>demonſtrator. </s> <s xml:id="echoid-s2303" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2304" xml:space="preserve">Planum F G H baſi B C D parallelum, <lb/>ſecet datam pyramidem, ejuſque axem A E in I; </s> <s xml:id="echoid-s2305" xml:space="preserve">deinde F K, G L, H M, axi <lb/>parallelæ terminentur in baſi B C D. </s> <s xml:id="echoid-s2306" xml:space="preserve">Similiter pyramis ſecundo interſecetur <lb/>plano N O P baſi parallelo, & </s> <s xml:id="echoid-s2307" xml:space="preserve">axis in Q, hinc ſimiliter centra A E eductis pa-<lb/>rallelis N R, O S, P T, comprehendatur pyramis N O P R S T.</s> <s xml:id="echoid-s2308" xml:space="preserve"/> </p> <pb o="71" file="527.01.071" n="71" rhead="*DE* S*TATICÆ PRINCIPIIS*."/> </div> <div xml:id="echoid-div336" type="section" level="1" n="240"> <head xml:id="echoid-head254" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2309" xml:space="preserve">Triangula NOP, RST, FG, KLM, ſimilia ſunt triangulo BCD, & </s> <s xml:id="echoid-s2310" xml:space="preserve"><lb/>puncta Q, I, E, in iſtis ſimili ſitu reſpondent puncto E in triangulo BCD, <lb/>quod ejuſdem gravitatis eſt centrum, ideo Q, I, E, ſuorum triangulorum <lb/> <anchor type="figure" xlink:label="fig-527.01.071-01a" xlink:href="fig-527.01.071-01"/> gravitatis ſunt centra, & </s> <s xml:id="echoid-s2311" xml:space="preserve">I E axis priſmatis <lb/>FGHKLM quem medium, per 15 pro-<lb/>poſ. </s> <s xml:id="echoid-s2312" xml:space="preserve">gravitatis centrum incîdit; </s> <s xml:id="echoid-s2313" xml:space="preserve">ſic item <lb/>Q I axis priſmatis NOPRST medius à <lb/>centro ſuo dividetur, quamobrem ſolidum <lb/>ex utroque priſmate compoſitum centrum <lb/>habet in Q E hoc eſt in A E, verumenim-<lb/>vero hujuſmodi priſmatum frequentiſſima <lb/>inſcriptio, componet ſolidum quod ad py-<lb/>ramidis ſoliditatem proximè accedat, cujus <lb/>tamen gravitatis centrum in axe A E ſem-<lb/>per hæreat. </s> <s xml:id="echoid-s2314" xml:space="preserve">Sed ſolidum tale poteſt intra py-<lb/>ramidem inſcribi ut ejus à pyramide diffe-<lb/>rentia quocunque dato corpore minor ſit, <lb/>unde efficitur, poſita diametro A E gravita-<lb/>tis ſitum unius partis à reliqua minori etiam <lb/>quam dari poſlit differentiâ abeſſe; </s> <s xml:id="echoid-s2315" xml:space="preserve">Quod <lb/>eodem quo ſupra ſyllogiſmo evincam.</s> <s xml:id="echoid-s2316" xml:space="preserve"/> </p> <div xml:id="echoid-div336" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.071-01" xlink:href="fig-527.01.071-01a"> <image file="527.01.071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.071-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s2317" xml:space="preserve">Ineæqualium ſitu gravium ponderum differentiâ minus pondus dari poteſt. <lb/></s> <s xml:id="echoid-s2318" xml:space="preserve">Sed borum ponderum differentiâ pondus minus exhiberi nullum poteſt. </s> <s xml:id="echoid-s2319" xml:space="preserve"><lb/>Itaque horum ponderum differentia nulla eſt.</s> <s xml:id="echoid-s2320" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2321" xml:space="preserve">Simillima demonſtratio erit in cæteris quorum baſes erunt quadrangulæ, <lb/>aut quomodocunque multangulæ, vel rotundæ denique.</s> <s xml:id="echoid-s2322" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2323" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2324" xml:space="preserve">Itaque, centrum gravitatis pyramidis eſt in axe.</s> <s xml:id="echoid-s2325" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div338" type="section" level="1" n="241"> <head xml:id="echoid-head255" xml:space="preserve">6 PROBLEMA. 17 PROPOSITIO.</head> <p> <s xml:id="echoid-s2326" xml:space="preserve">Pyramidís triangulæ baſis gravitatis centrum invenire.</s> <s xml:id="echoid-s2327" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2328" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2329" xml:space="preserve">Pyramidis ABC baſis ſit BCD.</s> <s xml:id="echoid-s2330" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2331" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2332" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2333" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div339" type="section" level="1" n="242"> <head xml:id="echoid-head256" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2334" xml:space="preserve">Duarum hedrarum BCD, ABC gravitatis <lb/> <anchor type="figure" xlink:label="fig-527.01.071-02a" xlink:href="fig-527.01.071-02"/> centra EF, oppoſitis verticibus connexa rectis AE, <lb/>BF ſeſe incîdent in G & </s> <s xml:id="echoid-s2335" xml:space="preserve">cum utraque ſit diameter, <lb/>Ajo G eſſe centrum optatum.</s> <s xml:id="echoid-s2336" xml:space="preserve"/> </p> <div xml:id="echoid-div339" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.071-02" xlink:href="fig-527.01.071-02a"> <image file="527.01.071-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.071-02"/> </figure> </div> </div> <div xml:id="echoid-div341" type="section" level="1" n="243"> <head xml:id="echoid-head257" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2337" xml:space="preserve">Etenim pyramidis gravitatis centrum eſt in AE, <lb/>itemq́ue in B F per 16 propoſ. </s> <s xml:id="echoid-s2338" xml:space="preserve">eſt itaque in G ipſa-<lb/>rum mutua interſectione.</s> <s xml:id="echoid-s2339" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2340" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2341" xml:space="preserve">Pyramidis igitur à triangula baſi aſſurgentis, centrum gra-<lb/>vitatis, ut petebatur, invenimus.</s> <s xml:id="echoid-s2342" xml:space="preserve"/> </p> <pb o="72" file="527.01.072" n="72" rhead="2 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div342" type="section" level="1" n="244"> <head xml:id="echoid-head258" xml:space="preserve">12 PROBLEMA. 18 PROPOSITIO.</head> <p> <s xml:id="echoid-s2343" xml:space="preserve">Centrum gravitatis pyramidis axem ita ſecat ut ſegmen-<lb/>tum vertici vicinius reliqui ſit triplum.</s> <s xml:id="echoid-s2344" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2345" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2346" xml:space="preserve">Pyramidis ABCD baſis triangulæ, vertex A, baſis BCD, axis <lb/>à B ad E centrum gravitatis trianguli ADC eſto BE, hinc ab A ad cen-<lb/>trum gravitatis oppoſitæ hedræ BCD eſto AF quæ per antecedentem pro-<lb/>poſ. </s> <s xml:id="echoid-s2347" xml:space="preserve">ſecet priorem BE in G centro gravitatis pyramidis.</s> <s xml:id="echoid-s2348" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div343" type="section" level="1" n="245"> <head xml:id="echoid-head259" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2349" xml:space="preserve">Recta AH, angulum A & </s> <s xml:id="echoid-s2350" xml:space="preserve">punctum H baſis me-<lb/> <anchor type="figure" xlink:label="fig-527.01.072-01a" xlink:href="fig-527.01.072-01"/> dium connectens, ita ſecatur ab E trianguli ADC <lb/>gravitatis centro per 4 propoſ. </s> <s xml:id="echoid-s2351" xml:space="preserve">ut AE ſegmentum <lb/>vertici conterminum reliqui E H ſit duplum, pari ra-<lb/>tione BF dupla erit rectæ FH. </s> <s xml:id="echoid-s2352" xml:space="preserve">Quod cum ita ſit, ra-<lb/>tio B F ad FH, per Ptolemaicam {δι}αςρεσιν lib. </s> <s xml:id="echoid-s2353" xml:space="preserve">1. </s> <s xml:id="echoid-s2354" xml:space="preserve">cap. </s> <s xml:id="echoid-s2355" xml:space="preserve">12. <lb/></s> <s xml:id="echoid-s2356" xml:space="preserve">μεγάλης σ{μν}θαξ. </s> <s xml:id="echoid-s2357" xml:space="preserve">componetur è ratione BG ad GE <lb/>& </s> <s xml:id="echoid-s2358" xml:space="preserve">EA ad AH, ſubducta igitur ratione EA 2 ad <lb/>AH 3 de ratione B F 2 ad FH 1 reliqua erit ratio <lb/>BG 3 ad GE 1.</s> <s xml:id="echoid-s2359" xml:space="preserve"/> </p> <div xml:id="echoid-div343" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.072-01" xlink:href="fig-527.01.072-01a"> <image file="527.01.072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.072-01"/> </figure> </div> <p> <s xml:id="echoid-s2360" xml:space="preserve">Verumenimverò in pyramide baſis quadrangulæ demonſtratio hinc deriva-<lb/>ta hujuſmodi erit: </s> <s xml:id="echoid-s2361" xml:space="preserve">Etenim ABCDE pyramis <lb/> <anchor type="figure" xlink:label="fig-527.01.072-02a" xlink:href="fig-527.01.072-02"/> aſſurgat à baſi BCDE, & </s> <s xml:id="echoid-s2362" xml:space="preserve">axis ſit AF. </s> <s xml:id="echoid-s2363" xml:space="preserve">Diviſa <lb/>igitur hac in pyramides componentes quarum <lb/>baſes ECB, ECD & </s> <s xml:id="echoid-s2364" xml:space="preserve">axes AG, AH, centra <lb/>item gravitatis I, K, etiam totius pyramidis cen-<lb/>trum fuerit per 16 propoſ. </s> <s xml:id="echoid-s2365" xml:space="preserve">in jugo IK, videlicet in <lb/>L cõmuni axis & </s> <s xml:id="echoid-s2366" xml:space="preserve">jugi interſectione, ſed in trian-<lb/>gulo AGH, recta IK baſi GH parallela eſt, la-<lb/>tera enim A G, AH proportionaliter ſecantur in <lb/>I & </s> <s xml:id="echoid-s2367" xml:space="preserve">K per priorem partem, itaque AL quoque <lb/>tripla erit ipſius L F nam ob ſimilitudinem ut AI <lb/>ad IG, ſic AL ad LF, ſimillima in ceteris à <lb/>quamlibet multangula baſi aſſurgentibus pyrami-<lb/>dibus ratio quoque fuerit.</s> <s xml:id="echoid-s2368" xml:space="preserve"/> </p> <div xml:id="echoid-div344" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.072-02" xlink:href="fig-527.01.072-02a"> <image file="527.01.072-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.072-02"/> </figure> </div> <p> <s xml:id="echoid-s2369" xml:space="preserve">Denique coni tum circularis quam ellipticæ baſis demonſtratio eodem re-<lb/>dit, cum enim ex antecedente parte pyramis baſis quoquomodo polygonæ <lb/>axem gravitatis incîdat ratione tripla, in cono verò baſis ellipticæ vel cir-<lb/>cularis pyramis poteſt inſcribi quæ à dato cono quamcunque minimi ſolidi <lb/>differentia abſit, itaque intervallum centrorum gravitatis dati & </s> <s xml:id="echoid-s2370" xml:space="preserve">inſcripti ſolidi <lb/>minus erit quâcunque minima diſtantia, unde ſyllogiſmus talis inſtituitur.</s> <s xml:id="echoid-s2371" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2372" xml:space="preserve">Duorum distantium punctorum intervallo minus intervallum dari poteſt. <lb/></s> <s xml:id="echoid-s2373" xml:space="preserve">Sed horum centrθrum intervallo minus dari nullum poteſt. </s> <s xml:id="echoid-s2374" xml:space="preserve"><lb/>Itaqueista puncta nullo intervallo à ſe mutuò abſunt.</s> <s xml:id="echoid-s2375" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2376" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2377" xml:space="preserve">Quamobrem axis pyramidis cujuſcunque ratione tripla <lb/>à gravitatis centro ſecatur, videlicet utſummum & </s> <s xml:id="echoid-s2378" xml:space="preserve">vertici vicinius imiſit tri-<lb/>plum. </s> <s xml:id="echoid-s2379" xml:space="preserve">Quod facere oportuit.</s> <s xml:id="echoid-s2380" xml:space="preserve"/> </p> <pb o="73" file="527.01.073" n="73" rhead="*DE* S*TATICÆ PRINCIPIIS.*"/> </div> <div xml:id="echoid-div346" type="section" level="1" n="246"> <head xml:id="echoid-head260" xml:space="preserve">7 PROBLEMA. 19 PROPOSITIO.</head> <p> <s xml:id="echoid-s2381" xml:space="preserve">Solidi gravitatis centro & </s> <s xml:id="echoid-s2382" xml:space="preserve">ſegmenti ſui cujus ad reli-<lb/>quum ratio ſit data cognitis, ejuſdem reliqui ſegmenti gra-<lb/>vitatis centrum quoque invenire.</s> <s xml:id="echoid-s2383" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2384" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2385" xml:space="preserve">Corporis ABCD centrum gravitatis ſit E, ſegmenti verò BDA <lb/>centrum F. </s> <s xml:id="echoid-s2386" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2387" xml:space="preserve">Reliqui ſegmenti BCD gravitatis centrum in-<lb/>venire.</s> <s xml:id="echoid-s2388" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div347" type="section" level="1" n="247"> <head xml:id="echoid-head261" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2389" xml:space="preserve">Fiat FE ad continuationem EG ut ſolidum BDC ad BDA; </s> <s xml:id="echoid-s2390" xml:space="preserve">ajo G eſſe <lb/>optatum gravitatis centrum reliqui ſegmenti BDC, cujus demonſtratio ſimil-<lb/>lima erit 9 propoſ. </s> <s xml:id="echoid-s2391" xml:space="preserve">Sed ſphæræ ſegmenti cujus ad re-<lb/> <anchor type="figure" xlink:label="fig-527.01.073-01a" xlink:href="fig-527.01.073-01"/> liquam ratio ſit data cum è ſecundo 9 propoſ. </s> <s xml:id="echoid-s2392" xml:space="preserve">exem-<lb/>plo ſatis plana ſit nullum paradigma proponam.</s> <s xml:id="echoid-s2393" xml:space="preserve"/> </p> <div xml:id="echoid-div347" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.073-01" xlink:href="fig-527.01.073-01a"> <image file="527.01.073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.073-01"/> </figure> </div> <p> <s xml:id="echoid-s2394" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2395" xml:space="preserve">Quamobrem datis gravitatis <lb/>centris totius ſolidi & </s> <s xml:id="echoid-s2396" xml:space="preserve">ſui ſegmĕti cujus ad reliquum <lb/>ratio inveniri poſſit, ejuſdem reliqui gravitatis cen-<lb/>trum, invenimus. </s> <s xml:id="echoid-s2397" xml:space="preserve">Quod feciſſe oportebat.</s> <s xml:id="echoid-s2398" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div349" type="section" level="1" n="248"> <head xml:id="echoid-head262" xml:space="preserve">8 PROBLEMA. 20 PROPOSITIO.</head> <p> <s xml:id="echoid-s2399" xml:space="preserve">Pyramidis curtæ gravitatis centrum invenire.</s> <s xml:id="echoid-s2400" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2401" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2402" xml:space="preserve">Eſto ABCDEF pyramis curta, cujus ſumma baſis ABC, <lb/>ima DEF. </s> <s xml:id="echoid-s2403" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2404" xml:space="preserve">Ejus gravitatis centrum invenire.</s> <s xml:id="echoid-s2405" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div350" type="section" level="1" n="249"> <head xml:id="echoid-head263" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2406" xml:space="preserve">Abſolvito curtam pyramidem, expleto defectu ABCG, & </s> <s xml:id="echoid-s2407" xml:space="preserve">axis ductus à <lb/>vertice G ad H centrum baſis D E F ſecet planum ABC in I, ſegmen verò <lb/>GL ita ſecetur in K ut GK ipſius KL ſit tripla, <lb/> <anchor type="figure" xlink:label="fig-527.01.073-02a" xlink:href="fig-527.01.073-02"/> & </s> <s xml:id="echoid-s2408" xml:space="preserve">L totum axem ita incîdat ut pars ſumma GL <lb/>imæ LH ſit item tripla, denique eadem ſit ratio <lb/>jugi KL ad LM quæ curtæ pyramidis ABCDEF <lb/>ad complementum ABCG; </s> <s xml:id="echoid-s2409" xml:space="preserve">ajo M eſſa gravita-<lb/>tis optatum centrum.</s> <s xml:id="echoid-s2410" xml:space="preserve"/> </p> <div xml:id="echoid-div350" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.073-02" xlink:href="fig-527.01.073-02a"> <image file="527.01.073-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.073-02"/> </figure> </div> </div> <div xml:id="echoid-div352" type="section" level="1" n="250"> <head xml:id="echoid-head264" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2411" xml:space="preserve">L centrum eſt totius, K verò ſegmenti, ut au-<lb/>tem imum ſegmentum ad ſummum, ſic KL ad <lb/>LM. </s> <s xml:id="echoid-s2412" xml:space="preserve">Quare per 1 propoſ. </s> <s xml:id="echoid-s2413" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2414" xml:space="preserve">M fuerit optatum <lb/>gravitatis centrum. </s> <s xml:id="echoid-s2415" xml:space="preserve">Demonſtratio in cæteris cur-<lb/>tis pyramidibus multangulæ vel circularis baſis huic affinis eſt.</s> <s xml:id="echoid-s2416" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2417" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2418" xml:space="preserve">Quamobrem datæ curta pyramidis gravitatis centrum, ut <lb/>decuit, invenimus.</s> <s xml:id="echoid-s2419" xml:space="preserve"/> </p> <pb o="74" file="527.01.074" n="74" rhead="2 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div353" type="section" level="1" n="251"> <head xml:id="echoid-head265" xml:space="preserve">9 PROBLEMA. 21 PROPOSITIO.</head> <p> <s xml:id="echoid-s2420" xml:space="preserve">Dato ſolido epipedoëdro quocunque; </s> <s xml:id="echoid-s2421" xml:space="preserve">gravitatis centrum <lb/>invenire.</s> <s xml:id="echoid-s2422" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2423" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2424" xml:space="preserve">Eſto epipedoëdrum A quotcunque planis ſuperficiebus com-<lb/>prehenſum. </s> <s xml:id="echoid-s2425" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s2426" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2427" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div354" type="section" level="1" n="252"> <head xml:id="echoid-head266" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2428" xml:space="preserve">Solidum ipſum tribuito in pyramides componentes, quam fieri poterit pau-<lb/>ciſſimas. </s> <s xml:id="echoid-s2429" xml:space="preserve">Summa autem eo caſu difficultas hucredit, utſi neceſſum ſit ſolidum <lb/>ipſum in totpyramides dirimatur quot hedris clauditur, pun-<lb/> <anchor type="figure" xlink:label="fig-527.01.074-01a" xlink:href="fig-527.01.074-01"/> cto quocunque intra corpus pro vertice aſſumpto; </s> <s xml:id="echoid-s2430" xml:space="preserve">quibus cõ-<lb/>ſtitutis, pyramidum centra ſigillatim per 17 propoſ. </s> <s xml:id="echoid-s2431" xml:space="preserve">invenian-<lb/>tur. </s> <s xml:id="echoid-s2432" xml:space="preserve">deinde duorum pyramidum centris rectâ linea connexis, <lb/>jugum hoc ſecetur ratione ipſorũ pyramidum, ut tamen mi-<lb/>nus ſegmentum ponderoſiori pyramidi ſit vicinum, deinde <lb/>centrum hoc inventum cum tertiæ pyramidis centro conjungatur, quarũ com-<lb/>mune centrum cum quarto connectes, atque eò in reliquis omnibus ordine <lb/>continuato, noviſſima jugi ſectio exhibebit optatum dati ſolidi gravitatis cen-<lb/>trum; </s> <s xml:id="echoid-s2433" xml:space="preserve">cujus demonſtratio ipſo operis ſucceſſu manifeſta eſt.</s> <s xml:id="echoid-s2434" xml:space="preserve"/> </p> <div xml:id="echoid-div354" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.074-01" xlink:href="fig-527.01.074-01a"> <image file="527.01.074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.074-01"/> </figure> </div> <p> <s xml:id="echoid-s2435" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2436" xml:space="preserve">Itaque, dato qualicunque ſolido planis hedris compre-<lb/>henſo, gravitatis centrum invenimus. </s> <s xml:id="echoid-s2437" xml:space="preserve">Quod feciſſe oport<unsure/>uit.</s> <s xml:id="echoid-s2438" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div356" type="section" level="1" n="253"> <head xml:id="echoid-head267" xml:space="preserve">13 PROBLEMA. 22 PROPOSITIO.</head> <p> <s xml:id="echoid-s2439" xml:space="preserve">Conoïdalis gravitatis centrum eſt in axe.</s> <s xml:id="echoid-s2440" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2441" xml:space="preserve">Conoïdalis recti centrum gravitatis eſſe in axe, per ſe & </s> <s xml:id="echoid-s2442" xml:space="preserve">communi quaſi no-<lb/>titiâ manifeſtum eſt, quamobrem duntaxat eo caſu cum axis baſi obliquus erit <lb/>demonſtrationem formabimus.</s> <s xml:id="echoid-s2443" xml:space="preserve"/> </p> <figure> <image file="527.01.074-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.074-02"/> </figure> <p> <s xml:id="echoid-s2444" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2445" xml:space="preserve">ABC conoïdale, baſis BC, <lb/>axis AD dictæ baſi obliquus.</s> <s xml:id="echoid-s2446" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2447" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2448" xml:space="preserve">Gravitatis centrum in <lb/>AD conſiſtere demonſtrandum.</s> <s xml:id="echoid-s2449" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2450" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s2451" xml:space="preserve">Conoïdale inter-<lb/>ſecetur planis duobus FF, GH baſi pa-<lb/>rallelis quæ axem AD incîdant in I & </s> <s xml:id="echoid-s2452" xml:space="preserve">K, <lb/>deinde ducantur rectæ EL, FM, GN, <lb/>HO, quare LM, NO, GH exipsâ ſe-<lb/>ctione ellipſes erunt ſimiles baſi BC: </s> <s xml:id="echoid-s2453" xml:space="preserve">& </s> <s xml:id="echoid-s2454" xml:space="preserve"><lb/>EM, GO cylindri baſis ellipticæ</s> </p> </div> <div xml:id="echoid-div357" type="section" level="1" n="254"> <head xml:id="echoid-head268" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2455" xml:space="preserve">LD ſemidiameter ellipſis LM, æquatur ſemidiametro DM, æquatur item <lb/>ipſis EI, IF; </s> <s xml:id="echoid-s2456" xml:space="preserve">Igitur ID axis fuerit cylindri EM in quo ejus gravitatis cen-<lb/>trum conſiſtit; </s> <s xml:id="echoid-s2457" xml:space="preserve">pari ratione cylindri GO gravitatis centrum erit in axe KI. <lb/></s> <s xml:id="echoid-s2458" xml:space="preserve">quamobrem centrum ſolidi ex utroque compoſiti erit in KD, atque adeò in <lb/>axe AD. </s> <s xml:id="echoid-s2459" xml:space="preserve">ſed quò crebriores cylindri in conoïdale inſcribentur, eò differentia <pb o="75" file="527.01.075" n="75" rhead="*DE* S*TATICÆ PRINCIPIIS*."/> ſolidi ex inſcriptis cylindris compoſiti à dato minus erit. </s> <s xml:id="echoid-s2460" xml:space="preserve">Itaque infinita hac in-<lb/>ſcriptione tandem eò adſcenditur ut ſolidum factitium à conoïdali abl<unsure/>it diffe-<lb/>rentiâ, quæ ſolido dato quocunque minor ſit, cui conſequens eſt AD dati co-<lb/>noïdalis gravitatis eſſe diametrum, itaque gravitas ſitus unius lateris à gravita-<lb/>te lateris alterius minus aberit, quam vel minimi ponderis differentiâ. </s> <s xml:id="echoid-s2461" xml:space="preserve">Quod <lb/>legittimo ſyllogiſmi judicio ita concludam.</s> <s xml:id="echoid-s2462" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2463" xml:space="preserve">Ponderum ſitu gravium differentiâ minus pondus dari poteſt. <lb/></s> <s xml:id="echoid-s2464" xml:space="preserve">Sed borum ſegmentorum ſitu gravium differentiâ pondus minus nullu dari poteſt. </s> <s xml:id="echoid-s2465" xml:space="preserve"><lb/>Itaque borum conoïdalis ſegmentorum ſitu gravium differentia nullaeſt.</s> <s xml:id="echoid-s2466" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2467" xml:space="preserve">Et AD gravitatis erit diameter. </s> <s xml:id="echoid-s2468" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2469" xml:space="preserve">Quamobrem conoïda-<lb/>lis gravitatis centrum eſt in axe. </s> <s xml:id="echoid-s2470" xml:space="preserve">quod demonſtraſſe oportuit.</s> <s xml:id="echoid-s2471" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div358" type="section" level="1" n="255"> <head xml:id="echoid-head269" xml:space="preserve">10 PROBLEMA. 23 PROPOSITIO.</head> <p> <s xml:id="echoid-s2472" xml:space="preserve">Conoïdalis gravitatis centrum invenire.</s> <s xml:id="echoid-s2473" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2474" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2475" xml:space="preserve">ABC conoïdale, A vertex, AD axis.</s> <s xml:id="echoid-s2476" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2477" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2478" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2479" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div359" type="section" level="1" n="256"> <head xml:id="echoid-head270" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2480" xml:space="preserve">A D axis ſecetur in E ratione dupla videlicet ut ſegmentum vertici conter-<lb/>minum reliqui ſit duplum, ajo E eſſe centrum quæſitum cujus demonſtrario-<lb/>nem ſolers & </s> <s xml:id="echoid-s2481" xml:space="preserve">ſubtilis Mathematicus Fredericus Commandinus de ſolidorũ cen-<lb/>trobaricis propoſ. </s> <s xml:id="echoid-s2482" xml:space="preserve">29 exhibet, quæ noſtro more & </s> <s xml:id="echoid-s2483" xml:space="preserve">modo digeſta ita habet.</s> <s xml:id="echoid-s2484" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div360" type="section" level="1" n="257"> <head xml:id="echoid-head271" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2485" xml:space="preserve">Conoïdale ſecetur plano FG axem in H biſecante, baſiq́ue BC parallelo, <lb/>atque planiſecantis & </s> <s xml:id="echoid-s2486" xml:space="preserve">ſuperficiei ſectio eſto in I, K, deinde BCGF, IKLM <lb/>cylindri circa conoïdale circumſcribantur, quorum gravitatis centra N, O: <lb/></s> <s xml:id="echoid-s2487" xml:space="preserve">præterea intra ipſum cylindri IKPQ inſcripti O itidem gravitatis erit centrũ. </s> <s xml:id="echoid-s2488" xml:space="preserve"><lb/>Cum per 20 prop. </s> <s xml:id="echoid-s2489" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2490" xml:space="preserve">Apoll. </s> <s xml:id="echoid-s2491" xml:space="preserve">& </s> <s xml:id="echoid-s2492" xml:space="preserve">2. </s> <s xml:id="echoid-s2493" xml:space="preserve">pr. </s> <s xml:id="echoid-s2494" xml:space="preserve">12. </s> <s xml:id="echoid-s2495" xml:space="preserve">lib. </s> <s xml:id="echoid-s2496" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-527.01.075-01a" xlink:href="fig-527.01.075-01"/> Eucl. </s> <s xml:id="echoid-s2497" xml:space="preserve">igitur ſit ut DA ad AH videlicet 2 <lb/>ad 1, ſic circulus BC ad circulũ IK, etiam <lb/>cylindri BC ad cylindrum IL (propter æ-<lb/>qualĕ altitudinem) ratio dupla erit, quam <lb/>obrem ſi BG 2 librarum ſtatu@ur IL erit <lb/>1 libræ, ſed centra gravitatis ſunt N, O, <lb/>ideoq́ue NO jugo in R ſecto ut NR <lb/>radii RO duplus ſit, ipſum circumſcripto-<lb/>rum cylindrorum gravitatis erit centrum, <lb/>ſed & </s> <s xml:id="echoid-s2498" xml:space="preserve">O inſcripti cylindri eſt centrum, E verò ab O & </s> <s xml:id="echoid-s2499" xml:space="preserve">ab R eodem intervallo <lb/>diſtat, videlicet {1/12} totius AD. </s> <s xml:id="echoid-s2500" xml:space="preserve">Acſimilis erit cæterorum ſimilium paradigma-<lb/>r<unsure/>um eventus. </s> <s xml:id="echoid-s2501" xml:space="preserve">Verumenimverò quo res ſit manifeſtior, altero exemplo idem ex-<lb/>plicabimus.</s> <s xml:id="echoid-s2502" xml:space="preserve"/> </p> <div xml:id="echoid-div360" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.075-01" xlink:href="fig-527.01.075-01a"> <image file="527.01.075-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.075-01"/> </figure> </div> <p> <s xml:id="echoid-s2503" xml:space="preserve">Denuò iſta axis biſegmenta AH, HD, bifariam dividantur, unde tres cy-<lb/>lindri inſcribantur & </s> <s xml:id="echoid-s2504" xml:space="preserve">quatuor circumſcribantur, ut in ſecundo diagrammate <lb/>ubi AD conoïdalis axis ſit, centra verò cylindrorum I, K, L, M, AE verò <lb/>dupla ſit ipſius ED ut ſupra. </s> <s xml:id="echoid-s2505" xml:space="preserve">Itaque cum ſit ut AD ad AN (nempe ut 4 ad 3) <lb/>ſic circulus BC ad circulum OP, erit quoque cylindrus BF ad OQ in ea- <pb o="76" file="527.01.076" n="76" rhead="2 L*IBER* S*TATICÆ*"/> dem ratione ſeſquitertia ſunt enim æquealti, ſimillima ratione BF cylindrus <lb/>rertii circumſcripti cujus centrum K erit du-<lb/> <anchor type="figure" xlink:label="fig-527.01.076-01a" xlink:href="fig-527.01.076-01"/> plus, quarti verò cujus centrum I quadru-<lb/>plus. </s> <s xml:id="echoid-s2506" xml:space="preserve">poſito itaque imo cylindro 4 librarum, <lb/>ſecundus erit 3 ℔, tertius 2 ℔, ſummus de-<lb/>nique I ℔: </s> <s xml:id="echoid-s2507" xml:space="preserve">Pari ratione ſi imus inſcriptorum <lb/>ſit 3 librarum, ſecundus erit 2 ℔, ultimus ver <lb/>tici proximus I ℔. </s> <s xml:id="echoid-s2508" xml:space="preserve">Quæ cum ita ſint, & </s> <s xml:id="echoid-s2509" xml:space="preserve">cen-<lb/>tra cylindrorum, & </s> <s xml:id="echoid-s2510" xml:space="preserve">ipſorum ponderoſitas <lb/>nota, centrum gravitatis circumſcriptorum <lb/>cadet in L ut LE occupet {1/24} totius AD; <lb/></s> <s xml:id="echoid-s2511" xml:space="preserve">Trium itidem inſcriptorum gravitatis centrum cadet in S, ut S E {1/24} totius A D <lb/>obtineat. </s> <s xml:id="echoid-s2512" xml:space="preserve">Quamobrem L & </s> <s xml:id="echoid-s2513" xml:space="preserve">S ab E rurſum æquidiſtant.</s> <s xml:id="echoid-s2514" xml:space="preserve"/> </p> <div xml:id="echoid-div361" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.076-01" xlink:href="fig-527.01.076-01a"> <image file="527.01.076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.076-01"/> </figure> </div> <p> <s xml:id="echoid-s2515" xml:space="preserve">Verumenimvero ſi biſectio & </s> <s xml:id="echoid-s2516" xml:space="preserve">cylindrorum iſta ſiguratio continuentur, ut <lb/>octo datum conoïdale ambiantac ſeptem induant, diſtantia centrorum & </s> <s xml:id="echoid-s2517" xml:space="preserve">in-<lb/>ſcriptorum & </s> <s xml:id="echoid-s2518" xml:space="preserve">circumſcriptorum ſecabunt axem æquidiſtanter à puncto E, ab-<lb/>erunt enim {1/48} totius A D.</s> <s xml:id="echoid-s2519" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2520" xml:space="preserve">Denique ſi ſectio iſta viciſſim duplicetur ut ſedecim cylindri circumſcriban-<lb/>tur, & </s> <s xml:id="echoid-s2521" xml:space="preserve">quindecim intra includantur, nihilo ſecius centra gravitatis ſolidorum <lb/>inſcriptorum & </s> <s xml:id="echoid-s2522" xml:space="preserve">circumſcriptorum pari diſtantia ab E puncto utrimque diſta-<lb/>bant, videlicet {1/96} axis A D. </s> <s xml:id="echoid-s2523" xml:space="preserve">Atque adeò ſequens biſectio antecedentem di-<lb/>ſtantiam continuò bipartito ſecat, cujus conſecutionis veritatem & </s> <s xml:id="echoid-s2524" xml:space="preserve">neceſſita-<lb/>tem inductione continuatâ demonſtrarem, niſi brevitatis ſtudio ductus, cum <lb/>cuilibet in promptu ſit, iſtud omitterem.</s> <s xml:id="echoid-s2525" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2526" xml:space="preserve">Quamobrem E gravitatis centrum dati conoïdalis erit: </s> <s xml:id="echoid-s2527" xml:space="preserve">Enimverò ſi cen-<lb/>trum aliud ſumatur in ipſa E L aut E S tandem continua biſectione & </s> <s xml:id="echoid-s2528" xml:space="preserve">cylin-<lb/>drorum circumſcriptione & </s> <s xml:id="echoid-s2529" xml:space="preserve">inſcriptione eò devenitur ut centrum ſolidi ex cir-<lb/>cumſcriptis conflati deſcĕdat infra conoïdalis centrum; </s> <s xml:id="echoid-s2530" xml:space="preserve">vel inſcripti ſupra ejuſ-<lb/>dem conoïdalis centrũ adſcendat. </s> <s xml:id="echoid-s2531" xml:space="preserve">Quod impoſſibile per ſe clarum fuerit; </s> <s xml:id="echoid-s2532" xml:space="preserve">cum <lb/>enim ſolidum tale è cylindris circumſcriptis componi poſſit ut ejus à conoï-<lb/>dali differentia minor ſit quocunque ſolido, poſſit item tale ſolidum inſcribi, <lb/>utriuſque ab E puncto differentia tantulo utrimque intervallo aberit ut minus <lb/>nullum effingi poſſit. </s> <s xml:id="echoid-s2533" xml:space="preserve">Quamobrem eodem coïbunt in E. </s> <s xml:id="echoid-s2534" xml:space="preserve">Vnde efficitur E dati <lb/>conoïdalis gravitatis eſſe centrum.</s> <s xml:id="echoid-s2535" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2536" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2537" xml:space="preserve">Itaque conoïdalis gravitatis centrum invenimus. </s> <s xml:id="echoid-s2538" xml:space="preserve">Quod <lb/>feciſſe oportuit.</s> <s xml:id="echoid-s2539" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div363" type="section" level="1" n="258"> <head xml:id="echoid-head272" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s2540" xml:space="preserve">Cum recta ab angulo trianguli ad medium oppoſitæ baſis educta per 4 pro-<lb/>poſ. </s> <s xml:id="echoid-s2541" xml:space="preserve">item ſecetur ratione dupla, conſequens eſt, ſimilem à centro æquidiſtan-<lb/>tiam iſtic ab inſcriptis & </s> <s xml:id="echoid-s2542" xml:space="preserve">circumſcriptis parallelogrammis argui, qualis hic in <lb/>cylindris adſcriptis demonſtrata eſt.</s> <s xml:id="echoid-s2543" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div364" type="section" level="1" n="259"> <head xml:id="echoid-head273" xml:space="preserve">11 THE OREMA. 24 PROPOSITIO.</head> <head xml:id="echoid-head274" xml:space="preserve">Conoïdalis curtigravitatis centrum invenire.</head> <p> <s xml:id="echoid-s2544" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2545" xml:space="preserve">ABCD conoïdale curtum, baſis ima D C, ſumma A B, axis <lb/>verò EF. </s> <s xml:id="echoid-s2546" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2547" xml:space="preserve">Gravitatis centrum invenire.</s> <s xml:id="echoid-s2548" xml:space="preserve"/> </p> <pb o="77" file="527.01.077" n="77" rhead="*DE* S*TATIGÆ PRINGIPIIS*."/> </div> <div xml:id="echoid-div365" type="section" level="1" n="260"> <head xml:id="echoid-head275" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2549" xml:space="preserve">Conoïdale curtum abſolvito, addito ſegmento A B G; </s> <s xml:id="echoid-s2550" xml:space="preserve">deinde ſtatuatur GH <lb/>dupla ipſius H E, item G I ipſius I F; </s> <s xml:id="echoid-s2551" xml:space="preserve">denique fiat ut conoïdale curtũ A B C D, <lb/>ad complementum A B G ſic H I ad I K; </s> <s xml:id="echoid-s2552" xml:space="preserve">Ajo K gravitatis cupitum eſſe cen-<lb/>trum.</s> <s xml:id="echoid-s2553" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div366" type="section" level="1" n="261"> <head xml:id="echoid-head276" xml:space="preserve">DEMONSTRATIO.</head> <figure> <image file="527.01.077-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.077-01"/> </figure> <p> <s xml:id="echoid-s2554" xml:space="preserve">Etenim I eſt centrum gravitatis totius <lb/>D C G, & </s> <s xml:id="echoid-s2555" xml:space="preserve">H ſegmenti A B G: </s> <s xml:id="echoid-s2556" xml:space="preserve">verum ut <lb/>reliquum ſegmentum A B C D, ad comple-<lb/>mentum A B G ſic H I ad I K; </s> <s xml:id="echoid-s2557" xml:space="preserve">ideo K per <lb/>19 propoſ. </s> <s xml:id="echoid-s2558" xml:space="preserve">optatum erit centrum, Quod de-<lb/>monſtraſſe oportuit.</s> <s xml:id="echoid-s2559" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2560" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2561" xml:space="preserve">Itaque curti conoïdalis centrum invenimus.</s> <s xml:id="echoid-s2562" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div367" type="section" level="1" n="262"> <head xml:id="echoid-head277" style="it" xml:space="preserve">Atque hic Liber ſecundus nobis explicitus esto.</head> <pb file="527.01.078" n="78"/> <pb file="527.01.079" n="79"/> </div> <div xml:id="echoid-div368" type="section" level="1" n="263"> <head xml:id="echoid-head278" xml:space="preserve">LIBER TERTIVS <lb/>DE <lb/>STATIC AE <lb/>PRAXI.</head> <pb file="527.01.080" n="80"/> <pb o="81" file="527.01.081" n="81" rhead="AD LECTOREM"/> <p style="it"> <s xml:id="echoid-s2563" xml:space="preserve">QVandoquidem nonnullis bujus Praxis <lb/>propoſitionibus de corporum motu dictu-<lb/>riſumus, paucis priuſquam in rem ingre-<lb/>dimur lectoriaperire viſum fuit Stati-<lb/>cem duntaxat potentiæ moventis & </s> <s xml:id="echoid-s2564" xml:space="preserve">pon-<lb/>derismotiſitus æquilibritatem ſeu æqua. <lb/></s> <s xml:id="echoid-s2565" xml:space="preserve">mentum docere; </s> <s xml:id="echoid-s2566" xml:space="preserve">quanta verò moventis <lb/>potentiæ vis præterea deſideretur, (nam <lb/>cuilibet movendoimpedimentũ ſola cogitatione tantũ ſepar abi-<lb/>le perpetuò inbæret, quod & </s> <s xml:id="echoid-s2567" xml:space="preserve">ipſum ſuperari quoque neceſſe eſt,) <lb/>quò datum pondus commoveatur & </s> <s xml:id="echoid-s2568" xml:space="preserve">impellatur, à Staticæ do-<lb/>ctrina alienũ fuerit; </s> <s xml:id="echoid-s2569" xml:space="preserve">cum via & </s> <s xml:id="echoid-s2570" xml:space="preserve">ratione mathematica iste po-<lb/>tentiæ exceſſus invenirineque explicari poſsit, etenim impedi-<lb/>menta & </s> <s xml:id="echoid-s2571" xml:space="preserve">res mot æ constantem analogiam nullam babent. </s> <s xml:id="echoid-s2572" xml:space="preserve">Ve-<lb/>rùm quò ratio iſta ſit magis perſpicua, eſto currus gravitatis <lb/>notæin cognitæ acclivitatis collem pertrabendus. </s> <s xml:id="echoid-s2573" xml:space="preserve">Ajo ſtaticam, <lb/>ut 4 exemplo 9 propaſ. </s> <s xml:id="echoid-s2574" xml:space="preserve">perſpicitur; </s> <s xml:id="echoid-s2575" xml:space="preserve">docere quæpotentiaſit currui <lb/>æqualis, boc eſt, ſitu æquiponderet non habita motus aut impe-<lb/>dimentorumratione, cujus generis ſunt illa ſingularium par-<lb/>tium ut axium in ſuis ſyringibus, rotarum in plateis, ac deni{q́ue} <lb/>totius currus in aere. </s> <s xml:id="echoid-s2576" xml:space="preserve">Impedimentorum inquam potentia cum <lb/>catholica non ſit à ſtaticæ præceptis rejicienda, quia eius ad po-<lb/>tentiam moventem ratio unica & </s> <s xml:id="echoid-s2577" xml:space="preserve">certa nulla apparet. </s> <s xml:id="echoid-s2578" xml:space="preserve">Quod <lb/>refutatis eorum argumentis qui in ponderibus deprimentibus <lb/>ſecus ſtatuunt, arguerem, niſi in ſtatica tantum doctrinæ præ-<lb/>cepta instituere ſatius ducerem, alibi priſtinũ & </s> <s xml:id="echoid-s2579" xml:space="preserve">inveteratum <lb/>de ponderũ affectionibus errorem firmis rationibus retecturus. </s> <s xml:id="echoid-s2580" xml:space="preserve"><lb/>Cæterum æquilibritatis cognitio hic ſufficit; </s> <s xml:id="echoid-s2581" xml:space="preserve">enimverò ſiin li-<lb/>bræ utrâque lance tantundem ponderis collocetur, quamvis <lb/>librile ſeu jugum ſua habeat motus impedimenta, levi tamen <lb/>momento lances huc at que illuc impelluntur: </s> <s xml:id="echoid-s2582" xml:space="preserve">idque in cæteris <lb/>omnibus evenire certum eſt.</s> <s xml:id="echoid-s2583" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s2584" xml:space="preserve">At que bæc quidem de motus impedimentis dicta ſunto, ne <pb o="82" file="527.01.082" n="82"/> quis uſu magistro eruditus potentiam moventem moto pon-<lb/>dere majorem eſſe, vitium boc arti adſcribat ſiquidem poten-<lb/>tia motrix movendo tanto ſit maior oportet, ut ipſum quoque <lb/>impedimentum ſuper are poſsit. </s> <s xml:id="echoid-s2585" xml:space="preserve">Neve quis hoc æquilibritatis <lb/>prætextu fiſus in errorem præceps ruat, quod his contingit <lb/>maximè quifalſas ſententias pro veris amplexantur.</s> <s xml:id="echoid-s2586" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div369" type="section" level="1" n="264"> <head xml:id="echoid-head279" xml:space="preserve">BREVIARIVM.</head> <p> <s xml:id="echoid-s2587" xml:space="preserve">PR*AXIS* Statices gravitatis planum diametrale, gravi-<lb/>tatis rectam diametrum & </s> <s xml:id="echoid-s2588" xml:space="preserve">centrum pragmaticè five <lb/>mechanicè invenire docebit, hinc libræ & </s> <s xml:id="echoid-s2589" xml:space="preserve">ſtateræ perfectif-<lb/>fimæ fabricas atque affectiones nõnullas, quibus vectium <lb/>genera, & </s> <s xml:id="echoid-s2590" xml:space="preserve">ponderum tum geftatorum tum attractorum <lb/>& </s> <s xml:id="echoid-s2591" xml:space="preserve">trochlearum formas, ac denique infinitam potentiam <lb/>ſubjunget.</s> <s xml:id="echoid-s2592" xml:space="preserve"/> </p> <pb o="83" file="527.01.083" n="83" rhead="I PROPOSITIO."/> <p> <s xml:id="echoid-s2593" xml:space="preserve">Dati cujuſcunque corporis planum diametrale, diame-<unsure/> <lb/>trum perpendicularem & </s> <s xml:id="echoid-s2594" xml:space="preserve">centrum pragmaticè invenire.</s> <s xml:id="echoid-s2595" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div370" type="section" level="1" n="265"> <head xml:id="echoid-head280" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s2596" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2597" xml:space="preserve">A B corpus figurâ qualicunque.</s> <s xml:id="echoid-s2598" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2599" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2600" xml:space="preserve">Planum diametrale, diametrum perpendicularem, cen-<lb/>trum denique mechanicè anquirere.</s> <s xml:id="echoid-s2601" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div371" type="section" level="1" n="266"> <head xml:id="echoid-head281" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2602" xml:space="preserve">Solidum datum ſuſpendatur è fune CD, acta deinde recta EF per ſum-<lb/>mum punctum C, ab utroque ejus termino perpendiculares E G, F H è plum-<lb/>bo & </s> <s xml:id="echoid-s2603" xml:space="preserve">filo ſuſpendantur, quæ corpus ipſum A B proximè ſtringant, planum <lb/> <anchor type="figure" xlink:label="fig-527.01.083-01a" xlink:href="fig-527.01.083-01"/> inter parallelas E G, F H comprehenſum, ſiſolidum ſecare intelligatur, dati <lb/>corporis diametrale planum erit, verùm ut ipſæ lineæ in ſolidi ſuperficie inſcri-<lb/>bantur, fila tenſa & </s> <s xml:id="echoid-s2604" xml:space="preserve">creta infecta impreſſionem faciant uti à materiariis fabris <lb/>fieri confuevit, illæ igitur ſunto I K, L M quæ connexæ optatum LIK M pla-<lb/>num continuebunt.</s> <s xml:id="echoid-s2605" xml:space="preserve"/> </p> <div xml:id="echoid-div371" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.083-01" xlink:href="fig-527.01.083-01a"> <image file="527.01.083-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.083-01"/> </figure> </div> <p> <s xml:id="echoid-s2606" xml:space="preserve">Sed ad diametri perpendicularis inventionem corpus ex eadem recta C D <lb/>dependens pauxillum convertito & </s> <s xml:id="echoid-s2607" xml:space="preserve">ſimile aliud diametrale planum delineato, <lb/>à quo prius infra in N ſupra autem in D interſecetur, recta D N, ipſorum com-<lb/>munis fectio, erit perpendicularis gravitatis diameter. </s> <s xml:id="echoid-s2608" xml:space="preserve">Denique ad centri inve-<lb/>ſtigationem corpus tranſverſim alicunde ex O ſuſpenſum exhibeat alteram <lb/>perpendicularem diametrum O P, quæ priorem incîdatin Q, idipſum gra-<lb/>vitatis erit centrum.</s> <s xml:id="echoid-s2609" xml:space="preserve"/> </p> <pb o="84" file="527.01.084" n="84" rhead="2 *LIBER STATICÆ*"/> </div> <div xml:id="echoid-div373" type="section" level="1" n="267"> <head xml:id="echoid-head282" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s2610" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2611" xml:space="preserve">Eſto A B ſolidum quodcunque datum.</s> <s xml:id="echoid-s2612" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2613" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2614" xml:space="preserve">Gravitatis planum diametrale, diametrum perpendicula-<lb/>rem, atque ipſum denique centrum mechanicè invenite.</s> <s xml:id="echoid-s2615" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div374" type="section" level="1" n="268"> <head xml:id="echoid-head283" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2616" xml:space="preserve">Oblatum corpus A B in acie C D verſato donec æquamentum utriuſque <lb/>partis adeptus videbere, ſitq́ue in E, planum igitur per E horizonti normale <lb/>erit quæſitum diametrale planum, quod altero ſimili interſectum gravitatis <lb/>perpendicularem diametrum in communiſe-<lb/>ctione deſcribet, denique tertium tranſver-<lb/>ſum diametrale planum eandem in gravitatis <lb/>centro incîdet. </s> <s xml:id="echoid-s2617" xml:space="preserve">Quarum demonſtratio ex an-<lb/>tecedentibus perſpicitur.</s> <s xml:id="echoid-s2618" xml:space="preserve"/> </p> <figure> <image file="527.01.084-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.084-01"/> </figure> <p> <s xml:id="echoid-s2619" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2620" xml:space="preserve">Itaque ſolidi cujuſcun-<lb/>quegravitatis diametrale planum, diametrum <lb/>perpendicularem, & </s> <s xml:id="echoid-s2621" xml:space="preserve">centrum invenimus. <lb/></s> <s xml:id="echoid-s2622" xml:space="preserve">Quodfacere oportebat.</s> <s xml:id="echoid-s2623" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div375" type="section" level="1" n="269"> <head xml:id="echoid-head284" xml:space="preserve">2 PROPOSITIO</head> <p> <s xml:id="echoid-s2624" xml:space="preserve">Libram perfectiſsimam fabricari.</s> <s xml:id="echoid-s2625" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div376" type="section" level="1" n="270"> <head xml:id="echoid-head285" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2626" xml:space="preserve">In medio ſcapiſeu librilis A B, cujus examen loco convenienti ſit, rectam <lb/>C D ſub examinis mediolateribus ſcapi perpendicularem inſcribito, tantum <lb/>deinde materię à parte præponderante auferto donecſcapus ſecundum rectam <lb/>C D aciei impoſitus utroque radio æquamentum nactus erit. </s> <s xml:id="echoid-s2627" xml:space="preserve">hinc E D duci-<lb/>to lateri perpĕdicularem, in qua perpendicularem diametrũ inquires, ſcapo ſeu <lb/>librili acutiſſimo mucroni impoſito atque huc illuc ſecundum rectam D E im-<lb/>pellens donec æquilibritas inventa ſit, ut puta in puncto F, punctum deinde <lb/>ſimile in oppoſito latere imprimito, recta ea connectens erit ſcapi perpendicu-<lb/>laris diameter notans aciem, axis tranſverſi quod eſtferramentum tranſverſim <lb/>librili infixum extremæ aginæ fibulis inhærens deinde quia lances è librili un-<lb/>cis ſuſpenduntur, horum confinia A, B & </s> <s xml:id="echoid-s2628" xml:space="preserve">tranſverſi axis terminus conſtituan-<lb/>tur in eàdem recta A F B, confinia inquam uncorum illa quâ librile contin-<lb/>gunt. </s> <s xml:id="echoid-s2629" xml:space="preserve">Sin verò lances non uncis ſed alius generis retinaculis è jugoſeu librili <lb/>dependeant, ipſorum ſimile confinium conſiderandum, quibus & </s> <s xml:id="echoid-s2630" xml:space="preserve">aginæ fibu-<lb/>lis loco convenienti conſtitutis, libra iſta ponderibus æqualibus in utramque <lb/>lancem impoſitis, ſervabit eum quem dederis ſitum, quamdiu axis tranſverſus <lb/>aciei ſuæ innitetur, cujus veritas è 10 propoſ. </s> <s xml:id="echoid-s2631" xml:space="preserve">I libri de Staticæ principiis clare <lb/>perſpicitur.</s> <s xml:id="echoid-s2632" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2633" xml:space="preserve">Verumenimverò libram hanc eſſe perfectiſſimam patet è I exemplo, 12 pro-<lb/>poſ. </s> <s xml:id="echoid-s2634" xml:space="preserve">I lib. </s> <s xml:id="echoid-s2635" xml:space="preserve">ubi demonſtratum eſt, poſito E firmitudinis puncto quantum pon- <pb o="85" file="527.01.085" n="85" rhead="DE S*TATICÆ PRAXI*."/> <anchor type="figure" xlink:label="fig-527.01.085-01a" xlink:href="fig-527.01.085-01"/> dusad D applicari oporteat utjugum in data theſi permaneat, ſi verò N iſtic <lb/>fuiſſet firmitudinis punctum, hoc eſt, dati corporis gravitatis centrum, quod-<lb/>libet vel minimum pondus, mathematicum videlicet, è D ſuſpenſum iſtud <lb/>latus omnino depreſſerit: </s> <s xml:id="echoid-s2636" xml:space="preserve">Idem hic evenire intelligas utrumlibet æquiponde-<lb/>rantium radiorum minimo pondere auctum ab æquamento deorſum vergere; <lb/></s> <s xml:id="echoid-s2637" xml:space="preserve">qui tamen in aliis nonnullis bilancibus tantulo pondere vix impellerentur.</s> <s xml:id="echoid-s2638" xml:space="preserve"/> </p> <div xml:id="echoid-div376" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.085-01" xlink:href="fig-527.01.085-01a"> <image file="527.01.085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.085-01"/> </figure> </div> <p> <s xml:id="echoid-s2639" xml:space="preserve">Sin autem fabrica hæc in iſta aciei tranſverſi axis & </s> <s xml:id="echoid-s2640" xml:space="preserve">retinaculorum confinii <lb/>tam accurata inveftigatione nimium negotii faceſſat, huc tamen omnia quam <lb/>poterit fieri accuratiſſimè dirigantur; </s> <s xml:id="echoid-s2641" xml:space="preserve">Et ſi pauxillum ab iſta perfectione abeſle <lb/>velint, librilis & </s> <s xml:id="echoid-s2642" xml:space="preserve">uncorum confinia infra rectam A B potius quam ſupra collo-<lb/>cantor, iecus enim hinc illud incommodi exſtiterit ut cuncta inter librandum <lb/>invertantur; </s> <s xml:id="echoid-s2643" xml:space="preserve">Imo quod ponderoſius fuerit leviſſimum quandoq́ue videbitur, <lb/>præcipuè ſi axis propter librilis longitudinem horizonti non ſit parallelus, <lb/>omnia ſiquidem eò redeunt unde motus ſumpſerint principium.</s> <s xml:id="echoid-s2644" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2645" xml:space="preserve">Librilis radios à medio axis tranſverſi pari intervallo abeſſe oportere, vel <lb/>hinc patet quod pondera ſuis radiis per I propoſ. </s> <s xml:id="echoid-s2646" xml:space="preserve">I lib. </s> <s xml:id="echoid-s2647" xml:space="preserve">ſunt reciproca, ideoq́ue <lb/>ſi radius alter centeſima parte reliquum excedat libram doloſam efficiet, quæ <lb/>enim pondera æquipondia viderentur revera centeſima parte differrent; </s> <s xml:id="echoid-s2648" xml:space="preserve">hinc <lb/>ſi radii parte quinta & </s> <s xml:id="echoid-s2649" xml:space="preserve">vigeſima longitudinis diſcreparent ponderum inter ſe <lb/>diſcrepantiæ ratio foret eadem quæ 4 ad 100 atque ita deinceps in cæteris.</s> <s xml:id="echoid-s2650" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2651" xml:space="preserve">Præterea ſcapi teretes & </s> <s xml:id="echoid-s2652" xml:space="preserve">oblongi ad accuratum librilis judicium non pa-<lb/>rum momenti afferunt. </s> <s xml:id="echoid-s2653" xml:space="preserve">Etenim è duobus ſcapis gravitate quidem pari, ſed <lb/>impari longitudine ut puta dupla, certum eſt unciæ, ſemunciæve, aut alterius <pb o="86" file="527.01.086" n="86" rhead="3 L*IBER* S*TATICE*"/> cujuſcun<unsure/>que ponderis potentiam in ſcapo longiore duplam eſſe ejus, quæin <lb/>minore. </s> <s xml:id="echoid-s2654" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2655" xml:space="preserve">Itaque perfectiſſimam libram deformavimus. <lb/></s> <s xml:id="echoid-s2656" xml:space="preserve">Quod feciſſe oportuit.</s> <s xml:id="echoid-s2657" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div378" type="section" level="1" n="271"> <head xml:id="echoid-head286" xml:space="preserve">3 PROPOSITIO.</head> <p> <s xml:id="echoid-s2658" xml:space="preserve">Data libra cujus librile horizonti maneat parallelum: <lb/></s> <s xml:id="echoid-s2659" xml:space="preserve">pondus invenire quod alteræ lanci impoſitum librile in <lb/>optato ſitu retineat.</s> <s xml:id="echoid-s2660" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2661" xml:space="preserve">Libræ nonnullæ difficilius impelluntur, cujus cauſa nulla nobis apparet, <lb/>cum tamen acies axis tranſverſi accuratâ ratione aptata ſit. </s> <s xml:id="echoid-s2662" xml:space="preserve">Cujus cauſam dein-<lb/>ceps detegemus, oſtendentes quantum pondus in hujuſmodi lancem imponi <lb/>debeat, quô optatum retineat ſitum.</s> <s xml:id="echoid-s2663" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2664" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2665" xml:space="preserve">Iugum libræ A B C D liberum neque impeditum, ad horizon-<lb/>tis paralleliſmum perpetuò redeat, cujus tranſverſi axis acies ſit E.</s> <s xml:id="echoid-s2666" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2667" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2668" xml:space="preserve">Pondus inveniendum quod lanci D impoſitum librile in <lb/>dato ſitu retineat.</s> <s xml:id="echoid-s2669" xml:space="preserve"/> </p> <figure> <image file="527.01.086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.086-01"/> </figure> <pb o="87" file="527.01.087" n="87" rhead="DE S*TATICÆ PRAXI*."/> </div> <div xml:id="echoid-div379" type="section" level="1" n="272"> <head xml:id="echoid-head287" xml:space="preserve">CONSTRVCTIO</head> <p> <s xml:id="echoid-s2670" xml:space="preserve">Remotis lancibus, uncis, & </s> <s xml:id="echoid-s2671" xml:space="preserve">agina tantum librilis cum examine gravitatis <lb/>diametrum illam, per 1 prop. </s> <s xml:id="echoid-s2672" xml:space="preserve">hujus lib. </s> <s xml:id="echoid-s2673" xml:space="preserve">inveſtigato, quæ aciei axis tranſverſi E <lb/>æquidiſtet, ſitq́ue F, deinde jugi & </s> <s xml:id="echoid-s2674" xml:space="preserve">uncorum confinia connectito recta G H, <lb/>ejusq́ue medium ſit F, hinc ſecato F I ratione quam gravitas librilis cum exa-<lb/>mine puta 1 ℔ habetad lances, uncos, & </s> <s xml:id="echoid-s2675" xml:space="preserve">funes è quibus dependent quæ item <lb/>1 ℔ ponderent, biſecta igitur F I in K, ipſum K erit punctum unde libra ſuſpen-<lb/>ſa quamcunque dederis theſin retinebit, tum jungito K G, hanc pendula dia-<lb/>metro E L per E ducta interſeces in M. </s> <s xml:id="echoid-s2676" xml:space="preserve">Ajo, pondus illud cujus ratio ad 2 ℔ <lb/>(etenim 1 ℔ pro jugo altera pro lancibus additæ 2 ℔ valent) eadem ſit, quæ <lb/>M K ad M G, lanci D impoſitum, libram tali ſitu retinere. </s> <s xml:id="echoid-s2677" xml:space="preserve">Dicis gratia fue-<lb/>rit M K pars quinta & </s> <s xml:id="echoid-s2678" xml:space="preserve">vigeſima radii M G, etiam {1/23} duarum ℔ efficiet ne li-<lb/>bra ab iſta theſi recedat, cujus cauſa è 12 propoſ. </s> <s xml:id="echoid-s2679" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2680" xml:space="preserve">evidens eſt; </s> <s xml:id="echoid-s2681" xml:space="preserve">attamen <lb/>majoris perſpicuitatis ergô paucula hic in medium afferemus.</s> <s xml:id="echoid-s2682" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div380" type="section" level="1" n="273"> <head xml:id="echoid-head288" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2683" xml:space="preserve">Siquidem K centrum gravitatis ſit dati, etiam perpendicularis per K edu-<lb/>cta erit ejuſdem gravitatis diameter, ſed perpendicularis per G eſt item ponde-<lb/>ris in lancem injecti diameter, ideoq́ue K G ipſorum jugum, quod in M pun-<lb/>cto ita partiti ſumus, ut ratio ſegmentorum M G, M K ea ſit quæ ponderum <lb/>reciprocè quare perpendicularis per M demiſſâ, erit utriuſque communis gra-<lb/>vitatis diameter ſeu anſa, atqueideo librile iſtum ſervabit ſitum.</s> <s xml:id="echoid-s2684" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2685" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2686" xml:space="preserve">Itaque data libra cujus librile horizonti æquidiſtet, pon-<lb/>dus invenimus quod alteræ lanci impoſitum ipſum librile in optata theſi con-<lb/>ſervet. </s> <s xml:id="echoid-s2687" xml:space="preserve">Quod facere oportebat.</s> <s xml:id="echoid-s2688" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div381" type="section" level="1" n="274"> <head xml:id="echoid-head289" xml:space="preserve">4 PROPOSITIO.</head> <p> <s xml:id="echoid-s2689" xml:space="preserve">Dato librili quod lancibus appenſis horizonti æquidi-<lb/>ſtet, ſine quibus aciei axis tranſverſi inniti non poſsit; </s> <s xml:id="echoid-s2690" xml:space="preserve">lan-<lb/>ces invenire ut ipſum librile datum retineat ſitum.</s> <s xml:id="echoid-s2691" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2692" xml:space="preserve">Experientia edocemur nonnulla librilia aciei axis tranſverſi inhærere non <lb/>poſſe, ſed declinare, additis autem lancibus rectè inniti, cujus rei cauſa pragma-<lb/>ticè nobis inveſtiganda eſt.</s> <s xml:id="echoid-s2693" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2694" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s2695" xml:space="preserve">Librile A B cujuſmodi propoſitio depoſcit, axis tranſverſi <lb/>acies C. </s> <s xml:id="echoid-s2696" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s2697" xml:space="preserve">Geminas lances (ſuis uncis & </s> <s xml:id="echoid-s2698" xml:space="preserve">funibus perfectas in-<lb/>telligito) ejus ponderis invenire, quæ jugum eo quem dederis ſitu conſervent.</s> <s xml:id="echoid-s2699" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div382" type="section" level="1" n="275"> <head xml:id="echoid-head290" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2700" xml:space="preserve">Etjugi, & </s> <s xml:id="echoid-s2701" xml:space="preserve">examinis gravitatis diametrum, aciei tranſverſi axis C parallelam <lb/>per 1 propoſ. </s> <s xml:id="echoid-s2702" xml:space="preserve">inquirito, hæc eſto D, ſupra ipſam C, nam neque in C necin-<lb/>t<unsure/>ra cadet, ſiquidem ex hypotheſi librile in C quieſcere nequit, atque adeò mul-<lb/>tò minus inſra C. </s> <s xml:id="echoid-s2703" xml:space="preserve">tum confinia uncorum & </s> <s xml:id="echoid-s2704" xml:space="preserve">jugi connectar recta E F, videli-<lb/>cet infra C, ſecus enim ſi vel in C vel ſupra caderet, nullæ lances quantum-<lb/>cunque graves jugi datum ſitum aut ad horizontem æquidiſtantiam tueri poſ-<lb/>ſent, deinde G biſecet E F & </s> <s xml:id="echoid-s2705" xml:space="preserve">jungatur recta D C G, & </s> <s xml:id="echoid-s2706" xml:space="preserve">fiat ut C G, ad C D, <lb/>ſic librile ad quæſitarum lancium gravitatem, dicis gratia ſi C D æquetur C G, <pb o="88" file="527.01.088" n="88" rhead="3 L*IBER* S*TATICÆ*"/> etiam pondus lancium librilis ponderi æquale fuerit. </s> <s xml:id="echoid-s2707" xml:space="preserve">Cujus demonſtrationem <lb/> <anchor type="figure" xlink:label="fig-527.01.088-01a" xlink:href="fig-527.01.088-01"/> mathematicam 10 propoſ. </s> <s xml:id="echoid-s2708" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2709" xml:space="preserve">exhibuimus, attamen majoris evidentiæ gra-<lb/>tiâ, nonnulla huc afferemus.</s> <s xml:id="echoid-s2710" xml:space="preserve"/> </p> <div xml:id="echoid-div382" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.088-01" xlink:href="fig-527.01.088-01a"> <image file="527.01.088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.088-01"/> </figure> </div> </div> <div xml:id="echoid-div384" type="section" level="1" n="276"> <head xml:id="echoid-head291" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2711" xml:space="preserve">Perpendicularis per D eſt librilis gravitatis diameter, & </s> <s xml:id="echoid-s2712" xml:space="preserve">quæ per G educitur <lb/>eſt diameter lancium, ideoq́ue G D jugum fuerit. </s> <s xml:id="echoid-s2713" xml:space="preserve">ſed ut radius C D, ad C G, <lb/>ſic pondera ſe habent reciprocè; </s> <s xml:id="echoid-s2714" xml:space="preserve">quare ex C datam quamcunque theſin reti-<lb/>nebit C*ONCLVSIO*. </s> <s xml:id="echoid-s2715" xml:space="preserve">Itaque. </s> <s xml:id="echoid-s2716" xml:space="preserve">Librilis quæ appenſis lancibus horizonti æqui-<lb/>diſtat, abſque his vero ab axis tranſverſi acie declinat, lancium gravitatem in-<lb/>venimus, quibus librile in dato ſitu permaneat. </s> <s xml:id="echoid-s2717" xml:space="preserve">Quod erat faciendum.</s> <s xml:id="echoid-s2718" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div385" type="section" level="1" n="277"> <head xml:id="echoid-head292" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s2719" xml:space="preserve">Ex quibus evidens eſt ſi lances gravitatem iſtam pauxillo ſuperent, vel utræq; <lb/></s> <s xml:id="echoid-s2720" xml:space="preserve">æquali pondere augeantur, librile in dato ſitu non perſiſtere, ſed duntaxat ho-<lb/>rizonti æquidiſtare. </s> <s xml:id="echoid-s2721" xml:space="preserve">Quamobrem iſtiuſmodi libra non fuerit omnibus nume-<lb/>ris perfectiffima.</s> <s xml:id="echoid-s2722" xml:space="preserve"/> </p> <pb o="89" file="527.01.089" n="89" rhead="DE S*TATICÆ PRAXI.*"/> </div> <div xml:id="echoid-div386" type="section" level="1" n="278"> <head xml:id="echoid-head293" xml:space="preserve">5 PROPOSITIO.</head> <p> <s xml:id="echoid-s2723" xml:space="preserve">Stateram perfectiſsimam fabricari.</s> <s xml:id="echoid-s2724" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div387" type="section" level="1" n="279"> <head xml:id="echoid-head294" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2725" xml:space="preserve">Scapi cujuſdam ſolidi ſummum latus A B continuetur in C, ſitq́ue acies <lb/>tranſverſorum axium D, E, in rectâ B C, & </s> <s xml:id="echoid-s2726" xml:space="preserve">quidem acies D deorſum E verò <lb/>ſurſum vergat; </s> <s xml:id="echoid-s2727" xml:space="preserve">deinde tantum materiæ præponderantis è ſcapo, B C verſum <lb/>auferto, donec ex agina F ſuſpenſus æquamentum nanciſcatur, ut tamen acies <lb/>tranſverſi axis D (dempta agina F) diameter ſit gravitatis ſcapi A C. </s> <s xml:id="echoid-s2728" xml:space="preserve">Quibus <lb/>conſtitutis, ſcapoq́ue ex agina F ſuſpenſo, quemcunque dederis ſitum (dum <lb/>tranſverſus axis D aciei ſuæ incumbet) perpetuò ſervabit. </s> <s xml:id="echoid-s2729" xml:space="preserve">Conſiderandum <lb/>deinde pondus unci H, & </s> <s xml:id="echoid-s2730" xml:space="preserve">G æquipondii per ſcapum mobilis, & </s> <s xml:id="echoid-s2731" xml:space="preserve">G quidem <lb/>libram, H unciam id eſt ipſius G ſextamdecimam partem valeat; </s> <s xml:id="echoid-s2732" xml:space="preserve">ſignato dein-<lb/>de I ut interſtitium inter ipſum & </s> <s xml:id="echoid-s2733" xml:space="preserve">aciem tranſverſi axis D interjectum {1/<unsure/>6} rectæ <lb/>D E obtineat, atque idem intervallum D E inter utroſque tranſverſos axes in-<lb/>termedium, initio ab I facto verſus A toties iterabis quoties commode de-<lb/> <anchor type="figure" xlink:label="fig-527.01.089-01a" xlink:href="fig-527.01.089-01"/> ſcribi poſſe animadvertes, ut in punctis K, L, M, N, O, P, Q, R, quæ ſingula <lb/>rurſum in tot æquales particulas ſubdividantur, quot ipſorum longitudo per-<lb/>mittet, duas, quatuor, octo ſedecim, atque ſtateræ fabrica fuerit perfecta.</s> <s xml:id="echoid-s2734" xml:space="preserve"/> </p> <div xml:id="echoid-div387" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.089-01" xlink:href="fig-527.01.089-01a"> <image file="527.01.089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.089-01"/> </figure> </div> <p> <s xml:id="echoid-s2735" xml:space="preserve">Attamen ſi tanta accuratio nimium afferat laboris, huc tamen (ut 2 propoſ. <lb/></s> <s xml:id="echoid-s2736" xml:space="preserve">monuimus) quam poterit fieri accuratiſſime omnia referantur, atque aciem <lb/>axis tranſverſi D potius pauxillum ſupra rectam A C attollito quàm deprimas.</s> <s xml:id="echoid-s2737" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2738" xml:space="preserve">Vſus verò hujuſmodi eſt, ſi σφαίρωμαa G ex O & </s> <s xml:id="echoid-s2739" xml:space="preserve">pondus huic ſitu æquili-<lb/>bre ex unico H dependeat, iſtud ipſum fuerit 5 librarum, ſi præterea unaquæq; <lb/></s> <s xml:id="echoid-s2740" xml:space="preserve">rectarum I K, K L, L M, in ſedecim particulas tributa eſſet quælibet particula <lb/>unciam adderet. </s> <s xml:id="echoid-s2741" xml:space="preserve">Exemplũ tale eſto, æquipondiũ G quintam partem PQ rectæ <lb/>obtineto, tum pondus H 6 ℔ & </s> <s xml:id="echoid-s2742" xml:space="preserve">5 unciarũ erit atq; </s> <s xml:id="echoid-s2743" xml:space="preserve">ita in cęteris. </s> <s xml:id="echoid-s2744" xml:space="preserve">C um igitur ſta-<lb/>tera hæc (ſi pondus mobile altera parte depreſſa haud deorſum prolabi fingas) <lb/>qualibet gravitate ſitu æquiponderante quemcunque dederisſitum conſervet, <lb/>ob cauſas antecedente propoſitione de perfectiſſimæ libræ fabrica, expoſitas; </s> <s xml:id="echoid-s2745" xml:space="preserve"><lb/>etiam hæc ſtatera fuerit perfectiſlima. </s> <s xml:id="echoid-s2746" xml:space="preserve">Demonſtratio verò è 2 propoſ. </s> <s xml:id="echoid-s2747" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2748" xml:space="preserve">evi-<lb/>dens eſt. </s> <s xml:id="echoid-s2749" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s2750" xml:space="preserve">Stateram itaque, ut decuit, perfectiſſimam con-<lb/>ſtruximus.</s> <s xml:id="echoid-s2751" xml:space="preserve"/> </p> <pb o="90" file="527.01.090" n="90" rhead="3 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div389" type="section" level="1" n="280"> <head xml:id="echoid-head295" xml:space="preserve">6 PROPOSITIO.</head> <p> <s xml:id="echoid-s2752" xml:space="preserve">Libram obliquam fabricari.</s> <s xml:id="echoid-s2753" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2754" xml:space="preserve">Quia pondera non perpetuò recta ſurſum aut deorſum, ſed in latus non-<lb/>nunquam & </s> <s xml:id="echoid-s2755" xml:space="preserve">obliquè moventur, cujus varia exempla, partim in antecedentibus <lb/>partim in conſequentibus exhibentur, quæque ideo libram non vulgarem il-<lb/>lam ſed peculiarem depoſcunt quam obliquam propterea dicimus, cuju præ-<lb/>cipuus ſeopus eſt, ut uſus & </s> <s xml:id="echoid-s2756" xml:space="preserve">experientia rationem proportionemq́; </s> <s xml:id="echoid-s2757" xml:space="preserve">hujuſmodi <lb/>ponderum 1 lib. </s> <s xml:id="echoid-s2758" xml:space="preserve">theorematicè propoſitam deinceps comprobet, ac fidem fa-<lb/>cíat, quo minus dubii ſimus & </s> <s xml:id="echoid-s2759" xml:space="preserve">certius iis fidem habere peſſimus quæ ad huma-<lb/>ni generis utilitatem hinc derivantur.</s> <s xml:id="echoid-s2760" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div390" type="section" level="1" n="281"> <head xml:id="echoid-head296" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2761" xml:space="preserve">Fiat baſis ſive ſuſtentamentum A cui inſerta regula B crebris locis perfore-<lb/>tur, hinc orbiculus C ambitu cavo ut ducta-<lb/> <anchor type="figure" xlink:label="fig-527.01.090-01a" xlink:href="fig-527.01.090-01"/> rium funem recipiat, pér cujus centrum indu-<lb/>ctus axiculus D cardinibus ſuis in tigno ex-<lb/>cavato verſetur, qui tylo E foraminibus regu-<lb/>læ B pro voto altius aut humiliusvé inſertari <lb/>poſſit; </s> <s xml:id="echoid-s2762" xml:space="preserve">orbiculus vero & </s> <s xml:id="echoid-s2763" xml:space="preserve">axiculus ipſi infixus ne <lb/>craſſi ſed quam tenuiſſimi ſunto, ne ambitus <lb/>orbiculi uſquã ſedem ſuam ſtringat, ſed ſpatiũ <lb/>axiculi inter orbiculum & </s> <s xml:id="echoid-s2764" xml:space="preserve">ſedem interjectum <lb/>paulò craſſius ſit cardinibus circa quos cõver-<lb/>titur: </s> <s xml:id="echoid-s2765" xml:space="preserve">Ipſe privato uſui orbiculum è buxo de-<lb/>formandum dedi ſemidiametro digitorum <lb/>quinque cujus ſpiſſitudo tenuiſsimi cultri ter-<lb/>gum non excedebat, axiculum autem ſolido <lb/>elephanto tornatili opere effectum craſsitie <lb/>acus, id eſt, quam potuit torno confieri te-<lb/>nuiſsimus.</s> <s xml:id="echoid-s2766" xml:space="preserve"/> </p> <div xml:id="echoid-div390" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.090-01" xlink:href="fig-527.01.090-01a"> <image file="527.01.090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.090-01"/> </figure> </div> </div> <div xml:id="echoid-div392" type="section" level="1" n="282"> <head xml:id="echoid-head297" xml:space="preserve">7 PROPOSITIO.</head> <p> <s xml:id="echoid-s2767" xml:space="preserve">Vectium rationem & </s> <s xml:id="echoid-s2768" xml:space="preserve">formas inquirere.</s> <s xml:id="echoid-s2769" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2770" xml:space="preserve">Poſtquam longioribus palangis, vectibusq́ue vim & </s> <s xml:id="echoid-s2771" xml:space="preserve">effectum majorem, con-<lb/>tra verò brevioribus minorem efficientiam ineſſe vulgò animadverſum fuit di-<lb/>verſæ machinæ & </s> <s xml:id="echoid-s2772" xml:space="preserve">diverſa inſtrumenta molitionibus variis utilia efformari coe-<lb/>pta ſunt; </s> <s xml:id="echoid-s2773" xml:space="preserve">quia verò ſolo uſu magiſtro confirmante experientia, cauſis tamen, ut <lb/>ratione & </s> <s xml:id="echoid-s2774" xml:space="preserve">proportione, ignotis expedita ſunt; </s> <s xml:id="echoid-s2775" xml:space="preserve">ideò majores & </s> <s xml:id="echoid-s2776" xml:space="preserve">novæ machina-<lb/>tiones pari ſucceſſu caruere, magno autorum incommodo, arque adeò infe-<lb/>lici eventu. </s> <s xml:id="echoid-s2777" xml:space="preserve">Quamobrem ut ante quanta vectium in perfecto opere potent<unsure/>ia <lb/>ſit perſpici comodè poſſit, etiam præter mathematicas 1 libri ratioaes, mecha-<lb/>nicè ſeu pragmaticè paucis exemplis eadem confirmare ſtatui. </s> <s xml:id="echoid-s2778" xml:space="preserve">Principiò quia <lb/>naves minori detrimento longioribus vectibus per aggeres tran ſverſos pertra-<lb/>hi nonnullis placuit quàm vel ſuculis vel ergatis, quæ uſurpantur, hoc ipſum <lb/>plenius paulo ſcrutabimur ut quid hinc ſequatur planum ſit, iſto qui ſequitur <lb/>modo.</s> <s xml:id="echoid-s2779" xml:space="preserve"/> </p> <pb o="91" file="527.01.091" n="91" rhead="*DE* S*TATICÆ PRAXI.*"/> </div> <div xml:id="echoid-div393" type="section" level="1" n="283"> <head xml:id="echoid-head298" xml:space="preserve">1 Exemplum@.</head> <p> <s xml:id="echoid-s2780" xml:space="preserve">Eſto agger A, B C ſuſtentamentum ligneum, cui D navis pondere 24000 ℔ <lb/>inſideat (quomodo aute@@ navis oneratæ gravitas in aquis inveniatur in hydro-<lb/>ſtatica dicetur) hujus medium E, medio aggeris A incumbat ſitq́ue B F ſcapus <lb/>& </s> <s xml:id="echoid-s2781" xml:space="preserve">ab altera parte ei æqualis C G, jam navi remota ſegmenta F E, E G ſuntæ-<lb/>quipondia, quamobrem utnavis aggerem trajiciat, ſcapus ex F deprimendus <lb/>velex G ſublevandus erit, velutræque vires conjungendæ, fitq́ue H I navis <lb/>gravitatis diameter, & </s> <s xml:id="echoid-s2782" xml:space="preserve">F E ſextupla ipſius E H, unde concludendum quanta <lb/>potentia vel ex F vel ex G navi ſit æquilibris.</s> <s xml:id="echoid-s2783" xml:space="preserve"/> </p> <figure> <image file="527.01.091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.091-01"/> </figure> </div> <div xml:id="echoid-div394" type="section" level="1" n="284"> <head xml:id="echoid-head299" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2784" xml:space="preserve">Cùm F G ſit librilis inſtar cujus firmitudinis punctum E, & </s> <s xml:id="echoid-s2785" xml:space="preserve">H I gravitatis <lb/>diameter, F E autem ſextupla ipſius E H, navis ſextupla erit ponderis ſibi ex F <lb/>puncto æquipondii, ſed ex hypotheſi navis librarum eſt 24000, itaque pondus <lb/>ab F dependens 4000 ℔, & </s> <s xml:id="echoid-s2786" xml:space="preserve">25 homines ſinguli 160 ℔ pendentes ex F navi æ-<lb/>quepõderabunt, atque id quidem hoc ſitu, verum ſi K navis ſtatuatur centrum, <lb/>& </s> <s xml:id="echoid-s2787" xml:space="preserve">E G attollatur, ab F minore quam 4000 ℔ pondere opus erit, nam ex K <lb/>perpendicularis, in planum E C horizonti parallelum, demiſlà gravitatis erit <lb/>diameter & </s> <s xml:id="echoid-s2788" xml:space="preserve">ipſi E vicinior, eſto igitur E F ſeptupla ſegminis E L, quare <lb/>3428 {4/5} ℔ pondus ab F ſuſpenſum iſtic ſitus æquilibritatem vindicabit.</s> <s xml:id="echoid-s2789" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div395" type="section" level="1" n="285"> <head xml:id="echoid-head300" xml:space="preserve">NOTATO.</head> <p> <s xml:id="echoid-s2790" xml:space="preserve">Exemplum hoc quidem nobis expoſitum eſt, unde machinationis exemplar <lb/>ad imitandum derivari queat, conſiderandum tamen ſcapum E F ſextuplum <lb/>ipſius E H nobis ſumi, qui ſanè longitudine ampla & </s> <s xml:id="echoid-s2791" xml:space="preserve">craſſitudine ſymmetra <lb/>fabricandus foret: </s> <s xml:id="echoid-s2792" xml:space="preserve">non tamen exiſtimo in majoribus navigiis (aliarum machi-<lb/>narum ratione habita) perinde felici eventu uſurpari, licet in minoribus moli-<lb/>tio hæcforſan uſui fuerit. </s> <s xml:id="echoid-s2793" xml:space="preserve">Et quamvis ſuculæ ergatævèad terminos F, G con-<lb/>ſtitutæ non parum ſubſidii huic machinationi attulerint, nunc tamen hic per <lb/>numeros expol<unsure/>uiſſe ſatis eſſe duxi, aliam commodiorem rationem 10 propoſ. <lb/></s> <s xml:id="echoid-s2794" xml:space="preserve">demonſtraturus.</s> <s xml:id="echoid-s2795" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div396" type="section" level="1" n="286"> <head xml:id="echoid-head301" xml:space="preserve">2 Exemplum@.</head> <p> <s xml:id="echoid-s2796" xml:space="preserve">Antecedente paradigmate exhibui rationem ſcaporum cum pondere tum <lb/>etiam longitudine æqualium inæquales deinceps ſuecedunt.</s> <s xml:id="echoid-s2797" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2798" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s2799" xml:space="preserve">* Sunto ſcapi ABC, ABD, ſubnixi hypomochlio E ſecundum <lb/>rectam A B, quam axis totius vectis ABCD ponderantis 400 ℔ interſecet <lb/>in F, hujus gravitatis centrum ſit G, (& </s> <s xml:id="echoid-s2800" xml:space="preserve">quamvis planum diametrale ex per- <pb o="92" file="527.01.092" n="92" rhead="3 L*IBER* S*TATICÆ*"/> pendicularetam hoc quam ſequentibus 3 & </s> <s xml:id="echoid-s2801" xml:space="preserve">4 exemplis in opere ſufficiat, atta-<lb/>men quò evidentius ſit centrum ipſum ſumpſimus) tumq́ue ſegmento ABD <lb/>incumbat pondus H 2000 ℔ cujus gravitatis diameter IK eſto ut K termi-<lb/>netur in axe C D; </s> <s xml:id="echoid-s2802" xml:space="preserve">quæritur potentia quæ ex C, attollat datum pondus H.</s> <s xml:id="echoid-s2803" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div397" type="section" level="1" n="287"> <head xml:id="echoid-head302" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2804" xml:space="preserve">Oneris H & </s> <s xml:id="echoid-s2805" xml:space="preserve">machinæ D A C B ſimulutriuſque gravitatis centrum inve-<lb/>nies, diviſa K G in L, ut ſegmentorum G L, L K ratio ſit eadem quæ 2000 ℔ <lb/>ad 400 ℔, hoc eſt, 5 ad 1, perpendicularis per L educta eſt gravitatis diameter; <lb/></s> <s xml:id="echoid-s2806" xml:space="preserve">præterea F C dicis gratia duodecupla aſſumatur ipſius F L, unde concludo, ut <lb/>FC 12 ad FL 1, ſic 2400 ℔ videlicet vectis & </s> <s xml:id="echoid-s2807" xml:space="preserve">oneris conjunctum pondus, ad <lb/>200 ℔ quæ ex C ſuſpenſæ iſtis æquipondiæ forent iſtoc duntaxat ſitu, poſito <lb/> <anchor type="figure" xlink:label="fig-527.01.092-01a" xlink:href="fig-527.01.092-01"/> enim M centro gravitatis H, & </s> <s xml:id="echoid-s2808" xml:space="preserve">aſſurgente ſegmento A B D, non opus erit <lb/>200 ℔ pondere, cujus illuſtrandi gratia perpendicularis N O per centrum M <lb/>plano A B D horizonti parallelo terminetur in O, tributa igitur O G in P ut <lb/>P G itidem quintupla ſit reliqui P O, atque eo ſitu perpendicularis per P edu-<lb/>cta totius ponderis erit pendula gravitatis diameter, eſto autem F C q́uintupla <lb/>rectæ F P, unde concludes, ut F C 15 ad F P 1, ſic 2400 ℔ ad 160 ℔ quæ tunc <lb/>ſitus æquiponderantiam tuebuntur.</s> <s xml:id="echoid-s2809" xml:space="preserve"/> </p> <div xml:id="echoid-div397" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.092-01" xlink:href="fig-527.01.092-01a"> <image file="527.01.092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.092-01"/> </figure> </div> </div> <div xml:id="echoid-div399" type="section" level="1" n="288"> <head xml:id="echoid-head303" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s2810" xml:space="preserve">Verumenimverò quia haſtarum ſimiliumve ſcaporum humeris geſtatorum <lb/>ratio antecedenti exemplo non ſit admodum adſimilis, hanc ipſam tertio para-<lb/>digmate perſequemur.</s> <s xml:id="echoid-s2811" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2812" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s2813" xml:space="preserve">* Palangrarius humero B geſtet haſtile librarum 12, cujus axis ſit <lb/>C D, centrum gravitatis E, atque ab haſtilis & </s> <s xml:id="echoid-s2814" xml:space="preserve">humeri contactu acta B F ho-<lb/>rizonti perpendicularis ſecet axem DC in G, manus haſtile rectà deprimens <lb/>ſtatuitior in puncto axis H, ſitq́ue G H dupla G E.</s> <s xml:id="echoid-s2815" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2816" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s2817" xml:space="preserve">* Quanta potentia manus ſit inquirere.</s> <s xml:id="echoid-s2818" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div400" type="section" level="1" n="289"> <head xml:id="echoid-head304" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2819" xml:space="preserve">Quandoquidem G H dupla eſt ſegmenti G E, etiam pondus ab E, quod <lb/>eſt haſtilis univerſi duplum erit ejus quod ad H collocatur, videlicet manus; </s> <s xml:id="echoid-s2820" xml:space="preserve">fed <pb o="93" file="527.01.093" n="93" rhead="*DE* S*TATICÆ PRAXI.*"/> haſtile eſt 12 ℔, itaque manus <lb/>potĕtia tanta erit quanta pon-<lb/> <anchor type="figure" xlink:label="fig-527.01.093-01a" xlink:href="fig-527.01.093-01"/> dus 6 ℔ ab H dependentium.</s> <s xml:id="echoid-s2821" xml:space="preserve"/> </p> <div xml:id="echoid-div400" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.093-01" xlink:href="fig-527.01.093-01a"> <image file="527.01.093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.093-01"/> </figure> </div> <p> <s xml:id="echoid-s2822" xml:space="preserve">Sed ſi ab haſtili ex K depen-<lb/>deat jucundum raptori præ-<lb/>mium gallus I 3 ℔ ut K G <lb/>ipſius G H ſit tripla palam eſt <lb/>prædam 9 ℔ pondus adjicere, <lb/>atque univerſim manus po-<lb/>tentiam 15 ℔ præmere.</s> <s xml:id="echoid-s2823" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2824" xml:space="preserve">Verùm hæc manu recta <lb/>deorſum premente intelligan-<lb/>tur, atqui ſi in obliquum duca-<lb/>tur, quæ ratio erit rectà deſcĕ-<lb/>dentis ad deſcendentem obliquè, ea erit per 21 propoſ. </s> <s xml:id="echoid-s2825" xml:space="preserve">I lib. </s> <s xml:id="echoid-s2826" xml:space="preserve">ſtatices ponde-<lb/>ris recti deſcendentis ad deſcendens obliquè, unde reliqua per ejuſdem lib. <lb/></s> <s xml:id="echoid-s2827" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s2828" xml:space="preserve">22. </s> <s xml:id="echoid-s2829" xml:space="preserve">perſpiciuntur.</s> <s xml:id="echoid-s2830" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div402" type="section" level="1" n="290"> <head xml:id="echoid-head305" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s2831" xml:space="preserve">Hactenus quidem affectiones expoſitæ nobis ſunt, ubi utrimque à ſirmitu-<lb/>dinis puncto ſcapi extenduntur. </s> <s xml:id="echoid-s2832" xml:space="preserve">ſequitur unicum unici ſcapi exemplum.</s> <s xml:id="echoid-s2833" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2834" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s2835" xml:space="preserve">* A B axis eſto ſcapi 10 pedes longi pondere 400 ℔, fixi termino <lb/>A cætera verò mobilis, cui pondus 1000 ℔ infideat & </s> <s xml:id="echoid-s2836" xml:space="preserve">ſcapi quidem ſeu vectis <lb/>gravitatis diameter ſit C D, ponderis vero F G. </s> <s xml:id="echoid-s2837" xml:space="preserve">Quæritur quantis viribus ad <lb/>B, pondus E commoveatur.</s> <s xml:id="echoid-s2838" xml:space="preserve"/> </p> <figure> <image file="527.01.093-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.093-02"/> </figure> </div> <div xml:id="echoid-div403" type="section" level="1" n="291"> <head xml:id="echoid-head306" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2839" xml:space="preserve">Inveſtigato conjunctim utriuſque gravitatis diametrum, jugo G D quæ <lb/>connectit diametros E G, C D ita diviſo in H, ut ſegmentorum H G, H D <lb/>ratio ſit quæ vectis 400 ℔ ad onus F 1000 ℔ ſeu quod idem ſit ratione ſubdu-<lb/>pla-ſubſeſquialtera, Si v. </s> <s xml:id="echoid-s2840" xml:space="preserve">g. </s> <s xml:id="echoid-s2841" xml:space="preserve">A H aſſumatur pedum 2, concludes ut A B 10 pe-<lb/>des ad A H 2, ſic onus ponderis & </s> <s xml:id="echoid-s2842" xml:space="preserve">vectis 1400 ℔, ad 280 ℔ potentiam videlicet <lb/>quaad B opus ſit ut cæteris vi æquipolleat, hoc eſt quaſi 280 ℔ attollendæ fo-<lb/>rent, cujus demonſtratio è 14 propoſ. </s> <s xml:id="echoid-s2843" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2844" xml:space="preserve">Statices manifeſta eſt.</s> <s xml:id="echoid-s2845" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2846" xml:space="preserve">Verumenimverò ſi Staticus ſimpliciſſima cauſæ cognitione computum in-<lb/>ſtituere malit, Iſorropica diagrammata ſibi effingat, ut Geometra ad juvandam <lb/>memoriam geometrica depingit. </s> <s xml:id="echoid-s2847" xml:space="preserve">Sit igitur IK vice vectis decempedalis A B, <lb/>hinc I L referat bipedalem rectam A H & </s> <s xml:id="echoid-s2848" xml:space="preserve">loco H quod centrum gravitatis <lb/>notat, hic erit L, unde M 1400 ℔ dependent: </s> <s xml:id="echoid-s2849" xml:space="preserve">deinde ab 1, quod firmitudinis <lb/>punctum intelligitor, deſcribatur I N æqualis priori IK, & </s> <s xml:id="echoid-s2850" xml:space="preserve">quantum pondus <pb o="94" file="527.01.094" n="94" rhead="3 L*IBER* S*TATICÆ*"/> ipſi M æquipondium hinc ſuſpendi opus ſit per 3 propoſ. </s> <s xml:id="echoid-s2851" xml:space="preserve">I lib. </s> <s xml:id="echoid-s2852" xml:space="preserve">inquirito, hoc <lb/>modo, cum I L ſubquintupla ſit rectæ IN, <lb/> <anchor type="figure" xlink:label="fig-527.01.094-01a" xlink:href="fig-527.01.094-01"/> quinta pars ponderis M 1400 ℔ ex N pun-<lb/>cto, ſcilicet O 280 ℔, priori æquipondia <lb/>concludetur, ſed ponderis O ab N deſcen-<lb/>dentis per 13 propoſ. </s> <s xml:id="echoid-s2853" xml:space="preserve">I lib. </s> <s xml:id="echoid-s2854" xml:space="preserve">Statices tanta po-<lb/>tentia eſt, quanta ejuſdem ad K attollentis, <lb/>I K enim & </s> <s xml:id="echoid-s2855" xml:space="preserve">I N ex hypotheſi æquantur. <lb/></s> <s xml:id="echoid-s2856" xml:space="preserve">Quamobrem vis qua K attollitur, tollet <lb/>pondus 280 ℔ in vecte ad punctum B. </s> <s xml:id="echoid-s2857" xml:space="preserve">Si-<lb/>milibus diagrammatis Staticus mechanicorum paradigmatum, quæ brevitatis <lb/>ſtudio in hæc paucula contraximus, cauſas inquiret.</s> <s xml:id="echoid-s2858" xml:space="preserve"/> </p> <div xml:id="echoid-div403" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.094-01" xlink:href="fig-527.01.094-01a"> <image file="527.01.094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.094-01"/> </figure> </div> </div> <div xml:id="echoid-div405" type="section" level="1" n="292"> <head xml:id="echoid-head307" xml:space="preserve">8 PROPOSITIO.</head> <p> <s xml:id="echoid-s2859" xml:space="preserve">Ponderum geſtatorum formas & </s> <s xml:id="echoid-s2860" xml:space="preserve">rationes inquirere.</s> <s xml:id="echoid-s2861" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2862" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s2863" xml:space="preserve">* Exponantur ſcalæ A B quarum extrema, ut ferè ſolent, diſpari <lb/>ſintgravitate, cujus onus duobus palangariis æquali pondere partiendum, ut <lb/>axis C D horizonti parallelus perpetuò maneat.</s> <s xml:id="echoid-s2864" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div406" type="section" level="1" n="293"> <head xml:id="echoid-head308" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s2865" xml:space="preserve">Scalas in acie aliqua huc illuc verſato donec partium æquamentum invene-<lb/>ris in E, & </s> <s xml:id="echoid-s2866" xml:space="preserve">ſi crebrò alio transferendę ſint notam iſtic incidito, deinde perpen-<lb/>dicularis ab E ducta occurrat axi C D in F, denique ſignato duo puncta G, H, <lb/>æquidiſtanter ab E F, ajo hic in G iſtic in Hæquale pondus à palangariis ſu-<lb/>ſtineri.</s> <s xml:id="echoid-s2867" xml:space="preserve"/> </p> <figure> <image file="527.01.094-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.094-02"/> </figure> <p> <s xml:id="echoid-s2868" xml:space="preserve">Sin autem alterum alterius duplum perferre opus ſit, unius ab E F diſtantia <lb/>reliqui fiat dupla ut A E ipſius E I; </s> <s xml:id="echoid-s2869" xml:space="preserve">atque is qui ad I duplo pondere premetur <lb/>reſpectu ejus qui ad H. </s> <s xml:id="echoid-s2870" xml:space="preserve">Simili via onera diſtribuentur palangariis pro ratio-<lb/>ne data.</s> <s xml:id="echoid-s2871" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2872" xml:space="preserve">QV *Æ* ſcalarum ſpeciali exemplo oſtendimus, de quolibet ſolido corpore <lb/>intelligas, ut in diagrammatis ſubjectis perſpicitur; </s> <s xml:id="echoid-s2873" xml:space="preserve">memineris autem li-<lb/>neas planorum quę per perinordinatorum corporum gravitatis centrum ducuntur <lb/>in ipſorum ſuperficie per I prop. </s> <s xml:id="echoid-s2874" xml:space="preserve">deſignari; </s> <s xml:id="echoid-s2875" xml:space="preserve">perpendiculares item per G, H actas <lb/>ipſi E F parallelas eſſe.</s> <s xml:id="echoid-s2876" xml:space="preserve"/> </p> <pb o="95" file="527.01.095" n="95" rhead="*DE* S*TATICÆ PRAXI.*"/> <figure> <image file="527.01.095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.095-01"/> </figure> <p> <s xml:id="echoid-s2877" xml:space="preserve">ATque axe quidem C D ad horizontem parallelo ratio hujuſmodi fuit, <lb/>at qui eo ad horiz ontem obliquo ut in adſcenſu montis ponderoſitatis ra-<lb/>tio quidem diverſa, ex antecedentibus tamen in promptu erit. </s> <s xml:id="echoid-s2878" xml:space="preserve">I gitur in clivi <lb/>adſcenſu qui ad G præcedat, alter ad H ſubſequatur.</s> <s xml:id="echoid-s2879" xml:space="preserve"/> </p> <figure> <image file="527.01.095-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.095-02"/> </figure> <pb o="96" file="527.01.096" n="96" rhead="3 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s2880" xml:space="preserve">Perpendiculares per G, H, eductæ ſecent axem C D in K, L, hîc pondus <lb/>non tribuitur, ut ante, æquis partibus, namque F K in duobus primis diagram-<lb/>matis major eſt quam F L, in reliquis verò minor, ſed ut F K ad F L ſic pon-<lb/>dus palangarii H, ad palangarium ſeu vectiarium G. </s> <s xml:id="echoid-s2881" xml:space="preserve">Vnde evidens eſt ſi firmi-<lb/>tudinis puncta G, H ſub axe C D conſtituantur antecedentem minus premi, <lb/>ſin verò ſupraſit, ſequentem leviore pondere urgeri. </s> <s xml:id="echoid-s2882" xml:space="preserve">Denique ſi firmitudinis <lb/>puncta in ipſo axe C D figantur ponde-<lb/>ris varietatem, neque in clivo neque in <lb/> <anchor type="figure" xlink:label="fig-527.01.096-01a" xlink:href="fig-527.01.096-01"/> planitie ullam eſſe. </s> <s xml:id="echoid-s2883" xml:space="preserve">Quarum demonſtra-<lb/>tiones è 14, 15, 16, 17, 18, 27, 28 propoſ. <lb/></s> <s xml:id="echoid-s2884" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2885" xml:space="preserve">repetantur.</s> <s xml:id="echoid-s2886" xml:space="preserve"/> </p> <div xml:id="echoid-div406" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.096-01" xlink:href="fig-527.01.096-01a"> <image file="527.01.096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.096-01"/> </figure> </div> <p> <s xml:id="echoid-s2887" xml:space="preserve">Veruntamen cum multorum inſti-<lb/>tutum non patiatur iſtas cognoſcere, <lb/>qui nihilominus id optant mechani-<lb/>ca ratione edoceri, ii ſumant baculum <lb/>quomodocunque incurvum A B, quod <lb/>funiculo C D ſuſpendant ex C. </s> <s xml:id="echoid-s2888" xml:space="preserve">De-<lb/>miſſis deinde à C D æquali diſtantia <lb/>duobus perpendiculis G H, I K, ut H L, <lb/>LK ęquales ſint, baculum eandem ſerva-<lb/>bit theſin; </s> <s xml:id="echoid-s2889" xml:space="preserve">idem erit ſi ſpatium N L di-<lb/>midium quidem ſit ipſius L K, pondus <lb/>verò M ponderis F duplum, atque ita <lb/>deinceps in cæteris. </s> <s xml:id="echoid-s2890" xml:space="preserve">Qua via experientia <lb/>comprobante, quæ ſupta nobis expoſi-<lb/>ta ſunt facillimè intelligentur.</s> <s xml:id="echoid-s2891" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2892" xml:space="preserve">REctas quibus in ſuperioribus dia-<lb/>grammatis corpora geſtari finxi-<lb/> <anchor type="figure" xlink:label="fig-527.01.096-02a" xlink:href="fig-527.01.096-02"/> mus, horizonti perpendiculares collo-<lb/>cavimus; </s> <s xml:id="echoid-s2893" xml:space="preserve">ſi vero obliquę ſumantur plus <lb/>virium deſiderabitur quam ſit corporis <lb/>ipſius gravitas, quantum verò unuſ-<lb/>quiſq; </s> <s xml:id="echoid-s2894" xml:space="preserve">ferat ductis perpendicularibus <lb/>I M, N O, evidens erit, namque per <lb/>27 propoſ. </s> <s xml:id="echoid-s2895" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2896" xml:space="preserve">ut M I ad I G ſic pon-<lb/>dus rectà ſublatum ad idem ſublatum <lb/>obliquè hoc eſt potentiam hominis in <lb/>G; </s> <s xml:id="echoid-s2897" xml:space="preserve">conſimili modo ut O N ad N H, <lb/>ſic ejus pondus cum rectà attollitur ad <lb/>idem obliquatum quæ efficientia eſt palangarii ad H, unde ſingulorum effe-<lb/>ctus per 22 propoſ. </s> <s xml:id="echoid-s2898" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2899" xml:space="preserve">concludetur.</s> <s xml:id="echoid-s2900" xml:space="preserve"/> </p> <div xml:id="echoid-div407" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.096-02" xlink:href="fig-527.01.096-02a"> <image file="527.01.096-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.096-02"/> </figure> </div> <p> <s xml:id="echoid-s2901" xml:space="preserve">A pluribus & </s> <s xml:id="echoid-s2902" xml:space="preserve">magis variis geſtatorum ponderum paradigmatis cum brevi-<lb/>tatis ſtudio ſuperſedemus, tum quia ex antecedentibus lucem & </s> <s xml:id="echoid-s2903" xml:space="preserve">demonſtra-<lb/>tionem accipiunt compendi facimus.</s> <s xml:id="echoid-s2904" xml:space="preserve"/> </p> <pb o="97" file="527.01.097" n="97" rhead="DE S*TATICÆ* PRAXI."/> </div> <div xml:id="echoid-div409" type="section" level="1" n="294"> <head xml:id="echoid-head309" xml:space="preserve">9 PROPOSITIO.</head> <p> <s xml:id="echoid-s2905" xml:space="preserve">Axium in peritrochio & </s> <s xml:id="echoid-s2906" xml:space="preserve">tractorum ponderum rationes <lb/>inquirere.</s> <s xml:id="echoid-s2907" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2908" xml:space="preserve">Pondus movens & </s> <s xml:id="echoid-s2909" xml:space="preserve">motum ſemidiametris tympani & </s> <s xml:id="echoid-s2910" xml:space="preserve">axis proportionales <lb/>ſunt, unde majoris evidentiæ gratia, theorema tale inſtituimus.</s> <s xml:id="echoid-s2911" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div410" type="section" level="1" n="295"> <head xml:id="echoid-head310" xml:space="preserve">THEOREMA.</head> <p style="it"> <s xml:id="echoid-s2912" xml:space="preserve">Si & </s> <s xml:id="echoid-s2913" xml:space="preserve">ab axe & </s> <s xml:id="echoid-s2914" xml:space="preserve">ab extremâ tympani ſemidiametro horizonti par allelâ. <lb/></s> <s xml:id="echoid-s2915" xml:space="preserve">ponder a ſitu æquipondia dependeant, hæc erunt ſemidiametris reciproce pro-<lb/>portionalia.</s> <s xml:id="echoid-s2916" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2917" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s2918" xml:space="preserve">* Eſto tympanum A B C D E F G, quod verſetur circa axem <lb/>E F G cujus diameter E F, centrum H, pondus I ab axe dependeat, A B C D <lb/>ſit ipſa rota ſeu tympanum, cujus ſemidiameter horizonti parallela A C, & </s> <s xml:id="echoid-s2919" xml:space="preserve">à <lb/>termino A pondus K dependeat ſitu æquipondium ipſi I; </s> <s xml:id="echoid-s2920" xml:space="preserve">L autem axis & </s> <s xml:id="echoid-s2921" xml:space="preserve"><lb/>ſuſtentaculi ſui infimus contactus. </s> <s xml:id="echoid-s2922" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s2923" xml:space="preserve">* Demonſtrato, H A, H F, <lb/>& </s> <s xml:id="echoid-s2924" xml:space="preserve">I, K, in eadem analogia eſſe.</s> <s xml:id="echoid-s2925" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div411" type="section" level="1" n="296"> <head xml:id="echoid-head311" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s2926" xml:space="preserve">Enimverò tympanum A B C D librilis iſtar eſto, cujus anſa L B, ut ſublatis<unsure/> <lb/>ponderibus K, I, partes tympa-<lb/> <anchor type="figure" xlink:label="fig-527.01.097-01a" xlink:href="fig-527.01.097-01"/> ni B D A, B D C ſint æquili-<lb/>bres. </s> <s xml:id="echoid-s2927" xml:space="preserve">Siigitur pondus I ab F de-<lb/>pendere fingas, quia hic tantæ <lb/>eſt potentiæ quantæ ſuo loco, K <lb/>autem è loco ſuo A, per 1 prop. <lb/></s> <s xml:id="echoid-s2928" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2929" xml:space="preserve">eadem ratio erit radii lon-<lb/>gioris H A, ad breviorem H F, <lb/>quæ ponderis majoris I ad mi-<lb/>nus K. </s> <s xml:id="echoid-s2930" xml:space="preserve">Quamobrem ſi H A ſex-<lb/>tuplus ſit ipſius H F, etiam I <lb/>ſextuplum fuerit ipſius K, ut ſi <lb/>I ponatur 600 librarum, K erit <lb/>100, ideo q́ue potentia hominis <lb/>ad A æquivalens 100 libris, ſitu <lb/>æquipondia foret 600 libris ad <lb/>I, & </s> <s xml:id="echoid-s2931" xml:space="preserve">quo attolli poſſit (propter <lb/>impedimenta contactus axis & </s> <s xml:id="echoid-s2932" xml:space="preserve">ſuſtentaculi C) paulo potentius quam 100 ℔ <lb/>agat neceſſe erit.</s> <s xml:id="echoid-s2933" xml:space="preserve"/> </p> <div xml:id="echoid-div411" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.097-01" xlink:href="fig-527.01.097-01a"> <image file="527.01.097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.097-01"/> </figure> </div> </div> <div xml:id="echoid-div413" type="section" level="1" n="297"> <head xml:id="echoid-head312" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s2934" xml:space="preserve">Hinc tympanorum qui Græcis γέζανθι ſimiliumq́ue rotarum ratio, quæ à <lb/>calcantibus hominibus verſantur, manifeſta eſt. </s> <s xml:id="echoid-s2935" xml:space="preserve">Exponatur enim tympanum <lb/>A B C D, diametro A C horizonti parallela, cujus axis craſſities ad tornum <lb/>rotundati ſit E F, centrum G, pondus ab axe H; </s> <s xml:id="echoid-s2936" xml:space="preserve">homo ad I ſitu æquilibris <lb/>ponderi H, & </s> <s xml:id="echoid-s2937" xml:space="preserve">I K gtavitatis ejus perpendicularis in A C. </s> <s xml:id="echoid-s2938" xml:space="preserve">Notumautem G K, <lb/>ad G F eſt<unsure/>e, ſicut pondus H ad gravitatem hominis qui ad I, ſi igitur G K qua- <pb o="98" file="527.01.098" n="98" rhead="3 L*IBER* S*TATICÆ*"/> drupla ſit ipſius G F etiam pondus H quadruplum erit ipſius hominis, quare <lb/>poſita gravitate hominis I librarum 150, atque H 600 ℔, eo ſitu æquiponde-<lb/> <anchor type="figure" xlink:label="fig-527.01.098-01a" xlink:href="fig-527.01.098-01"/> r<unsure/>abunt, neque H quoquam impelletur, ſed ſi ulterius tendat verſus A, etiam <lb/>H neceſſariò attolletur, quia ratio ipſius G K ad G F tum major foret quam in <lb/>theſi. </s> <s xml:id="echoid-s2939" xml:space="preserve">Si plures tympanum calcent, vicinior ipſi A aget potentius remotiore; <lb/></s> <s xml:id="echoid-s2940" xml:space="preserve">atque per 3 prop. </s> <s xml:id="echoid-s2941" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2942" xml:space="preserve">cum ſingulorum tum etiam univerſorum potentia con-<lb/>cludetur.</s> <s xml:id="echoid-s2943" xml:space="preserve"/> </p> <div xml:id="echoid-div413" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.098-01" xlink:href="fig-527.01.098-01a"> <image file="527.01.098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.098-01"/> </figure> </div> </div> <div xml:id="echoid-div415" type="section" level="1" n="298"> <head xml:id="echoid-head313" xml:space="preserve">3 Exemplums.</head> <p> <s xml:id="echoid-s2944" xml:space="preserve">Atqui ponderum quæ rectà attolluntur, ut ſunt onera quæcunque, ſarcinæ, <lb/>vaſa, quæ tympanorum ſubſidio è navibus eximuntur, ratio hujuſmodi eſt. <lb/></s> <s xml:id="echoid-s2945" xml:space="preserve">Pondera verò obliquè adſcendentia, cujus generis ſunt naves quæ in Belgio ſæ-<lb/>pe trans aggeres & </s> <s xml:id="echoid-s2946" xml:space="preserve">aquarum obices traducuntur, rationem non paulò diver-<lb/>ſam habent. </s> <s xml:id="echoid-s2947" xml:space="preserve">Enimverò ſit agger A, B navis trans aggerem pertrahenda, C D <lb/>tympanum, diameter C D horizonti parallela, homo navi B ſitu ἰ{σό}ῤῥοπ & </s> <s xml:id="echoid-s2948" xml:space="preserve"><lb/>ſeu æquilibris habeat gravitatis diametrum F E, funis ductarius G H, axis ſo-<lb/>liditas I K. </s> <s xml:id="echoid-s2949" xml:space="preserve">& </s> <s xml:id="echoid-s2950" xml:space="preserve">centrum L: </s> <s xml:id="echoid-s2951" xml:space="preserve">deinde eſto M N normalis aggeris clivo, horizonti <lb/>autem perpendicularis ad idem punctum N ſit N O. </s> <s xml:id="echoid-s2952" xml:space="preserve">præterea L F ſextupla <lb/>eſt ſemidiametri L K; </s> <s xml:id="echoid-s2953" xml:space="preserve">& </s> <s xml:id="echoid-s2954" xml:space="preserve">N O tripla ipſius M O, pondus autem hominis tym-<lb/>panum calcantis ℔ 150. </s> <s xml:id="echoid-s2955" xml:space="preserve">Quæ cum ita ſint, erit ut L F ad L K, ſic ex antece-<lb/>dente I<unsure/> heoremate pondus è fune ductario ſuſpenſum rectà deſcendens, ad <lb/>gravitatem hominis 150 ℔, L F autem ex hypotheſi ſextupla eſt ipſius L K, <lb/>quamobrem pondus è fune H G recta deſcendens ſextuplum quoque eſſet <lb/>150 ℔, quæ ſunt 900 libræ, & </s> <s xml:id="echoid-s2956" xml:space="preserve">homo tympanum verſans tam potenter agit <lb/>quam 900 ℔ obliquâ trutinâ ſuſpenſæ. </s> <s xml:id="echoid-s2957" xml:space="preserve">Itaque per 20 propoſ. </s> <s xml:id="echoid-s2958" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2959" xml:space="preserve">pondus na-<lb/>vis B ita ſe habet ad 900 ℔, ut N O ad O M, ſed N O tripla eſt ipſius O M <pb o="99" file="527.01.099" n="99" rhead="DE S*TATICÆ PRAXI.*"/> <anchor type="figure" xlink:label="fig-527.01.099-01a" xlink:href="fig-527.01.099-01"/> ex theſi, itaque navis eſt tripla 900 ℔, hoc eſt pendet 2700 ℔, cujus ratio ad gra-<lb/>vitatem hominis verſantis eſt octupla. </s> <s xml:id="echoid-s2960" xml:space="preserve">Atque hoc-quidem tali ſitu, ſed ſi navis <lb/>promoveatur & </s> <s xml:id="echoid-s2961" xml:space="preserve">ſurſum attollatur, ductarius funis eò adſcendet rectius (niſi for-<lb/>tè alibi in navi firmetur) & </s> <s xml:id="echoid-s2962" xml:space="preserve">conſequenter recta M O ad N O majorem habue-<lb/>rit rationem, & </s> <s xml:id="echoid-s2963" xml:space="preserve">propterea ſitus æquamentum paulò majus foret quam 900 ℔. <lb/></s> <s xml:id="echoid-s2964" xml:space="preserve">Qui igitur & </s> <s xml:id="echoid-s2965" xml:space="preserve">axem & </s> <s xml:id="echoid-s2966" xml:space="preserve">tympanum juſtæ quantitatis fabricari cupiet, quod nec <lb/>mole ſua excedat, aut exilitate deficiat, rationem inibit ſitus, quo navis graviſ-<lb/>ſima cum erit, maxima potentia opus habebit.</s> <s xml:id="echoid-s2967" xml:space="preserve"/> </p> <div xml:id="echoid-div415" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.099-01" xlink:href="fig-527.01.099-01a"> <image file="527.01.099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.099-01"/> </figure> </div> <p> <s xml:id="echoid-s2968" xml:space="preserve">Advertendum autem è 24 propoſ. </s> <s xml:id="echoid-s2969" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2970" xml:space="preserve">hominis E potentiam, ſe tum exe-<lb/>rere maximè cum funis ductarius G H plano aggeris P N parallelus erit, tum <lb/>enim H G navis axi perpendicularis inſiſtit, hoc eſt, rectæ per navis gravitatis <lb/>centrum plano P N perpendiculari. </s> <s xml:id="echoid-s2971" xml:space="preserve">Quamobrem quanto G H, P N rectæ <lb/>ad paralleliſmum magis accedunt, tantò faciliús, & </s> <s xml:id="echoid-s2972" xml:space="preserve">ſi recedant difficilius ponde-<lb/>ra movebuntur.</s> <s xml:id="echoid-s2973" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div417" type="section" level="1" n="299"> <head xml:id="echoid-head314" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s2974" xml:space="preserve">Indidem planum eſt, quanto majori pondere æquus curru junctus clivumq́; <lb/></s> <s xml:id="echoid-s2975" xml:space="preserve">adſcendens afficiatur, quam ſi eundem in planitie trahat. </s> <s xml:id="echoid-s2976" xml:space="preserve">Exponatur enim A B <lb/>montis clivus, currus C D 2000 ℔, E F funis eſto, G equus currui hoc ſitu <lb/>æquivalens, tum H I, I K perpendiculares plano A B, & </s> <s xml:id="echoid-s2977" xml:space="preserve">H I quadrupla ipſius <lb/>H K; </s> <s xml:id="echoid-s2978" xml:space="preserve">his poſitis, per 20 propoſ. </s> <s xml:id="echoid-s2979" xml:space="preserve">1 libri erit, ut K H ad H I, ſic pondus obliquè <lb/>attollens cujus vicem equus explet, ad gravitatem currus, ſed K H quarta pars <lb/>eſt H I ex theſi; </s> <s xml:id="echoid-s2980" xml:space="preserve">quamobrem pondus obliquè tollens foret 500 ℔ nimirum <lb/>quarta pars currus; </s> <s xml:id="echoid-s2981" xml:space="preserve">itaque antilena pectus equi non tam præmit, quam onus <lb/> <anchor type="figure" xlink:label="fig-527.01.099-02a" xlink:href="fig-527.01.099-02"/> @0 ℔ dorſum, atque hoc quidem (videlicet cum promovebitur) præter im-<lb/>preſtionem iſtam quâ afficitur in campi planitie trahens.</s> <s xml:id="echoid-s2982" xml:space="preserve"/> </p> <div xml:id="echoid-div417" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.099-02" xlink:href="fig-527.01.099-02a"> <image file="527.01.099-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.099-02"/> </figure> </div> <pb o="100" file="527.01.100" n="100" rhead="3 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s2983" xml:space="preserve">Præterea per 24 prop. </s> <s xml:id="echoid-s2984" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s2985" xml:space="preserve">& </s> <s xml:id="echoid-s2986" xml:space="preserve">per ea quæ de navi paulo ante diſſeruimus pater, <lb/>tum equorũ potĕtiam eſſe maximam cum lora currus parallela erunt viæ æqua-<lb/>bili videlicet & </s> <s xml:id="echoid-s2987" xml:space="preserve">duræ, nam in aſperis, inæqualibus, & </s> <s xml:id="echoid-s2988" xml:space="preserve">arenoſis ſatius eſt lora po-<lb/>nè paulò inferiora eſſe quam ante. </s> <s xml:id="echoid-s2989" xml:space="preserve">Quod Batavicis ciſiariis experientia eruditis <lb/>uſitatũ eſt, qui currubus ita aptatis lora cum per littus maris aut per alias planas <lb/>duraſq́ue vias vehuntur ponealtius firmant quàm in aſperis arenoſisq́ue, hic <lb/>enim quamvis horizonti parallela ſint ſalebris tamen non ſunt, quod in pertra-<lb/>hendo ponderoſius & </s> <s xml:id="echoid-s2990" xml:space="preserve">difficilius eſt quam ſi poſtilenæ pone humiliores fo-<lb/>rent quiatum ad ſalebrarum paralleliſmum propius accedunt. </s> <s xml:id="echoid-s2991" xml:space="preserve">In locis areno-<lb/>ſis ubi currus altius arenæ inſidet, etiam rotæ deſcendunt, ac propterea quoq; <lb/></s> <s xml:id="echoid-s2992" xml:space="preserve">ſilora horizonti parallela ſint plus negotii faceſſunt quam ſi pone depreſſiora <lb/>fuerint.</s> <s xml:id="echoid-s2993" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div419" type="section" level="1" n="300"> <head xml:id="echoid-head315" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s2994" xml:space="preserve">Objiciat autem quis primò, Cur dixerimus H K eſſe ad H I, ſicut pondus libræ <lb/>obliquæ ſi quâ hîc eſſet (cujus vicem equus ſubit) ad currus gravitatem; </s> <s xml:id="echoid-s2995" xml:space="preserve">Fortean ratus <lb/>im proprie uſurpari gravitatem currus pro pondere rectâ attollĕte gravitatem currus.</s> <s xml:id="echoid-s2996" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s2997" xml:space="preserve">Secundò cur funis E F nul-<lb/> <anchor type="figure" xlink:label="fig-527.01.100-01a" xlink:href="fig-527.01.100-01"/> lam ſitus differentiam annota-<lb/>verim, qui per currus gravitatis <lb/>centrum trajectus equum fortè <lb/>leviore aut ponderoſiore gra-<lb/>vitate afficiat, quam ſi vel ſupra <lb/>id vel infra meet. </s> <s xml:id="echoid-s2998" xml:space="preserve">His igitur ra-<lb/>tionibus quî commodè occurri <lb/>poſſit, & </s> <s xml:id="echoid-s2999" xml:space="preserve">ſimul {νρ}αμμιχῶς evin-<lb/>ci proportionem expoſitam eſſe <lb/>legittimam, A B C currus eſto <lb/>è mathematicis lineis efforma-<lb/>tus & </s> <s xml:id="echoid-s3000" xml:space="preserve">quaſi μουό{νρ}αμμ&</s> <s xml:id="echoid-s3001" xml:space="preserve">, rotæ <lb/>D, E, via cui inſiſtit F G, de-<lb/>nique funis ductarius libræ obliquè attollentis poſt paulò exprim endæ ſit A H.</s> <s xml:id="echoid-s3002" xml:space="preserve"/> </p> <div xml:id="echoid-div419" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.100-01" xlink:href="fig-527.01.100-01a"> <image file="527.01.100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.100-01"/> </figure> </div> <p> <s xml:id="echoid-s3003" xml:space="preserve">HVic currui μουο{νρ}ἀμμω& </s> <s xml:id="echoid-s3004" xml:space="preserve">imponito priſima I K, quale in ſubjecta figurâ vi-<lb/>dere licet, ut A H continuatus, occurrat priſmatis centro L: </s> <s xml:id="echoid-s3005" xml:space="preserve">Eſto item <lb/> <anchor type="figure" xlink:label="fig-527.01.100-02a" xlink:href="fig-527.01.100-02"/> abliquè attollens pondus M priſmati ſitu æquilibre; </s> <s xml:id="echoid-s3006" xml:space="preserve">centroq́ue L annectitor <pb o="101" file="527.01.101" n="101" rhead="*DE STATICÆ PRAXI.*"/> aliud pondus N rectâ attollens priſmati expoſito ſitu quoque æquipondium; <lb/></s> <s xml:id="echoid-s3007" xml:space="preserve">horizonti autem perpendicularis B O rectam A H interſecet in O: </s> <s xml:id="echoid-s3008" xml:space="preserve">quæ cum <lb/>ita ſint, Ajo per 20 propoſ. </s> <s xml:id="echoid-s3009" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s3010" xml:space="preserve">A O, O B ponderibus M, N, proportionales <lb/>eſſe, ſed quia N connexum eſt cum L expoſiti priſmatis I K gravitatis centro, <lb/>ipſum N per 14 propoſ 1 lib. </s> <s xml:id="echoid-s3011" xml:space="preserve">priſmati erit æquipondium, unde efficitur A O <lb/>eſſe ad O B, ſicut M ad priſma I K. </s> <s xml:id="echoid-s3012" xml:space="preserve">Atqueita primæ objectioni occurrimus ſi <lb/>A H ductarius funis per L gravitatis centrum tranſeat.</s> <s xml:id="echoid-s3013" xml:space="preserve"/> </p> <div xml:id="echoid-div420" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.100-02" xlink:href="fig-527.01.100-02a"> <image file="527.01.100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.100-02"/> </figure> </div> <p> <s xml:id="echoid-s3014" xml:space="preserve">Secundæ objectioni autem <lb/> <anchor type="figure" xlink:label="fig-527.01.101-01a" xlink:href="fig-527.01.101-01"/> cum AH infra ſuprave centrum <lb/>L conſiſtet hoc modo obviam <lb/>imus. </s> <s xml:id="echoid-s3015" xml:space="preserve">Priſma I K pendulæ ſuæ <lb/>gravitatis diametro L P innixũ <lb/>recta ſurſum tollitor. </s> <s xml:id="echoid-s3016" xml:space="preserve">jam per 3 <lb/>poſtulatum ipſum currui A B C <lb/>non eſt majori pondere onero <lb/>ſum quam fuerat in priore ſitu, <lb/>atque ideo non erit opus ut M <lb/>nunc potentius agat quam ante; <lb/></s> <s xml:id="echoid-s3017" xml:space="preserve">ſed H A continuatus meat ſub <lb/>centro L, quamobrem M idem <lb/>pondus ducit quod ante, cum <lb/>H A ad ipſum gravitatis cen-<lb/>trum pertineret. </s> <s xml:id="echoid-s3018" xml:space="preserve">Demonſtratio huicaffinis erit quando AH continuatus ſupra <lb/>gravitatis centrum L meabit, boc eſt, cum priſma IK recta deorſum ſub curru <lb/>trahetur. </s> <s xml:id="echoid-s3019" xml:space="preserve">Quamobrem utramque iſtam {απο}ρίαυ rationibus mathematicis ex-<lb/>plicavimus.</s> <s xml:id="echoid-s3020" xml:space="preserve"/> </p> <div xml:id="echoid-div421" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.101-01" xlink:href="fig-527.01.101-01a"> <image file="527.01.101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.101-01"/> </figure> </div> </div> <div xml:id="echoid-div423" type="section" level="1" n="301"> <head xml:id="echoid-head316" xml:space="preserve">10 PROPOSITIO.</head> <p> <s xml:id="echoid-s3021" xml:space="preserve">Infinitæ potentiæ formas & </s> <s xml:id="echoid-s3022" xml:space="preserve">accidentia exponere.</s> <s xml:id="echoid-s3023" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3024" xml:space="preserve">Variæ machinæ humano ingenio ad effectiones mechanicas excogitantur, <lb/>quarũ potentia infinite augeri poſſit unde & </s> <s xml:id="echoid-s3025" xml:space="preserve">nomen ſortitas infinitas potentias <lb/>appellant, quarũ hic ſcopus eſt, videlicet quę potĕtia effectrix dato inſtrumento, <lb/>æquivalens ſit põderi mobili. </s> <s xml:id="echoid-s3026" xml:space="preserve">Vel quanto tempore pondus ad datum interval-<lb/>lum promoveatur atq; </s> <s xml:id="echoid-s3027" xml:space="preserve">his conſimilia; </s> <s xml:id="echoid-s3028" xml:space="preserve">huic negotio exprimam inſtrumentum <lb/>ſimpliciſſimũ quidem quantũ res ipſa feret, quo tamen commodiſſimè inſtitu-<lb/>tum meum explicem, ubi ante de machina Archimedis infinitæ potentiæ, cujus <lb/>Plutarchus aliiq́ meminerunt, pauca retulero. </s> <s xml:id="echoid-s3029" xml:space="preserve">Cum enim Hiero Siciliæ Tyran-<lb/>nus navem immenſæ magnitudinis, formaq́ue in ſpeciem perquam eleganti, <lb/>cõſtruxiſſet uteam Ptolomæo Æ gypti Regi dono mitteret; </s> <s xml:id="echoid-s3030" xml:space="preserve">Hanc univerſæ Sira-<lb/>cuſæ ſummo conamine è littore in altum deducere nequibant; </s> <s xml:id="echoid-s3031" xml:space="preserve">ſed poſtquam <lb/>Archimedes inſtrumenta machinaſq́ue admoviſſet Rex Hiero ſola manu navem <lb/>impulit. </s> <s xml:id="echoid-s3032" xml:space="preserve">Machina autem hæc Archimedea Charistion dicta, (cujus formam & </s> <s xml:id="echoid-s3033" xml:space="preserve"><lb/>deſcriptionem in Regia Bibliotheca inventam Iacobus Beſſonius publicavit) axes <lb/>habebat cum infinitis cochleis, inventum ſane dignum quod ad poſteritatem <lb/>tranſmittatur, cujus deſcriptio quia præſenti inſtituto aſſidet, huic loco conve-<lb/>niret, in ipſi ſubſtituerem infinitam hanc potentiam, quâ communis cæterarum <lb/>ſimilium infinitæ potentiæ machinarum affectio commodiffime explicabitur, <lb/>quæque, ut mihi quidem videtur, iſti operi ſit aptior. </s> <s xml:id="echoid-s3034" xml:space="preserve">Cum ſit materiâ firmiore <pb o="102" file="527.01.102" n="102" rhead="*3 LIBER STATICÆ*"/> folidioreq̀<unsure/>ue & </s> <s xml:id="echoid-s3035" xml:space="preserve">impenſis minoribus parabilis, quâque eodem temporis ſpatio <lb/>plus efficiatur, potentiæ tamen infinitæ conſimiliter ipſi chariſtio.</s> <s xml:id="echoid-s3036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3037" xml:space="preserve">Sumito trabem A B ſoliditate & </s> <s xml:id="echoid-s3038" xml:space="preserve">magnitudine operi inſtituto & </s> <s xml:id="echoid-s3039" xml:space="preserve">effectioni-<lb/>bus mechanicis congruente; </s> <s xml:id="echoid-s3040" xml:space="preserve">efformato deinde tympanum ferreum C per am-<lb/>bitum dentatum, cujus diameter ſit dicis gratiâ, digitorum trium, inque ambi-<lb/>tu dentes ſex, per centrum tranſidetur axis ferreus CD ad rerminos C & </s> <s xml:id="echoid-s3041" xml:space="preserve">D <lb/>quadrangulus, ſcapus autem intermedius rotondus eſto; </s> <s xml:id="echoid-s3042" xml:space="preserve">hinc tympana E v. </s> <s xml:id="echoid-s3043" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s3044" xml:space="preserve">dentibus 18, F dentibus ſex traji-<lb/> <anchor type="figure" xlink:label="fig-527.01.102-01a" xlink:href="fig-527.01.102-01"/> ciantur eodem axe EF qui ſimilis <lb/>ſit antecedenti CD, terminis vi-<lb/>delicet quadrangulis teretem ſca-<lb/>pum claudentibus. </s> <s xml:id="echoid-s3045" xml:space="preserve">huic EF effin-<lb/>gantur ſimiles axes GH, IK, ut <lb/>tympana G, K, ambiantur denti-<lb/>bus ſex, H, I, 18. </s> <s xml:id="echoid-s3046" xml:space="preserve">Et quia tympana <lb/>ſuperiora plus ponderis ſuſtine-<lb/>bunt, ut poſtea intelligetur, etiam <lb/>firmiora majoraq́; </s> <s xml:id="echoid-s3047" xml:space="preserve">deformantor, <lb/>atque ideo cum axes ordinabun-<lb/>tur paralleli H quidem mordicus <lb/>implicabitur ipſi F, diſtabit autem <lb/>à tympano K, item G implicabi-<lb/>tur tympano I non autem tym-<lb/>pano E, quod ipſum partiũ diſpo-<lb/>ſitio quo que poſtulabat.</s> <s xml:id="echoid-s3048" xml:space="preserve"/> </p> <div xml:id="echoid-div423" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.102-01" xlink:href="fig-527.01.102-01a"> <image file="527.01.102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.102-01"/> </figure> </div> <p> <s xml:id="echoid-s3049" xml:space="preserve">Fiat deinde manubriũ L M N, <lb/>cavo quadrangulo cui quadran-<lb/>gula axium capitula D, F, H, K <lb/>exactè congruant, ſitq́ue flexura <lb/>M L pedem longa, ſchapus autem <lb/>M N fiat longitudine mox paulò infra definien-<lb/> <anchor type="figure" xlink:label="fig-527.01.102-02a" xlink:href="fig-527.01.102-02"/> dâ. </s> <s xml:id="echoid-s3050" xml:space="preserve">Denique in trabe A B tranſverſim quatuor <lb/>foramina terebrĕtur eâdem inter ſe diſtantia qua <lb/>axes, ut tympanorum dentes commodè inter ſe <lb/>implicati & </s> <s xml:id="echoid-s3051" xml:space="preserve">mutuo ſeſe impellere poſſint, cujuſ-<lb/>modihic ſunt O, P, Q, R quibus axes IK, GH, <lb/>EF, CD inſerti congruant, axium autem ſcapi, <lb/>teretes inter duos tympanos intermedii longitu-<lb/>dine ſuâ trabis craſſitiem æquent; </s> <s xml:id="echoid-s3052" xml:space="preserve">quadrati autĕ <lb/>iſti axium termini K, H, F, D, tres aut ſum-<lb/>mum quatuor digitos extra tympanos promi-<lb/>neant. </s> <s xml:id="echoid-s3053" xml:space="preserve">His ita deformatis detracto tympano I <lb/>axem I K inſerito in foramen O, ſimiliter GH <lb/>in P, EF in Q, CD in R, affigantur deinde <lb/>cuique axi ſua tympana, dentesq́ue tympani F <lb/>in fronte mordicus impliciti dentibus H ipſum <lb/>impellant; </s> <s xml:id="echoid-s3054" xml:space="preserve">ſimiliter in tergo ut C impellat tym-<lb/>panum E, & </s> <s xml:id="echoid-s3055" xml:space="preserve">G ipſum I, formaq́ue perfecti Pan-<lb/>cratii (ſic enim inſtrumentum hoc à ſuâ efficien- <pb o="103" file="527.01.103" n="103" rhead="*DE STATICÆ PRAXI.*"/> tia appellare liceat) erit qualem in ſubjectâ figurà expreſſimus.</s> <s xml:id="echoid-s3056" xml:space="preserve"/> </p> <div xml:id="echoid-div424" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.102-02" xlink:href="fig-527.01.102-02a"> <image file="527.01.102-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.102-02"/> </figure> </div> <p> <s xml:id="echoid-s3057" xml:space="preserve">Exemplum verò hoc quod quatuor duntaxat axibus effinximus augeri mi-<lb/>nuive, & </s> <s xml:id="echoid-s3058" xml:space="preserve">ratio dentium 18 majoris tympani quæ minori comparata tripla eſt, <lb/>ad libitum item mutari poterit pro modo mechanicis effectionibus quibus <lb/>Pancration iſtud deformabis congruente.</s> <s xml:id="echoid-s3059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div426" type="section" level="1" n="302"> <head xml:id="echoid-head317" xml:space="preserve">DE VSV ALIISQVE PANCRATIO <lb/>ACCIDENTIBVS.</head> <p> <s xml:id="echoid-s3060" xml:space="preserve">PAncratii noſtri uſum unico duntaxat exemplo quod cæteris facem prælu-<lb/>ceat, reddemus illuſtriorem; </s> <s xml:id="echoid-s3061" xml:space="preserve">videlicet quâ ratione ipſo in ſuculam transfor-<lb/>mato ejus adjumento naves trans aggeres ſeptaq́ue ſeu aquarum obices tradu-<lb/>cantur; </s> <s xml:id="echoid-s3062" xml:space="preserve">neque enim hinc parum utilitatis ad regiones has, præcipuè verò Hol-<lb/>landos redundabit. </s> <s xml:id="echoid-s3063" xml:space="preserve">Eſto igitur Pancration A B, tympana dentata K, H, F in <lb/>fronte trabis; </s> <s xml:id="echoid-s3064" xml:space="preserve">I, G, E, C in tergo, L M N manubrium ſeu ut Pappus in me-<lb/>chanicis loquitur λάβη, axis autem cui funis ductarius obvolvitur S craſſitie <lb/>ſeſquipedali per trabem trajectus emineat capitulo quadrato cui affigatur den-<lb/> <anchor type="figure" xlink:label="fig-527.01.103-01a" xlink:href="fig-527.01.103-01"/> tatum tympanum T, diametri bipedalis per ambitum 36 dentibus diſtinctus, <lb/>arque aſteriſcus hic minimum tam amplus formandus axe S ne rotatus ejus ab <lb/>I tympano impediatur. </s> <s xml:id="echoid-s3065" xml:space="preserve">Deniq; </s> <s xml:id="echoid-s3066" xml:space="preserve">ſummitas aggeris V iuperet imam carinam na-<lb/>vis X aquę innatantis pedibus quatuor, hoc eſt, ut perpendicularis à ſummo <lb/>aggere demiſſa in planum horizonti parallelum per imam carinam actum, ſit <lb/>pedum quatuor. </s> <s xml:id="echoid-s3067" xml:space="preserve">Iam ut navis trans aggerem traducatur circumducito manu-<lb/>brium LMN, oportebit igitur ſcapum MN tantâ longitudine conſtrui ut <lb/>omnes quorum opera opus erit commodè hinc inde conſiſtant.</s> <s xml:id="echoid-s3068" xml:space="preserve"/> </p> <div xml:id="echoid-div426" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.103-01" xlink:href="fig-527.01.103-01a"> <image file="527.01.103-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.103-01"/> </figure> </div> <pb o="104" file="527.01.104" n="104" rhead="*3 LIBER STATICÆ*"/> </div> <div xml:id="echoid-div428" type="section" level="1" n="303"> <head xml:id="echoid-head318" xml:space="preserve">RATIO VERSATIONVM MANVBRII <lb/>AD AXEM.</head> <p> <s xml:id="echoid-s3069" xml:space="preserve">QVia manubrum L M N ter rotatum ſemel circumagit tympanum F, at no-<lb/>vies ſemel in orbem aget H, & </s> <s xml:id="echoid-s3070" xml:space="preserve">27 tympanum K, hinc 162 circumductum <lb/>convertet ipſum T hoc eſt axem S. </s> <s xml:id="echoid-s3071" xml:space="preserve">Eodemq́ue modo manubrium L M N <lb/>affixum ad F id tympanum 54 circumducendum erit ut axis S ſemel converta-<lb/>tur. </s> <s xml:id="echoid-s3072" xml:space="preserve">ad H 18, K ſexies, T autem toties in orbem vertitur quoties ipſemet axis S. <lb/></s> <s xml:id="echoid-s3073" xml:space="preserve">Cum autem manubrium altius inſeres quam in D, verbi gratiâ in K, ne infe-<lb/>riora tympana, quæ difficultatem operi inducunt, unà convertere ſit opus, pro-<lb/>ximè inferius quod in expoſito caſu eſt G loco ſuo depelles ne dentes ejus den-<lb/>tibus I amplius implicentur, atque ita inferior machinæ pars univerſa immota <lb/>conſiſtet.</s> <s xml:id="echoid-s3074" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div429" type="section" level="1" n="304"> <head xml:id="echoid-head319" xml:space="preserve">RATIO POTENTIÆ MANVBRIVM VER-<lb/>SANTIS AD PONDVS TRACTVM, <lb/>QUALE EST NAVIS X.</head> <p> <s xml:id="echoid-s3075" xml:space="preserve">CVm flexura LM ex hypotheſi pedem longa, octupla ſit aſteriſci C, etiam <lb/>potentia quâ C agit in E æquivalens potentiæ impellentis manubrium <lb/>LM erit ut 8 ad I, eodemq́ue modo propter efficientiam H in F, ut 24 ad I <lb/>atque deinceps ab I in G, ut 72 ad I, denique à T in K ut 216 ad I. </s> <s xml:id="echoid-s3076" xml:space="preserve">Sed or-<lb/>biculus T æquivalet axi S, (æquivalere dixi, reverâ enim diameter axis <lb/>S eſt ſeſquipedalis, T autem bipedalis ex hypotheſi, ſed quia dentes aſte-<lb/>riſci T ſextupli ſunt ipſius K, etiam diameter ſextuplum peterit diametri K <lb/>quæ propterea erit 3 digitorum, & </s> <s xml:id="echoid-s3077" xml:space="preserve">quiad T digitorum 18, ſive ſeſquipedalis, <lb/>quemadmodum diameter axis S) quare pondus ab axe S rectà deſcendens <lb/>eandem habebit rationem ad pondus ſitu ſibi æquipondium, ſeu potentiam <lb/>æquivalentem in M N quam 216 ad I. </s> <s xml:id="echoid-s3078" xml:space="preserve">Ratiocinium hoc ab axe S deſcenden-<lb/>do deorſum inire licebit, eodem modo quo ſurſum adſcendendo nunc nobis <lb/>inſtitutum fuit.</s> <s xml:id="echoid-s3079" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3080" xml:space="preserve">Sedidem etiam hoc pacto explicari poterit. </s> <s xml:id="echoid-s3081" xml:space="preserve">Cum manubrum M N 162 in <lb/>orbem gyratum ſemel circumducat axem S, ut ſupra jam aſſertum eſt, & </s> <s xml:id="echoid-s3082" xml:space="preserve">ra-<lb/>tio diametri circuli à verſura manubrii M N deſcripti ad radium axis S ſit ſeſ-<lb/>quitertia (namque LM ſcapus pedalis eſt, ſemidiameter autem axis S {3/4} pedis) <lb/>exporrecti ambitus 162 verſationum manubrii M N, ad ambitum circuli in <lb/>axe S ſe habet ut 216 ad I, atque in iſta ratione quoque ſunt ſemidiametri 216 <lb/>iſtius circuli ad hujus circuli ſemidiametrum. </s> <s xml:id="echoid-s3083" xml:space="preserve">quare per I propoſ. </s> <s xml:id="echoid-s3084" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s3085" xml:space="preserve">pondus <lb/>iſtius ad pondus hujus rationem habebit eandem quam 216 ad I. </s> <s xml:id="echoid-s3086" xml:space="preserve">Vnde effici-<lb/>tur ſi M N tantâ verſetur potentiâ quanta eſt 25 librarum deſcendendo, quæ <lb/>nobis v. </s> <s xml:id="echoid-s3087" xml:space="preserve">g. </s> <s xml:id="echoid-s3088" xml:space="preserve">hominis viribus æſtimetur, & </s> <s xml:id="echoid-s3089" xml:space="preserve">major ubi collibitum erit (quamvis <lb/>enim longè infra viri vires ſubſiſtat expoſita potentia, ita tamen exempli gra-<lb/>tia ſumpſiſſe placuit) iſta inquam potentia æquivalebit, 5400 libris (nam 216 <lb/>ſumpta 25 hanc ſummam efficiunt) ab axerectà deorſum tendentibus. </s> <s xml:id="echoid-s3090" xml:space="preserve">Iam ve-<lb/>rò navis X ſit ſextupla ponderis ab axe S rectà dimiſſi, itaque X navis 32400 <lb/>librarum (quod pondus eſt 9 modiorum, ſi 3600 libras ſingulis modiis tribua-<lb/>mus) æquiponderabit priori ponderi ſeu quod idem eſt potentiæ manubrium <lb/>MN continuè verſantis.</s> <s xml:id="echoid-s3091" xml:space="preserve"/> </p> <pb o="105" file="527.01.105" n="105" rhead="*DE STATICÆ PRAXI.*"/> </div> <div xml:id="echoid-div430" type="section" level="1" n="305"> <head xml:id="echoid-head320" xml:space="preserve">DE VERSATIONVM MANVBRII MVLTI-<lb/>TVDINE, ET TEMPORIS SPATIO QVEIS OPVS <lb/>EST AD NAVEM TRANS AGGEREM <lb/>PERTRAHENDVM.</head> <p> <s xml:id="echoid-s3092" xml:space="preserve">SEd quand oquidem ex hypotheſi navis ſextupla ſit ponderis ab axe S ſuſpen-<lb/>ſi, etiam inclinatio aggeris quà navis adſcendit ſextupla erit altitudinis ſuæ, <lb/>quod è per 19 prop. </s> <s xml:id="echoid-s3093" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s3094" xml:space="preserve">cognoſcere eſt; </s> <s xml:id="echoid-s3095" xml:space="preserve">atqui altitudinem iſtam 4 pedum ſu-<lb/>pra ſtatuimus, hæcigitur ſexies ſumpta efficit 24 pedum inclinatam longitudi-<lb/>nem. </s> <s xml:id="echoid-s3096" xml:space="preserve">Ea longitudo axi S obvolvi debet ut navis gravitatis centrum trans agge-<lb/>ris medium traduci poſſit. </s> <s xml:id="echoid-s3097" xml:space="preserve">Quare ſi ut ante ambitus circuli in axe triplus ſtatua-<lb/>tur ſuæ diametri (nam in expoſito caſu ratio hæc verò ſatis vicina eſt) ſeſquipe-<lb/>dalis, is eritipſe 4 {1/2} pedum, qui in dictis 24 pedibus cõtinentur 5 {1/3}. </s> <s xml:id="echoid-s3098" xml:space="preserve">quamobrem <lb/>axis S 5 {1/3} verſandus; </s> <s xml:id="echoid-s3099" xml:space="preserve">ſingulis autem ejus verſationibus reſpondent 162 verſatio-<lb/>nes manubrii M N, utſupra patuit, itaque in univerſum manubrium M N 864 <lb/>circumducendum erit.</s> <s xml:id="echoid-s3100" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3101" xml:space="preserve">Vel ſie. </s> <s xml:id="echoid-s3102" xml:space="preserve">Singulæ verſationes manubrii promovĕt 25 ℔ per intervallum 6 pe-<lb/>dum, hoc eſt, ſi pondus 5400 ℔ dependeat ab axe S ſingulæ verſationes manu-<lb/>brii MN tantundem efficiunt ac ſi 25 ℔ 6 pedes attolleret, & </s> <s xml:id="echoid-s3103" xml:space="preserve">conſequenter <lb/>quaſi ſexies 25 ℔ navis quę ſunt 150 ℔ iſtos 6 pedes attolleret, diviſisq́; </s> <s xml:id="echoid-s3104" xml:space="preserve">32400 ℔ <lb/>per 150 ℔ quotus erit 216, navis igitur univerſæmoles 216 verſationibus manu-<lb/>brii MN 6 pedes promovebitur, atqui quater ſenos pedes promovenda eſt, <lb/>itaque quater 216, hoc eſt 864 verſationibus opus eſt. </s> <s xml:id="echoid-s3105" xml:space="preserve">Vel (quia navis 4 pedes <lb/>attollendaſit) proportione hujuſmodi concludere poteris. </s> <s xml:id="echoid-s3106" xml:space="preserve">una verſatione at-<lb/>tolluntur 25 ℔ 6 pedes. </s> <s xml:id="echoid-s3107" xml:space="preserve">quotigitur opus erit ut 32400 ℔ attollantur pedes 4-<lb/>concludes ut ante 864.</s> <s xml:id="echoid-s3108" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3109" xml:space="preserve">Sed millies manubrium unico unius horæ quadrante circumagi poſſe ſta-<lb/>tuimus, quamobrem ſi omnia ſupra expoſitis congruant, univerſam molem na-<lb/>vis ſarcinarumq́ue, pondere 9 modiorum, homo unicus minori quàm quadran-<lb/>tis ſpatio trans aggerem pertraxerit. </s> <s xml:id="echoid-s3110" xml:space="preserve">Si verò à tribus hominibus verſetur manu-<lb/>briumq́ue inſertent ipſi F intra quadrantis trientem, hoc eſt, {1/12} horæ partem ab-<lb/>ſolverint. </s> <s xml:id="echoid-s3111" xml:space="preserve">ſiab hominibus 9 manubrium ipſi H affixum circumducatur {1<unsure/>/16} ho-<lb/>ræ navis pertrahetur. </s> <s xml:id="echoid-s3112" xml:space="preserve">Poterit autem ſi ſit opus Pancratium hoc geminari atque <lb/>alterum oppoſito ſcapo Y inſeri quale ipſi AB inſertum vides, atque ita ipſos <lb/>verſantes bipartiri ne tanta hominum manus mutua opera interturbentur.</s> <s xml:id="echoid-s3113" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div431" type="section" level="1" n="306"> <head xml:id="echoid-head321" xml:space="preserve">NOTATO</head> <p> <s xml:id="echoid-s3114" xml:space="preserve">Navem hoc exemplo nobis ita conſiderari tanquam in traductione æquali <lb/>ſemper pondere operas afficeret, cum tamen ipſum pro ſitu variari per tertium <lb/>exemplum 9 propoſ. </s> <s xml:id="echoid-s3115" xml:space="preserve">in confeſſo ſit, ponderoſius enim moleſtiusq́ue in fine <lb/>quàm ſub initium navis adducitur. </s> <s xml:id="echoid-s3116" xml:space="preserve">Quamobrem hoc tanquam exemplo dun-<lb/>taxat, quomodo in ſingulis ratiocinium ineundum ſit, uſos nos eſſe cogitabis.</s> <s xml:id="echoid-s3117" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3118" xml:space="preserve">Tympana autem iſta dentata ſeu aſtericos quos in Pancratio eadem ſerieſur-<lb/>ſum collocavimus, etiam tranſversè, etiam binos binisjunctos, prout operis <lb/>commoditas concinnitasq́ue exiget, conſtituere licebit.</s> <s xml:id="echoid-s3119" xml:space="preserve"/> </p> <pb o="106" file="527.01.106" n="106" rhead="*3 LIBER STATICÆ*"/> </div> <div xml:id="echoid-div432" type="section" level="1" n="307"> <head xml:id="echoid-head322" xml:space="preserve">ASSERTIO EORVM QVÆ SVPRA <lb/>DEMONSTRATVROS NOS RECEPIMVS.</head> <p> <s xml:id="echoid-s3120" xml:space="preserve">PAncration hoc firmius ſolidiusq́ue & </s> <s xml:id="echoid-s3121" xml:space="preserve">impenſæ minoris quam Chariſtion <lb/>Archimedeum mihi videri initio hujus propoſitionis dixeram; </s> <s xml:id="echoid-s3122" xml:space="preserve">quoque mi-<lb/>nori temporis ſpatio plus efficeretur; </s> <s xml:id="echoid-s3123" xml:space="preserve">potentiamq́; </s> <s xml:id="echoid-s3124" xml:space="preserve">infinitam ipſi ad extremum <lb/>tribueram.</s> <s xml:id="echoid-s3125" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3126" xml:space="preserve">Operis quidem firmitudinem per ſe claram (meliora tamen non cõtemnens) <lb/>cuilibet eſſe exiſtimo, quid enim huic machinationi ſolidius excogitari poteſt <lb/>ſolido arboris trunco? </s> <s xml:id="echoid-s3127" xml:space="preserve">cujus partes arctius longe tenaciusq́ue cohærent, quam <lb/>trabes inter ſe compactæ & </s> <s xml:id="echoid-s3128" xml:space="preserve">coagmentatæ. </s> <s xml:id="echoid-s3129" xml:space="preserve">Impenſæ quantulæ? </s> <s xml:id="echoid-s3130" xml:space="preserve">quas ne verbis <lb/>quidem explicare opus eſt.</s> <s xml:id="echoid-s3131" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3132" xml:space="preserve">Quid, temporis ſpatium cujuſmodi?</s> <s xml:id="echoid-s3133" xml:space="preserve">. manubrium, quod proratione opera-<lb/>rum ipſum verſantium cuilibetaſteriſco inſertare licet, nobis argumento erit; </s> <s xml:id="echoid-s3134" xml:space="preserve">le-<lb/>vioribus enim oneribus ducendis altius, ponderoſioribus verò humilius inſer-<lb/>to manubrio, onus trahendum quamlibet ponderofum convenienti labore ju-<lb/>giq́ue motu commovetur. </s> <s xml:id="echoid-s3135" xml:space="preserve">Cui abſimilis eſt iſte Chariſtii & </s> <s xml:id="echoid-s3136" xml:space="preserve">tympanorũ motus, <lb/>nam leviculæ navi etiam adhibetur potentia majoribus ponderibus congrua, <lb/>quæque ideo temporis moram prolatat; </s> <s xml:id="echoid-s3137" xml:space="preserve">imò cum trahendum pondus majus <lb/>ſit quam ut commodè commoveri poſſit, magna hominum equorumq́ue ma-<lb/>nus adhibenda, quæ nonnunquam laboriosè procedit, quandoq́; </s> <s xml:id="echoid-s3138" xml:space="preserve">ſubſiſtit atq; <lb/></s> <s xml:id="echoid-s3139" xml:space="preserve">ita tempus ducitur, navibus autem damnum infertur: </s> <s xml:id="echoid-s3140" xml:space="preserve">nam una è maximis tre-<lb/>decem aut quatuordecim modiorum capax, quę per <anchor type="note" xlink:href="" symbol="*"/> Lugodinenſem aquarum <anchor type="note" xlink:label="note-527.01.106-01a" xlink:href="note-527.01.106-01"/> obicem traducuntur, viginti hominibus tympanum calcando verſantibus opus <lb/>habet, qui ſæpè facta ſtatione ſimul ſe demittunt, navesq́ue ſubjecto ſtrato gra-<lb/>viter illidunt. </s> <s xml:id="echoid-s3141" xml:space="preserve">Quod in Pancratio ſecus eſt, quia navis jugiter æquabiliterq́ue <lb/>impellitur.</s> <s xml:id="echoid-s3142" xml:space="preserve"/> </p> <div xml:id="echoid-div432" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.106-01" xlink:href="note-527.01.106-01a" xml:space="preserve">Iepbtfchen <lb/>Dam.</note> </div> <p> <s xml:id="echoid-s3143" xml:space="preserve">Sed potentia, quam pancratio infinitam aſſignavimus, hic tanta eſt in D, <lb/>quanta Tympani diametro 324 pedum. </s> <s xml:id="echoid-s3144" xml:space="preserve">Enimverò per centrum tympani tan-<lb/>iæ diametri inducatur axis æqualis ipſi S diametri ſeſquipedalis, unde ratio ſe-<lb/>midiametrorum exiſtet ea quæ eſt 216 ad 1, atque ideò per 9 propof. </s> <s xml:id="echoid-s3145" xml:space="preserve">pondus ab <lb/>axe ęquilibre ponderi, è tympani diametro horizonti parallela dependenti, ha-<lb/>bebit ad ipſum rationem eandem quam 216 ad 1; </s> <s xml:id="echoid-s3146" xml:space="preserve">atqui eadent ratio eſt poten-<lb/>tiæ manubrii M N ad pondus ab axe S: </s> <s xml:id="echoid-s3147" xml:space="preserve">itaque manubrii ad D efficientia in <lb/>axem S tanta erit, quanta tympani diametri 324 pedum, qui ne in maximo qui-<lb/>dem tympano 30 pedes æquat nedum excedit, unde expeditè concludere li-<lb/>cet quanto Pancratii efficientia potior ſit potentia tympanorum, & </s> <s xml:id="echoid-s3148" xml:space="preserve">quamvis <lb/>iympanum minorelabore à calcante verſetur, non tamen, ut jam ediſſeruimus, <lb/>ideo utiliſſimum quoque fuerit. </s> <s xml:id="echoid-s3149" xml:space="preserve">At ſi quis commoditatem iſtam calcandi in <lb/>pancratio majoribus molibus ſubvectantis promovendisq́; </s> <s xml:id="echoid-s3150" xml:space="preserve">deſideret, a<unsure/>eriſco-<lb/>rum D, F, H, K, T axiculo cuicunque libebit pro manubrio rotulam, quam <lb/>majus tympanum dentanti ambitus verſet affigere quidem licebit, ſed utilita-<lb/>tem ſuperante impenſa.</s> <s xml:id="echoid-s3151" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3152" xml:space="preserve">Atquiſi potentia jam deformati pancratii oneri trahendo cedat, non tamen <lb/>labor, non impenſæ perierint: </s> <s xml:id="echoid-s3153" xml:space="preserve">alter enim aſteriſcus triplicans numerum poten-<lb/>tiæ dictæ 216 ipſi D ſubjiciatur, tum manubrium ejus axiculo affixum efficiet <lb/>hic potentiam unius libræ ponderi 648 librarum ab S dependentium æquili-<lb/>brem, qua via ad optatam efficientiam tandem adſcenditur.</s> <s xml:id="echoid-s3154" xml:space="preserve"/> </p> <pb o="107" file="527.01.107" n="107" rhead="*DE STATICÆ PRAXI*"/> <p> <s xml:id="echoid-s3155" xml:space="preserve">Quamobrem ſi in ſtitueretur pancration 30 orbiculis axiculisq́ue, cujus maxi-<lb/>mus aſteriſcus minimi eſſet decuplus, tum manubrii flexura maximi aſteriſci ſe-<lb/>midiametro, & </s> <s xml:id="echoid-s3156" xml:space="preserve">ſucula S minimo aſteriſco æqualis, quę in ſpeciem machina ad-<lb/>modũ magnę molis nõ foret, põdus ab iſtiuſmodi ſucula ad alterũ ſibi æquilibre <lb/>è manubrio ſuſpensũ eſſet in ratione 1000000000000000000000000000000 <lb/>ad 1. </s> <s xml:id="echoid-s3157" xml:space="preserve">Poſita igitur perimetro minimi aſteriſci pedis unius tum tympani perime-<lb/>ter (per cujus centrum inductus eſt axis perimetri pedalis) inſtituti pancratii <lb/>manubrio æquepotĕtis eſſet pedũ 1000000000000000000000000000000, <lb/>Atqui maximi in terreno globo circuli ambitus nõ excedit pedes 108000000, <lb/>gradu æſtimato ſtadiis 480, ſtadio paſſibus geometricis 12@, paſſu denique pe-<lb/>dibus 5. </s> <s xml:id="echoid-s3158" xml:space="preserve">Itaque ambitus tympani huic tantulo pancratio æquivalentis aliquot <lb/>millies millenis modis maximum terreſtrem circulũ excederet. </s> <s xml:id="echoid-s3159" xml:space="preserve">Iam verò pan-<lb/>cratii manubrium infimo aſteriſco affixum, verſet aliquis puſio qui paulò effi-<lb/>cacius in id agat quam unius libræ pondus, is ſummo axi abſolvet funem <lb/>100000000000000000000000000000000 libras ducentem (adverte tamen <lb/>ſummum iſtum axem die uno ne ſemel quidem integrè circumductum iri) quę <lb/>univerſæ terræ molem cæteraq́ue hac comprehenſa quater millies ſuo pondere <lb/>excederent, cujus demonſtratio hujuſmodi affertur. </s> <s xml:id="echoid-s3160" xml:space="preserve">Ambitus maximi terreſtris <lb/>circuli eſt pedum 108000000, ut jam ſupra docuimus, unde efficitur aream mi-<lb/>norem eſſe pedibus 1000000000000000, at que ſphæricam terræ ſuperſiciem <lb/>minorem 4000000000000000 pedibus; </s> <s xml:id="echoid-s3161" xml:space="preserve">ſextans autem ſemidiametri globi <lb/>terreſtris minor eſt pedibus 6000000 quæ multiplicata cũ 4000000000000000 <lb/>efficiunt 24000000000000000000000 cubicos pedes lõgè majores univer-<lb/>ſæterræ ſoliditate, jam ſingulorũ pedum pondere (liceteò neutiquã pertingat) <lb/>100 libris taxato, univerſiterrarum orbis põdus 2400000000000000000000000 <lb/>libris cedere intelliges qui numerus per 4000 multiplicatus longè infra expo-<lb/>ſitum 1000000000000000000000000000000 conſiſtit.</s> <s xml:id="echoid-s3162" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3163" xml:space="preserve">Meritò igitur machinæ huic infinitam potentiam tribuimus. </s> <s xml:id="echoid-s3164" xml:space="preserve">Neque Archi-<lb/>medis exclamatio, δός μοι π{omi;</s> <s xml:id="echoid-s3165" xml:space="preserve">ῦ} ςῶ {καὶ}, κινῶ τ{ὴν} γ{ήν} da mihi ubi conſiſtam & </s> <s xml:id="echoid-s3166" xml:space="preserve">ter-<lb/>ram hanc eo pertraham, in quam ob inventũ à ſe Chariſtion quondam ovans <lb/>erupit, impoſſibilis autabſona videri debet. </s> <s xml:id="echoid-s3167" xml:space="preserve">imò rationi conſentaneam jam evi-<lb/>cimus, ſecus radiorum & </s> <s xml:id="echoid-s3168" xml:space="preserve">ponderum reciproca proportio quæ prima primi li-<lb/>bri propoſitionenobis demonſtrata ſuit, à veritate diſſideret, quod ipſum & </s> <s xml:id="echoid-s3169" xml:space="preserve">fal-<lb/>ſum & </s> <s xml:id="echoid-s3170" xml:space="preserve">impoſſibile eſt. </s> <s xml:id="echoid-s3171" xml:space="preserve">Vt tamen etiam exemplo veritas ejus eluceſcat; </s> <s xml:id="echoid-s3172" xml:space="preserve">demus <lb/>Chariſtio, vel huic Pancratio locum extra hanc terram ubi defigi poſſit, glo-<lb/>bumq́; </s> <s xml:id="echoid-s3173" xml:space="preserve">hunc terrenũ ponderi jam dicto 2400000000000000000000000 ℔ <lb/>æqualem, atque hominem ſingulis manubrii verſuris 100 libras tres pedes ſu-<lb/>blimè attollere, horisq́ue ſingulis 4000 verſationes conficere idque decem cõ-<lb/>tinuorum annorum jugi motu, annis ſingulis 365 diebus definitis, manifeſtum <lb/>eſt terram iſto temporis ſpatio {10512/24000000000000000000} unius pedis ſua ſede emotum <lb/>iri, & </s> <s xml:id="echoid-s3174" xml:space="preserve">quamvis quantitas iſta nec cerni nec notari poſſit, potentia tamen infinita <lb/>nobis ita demonſtratur & </s> <s xml:id="echoid-s3175" xml:space="preserve">animo concipitur, quæ multis annorum ſeculis con-<lb/>tinuata motum oculis quoque notabilem produceret.</s> <s xml:id="echoid-s3176" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3177" xml:space="preserve">Deinceps verò in cæteris Pancratii formis humano uſui accommodis, quo-<lb/>modo in navi machina impenſa exigua loco γepávß ſeu tympani qua onera <lb/>imponantur atque eximantur, qua anchoræ majores adducantur cõſtitui poſ-<lb/>ſit; </s> <s xml:id="echoid-s3178" xml:space="preserve">Quomodo præla olearea vinaria atque hujus generis cætera vehementiffi-<lb/>mè comprimantur; </s> <s xml:id="echoid-s3179" xml:space="preserve">quomodo ſplendidis ſtructionibus maximi lapides ſubve- <pb o="108" file="527.01.108" n="108" rhead="3 L*IBER* S*TATICÆ*"/> ctari attolliq́ue poſſint, aliaq́ueid genus, non exhibemus quia quilibet Pan-<lb/>cration hoc ſibi ſuoq́ue uſui commodius reapſe aptabit quam nos longis ver-<lb/>borum ambagibus explicare poſſemus. </s> <s xml:id="echoid-s3180" xml:space="preserve">Atque hic eſto</s> </p> </div> <div xml:id="echoid-div434" type="section" level="1" n="308"> <head xml:id="echoid-head323" xml:space="preserve">T*ERTII* L*IBRI* <lb/>FINIS.</head> <pb file="527.01.109" n="109"/> </div> <div xml:id="echoid-div435" type="section" level="1" n="309"> <head xml:id="echoid-head324" xml:space="preserve">LIBER QVARTVS <lb/>STATICAE <lb/>DE <lb/>HYDROSTATICES <lb/>ELEMENTIS.</head> <pb o="110" file="527.01.110" n="110" rhead="BREVIARIVM."/> <p style="it"> <s xml:id="echoid-s3181" xml:space="preserve">VO*CES* at que <anchor type="note" xlink:href="" symbol="*"/> ἄιτημα{ρα} huic arti propria pri- <anchor type="note" xlink:label="note-527.01.110-01a" xlink:href="note-527.01.110-01"/> mum exponemus: </s> <s xml:id="echoid-s3182" xml:space="preserve">tum ſuccedentibus novem <lb/>primis propoſitionibus corporum in aquis <anchor type="note" xlink:href="" symbol="*"/> ςάτικα <anchor type="note" xlink:label="note-527.01.110-02a" xlink:href="note-527.01.110-02"/> ἰ{δι}ώμα{ρα}: </s> <s xml:id="echoid-s3183" xml:space="preserve">6 alteris aquæ contra ſubjectum fun-<lb/>dum preſsionis potentiam: </s> <s xml:id="echoid-s3184" xml:space="preserve">duabus ſequentibus <lb/>fundi laterum longitudinem ut habeatur aquæ contra ipſum <lb/>optata preſsio: </s> <s xml:id="echoid-s3185" xml:space="preserve">18, 19 & </s> <s xml:id="echoid-s3186" xml:space="preserve">20 preſſuum aquæ gravitatis centra per <lb/>ſua funda collectorum: </s> <s xml:id="echoid-s3187" xml:space="preserve">21 ex aquæ pondere, ejus magnitudinis <lb/>inventionem: </s> <s xml:id="echoid-s3188" xml:space="preserve">22 denique proportiones quæ ſunt inter corporum <lb/>magnitudines, materiæ gravitatem, & </s> <s xml:id="echoid-s3189" xml:space="preserve">ipſorum pondera. <lb/></s> <s xml:id="echoid-s3190" xml:space="preserve">Quibus tandem ad pleniorem intelligentiam Appendicem de <lb/>by drostatices praxi attexemus.</s> <s xml:id="echoid-s3191" xml:space="preserve"/> </p> <div xml:id="echoid-div435" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-527.01.110-01" xlink:href="note-527.01.110-01a" xml:space="preserve">Poſtulata.</note> <note symbol="*" position="right" xlink:label="note-527.01.110-02" xlink:href="note-527.01.110-02a" xml:space="preserve">Ponderita-<lb/>tu<unsure/> proprie-<lb/>tates.</note> </div> <pb o="111" file="527.01.111" n="111" rhead="DEFINITIONES."/> </div> <div xml:id="echoid-div437" type="section" level="1" n="310"> <head xml:id="echoid-head325" xml:space="preserve">1 DEFINITIO.</head> <p> <s xml:id="echoid-s3192" xml:space="preserve">Notam gravitatem hîc dicimus, cujus nota magnitudo <lb/>cognitâ ponderitate exprimitur.</s> <s xml:id="echoid-s3193" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div438" type="section" level="1" n="311"> <head xml:id="echoid-head326" xml:space="preserve">2 DEFINITIO.</head> <p> <s xml:id="echoid-s3194" xml:space="preserve">Materie æquiponderantia corpora ſunt, quæinaërema-<lb/>gnitudine & </s> <s xml:id="echoid-s3195" xml:space="preserve">ponderitate æquantur.</s> <s xml:id="echoid-s3196" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div439" type="section" level="1" n="312"> <head xml:id="echoid-head327" xml:space="preserve">3 DEFINITIO.</head> <p> <s xml:id="echoid-s3197" xml:space="preserve">Materiâ ponderoſius corpus, quod magnitudine æqua-<lb/>libus præponderat.</s> <s xml:id="echoid-s3198" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div440" type="section" level="1" n="313"> <head xml:id="echoid-head328" xml:space="preserve">4 DEFINITIO.</head> <p> <s xml:id="echoid-s3199" xml:space="preserve">Materiâ levius corpus, quod æqualibus magnitudine, <lb/>pondere cedit.</s> <s xml:id="echoid-s3200" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div441" type="section" level="1" n="314"> <head xml:id="echoid-head329" xml:space="preserve">5 DEFINITIO.</head> <p> <s xml:id="echoid-s3201" xml:space="preserve">Æqualium magnitudine corporum pondere majus, <lb/>materiâ eſt ponderoſiore.</s> <s xml:id="echoid-s3202" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div442" type="section" level="1" n="315"> <head xml:id="echoid-head330" xml:space="preserve">6 DEFINITIO.</head> <p> <s xml:id="echoid-s3203" xml:space="preserve">Solidum corpus eſt cujus materia non ſit fluxa, quodq; <lb/></s> <s xml:id="echoid-s3204" xml:space="preserve">necaqua necaër penetrat.</s> <s xml:id="echoid-s3205" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div443" type="section" level="1" n="316"> <head xml:id="echoid-head331" xml:space="preserve">7 DEFINITIO.</head> <p> <s xml:id="echoid-s3206" xml:space="preserve">Vas ſuperficiarium eſt ſuperficies corporis cogitatione <lb/>ab eo ſeparabilis.</s> <s xml:id="echoid-s3207" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div444" type="section" level="1" n="317"> <head xml:id="echoid-head332" xml:space="preserve">8 DEFINITIO.</head> <p> <s xml:id="echoid-s3208" xml:space="preserve">Fundum eſt ſuperficies quævis, quâ aqua ſubnixa eſt.</s> <s xml:id="echoid-s3209" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div445" type="section" level="1" n="318"> <head xml:id="echoid-head333" xml:space="preserve">9 DEFINITIO.</head> <p> <s xml:id="echoid-s3210" xml:space="preserve">Regulare fundum, eſt planum omni diametro biſe-<lb/>ctile.</s> <s xml:id="echoid-s3211" xml:space="preserve"/> </p> <pb o="112" file="527.01.112" n="112" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div446" type="section" level="1" n="319"> <head xml:id="echoid-head334" xml:space="preserve">INTERPRETAMENTVM.</head> <p> <s xml:id="echoid-s3212" xml:space="preserve">Hujus generis ſunt circuli, ellipſes, parallelogramma, cæteræq́ue figuræ or-<lb/>dinatæ quas circulus recipit laterum numero pari, denique quâcunque ſint <lb/>forma, ut ſubjectæ A, B, atque iſtis ſimi-<lb/> <anchor type="figure" xlink:label="fig-527.01.112-01a" xlink:href="fig-527.01.112-01"/> les infinitæ, quæ rectâ quâcunque per <lb/>centrum acta perpetuò bipartito divi-<lb/>duntur. </s> <s xml:id="echoid-s3213" xml:space="preserve">contra igitur irregulares dicen-<lb/>tur illæ quas quælibet ſua diametros bi-<lb/>fariam non partitur, huc referes triangu-<lb/>la reliquaque polygona laterum numero <lb/>impari, atque id genus cætera. </s> <s xml:id="echoid-s3214" xml:space="preserve">Defini-<lb/>tionem iſtam propoſui, quia ut in ſe-<lb/>quentibus perſpicitur, columna cujus baſis ſit fundum regulare, plano per pun-<lb/>cta in ambitu oppoſitarum hedrarum tranſverſim ſimiliter ſita trajecto, in duas <lb/>æquas partes perpetuò diſſecatur.</s> <s xml:id="echoid-s3215" xml:space="preserve"/> </p> <div xml:id="echoid-div446" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.112-01" xlink:href="fig-527.01.112-01a"> <image file="527.01.112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.112-01"/> </figure> </div> </div> <div xml:id="echoid-div448" type="section" level="1" n="320"> <head xml:id="echoid-head335" xml:space="preserve">10 DEFINITIO.</head> <p> <s xml:id="echoid-s3216" xml:space="preserve">Inane eſt locus corporis expers.</s> <s xml:id="echoid-s3217" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div449" type="section" level="1" n="321"> <head xml:id="echoid-head336" xml:space="preserve">11 DEFINITIO.</head> <p> <s xml:id="echoid-s3218" xml:space="preserve">Vacuum in quo aër duntaxa tineſt.</s> <s xml:id="echoid-s3219" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div450" type="section" level="1" n="322"> <head xml:id="echoid-head337" xml:space="preserve">*POSTVLATA.*</head> <head xml:id="echoid-head338" xml:space="preserve">1 POSTVLATVM.</head> <p> <s xml:id="echoid-s3220" xml:space="preserve">Ponderitatem corporum in aëre appellari propriè, in <lb/>aqua autem ſecundum hy potheſin.</s> <s xml:id="echoid-s3221" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div451" type="section" level="1" n="323"> <head xml:id="echoid-head339" xml:space="preserve">2 POSTVLATVM.</head> <p> <s xml:id="echoid-s3222" xml:space="preserve">Aquam propoſitam omnibus partibus eſſe ponderitatis <lb/>homogeneæ.</s> <s xml:id="echoid-s3223" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div452" type="section" level="1" n="324"> <head xml:id="echoid-head340" xml:space="preserve">3 POSTVLATVM.</head> <p> <s xml:id="echoid-s3224" xml:space="preserve">Pondus à quo vas minus altè deprimitur, levius: </s> <s xml:id="echoid-s3225" xml:space="preserve">quo al-<lb/>tiùs, graviùs: </s> <s xml:id="echoid-s3226" xml:space="preserve">quo æquèaltè, æquipondium eſſe.</s> <s xml:id="echoid-s3227" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div453" type="section" level="1" n="325"> <head xml:id="echoid-head341" xml:space="preserve">4 POSTVLATVM.</head> <p> <s xml:id="echoid-s3228" xml:space="preserve">Vas ſuperficiarium aquam, aliamq́ue materiem conti-<lb/>nere utipſum nec frangatur, nec flectatur.</s> <s xml:id="echoid-s3229" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div454" type="section" level="1" n="326"> <head xml:id="echoid-head342" xml:space="preserve">5 POSTVLATVM.</head> <p> <s xml:id="echoid-s3230" xml:space="preserve">Vas ſuperficiarum effuſa aqua vacuum eſſe.</s> <s xml:id="echoid-s3231" xml:space="preserve"/> </p> <pb o="113" file="527.01.113" n="113" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div455" type="section" level="1" n="327"> <head xml:id="echoid-head343" xml:space="preserve">INTERPRETAMENTVM.</head> <p> <s xml:id="echoid-s3232" xml:space="preserve">Vacuitatem non inanitatem dico, cæteroquin aëris adhuc pondus deeſſet.</s> <s xml:id="echoid-s3233" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div456" type="section" level="1" n="328"> <head xml:id="echoid-head344" xml:space="preserve">6 POSTVLATVM.</head> <p> <s xml:id="echoid-s3234" xml:space="preserve">Cujuſvis aquæ ſuperficiem ſummam, planam & </s> <s xml:id="echoid-s3235" xml:space="preserve">hori-<lb/>zonti parallelam eſſe.</s> <s xml:id="echoid-s3236" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div457" type="section" level="1" n="329"> <head xml:id="echoid-head345" xml:space="preserve">INTERPRETAMENTVM.</head> <p> <s xml:id="echoid-s3237" xml:space="preserve">Licet enim quatenus pars eſt ſphæricæ ſive mundanæ ſuperficiei, mundanam <lb/>autem ſuperficiem dicimus ſphæræ cujuſvis mundo concentricæ, itemq́ue in <lb/>guttula, aut aqua qua corpus aliquod oblitum ſit, hocipſum àverò diſſentiat; <lb/></s> <s xml:id="echoid-s3238" xml:space="preserve">tamen nec tantilla hæc quantitas, nec illa copia poſtulata noſtra evertunt. </s> <s xml:id="echoid-s3239" xml:space="preserve">Cer-<lb/>tè aquæ ſuperficiem ſummam, ſecundum Archimedis demonſtrata, pro parte <lb/>mundanæ ſuperficiei adſumere atque hac forma cuncta eſſerre poſſemus, ſed <lb/>quia majore tædio atque adeò nullo Hydroſtatices compendio hoc fieret, ſim-<lb/>pliciter & </s> <s xml:id="echoid-s3240" xml:space="preserve">apertè poſtulamus, omnis aquæ ſuperficiem ſummam eſſe planam & </s> <s xml:id="echoid-s3241" xml:space="preserve"><lb/>horizonti parallelam.</s> <s xml:id="echoid-s3242" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div458" type="section" level="1" n="330"> <head xml:id="echoid-head346" xml:space="preserve">7 POSTVLATVM.</head> <p> <s xml:id="echoid-s3243" xml:space="preserve">Rectas connecten tes aqueæ columnæ ſummæ imæque <lb/>hedræ horizonti parallelæ puncta <anchor type="note" xlink:href="" symbol="*"/> ὸμο{ρα}γῆ & </s> <s xml:id="echoid-s3244" xml:space="preserve">infinitum <anchor type="note" xlink:label="note-527.01.113-01a" xlink:href="note-527.01.113-01"/> continuatas in mundi centro concurrere; </s> <s xml:id="echoid-s3245" xml:space="preserve">ipſasq́ue adeò <lb/>hedras mundanæ ſuperficiei eſſe partes.</s> <s xml:id="echoid-s3246" xml:space="preserve"/> </p> <div xml:id="echoid-div458" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-527.01.113-01" xlink:href="note-527.01.113-01a" xml:space="preserve">ſimiliter <lb/>ſita.</note> </div> </div> <div xml:id="echoid-div460" type="section" level="1" n="331"> <head xml:id="echoid-head347" xml:space="preserve">INTERPRETAMENTVM.</head> <p> <s xml:id="echoid-s3247" xml:space="preserve">Columnæ A B C D hedræ ſumma imaquè ſunto A B, C D horizonti pa-<lb/>rallelæ atque B, C duo puncta ὸμο{ρα}γῆ, ut cum Ptolomæo loquar, connectat re-<lb/>cta B C horizonti perpendicularis, mundi autem centrum E ad quod adjun-<lb/>gantur A E, B E, baſin D C ſecantes in F & </s> <s xml:id="echoid-s3248" xml:space="preserve">G inter hæc puncta deſcribito <lb/>figuram F G ipſe D C ſimilem ſimiliterq́, ſitam. </s> <s xml:id="echoid-s3249" xml:space="preserve">Quæ cum ita ſint, manifeſtum <lb/>eſt nequerectas B C continuatas coïre in centro E, cum A F & </s> <s xml:id="echoid-s3250" xml:space="preserve">B G iſtic con-<lb/>currant, neque plana A B, D C mundanæ ſuperficies par-<lb/> <anchor type="figure" xlink:label="fig-527.01.113-01a" xlink:href="fig-527.01.113-01"/> tes eſſe: </s> <s xml:id="echoid-s3251" xml:space="preserve">poſtulamus tamen A D, B D iſtic concurrere, pla-<lb/>naq́ue A B, C D mundanæ ſuperficiei partes eſſe, quia in <lb/>hydroſtatices praxi hujus differentiæ ratio nulla erit, quæq; <lb/></s> <s xml:id="echoid-s3252" xml:space="preserve">nullo modo inter columnam A B C D & </s> <s xml:id="echoid-s3253" xml:space="preserve">curtam pyrami-<lb/>dem A B G F notari poſſit, etiamſi A B, F G tanquam mun-<lb/>danę ſuperficiei partes aſiumãtur. </s> <s xml:id="echoid-s3254" xml:space="preserve">Et quamvis pro columna <lb/>A B C D curtam pyramidem A B F G uſurpare poſſimus <lb/>atque ita ſuccedentia theoremata enuntiare, tamen eadem <lb/>cauia, quam 6 poſtul. </s> <s xml:id="echoid-s3255" xml:space="preserve">retulimus, nos ab iſto labore deterruit. </s> <s xml:id="echoid-s3256" xml:space="preserve"><lb/>Et profectò tam ineptum fuerit hæc ipſa non admittere, <lb/>quàm poſtulantibus Aſtrologis, terram eſſe mundi centrum, fidem derogare.</s> <s xml:id="echoid-s3257" xml:space="preserve"/> </p> <div xml:id="echoid-div460" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.113-01" xlink:href="fig-527.01.113-01a"> <image file="527.01.113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.113-01"/> </figure> </div> <pb o="114" file="527.01.114" n="114" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div462" type="section" level="1" n="332"> <head xml:id="echoid-head348" xml:space="preserve">PROPOSITIONES.</head> <head xml:id="echoid-head349" xml:space="preserve">1 THEOREMA. 1 PROPOSITIO.</head> <p> <s xml:id="echoid-s3258" xml:space="preserve">Aquam datam, datum ſibi intra aquam locum ſervare.</s> <s xml:id="echoid-s3259" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3260" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3261" xml:space="preserve">Aqua data vaſe ſuperficiario contenta collocator in aqua B C. <lb/></s> <s xml:id="echoid-s3262" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3263" xml:space="preserve">Aquam A eo loco ſubſiſtere.</s> <s xml:id="echoid-s3264" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div463" type="section" level="1" n="333"> <head xml:id="echoid-head350" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3265" xml:space="preserve">A igitur (ſi ullo modo per naturam ſieri poſſit) locum ſibi tributum non fer-<lb/>vato, ac delabatur in D; </s> <s xml:id="echoid-s3266" xml:space="preserve">quibus poſitis aqua quæ ipſi A ſucceffit eandem ob <lb/>cauſam deſluet in D, eademq́ue ab alia iſtinc expelletur, atque adeo aqua bæc <lb/>(cum ubique eadem ratio ſit) motũ inſtituet perpetuurn, quod <lb/>abſurdum fuerit. </s> <s xml:id="echoid-s3267" xml:space="preserve">Similiter demonſtratur A nec adſcendere <lb/> <anchor type="figure" xlink:label="fig-527.01.114-01a" xlink:href="fig-527.01.114-01"/> neque in latus moveri. </s> <s xml:id="echoid-s3268" xml:space="preserve">Proinde manifeſtum jam eſt, ſi A ſtatua-<lb/>tur alicubi intra aquam in D, E, F vel G iſtic in loco ſibi tribu-<lb/>to ſeſe ſiſtere.</s> <s xml:id="echoid-s3269" xml:space="preserve"/> </p> <div xml:id="echoid-div463" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.114-01" xlink:href="fig-527.01.114-01a"> <image file="527.01.114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.114-01"/> </figure> </div> <p> <s xml:id="echoid-s3270" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3271" xml:space="preserve">Itaque data aqua quemlibet datum ſibilo-<lb/>cum ſervat. </s> <s xml:id="echoid-s3272" xml:space="preserve">Quod demonſtrare oportebat.</s> <s xml:id="echoid-s3273" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div465" type="section" level="1" n="334"> <head xml:id="echoid-head351" xml:space="preserve">2 THE OREMA. 2 PROPOSITIO.</head> <p> <s xml:id="echoid-s3274" xml:space="preserve">Solidum corpus materia leviore quam ſit aqua non <lb/>omnino mergitur, ſed eminet aliqua ſui parte.</s> <s xml:id="echoid-s3275" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3276" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3277" xml:space="preserve">Solidum corpus A materiam habeat leviorem aquâ B C, cujus <lb/>ſumma ſuperficies B D. </s> <s xml:id="echoid-s3278" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3279" xml:space="preserve">A aquæ impoſitum non universè <lb/>mergi ſed aliqua parte eminere demonſtrandum eſto.</s> <s xml:id="echoid-s3280" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3281" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3282" xml:space="preserve">EF vas ſuperſiciarium, cujus pars G F demerſa & </s> <s xml:id="echoid-s3283" xml:space="preserve"><lb/>plena aquæ æqualis ſimilisq́ue ſit ipſi A, itaque aquæ ejus ſuperſicies G H erit <lb/>in plano B D, quia ſuperſiciarium vas E F gravitatis levitatisq́ue ſit expers.</s> <s xml:id="echoid-s3284" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div466" type="section" level="1" n="335"> <head xml:id="echoid-head352" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3285" xml:space="preserve">Cum igitur A ex hypotheſi materiam habeat leviorem aquâ G F, & </s> <s xml:id="echoid-s3286" xml:space="preserve">hæc <lb/>dato corpori A ſit æ qualis, G F neceſſariò ipſi A præponderabit, jam in vaſe <lb/>ſuperſiciario locoaquæ ſubſtituitor corpus A ipſi congruum, nam ex fabrica <lb/>parti G F æquale eſt & </s> <s xml:id="echoid-s3287" xml:space="preserve">ſimile; </s> <s xml:id="echoid-s3288" xml:space="preserve">cumq́ue A corpus levius ſit aqua effusâ pro-<lb/>pterea vas ſuperficiarium E F per 3 poſtul non tam alte <lb/> <anchor type="figure" xlink:label="fig-527.01.114-02a" xlink:href="fig-527.01.114-02"/> mergetur ab A atque ab F G: </s> <s xml:id="echoid-s3289" xml:space="preserve">atqui quantò minus altè <lb/>corpus ſuperſiciarium E F ſubſidit, tantum corporis A <lb/>ſupra aquam extare neceſſe eſt.</s> <s xml:id="echoid-s3290" xml:space="preserve"/> </p> <div xml:id="echoid-div466" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.114-02" xlink:href="fig-527.01.114-02a"> <image file="527.01.114-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.114-02"/> </figure> </div> <p> <s xml:id="echoid-s3291" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3292" xml:space="preserve">Quamobrem corpus ſolidum ma-<lb/>teriæ levioris quàm aqua non totum mergitur, ſed ali-<lb/>qua ſui parte ſupereminet. </s> <s xml:id="echoid-s3293" xml:space="preserve">Quod demonſtraſſe oportuit.</s> <s xml:id="echoid-s3294" xml:space="preserve"/> </p> <pb o="115" file="527.01.115" n="115" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div468" type="section" level="1" n="336"> <head xml:id="echoid-head353" xml:space="preserve">3 THEOREMA. 3 PROPOSITIO.</head> <p> <s xml:id="echoid-s3295" xml:space="preserve">Corpus ſolidum materiæ ponderoſioris quàm aqua ad <lb/>fundum uſque demergitur.</s> <s xml:id="echoid-s3296" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3297" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3298" xml:space="preserve">Efto corpus ſolidum A materiâ ponderoſiore quàm aqua B C, <lb/>cujus ſuperſicies ſumma B D, ima E C. </s> <s xml:id="echoid-s3299" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3300" xml:space="preserve">A aquæ B C im-<lb/>poſitum ad fundum E C uſque deſcendere demonſtrator.</s> <s xml:id="echoid-s3301" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3302" xml:space="preserve">P*RÆPARATIO*. </s> <s xml:id="echoid-s3303" xml:space="preserve">Vas ſuperſiciatium F G aqua plenum ipſi A ſimile & </s> <s xml:id="echoid-s3304" xml:space="preserve"><lb/>æquale deſormato, cujus ſuperſicies ſumma F H in plano B D collocetur.</s> <s xml:id="echoid-s3305" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div469" type="section" level="1" n="337"> <head xml:id="echoid-head354" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3306" xml:space="preserve">Cum A materiæ quidem ponderoſioris ſit ex hy potheſi quam aqua F G, <lb/>magnitudinis verò æ qualis; </s> <s xml:id="echoid-s3307" xml:space="preserve">A quoque gravius erit quam F G. </s> <s xml:id="echoid-s3308" xml:space="preserve">Iam ſi in locum <lb/>aquæ F G ſubſtituatur ipſi pro pter fab team congruum corpu; </s> <s xml:id="echoid-s3309" xml:space="preserve">A attamen, ut <lb/>patet, illo ponderoſius, vas ſuperſiciarium F G per 3 poſtul. </s> <s xml:id="echoid-s3310" xml:space="preserve">ab A corpore al-<lb/>tius demergetur quam ab aqua F G. </s> <s xml:id="echoid-s3311" xml:space="preserve">demonſiravimus igitur corpus A mergi; <lb/></s> <s xml:id="echoid-s3312" xml:space="preserve">fed ad fundum E C uſque demergi ita evincam. </s> <s xml:id="echoid-s3313" xml:space="preserve">Si enim fiem poſſit ne deſcen-<lb/>dat ad E C, ſed interinedio loco hæreat ut in I. </s> <s xml:id="echoid-s3314" xml:space="preserve">Iam ſi vice ſolidi corporis I <lb/>vas ejus ſuperſiciarium aquâ compleatur, eodem loco per I propoſ. </s> <s xml:id="echoid-s3315" xml:space="preserve">ſubſiſtet: </s> <s xml:id="echoid-s3316" xml:space="preserve"><lb/>fed æ qua hæc levior eſt iſto ſolido; </s> <s xml:id="echoid-s3317" xml:space="preserve">quamobrem gravius <lb/> <anchor type="figure" xlink:label="fig-527.01.115-01a" xlink:href="fig-527.01.115-01"/> leviusq́ue in humido eodem hærebuntloco, quod abſur <lb/>dum 3 poſtulato repugnat. </s> <s xml:id="echoid-s3318" xml:space="preserve">Itaque eorpus A inter B D, <lb/>E C, ſummam imamq́ue ſuperſiciem conſiſtere nequit, <lb/>ac neceſſariò deſcendet donec ſundo ſubjecto E C im-<lb/>pactum quieſcat. </s> <s xml:id="echoid-s3319" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s3320" xml:space="preserve">* Solidum corpus igi-<lb/>tur materiæ ponderoſioris quam aqua ad fundum uſque demergitur. </s> <s xml:id="echoid-s3321" xml:space="preserve">Quod <lb/>demonſtraſſe oportuit.</s> <s xml:id="echoid-s3322" xml:space="preserve"/> </p> <div xml:id="echoid-div469" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.115-01" xlink:href="fig-527.01.115-01a"> <image file="527.01.115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.115-01"/> </figure> </div> </div> <div xml:id="echoid-div471" type="section" level="1" n="338"> <head xml:id="echoid-head355" xml:space="preserve">4 THEOREMA. 4 PROPOSITIO.</head> <p> <s xml:id="echoid-s3323" xml:space="preserve">Corpus ſolidum materiâ aquæ æquiponderante, datum <lb/>inaqua locum ſervat.</s> <s xml:id="echoid-s3324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3325" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3326" xml:space="preserve">Corpus ſolidum A materiam habeat gravitate æqualem aquæ <lb/>BC. </s> <s xml:id="echoid-s3327" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3328" xml:space="preserve">Corpus A ubicunque in aqua ſtatuetur ſubſiſtere de-<lb/>monſtrato. </s> <s xml:id="echoid-s3329" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3330" xml:space="preserve">Eſto vas ſuperſiciarium D, ipſi A æquale & </s> <s xml:id="echoid-s3331" xml:space="preserve"><lb/>ſimile.</s> <s xml:id="echoid-s3332" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div472" type="section" level="1" n="339"> <head xml:id="echoid-head356" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3333" xml:space="preserve">Cum A materię ponderitate & </s> <s xml:id="echoid-s3334" xml:space="preserve">corporis mole æquale ſit aquæ D, ideo <lb/>D ipſi A pondere quoque æquatur. </s> <s xml:id="echoid-s3335" xml:space="preserve">Iam in vaſe ſuperſiciario D loco aquæ <lb/>ſubſtituitor corpus A ipſi propter fabricam per omnia congruum, quare per <lb/>3 poſſul. </s> <s xml:id="echoid-s3336" xml:space="preserve">vas ſuperficiarium D ab corpore A nec ma-<lb/> <anchor type="figure" xlink:label="fig-527.01.115-02a" xlink:href="fig-527.01.115-02"/> gis nec minus demergetur quam ab aqua D; </s> <s xml:id="echoid-s3337" xml:space="preserve">ſed aqua <lb/>D per 1 propoſ datum ſibi locum ſervat. </s> <s xml:id="echoid-s3338" xml:space="preserve">Itaque etiam <lb/>ſolidum corpus A in aqua B C tributum ſibi locum <lb/>retinebit, neque aliò delabetur.</s> <s xml:id="echoid-s3339" xml:space="preserve"/> </p> <div xml:id="echoid-div472" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.115-02" xlink:href="fig-527.01.115-02a"> <image file="527.01.115-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.115-02"/> </figure> </div> <p> <s xml:id="echoid-s3340" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3341" xml:space="preserve">Quamobrem corpus ſolidum, cujus <lb/>materies ponderitate aquæ eſt æqualis, datum ſervat locum. </s> <s xml:id="echoid-s3342" xml:space="preserve">Quod erat de-<lb/>monſtrandum.</s> <s xml:id="echoid-s3343" xml:space="preserve"/> </p> <pb o="116" file="527.01.116" n="116" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div474" type="section" level="1" n="340"> <head xml:id="echoid-head357" xml:space="preserve">5 THE OREMA. 5 PROPOSITIO.</head> <p type="title"> <s xml:id="echoid-s3344" xml:space="preserve">Corpus ſolidum materiæ levioris quamaqua cuiinnatat, <lb/>ponderitate æquale eſt tantæ aqueæ moli, quantâ ſui <lb/>parte demergitur.</s> <s xml:id="echoid-s3345" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3346" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3347" xml:space="preserve">* AB corpus ſolidum materiam habeat leviorem aqua BC in <lb/>quam immittiur, ſolidi autem ſuperficies AB, pars in aquam immerſa EB.</s> <s xml:id="echoid-s3348" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3349" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3350" xml:space="preserve">Demonſtrato ſolidum AB pondere æquari aquæ EB <lb/>quæ æqualis eſt parti in aquam CD immerſæ.</s> <s xml:id="echoid-s3351" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div475" type="section" level="1" n="341"> <head xml:id="echoid-head358" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3352" xml:space="preserve">In vaſe ſuperficiario AB loco ſolidi corporis AB ſubſtituitor aquæ copia <lb/>tanta, quę ipſum eadem altitudine qua priùs in aquam immer-<lb/>gat. </s> <s xml:id="echoid-s3353" xml:space="preserve">quibus poſitis, aqua EB (cujus ſuperſicies eſt in ſuperfi-<lb/> <anchor type="figure" xlink:label="fig-527.01.116-01a" xlink:href="fig-527.01.116-01"/> cie reliquæ aquæ cum vas ſuperficiarium nullius ſit ponderita-<lb/>tis) contenta vaſe ſuperficiario EB per 3 poſtul. </s> <s xml:id="echoid-s3354" xml:space="preserve">æquiponde-<lb/>rabit ſolido corpori AB quia vas idem eadem altitudine im-<lb/>mergunt. </s> <s xml:id="echoid-s3355" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s3356" xml:space="preserve">* Itaque ſolidum corpus materia <lb/>leviore quam aqua in quam immittitur æquiponderat tantæ <lb/>aqueæ moli, quantâ ſui parteaquæ immergitur.</s> <s xml:id="echoid-s3357" xml:space="preserve"/> </p> <div xml:id="echoid-div475" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.116-01" xlink:href="fig-527.01.116-01a"> <image file="527.01.116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.116-01"/> </figure> </div> </div> <div xml:id="echoid-div477" type="section" level="1" n="342"> <head xml:id="echoid-head359" xml:space="preserve">1 PROBLEMA. 6 PROPOSITIO.</head> <p> <s xml:id="echoid-s3358" xml:space="preserve">*Corpore ſolido ſui partenotæ magnitudinis in aquam* <lb/>*cognitæ ponderitatis immerſo, totius ſolidi pondus in-* <lb/>*venire.</s> <s xml:id="echoid-s3359" xml:space="preserve">*</s> </p> <p> <s xml:id="echoid-s3360" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3361" xml:space="preserve">* ABCD ſolidum corpus formæ contingentis, EF aqua cujus <lb/>pes cubicus ponderet 65 ℔, nam experientia edoctus Delfenſem pedem Del-<lb/>fenſis aquæ tanti ponderis eſſe cognovi, ſicq́ue deinceps in ſequentibus exem-<lb/>plis uſurpabimus; </s> <s xml:id="echoid-s3362" xml:space="preserve">ſolidi corporis pars aquæ immerſa ACD, cujus magni-<lb/>tudo ſit 10000 pedum cubicorum. </s> <s xml:id="echoid-s3363" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3364" xml:space="preserve">Quantum pondus ſit <lb/>univerſi corporis ABCD quæq́; </s> <s xml:id="echoid-s3365" xml:space="preserve">illo vel continentur velinnituntur invenien-<lb/>dum eſto.</s> <s xml:id="echoid-s3366" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div478" type="section" level="1" n="343"> <head xml:id="echoid-head360" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3367" xml:space="preserve">Multiplicatis 10000 cum 65 ℔ efficiuntur pro quęſito 650000 ℔.</s> <s xml:id="echoid-s3368" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div479" type="section" level="1" n="344"> <head xml:id="echoid-head361" xml:space="preserve">DEMONSTRATIO.</head> <figure> <image file="527.01.116-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.116-02"/> </figure> <p> <s xml:id="echoid-s3369" xml:space="preserve">Vniverſum ſolidum ABCD æquiponderat per <lb/>5 propoſ. </s> <s xml:id="echoid-s3370" xml:space="preserve">aquæ æquali corpori ACD, ſed tanta <lb/>aquæ moles pendet 650000 ℔, itaque etiam corpus <lb/>ABCD erit librarum totidem.</s> <s xml:id="echoid-s3371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3372" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s3373" xml:space="preserve">* Quamobrem corpore ſolido <lb/>ſui parte notæ magnitudinis in aquam cognitæ pon-<lb/>deritatis immerſo, etiam totius ponderitatem inveni-<lb/>mus. </s> <s xml:id="echoid-s3374" xml:space="preserve">Quod oportuit.</s> <s xml:id="echoid-s3375" xml:space="preserve"/> </p> <pb o="117" file="527.01.117" n="117" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> </div> <div xml:id="echoid-div480" type="section" level="1" n="345"> <head xml:id="echoid-head362" xml:space="preserve">6 THE OREMA. 7 PROPOSITIO.</head> <p type="title"> <s xml:id="echoid-s3376" xml:space="preserve">In aquis ponderitatis heterogeneæ erit ut ponderitas ma-<lb/>teriæ aquæ ponderoſioris ad ponderitatem materiæ aquæ <lb/>levioris, ſic pars corporis ſolidi in aquam materiæ levioris <lb/>immerſa ad partem ſolidi ejuſdem in aqua graviore de-<lb/>merſam.</s> <s xml:id="echoid-s3377" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3378" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3379" xml:space="preserve">* Aqua AB materiæ ſit gravioris quam DC, corpus ſolidum EF <lb/>materiâ utrâlibet ipſarum leviore, quod in aquam AB immiſſum parte GF <lb/>aquæ inſideat, idemq́ue immiſſum in aquam CD mergatur parte KI.</s> <s xml:id="echoid-s3380" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3381" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s3382" xml:space="preserve">* Demonſtrandum materiæ gravitatem aquæ AB, adgra-<lb/>vitatem materiæ aquæ CD eſſe, ſicut KI ad GF.</s> <s xml:id="echoid-s3383" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div481" type="section" level="1" n="346"> <head xml:id="echoid-head363" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3384" xml:space="preserve">Moles aquæ ex AB æqualis ipſi GF, ponderitate æquabit corpus EF; </s> <s xml:id="echoid-s3385" xml:space="preserve">ſic <lb/>item moles aquea ex CD æqualis ipſi KI æquiponderat p<unsure/>ci 5 propoſ. </s> <s xml:id="echoid-s3386" xml:space="preserve">corpo-<lb/>ri HI ſive EF cum ex hypotheſi ſit unum <lb/>idemq́ue corpus; </s> <s xml:id="echoid-s3387" xml:space="preserve">& </s> <s xml:id="echoid-s3388" xml:space="preserve">moles igitur aquæ ex AB <lb/> <anchor type="figure" xlink:label="fig-527.01.117-01a" xlink:href="fig-527.01.117-01"/> ęqualis ipſi GF ponderitate quoque æquat <lb/>molem aqueam ex CD ęqualem ipſi KI. </s> <s xml:id="echoid-s3389" xml:space="preserve">Sed <lb/>poſitis duabus aquis gravitatis æqualis erit ma-<lb/>gnitudinum & </s> <s xml:id="echoid-s3390" xml:space="preserve">materię ponderitatum propor-<lb/>tio reciproca, quod neceſſariò deduciturè de-<lb/>finitione 5. </s> <s xml:id="echoid-s3391" xml:space="preserve">quare ratio ponderoſitatis materiæ <lb/>aquæ AB ad ponderitatem materię aquæ CD eadem erit, quæ magnitudinis <lb/>KI ad mag<unsure/> itudinem GF. </s> <s xml:id="echoid-s3392" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s3393" xml:space="preserve">* Itaque in aquis ponderitatis <lb/>heterogeneæ ut ponderitas materiæ, &</s> <s xml:id="echoid-s3394" xml:space="preserve">c.</s> <s xml:id="echoid-s3395" xml:space="preserve"/> </p> <div xml:id="echoid-div481" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.117-01" xlink:href="fig-527.01.117-01a"> <image file="527.01.117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.117-01"/> </figure> </div> </div> <div xml:id="echoid-div483" type="section" level="1" n="347"> <head xml:id="echoid-head364" xml:space="preserve">7 THE OREMA. 8 PROPOSITIO.</head> <p> <s xml:id="echoid-s3396" xml:space="preserve">Corpus ſolidum in aqua levius eſt quàm in aëre, pon-<lb/>dere aquæ magnitudine ſibiæqualis.</s> <s xml:id="echoid-s3397" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3398" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3399" xml:space="preserve">* Eſto A corpus ſolidum, aqua BC.</s> <s xml:id="echoid-s3400" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3401" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3402" xml:space="preserve">Demonſtrato corpus A in aquam immiſſum iſticlevius <lb/>eſſe quam in aëre gravitate aqueę molis magnitudine ſibiæqualis.</s> <s xml:id="echoid-s3403" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3404" xml:space="preserve">P*RAEPARATIO.</s> <s xml:id="echoid-s3405" xml:space="preserve">* Sit D vas ſuperficiarium aquæ plenum, ſimile & </s> <s xml:id="echoid-s3406" xml:space="preserve">æqua-<lb/>leipſi A.</s> <s xml:id="echoid-s3407" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div484" type="section" level="1" n="348"> <head xml:id="echoid-head365" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3408" xml:space="preserve">Vas ſuperficiarium D aqua repletum per 1 propoſ. </s> <s xml:id="echoid-s3409" xml:space="preserve">in ipſa aqua nec gravi-<lb/>tatis neque levitatis habet momentum, quare eſſusâ aquâ D vastanto aquę <lb/>pondere erit levius, id eſt, perſectè leve; </s> <s xml:id="echoid-s3410" xml:space="preserve">in hocſubſti-<lb/>tuito corpus A ipſi congruum, jam vas ſuperficia-<lb/> <anchor type="figure" xlink:label="fig-527.01.117-02a" xlink:href="fig-527.01.117-02"/> rium cum corpore A ſibi inſerto pendet pódus ipſius <lb/>A ſimul cum dicta levitate, hoc eſt, pendet pondus <lb/>A dempto pondere aquæ prius effuſę, ſed aqua iſta <lb/>mole æquat corpus A. </s> <s xml:id="echoid-s3411" xml:space="preserve">Quamobrem A ponderitate <pb o="118" file="527.01.118" n="118" rhead="4 L*IBER* S*TATICÆ*"/> aquæ magnitudine ſibi æqualis levius eſt in aqua BC quam foret in aëre. </s> <s xml:id="echoid-s3412" xml:space="preserve">Quod <lb/>demonſtraſſe oportuit.</s> <s xml:id="echoid-s3413" xml:space="preserve"/> </p> <div xml:id="echoid-div484" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.117-02" xlink:href="fig-527.01.117-02a"> <image file="527.01.117-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.117-02"/> </figure> </div> </div> <div xml:id="echoid-div486" type="section" level="1" n="349"> <head xml:id="echoid-head366" xml:space="preserve">2 PROBLEMA. 9 PROPOSITIO.</head> <p> <s xml:id="echoid-s3414" xml:space="preserve">Data corporis ſolidi gravitate, ejusque materiæ ponde-<lb/>ritatis ratione ad ponderitatem aqueam; </s> <s xml:id="echoid-s3415" xml:space="preserve">ejuſdem in aqua <lb/>ſitus gravitatem invenire.</s> <s xml:id="echoid-s3416" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div487" type="section" level="1" n="350"> <head xml:id="echoid-head367" style="it" xml:space="preserve">1 Exemplum cum corpus ſolidum materiæ levioris erit <lb/>quam aqua.</head> <p> <s xml:id="echoid-s3417" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3418" xml:space="preserve">* Aqua AB, corpus ſolidum Cpendens 2 ℔, hinc ponderitatis <lb/>aqueæ ad corporis ſolidi materiæ ponderitatem ratio quintupla eſto.</s> <s xml:id="echoid-s3419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3420" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s3421" xml:space="preserve">* Invenire ſolidi C ſitus gravitatem in aqua AB.</s> <s xml:id="echoid-s3422" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div488" type="section" level="1" n="351"> <head xml:id="echoid-head368" xml:space="preserve">CONSTRVCTIO.</head> <figure> <image file="527.01.118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.118-01"/> </figure> <p> <s xml:id="echoid-s3423" xml:space="preserve">Quanta gravitas ſit aqueæ molis ipſi C æqualis ex-<lb/>pendito, ea erit bis quinarũ librarú hoc eſt 10 ℔ quibus <lb/>deductis de 2 ℔, relinquentur -- 8 ℔ ſolidi corporis <lb/>C, quæ ſunt levitas ſeu aſcenſus corporis C in data <lb/>aqua AB.</s> <s xml:id="echoid-s3424" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3425" xml:space="preserve">Sed ut clarius exprimam, cogitatione fingito C in <lb/>aquam AB immiſſum, contra quod, ut in diagramma-<lb/>te perſpicis, ſuſpenſum ſit pondus D 8 ℔ hocſitu D & </s> <s xml:id="echoid-s3426" xml:space="preserve"><lb/>C æquiponderabunt.</s> <s xml:id="echoid-s3427" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div489" type="section" level="1" n="352"> <head xml:id="echoid-head369" xml:space="preserve">2 Exemplum cum ſolidum corpus materiæ gravioris erit <lb/>quam aqua, cujus pragmatia antecedenti affinis eſt.</head> <p> <s xml:id="echoid-s3428" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3429" xml:space="preserve">* Ratio ponderitatis aqueæ, ut ſupra AB, ad ponderitatem ma-<lb/>teriæ ſolidi C nunc ſumitor ſubquadrupla, atque ipſum corpus C 12 ℔.</s> <s xml:id="echoid-s3430" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3431" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s3432" xml:space="preserve">* Invenire ſolidi ſitus gravitatem in aqua AB.</s> <s xml:id="echoid-s3433" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div490" type="section" level="1" n="353"> <head xml:id="echoid-head370" xml:space="preserve">CONSTRVCTIO.</head> <figure> <image file="527.01.118-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.118-02"/> </figure> <p> <s xml:id="echoid-s3434" xml:space="preserve">Conſiderato quanta ſit gravitas aquæ magnitudinis <lb/>ipſi C æqualis, eaque deprehendetur {1/3} 12 librarum quas <lb/>C pendet, eæ igitur erunt 3 ℔, quæ deductæ de 12 ℔ ſo-<lb/>lidi C, reliquas faciunt 9 ℔ pro pondere C in aqua AB.</s> <s xml:id="echoid-s3435" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3436" xml:space="preserve">Cujus illuſtrandi gratia aquæ AB immergatur cor-<lb/>pus C, cui ex oppoſito ſuſpendatur D 9 ℔; </s> <s xml:id="echoid-s3437" xml:space="preserve">hoc ſitu D <lb/>& </s> <s xml:id="echoid-s3438" xml:space="preserve">C æquiponderabunt.</s> <s xml:id="echoid-s3439" xml:space="preserve"/> </p> <pb o="119" file="527.01.119" n="119" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> <p> <s xml:id="echoid-s3440" xml:space="preserve">Tertium quoddam exemplum excogitari potuit, cum ratio ponderitatum <lb/>utriuſque materiæ aqueæ ſcilicet & </s> <s xml:id="echoid-s3441" xml:space="preserve">ſolidę ęqualis erit: </s> <s xml:id="echoid-s3442" xml:space="preserve">ſed eo caſu, normam for-<lb/>mamq́ue antecedentis pragmatiæ ſecutus, deprehendes ſolidum corpus in tali <lb/>aqua nec grave eſſe neque leve. </s> <s xml:id="echoid-s3443" xml:space="preserve">demonſtratio autem omniũ horum per 8 prop. <lb/></s> <s xml:id="echoid-s3444" xml:space="preserve">manifeſta eſt. </s> <s xml:id="echoid-s3445" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s3446" xml:space="preserve">* Itaq; </s> <s xml:id="echoid-s3447" xml:space="preserve">corporis ſolidi gravitate, ejuſdemq́; </s> <s xml:id="echoid-s3448" xml:space="preserve">ma-<lb/>teriæ ponderitatis ad ponderitatem aqueam ratione data; </s> <s xml:id="echoid-s3449" xml:space="preserve">ejus ſitus gravitatem <lb/>in aqua invenimus. </s> <s xml:id="echoid-s3450" xml:space="preserve">Quod faciendum erat.</s> <s xml:id="echoid-s3451" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div491" type="section" level="1" n="354"> <head xml:id="echoid-head371" xml:space="preserve">8 THE OREMA. 10 PROPOSITIO.</head> <p> <s xml:id="echoid-s3452" xml:space="preserve">Aquæ fundo horizonti parallelo tantum inſidet pon-<lb/>dus, quantum eſt aqueæ columnæ cujus baſis fundo, alti-<lb/>tudo perpendiculari ab aquæ ſuperſicie ſumma ad imam <lb/>demiſſæ æqualis ſit.</s> <s xml:id="echoid-s3453" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3454" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s3455" xml:space="preserve">* ABCD aquæ figura ſolida rectangula, AB ſuperficies ſumma, <lb/>EF pars fundi horizonti paralleli, GE perpendicularis à ſumma ad imam aquæ <lb/>ſuperficiem, columna GHFE comprehenſa ſub baſi EF & </s> <s xml:id="echoid-s3456" xml:space="preserve">altitudine EG.</s> <s xml:id="echoid-s3457" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3458" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s3459" xml:space="preserve">* Demonſtrato baſe ſeu fundo EF ſuſtineri pondus æqua-<lb/>le columnæ aqueæ GHFE.</s> <s xml:id="echoid-s3460" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div492" type="section" level="1" n="355"> <head xml:id="echoid-head372" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3461" xml:space="preserve">Sifundo EF plus ponderis inſideat quàm aquæ GHFE, id erit ab aqua <lb/>finitima, atque ideò ſi ſieri poſſit eſto ab AGED & </s> <s xml:id="echoid-s3462" xml:space="preserve">HBCF; </s> <s xml:id="echoid-s3463" xml:space="preserve">quibus poſitis <lb/>fundo DE quoque, propter aquam finitimam GHFE <lb/>(cum utrobiq; </s> <s xml:id="echoid-s3464" xml:space="preserve">ſit parratio) plus pó deris incumbet quàm <lb/> <anchor type="figure" xlink:label="fig-527.01.119-01a" xlink:href="fig-527.01.119-01"/> ſit aquæ AGED, perinde quoque baſi FC plusinſi-<lb/>det ponderis quam aquæ HBCF; </s> <s xml:id="echoid-s3465" xml:space="preserve">quare toti fundo <lb/>DC majus quoddam pondus inſidet quam aquæ totius <lb/>ABCD, quod tamen, cum ABCD corpus rectan-<lb/>gulum ſit, abſurdum ſuerit. </s> <s xml:id="echoid-s3466" xml:space="preserve">Eadem ratione evinces fun-<lb/>do EF non minus pondus ſuſtentari quam ſit aquæ GHFE; </s> <s xml:id="echoid-s3467" xml:space="preserve">quare tantun-<lb/>dem duntaxat ponderis neceſſario ipſi incumbet.</s> <s xml:id="echoid-s3468" xml:space="preserve"/> </p> <div xml:id="echoid-div492" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.119-01" xlink:href="fig-527.01.119-01a"> <image file="527.01.119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.119-01"/> </figure> </div> </div> <div xml:id="echoid-div494" type="section" level="1" n="356"> <head xml:id="echoid-head373" xml:space="preserve">1 C*ONSECTARIVM.*</head> <p> <s xml:id="echoid-s3469" xml:space="preserve">Immittito in aquam ABCD hujus propoſitionis corpus ſolidum IKLM, <lb/>materiæ levioris quam aqua, quodque ideo ipſi innatet parte NOLM <lb/>immersâ, reliquâ NOKI ſupereminente, ut in ſubjecta ſigura apparet. </s> <s xml:id="echoid-s3470" xml:space="preserve">Iam <lb/>ſolidum IKLM per 5 propoſ. </s> <s xml:id="echoid-s3471" xml:space="preserve">gravitate æquale eſt tan-<lb/> <anchor type="figure" xlink:label="fig-527.01.119-02a" xlink:href="fig-527.01.119-02"/> tæ aqueæ moli, quanta eſt pars ſui demerſa NOLM; <lb/></s> <s xml:id="echoid-s3472" xml:space="preserve">quare ſolidum IKLM cum reliqua ipſum ambiente <lb/>aqua pondere æquat corpus aqueum magnitudinis <lb/>ABCD. </s> <s xml:id="echoid-s3473" xml:space="preserve">Itaque etiamnum aſſerimus ſecundùm pro-<lb/>poſitionis ſententiam, ſundo EF inniti pondus æquale <lb/>corpori aqueo magnitudinis columnæ, cujus baſis ſit <lb/>EF, altitudo perpendicularis GE à ſumma ſuperficie <lb/>aquæ AB adimum fundum EF demiſſa. </s> <s xml:id="echoid-s3474" xml:space="preserve">Vnde efficitur à materia qualibct <lb/>aquæ innatante fundum nec magis nec minus affici, quam ab aqua in eadem <lb/>altitudine conſtituta.</s> <s xml:id="echoid-s3475" xml:space="preserve"/> </p> <div xml:id="echoid-div494" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.119-02" xlink:href="fig-527.01.119-02a"> <image file="527.01.119-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.119-02"/> </figure> </div> <pb o="120" file="527.01.120" n="120" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div496" type="section" level="1" n="357"> <head xml:id="echoid-head374" xml:space="preserve">2 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s3476" xml:space="preserve">Secundò in aquam ABCD immittitor corpus ſolidum, ſolidavé quot-<lb/>cunque materiâ aquæ æquipondiâ, inter quæ, reliqua omnia aqua expulſa, <lb/>tantùm comprehendatur IKFELM; </s> <s xml:id="echoid-s3477" xml:space="preserve">quæ cum ita ſint, hæc corpora fundum <lb/>EF nec aggravant neque relevant <lb/>ab eo pódere quo aqua prius ipſum <lb/> <anchor type="figure" xlink:label="fig-527.01.120-01a" xlink:href="fig-527.01.120-01"/> afficiebat. </s> <s xml:id="echoid-s3478" xml:space="preserve">quare etiamnum ex ſen-<lb/>tentia propoſitionis dicimus, fun-<lb/>do EF inſidere pondus æquale <lb/>aqueo ponderi, magnitudine co-<lb/>lumnam æquante, cujus baſis EF, <lb/>altitudo perpendicularis GE abaquæ ſummo AB ſeu MI ad imum EF de-<lb/>miſſa.</s> <s xml:id="echoid-s3479" xml:space="preserve"/> </p> <div xml:id="echoid-div496" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.120-01" xlink:href="fig-527.01.120-01a"> <image file="527.01.120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.120-01"/> </figure> </div> </div> <div xml:id="echoid-div498" type="section" level="1" n="358"> <head xml:id="echoid-head375" xml:space="preserve">3 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s3480" xml:space="preserve">Sittertiùm ABCD mera aqua, & </s> <s xml:id="echoid-s3481" xml:space="preserve">in ipſa EF fundum horizonti parallelum. <lb/></s> <s xml:id="echoid-s3482" xml:space="preserve">Quibus poſitis, aqua ſub fundo EF tam potenter ipſum ſurſum premit, quam <lb/>aqua ſupra inſidens deorſum; </s> <s xml:id="echoid-s3483" xml:space="preserve">ſecus enim per 1 propoſ. </s> <s xml:id="echoid-s3484" xml:space="preserve">infirmius validiori con-<lb/>cederet, quod hîc non fit quia utrumque loco ſuo permanet. </s> <s xml:id="echoid-s3485" xml:space="preserve">lam corpus ſoli-<lb/>dum iſti aquæ pondere homogeneum ita collocator ut <lb/> <anchor type="figure" xlink:label="fig-527.01.120-02a" xlink:href="fig-527.01.120-02"/> aqua IKEFLM ab inferiori parte preſſet fundum EF, <lb/>ut hîc. </s> <s xml:id="echoid-s3486" xml:space="preserve">Quibus poſitis, aqua ſubter EF nunc tam validè <lb/>premit fundum EF, ſive ipſum ſolidum, quàm prius <lb/>ipſam aquam oppoſitam: </s> <s xml:id="echoid-s3487" xml:space="preserve">ſed impreſſio tanta tunc erat <lb/>quanta ſuperioris aquæ ad EF depreſſio, ut ſupra pa-<lb/>tuit, ſuperioris autem aquæ depreſſio æqualis erat ponderi columnæ aqueæ <lb/>cujus baſis EF, altitudo perpendicularis GE, à ſuperficie AB ſeu MI ad <lb/>fundum EF demiſſa. </s> <s xml:id="echoid-s3488" xml:space="preserve">Itaque aquæ ſubter EF conſtitutæ impreſſio erit quoque <lb/>tanta.</s> <s xml:id="echoid-s3489" xml:space="preserve"/> </p> <div xml:id="echoid-div498" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.120-02" xlink:href="fig-527.01.120-02a"> <image file="527.01.120-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.120-02"/> </figure> </div> </div> <div xml:id="echoid-div500" type="section" level="1" n="359"> <head xml:id="echoid-head376" xml:space="preserve">4 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s3490" xml:space="preserve">Corpora ſolida ſecundi tertiiq́ue conſectatii iſtic ita firmentur, effuſaq́ue <lb/>aqua ſpatium IKFELM vacuum nullo amplius pondere afficiet fundũ EF; <lb/></s> <s xml:id="echoid-s3491" xml:space="preserve">unde apparetaqua in vacuum locum rurſum infusâ fundũ EF tam validè pre-<lb/>mi, ac ſi integrũ vas ABCD, ſublato iſto corpore ſolido, aquâ repletum foret.</s> <s xml:id="echoid-s3492" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div501" type="section" level="1" n="360"> <head xml:id="echoid-head377" xml:space="preserve">5 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s3493" xml:space="preserve">Atverò quia immiſſa ſolida 2 & </s> <s xml:id="echoid-s3494" xml:space="preserve">3 conſectarii ſunt ſuo loco defixa, ipſorum <lb/>materia extrema nec gravitate nec levitate ulla afficiet fundum EF, quam-<lb/>obrem ſublata omni aquam ambiente materia, relinquentur internæ iſtæ aqueæ <lb/>figuræ MIKFEL, quales hic vides.</s> <s xml:id="echoid-s3495" xml:space="preserve"/> </p> <figure> <image file="527.01.120-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.120-03"/> </figure> <p> <s xml:id="echoid-s3496" xml:space="preserve">Atque hic etiam ex ſententia propoſitionis dicimus baſe EF ſubnixum eſſe <pb o="121" file="527.01.121" n="121" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> pondus, æquale ponderi aqueæ columnæ cujus baſis E F, altitudo perpendicu-<lb/>laris ab M I aquæ ſummo in fundum E F demiſſa. </s> <s xml:id="echoid-s3497" xml:space="preserve">Atque ita in cæteris omni-<lb/>bus figuris quarum fundum fit in plano horizonti parallelo.</s> <s xml:id="echoid-s3498" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3499" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3500" xml:space="preserve">Itaque in fundo hofizonti parallelo, &</s> <s xml:id="echoid-s3501" xml:space="preserve">c.</s> <s xml:id="echoid-s3502" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3503" xml:space="preserve">Inſpice exorſum<unsure/> Praxis Hydroſtatices ubi experientia hæc clarius compro-<lb/>bantur.</s> <s xml:id="echoid-s3504" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div502" type="section" level="1" n="361"> <head xml:id="echoid-head378" xml:space="preserve">NOTATO</head> <p> <s xml:id="echoid-s3505" xml:space="preserve">Propoſitionem 10 magis propriè efferri hoc modo:</s> <s xml:id="echoid-s3506" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3507" xml:space="preserve">Aquæfundo in ſuperficie mundana cõſtitu to inſidet pon-<lb/>dus æquipondiũ aquæ cujus magnitudo ſit ęqualis ſegmĕ-<lb/>to ſphærę comprehenſæ à fundo & </s> <s xml:id="echoid-s3508" xml:space="preserve">mundana ſuperficie per <lb/>ſummitatem aquæ eductę, quæ cõjungat ſuperficies inter <lb/>ipſa interjecta, deſcripta à linea infinita in mundi centro <lb/>fixa & </s> <s xml:id="echoid-s3509" xml:space="preserve">circa fundi ambitum obvoluta.</s> <s xml:id="echoid-s3510" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3511" xml:space="preserve">Cujus demonſtratio eadem cum antecedente; </s> <s xml:id="echoid-s3512" xml:space="preserve">ſed propter cauſas 7 poſtulato <lb/>expoſitas, iſto modo proponere non placuit.</s> <s xml:id="echoid-s3513" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div503" type="section" level="1" n="362"> <head xml:id="echoid-head379" xml:space="preserve">9 THEOREMA. 11 PROPOSITIO.</head> <p> <s xml:id="echoid-s3514" xml:space="preserve">Si fundi regularis punctum altiſsimum in aquę ſuperfi-<lb/>cie ſumma conſiſtat, inſidens ipſi pondus æquatur ſemiſsi <lb/>aqueæ columnæ cujus baſis fundo, altitudo autem per-<lb/>pendicularì, à ſummo fundi puncto in planum per ejuſ-<lb/>dem imum punctum horizonti æquidiſtanter eductum, <lb/>demiſſæ æqualis ſit.</s> <s xml:id="echoid-s3515" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div504" type="section" level="1" n="363"> <head xml:id="echoid-head380" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s3516" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3517" xml:space="preserve">A B vasaqua plenum, A C D E fundum inclinatum ad hori-<lb/>zontem & </s> <s xml:id="echoid-s3518" xml:space="preserve">primò quidem in angulo recto, cujus ſupremum latus A C ſit in ſu-<lb/>perficie ſumma aquæ A C F G; </s> <s xml:id="echoid-s3519" xml:space="preserve">unde perpendicularis A E demiſſa in planum <lb/>per fundi imum punctum, ut E D, horizonti æquidiſtanter eductum. </s> <s xml:id="echoid-s3520" xml:space="preserve">Sitq́ue <lb/>recta D B horizonti parallela, à qua abſumatur D H ipſi D C æqualis, & </s> <s xml:id="echoid-s3521" xml:space="preserve">con-<lb/>nectatur C H; </s> <s xml:id="echoid-s3522" xml:space="preserve">atq; </s> <s xml:id="echoid-s3523" xml:space="preserve">A C H D E fit dimidia illa columna, cujus fundũ A C D E, <lb/>altitudo D H æqualis ipſi A E.</s> <s xml:id="echoid-s3524" xml:space="preserve"/> </p> <figure> <image file="527.01.121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.121-01"/> </figure> <p> <s xml:id="echoid-s3525" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3526" xml:space="preserve">Demonſtrato <lb/>impreſſionem gravitatis aquę cõ-<lb/>tra fundũ A C D E æquari expoſi-<lb/>tæ dimidiæ columnæ A C H D E. <lb/></s> <s xml:id="echoid-s3527" xml:space="preserve">Vel ut clariùs dicam: </s> <s xml:id="echoid-s3528" xml:space="preserve">ſi I ſit pon-<lb/>dus obliquè ducens gravitate ipſi <lb/>A C H D E æquale, funisq́; </s> <s xml:id="echoid-s3529" xml:space="preserve">du-<lb/>ctorius K L parallelus cõtra D H, <lb/>K autem preſſionis potentiæ cen-<lb/>trum in fundo collectæ (cujus in- <pb o="122" file="527.01.122" n="122" rhead="4 L*IBER* S*TATICÆ*"/> ventio 18 propoſ. </s> <s xml:id="echoid-s3530" xml:space="preserve">inſtituitur) pondus I aquæ preſſui erit æquilibre, fundum <lb/>A C D E (ſilabi poſſe fingas) eo ſtatu conſervans.</s> <s xml:id="echoid-s3531" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3532" xml:space="preserve">Vel, ut idem adhuc clarius illuſtrem. </s> <s xml:id="echoid-s3533" xml:space="preserve">M N O P fundum eſto, ipſi A C D E <lb/>æquale & </s> <s xml:id="echoid-s3534" xml:space="preserve">ſimile, lateribus M P, A C, M N, A E, homologis, cui inſidet ſolidum <lb/>M N O P Q ponderitate & </s> <s xml:id="echoid-s3535" xml:space="preserve">magnitudine dimidiæ <lb/> <anchor type="figure" xlink:label="fig-527.01.122-01a" xlink:href="fig-527.01.122-01"/> columnæ A C H D E æquale ipſiq́ue ſimile, ac <lb/>recta Q O æqualis D H horizonti ad perpendi-<lb/>culum normata inſiſtat. </s> <s xml:id="echoid-s3536" xml:space="preserve">Ajo, quemadmodum ſo-<lb/>lidum M N O P Q baſin M N O P premit pon-<lb/>deroſiùs verſus N O quam ad M P, quia iſtic <lb/>corpus ipſum ſpiſſius graviuſq́ue ſit; </s> <s xml:id="echoid-s3537" xml:space="preserve">ita quoque <lb/>aquam A B ponderoſiore validioreq́ preſſu con-<lb/>niti contra E D quàm contra A C.</s> <s xml:id="echoid-s3538" xml:space="preserve"/> </p> <div xml:id="echoid-div504" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.122-01" xlink:href="fig-527.01.122-01a"> <image file="527.01.122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.122-01"/> </figure> </div> <p> <s xml:id="echoid-s3539" xml:space="preserve">P*RÆPARATIO*. </s> <s xml:id="echoid-s3540" xml:space="preserve">Dirimito latus A E in qua<unsure/>-<lb/>tuor quadrantes, in R, S, T, unde parallelæ ſint <lb/>R V, S X, T Y contra A C; </s> <s xml:id="echoid-s3541" xml:space="preserve">ſint item V Z, X α, Y β parallelæ contra D H, <lb/>ſecentq́ue C H in γ, δ, ε, ut quælibet eductarum γ Z, δ α, ε β æquent rectam <lb/>V γ; </s> <s xml:id="echoid-s3542" xml:space="preserve">tum ζ η per γ parallela contra C D interſecet X α in θ & </s> <s xml:id="echoid-s3543" xml:space="preserve">V β in 1, ſi-<lb/>militer Z κ per δ educta ſecet Y β in λ, ad extremum eodem ordine ducan-<lb/>tur parallelæ α μ per ε, & </s> <s xml:id="echoid-s3544" xml:space="preserve">β H per H.</s> <s xml:id="echoid-s3545" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div506" type="section" level="1" n="364"> <head xml:id="echoid-head381" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3546" xml:space="preserve">Primùm fundo A C V R aliquod pondus incumbit, quia tuncſolum one-<lb/>re vacaret ſi in aquæ ſuperficie ſumma conſiſteret; </s> <s xml:id="echoid-s3547" xml:space="preserve">at infra eſt; </s> <s xml:id="echoid-s3548" xml:space="preserve">non igitur ponde-<lb/>ris preſſu vacat. </s> <s xml:id="echoid-s3549" xml:space="preserve">Secundò minore quàm A C ζ γ V R, aquei corporis pondere <lb/>afficitur, etenim per 10 propoſ. </s> <s xml:id="echoid-s3550" xml:space="preserve">ſi horizonti æquidiſtaret iſtud pondus ſuſtine-<lb/>ret, at nunc altiorem locum obtinet, minus igitur ſuſtinet. </s> <s xml:id="echoid-s3551" xml:space="preserve">Conſimiliter fundo <lb/>R V X S majus quoddam pondus incumbit quàm corporis A C ζ γ V R; </s> <s xml:id="echoid-s3552" xml:space="preserve">ete-<lb/>nim ſi fundum iſtud per R V horizonti æquidiſtaret iſtic per 10 propoſ. </s> <s xml:id="echoid-s3553" xml:space="preserve">tan-<lb/>tum corpus ſuſtineret: </s> <s xml:id="echoid-s3554" xml:space="preserve">at nunc cùm loco ſit inferiore plus quoq; </s> <s xml:id="echoid-s3555" xml:space="preserve">ſufferet, quam <lb/>pondus corporis A C ζ γ V S hoceſt ſibi æqualis R V γ θ X S. </s> <s xml:id="echoid-s3556" xml:space="preserve">Et rurſum mi-<lb/>nus ipſi inſidet quam corpus A C ζ θ X S, quia per 10 propof. </s> <s xml:id="echoid-s3557" xml:space="preserve">opus eſſet fun-<lb/>dum id, per S X ad horizontis paralleliſmum eductum eſſe; </s> <s xml:id="echoid-s3558" xml:space="preserve">jam verò cum fun-<lb/>dum R V X S ſublimius ſit, minus ponderis perpetitur quàm A C ζ θ X S, hoc <lb/>eſt, ipſi æquale R V Z δ X S. </s> <s xml:id="echoid-s3559" xml:space="preserve">Eodem raticinio, adhibito 10 propof. </s> <s xml:id="echoid-s3560" xml:space="preserve">& </s> <s xml:id="echoid-s3561" xml:space="preserve">plano per <lb/>X S horizonti parallelo, cõcludes fundo S X Y T plus ponderis inſidere quàm <lb/>corporis A C ζ θ X S, hoc eſt ipſi æqualis S X δ λ Y T; </s> <s xml:id="echoid-s3562" xml:space="preserve">& </s> <s xml:id="echoid-s3563" xml:space="preserve">minus tamen (pro-<lb/>pter eandem 10 prop. </s> <s xml:id="echoid-s3564" xml:space="preserve">& </s> <s xml:id="echoid-s3565" xml:space="preserve">planũ per T Y horizonti parallelũ) quam A C ζ. </s> <s xml:id="echoid-s3566" xml:space="preserve">Y T <lb/>hoc eſt quam ipſi æquale S X α ε Y T. </s> <s xml:id="echoid-s3567" xml:space="preserve">Denique eadem via, ſubſidio 10 propof. <lb/></s> <s xml:id="echoid-s3568" xml:space="preserve">& </s> <s xml:id="echoid-s3569" xml:space="preserve">plano per T Y horizonti parallelo, evincesfundo T Y D E inſidere pondus <lb/>majus corpore A C ζ. </s> <s xml:id="echoid-s3570" xml:space="preserve">Y T ſeu ipſi æquali T Y ε μ D E: </s> <s xml:id="echoid-s3571" xml:space="preserve">attamen (propter ean-<lb/>dem 10 propoſ. </s> <s xml:id="echoid-s3572" xml:space="preserve">& </s> <s xml:id="echoid-s3573" xml:space="preserve">planum per E D horizonti parallelum) minus corpore <lb/>A C ζ η D E hoc eſt ipſo T Y β H D E. </s> <s xml:id="echoid-s3574" xml:space="preserve">Iam autem cum his demonſtrationi-<lb/>bus effectum ſit fundo A C V R aliquod pondus inſidere neque vacare omni-<lb/>no, fundo R V X S plus corpore R V γ θ X S; </s> <s xml:id="echoid-s3575" xml:space="preserve">item fundo S X Y T plus cor-<lb/>pore S X δ λ Y T, ultimùm fundo T Y D E plus corpore T Y ε μ D E, toti <lb/>quoque fundo A C D E plus inſidet quàm pondus omnium iſtorum corpo-<lb/>rum, quæ addita cõſtituunt corpus R V γ θ δ λ ε μ D E in dimidiam columnam <lb/>inſcriptum. </s> <s xml:id="echoid-s3576" xml:space="preserve">Et cum iiſdem demonſtrationibus cõcl<unsure/>uſerimus fundo A C V R <pb o="123" file="527.01.123" n="123" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> minus inſidere pondus quàm A C ζ γ V R, fundo R V X S minus quàm <lb/>R V Z δ X S, item fundo S X Y T minus quam S X α ε Y T, denique ſundo <lb/>T Y D E minus quàm T Y β H D E, toti quoq; </s> <s xml:id="echoid-s3577" xml:space="preserve">fundo A C D E minus inſide-<lb/>bit ponere omniũ horũ, hoc eſt, corpore circumſcripto A C ζ γ Z δ α ε β H D E. <lb/></s> <s xml:id="echoid-s3578" xml:space="preserve">Atqui fundo A C D E, qui in diagrammate quadrãtibus diſtinguitur, ſic in octo <lb/>æqualia ſegmenta divio palam eſt corporum dimidiæ columnæ A C H E D <lb/>hujus inſcripti illius circumſcripti ab ipſa differentiam dimidio minorem fore <lb/>quàm nunc ſit: </s> <s xml:id="echoid-s3579" xml:space="preserve">quare hujuſmodi fundi ſectione infinita eo devenitur, ut differĕ-<lb/>tia ponderis (ſi qua tamen hîc ſit) fundo A C D E incumbĕtis a põdere dimidiæ <lb/>columnæ A C D E quolibet minimo põdere adhuc minor ſit. </s> <s xml:id="echoid-s3580" xml:space="preserve">Vnde ita ediſſero.</s> <s xml:id="echoid-s3581" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3582" xml:space="preserve">Gravitas cujus à pondere fundo A C D E inſidente differentia minor eſt quolibet <lb/>pondere dato, æquatur ponderi fundo A C D E inſidenti.</s> <s xml:id="echoid-s3583" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3584" xml:space="preserve">Sed pondus dimidiæ columnæ A C H D E eſt gravitas minus differens à pondere <lb/>fundo A C D E inſidente quam quodlibet datum.</s> <s xml:id="echoid-s3585" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3586" xml:space="preserve">Itaque pondus dimidiæ columnæ A C H D E æquatur ponderi in baſe A C D E.</s> <s xml:id="echoid-s3587" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div507" type="section" level="1" n="365"> <head xml:id="echoid-head382" style="it" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s3588" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3589" xml:space="preserve">Exponatur ſecundo A B vas plenum aquæ, fundumq́ue A C D E <lb/>quadrangulum ad horizontem in angulo obliquo inclinatum, ejusq́ue ſupre-<lb/>mum latus A C conſiſtar in A C F G aquæ ſuperficie ſumma. </s> <s xml:id="echoid-s3590" xml:space="preserve">Iam aqua <lb/>ipſiusq́ue fundum dividatur conſimiliter antecedenti 1 exemplo, & </s> <s xml:id="echoid-s3591" xml:space="preserve">A υ per-<lb/>pendicularis ſit àſummo fundi latere in planum, per inſimum latus E D ad ho-<lb/>rizontis paralleliſmum eductum, demiſſa. </s> <s xml:id="echoid-s3592" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3593" xml:space="preserve">Pondus aquæ <lb/>fundo A C D E ſubnixum dimidiæ columnæ cujus baſis A C D E, altitudo <lb/>A υ, æquari demonſtrato. </s> <s xml:id="echoid-s3594" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3595" xml:space="preserve">Perpendicularis A υ à tribus <lb/>punctis ο, π, ρ in quatuor æquas partes diſſecator.</s> <s xml:id="echoid-s3596" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div508" type="section" level="1" n="366"> <head xml:id="echoid-head383" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3597" xml:space="preserve">Fundo A C V R, cum nõ ſit in <lb/> <anchor type="figure" xlink:label="fig-527.01.123-01a" xlink:href="fig-527.01.123-01"/> aquę ſummitate, inſidet aliquod <lb/>pondus, minus tamen quàm co-<lb/>lumna aquea baſis A C V R, alti-<lb/>tudinis A ο, nam ſi per R V planũ <lb/>horizonti æquidiſtanter duce-<lb/>retur per 10 propof id hoc pon-<lb/>deris ſuſtineret, nuncverò cum <lb/>ſublimiori ſit loco minus ſuffert <lb/>quam columnam iſta baſi & </s> <s xml:id="echoid-s3598" xml:space="preserve">al-<lb/>titudine, hoc eſt, A C ζ γ V R. <lb/></s> <s xml:id="echoid-s3599" xml:space="preserve">Simili deductione ut in primo exemplo <lb/>cætera proſequeris; </s> <s xml:id="echoid-s3600" xml:space="preserve">unde tandem con-<lb/> <anchor type="figure" xlink:label="fig-527.01.123-02a" xlink:href="fig-527.01.123-02"/> cludes fundo A C D E inſidere corpus <lb/>æquale ipſi A C H D E, hoc eſt, colum-<lb/>næ baſis A C D E, altitudinis A υ (nam <lb/>A υ æqualis eſt perpendiculari ab H <lb/>in planum A C D E) tandem inquam <lb/>concludes fundo A C D E inſidere a-<lb/>queam molem magnitudine æqualem <lb/>columnæ cujus baſis A C D E, altitu-<lb/>do A υ.</s> <s xml:id="echoid-s3601" xml:space="preserve"/> </p> <div xml:id="echoid-div508" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.123-01" xlink:href="fig-527.01.123-01a"> <image file="527.01.123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.123-01"/> </figure> <figure xlink:label="fig-527.01.123-02" xlink:href="fig-527.01.123-02a"> <image file="527.01.123-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.123-02"/> </figure> </div> <pb o="124" file="527.01.124" n="124" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div510" type="section" level="1" n="367"> <head xml:id="echoid-head384" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s3602" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3603" xml:space="preserve">Fundum regulare A B ellipſis eſto, cujus ſupremum punctum <lb/>A ſit in aquæ ſuperficie ſumma, B in ima, A C perpendicularis à ſummo A in <lb/>planum horizonti parallelum per imum B.</s> <s xml:id="echoid-s3604" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3605" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3606" xml:space="preserve">Pondus aquæ fundo A B <lb/> <anchor type="figure" xlink:label="fig-527.01.124-01a" xlink:href="fig-527.01.124-01"/> incumbentis æquari dimidiæ columnæ, cu-<lb/>jus baſis A B, altitudo A C.</s> <s xml:id="echoid-s3607" xml:space="preserve"/> </p> <div xml:id="echoid-div510" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.124-01" xlink:href="fig-527.01.124-01a"> <image file="527.01.124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.124-01"/> </figure> </div> <p> <s xml:id="echoid-s3608" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3609" xml:space="preserve">Circumſcribito ellipſi <lb/>A B parallelogrammum quadrangulum <lb/>D E F G ut D E in aquæ ſummo tangat ejus <lb/>ſummum A, & </s> <s xml:id="echoid-s3610" xml:space="preserve">G F imum B; </s> <s xml:id="echoid-s3611" xml:space="preserve">ſitq́ue F I <lb/>perpendicularis in F G æqualis lateri F E, & </s> <s xml:id="echoid-s3612" xml:space="preserve"><lb/>horizonti parallela; </s> <s xml:id="echoid-s3613" xml:space="preserve">jam reliqua latera G H, <lb/>H I claudant parallelogrammum F G H I & </s> <s xml:id="echoid-s3614" xml:space="preserve"><lb/>connecto E I, D H.</s> <s xml:id="echoid-s3615" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3616" xml:space="preserve">Conſtruito deinde alteram figuram non tan-<lb/>tum forma ſimilem, ſed etiam magnitudine & </s> <s xml:id="echoid-s3617" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-527.01.124-02a" xlink:href="fig-527.01.124-02"/> ponderitate ipſi æqualem, cujus latus F I hori-<lb/>zonti ad perpendiculum inſiſtat, ut in ſubjecto <lb/>diagrammate. </s> <s xml:id="echoid-s3618" xml:space="preserve">ſitq́ue corpus hoc ſolidum ſub-<lb/>nixum fundo D E F G.</s> <s xml:id="echoid-s3619" xml:space="preserve"/> </p> <div xml:id="echoid-div511" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.124-02" xlink:href="fig-527.01.124-02a"> <image file="527.01.124-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.124-02"/> </figure> </div> </div> <div xml:id="echoid-div513" type="section" level="1" n="368"> <head xml:id="echoid-head385" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3620" xml:space="preserve">Quanto preſſu ſolidum D E F G H I afficit <lb/>ſuam hedram D E F G, tanto quoq, afficitaqua <lb/>primæ ſiguræ ſuum fundum D E F G, quod <lb/>paulò ante nobis demonſtratum eſt, & </s> <s xml:id="echoid-s3621" xml:space="preserve">conſe-<lb/>quenter quantâ preſſione ellipſis A B ſecundæ <lb/>formæ afficitur, tantâ q<unsure/>uoque omninò inerit ellipſi A B primæ formæ: </s> <s xml:id="echoid-s3622" xml:space="preserve">Atqui <lb/>preſſio quam ellipſis ſecunda perpetitur, eſt ſemiſſis columnæ (ut jam mox de-<lb/>monſtraturi ſumus) cujus baſis ellipſis, altitudo æqualis rectę A C, nam per-<lb/>pendicularis à K in planum ellipſis A B demiſſa æqualis foret dictę A C; </s> <s xml:id="echoid-s3623" xml:space="preserve">qua-<lb/>re aquę in primam ellipſin A B impreſſio, æquatur dimidiæ columnæ cujus ba-<lb/>ſis ipſa ellipſis ſit, altitudo autem A C.</s> <s xml:id="echoid-s3624" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3625" xml:space="preserve">Pondus autem ſecundæ ſiguræ inſidens ellipſi A B, æquari dimidiæ colum-<lb/>næ, cujus baſis iſta ipſa ſit ellipſis, & </s> <s xml:id="echoid-s3626" xml:space="preserve">altitudo æqualis A C, hoc pacto arguo. <lb/></s> <s xml:id="echoid-s3627" xml:space="preserve">Ducito B K æqualem & </s> <s xml:id="echoid-s3628" xml:space="preserve">parallelam rectæ F I, & </s> <s xml:id="echoid-s3629" xml:space="preserve">ípſam ita circum ellipſin A B <lb/>circumducito ut tamen perpetuò contra F I parallela ſit, eâque converſione in-<lb/>ter duas hedras oppoſitas figurabit columnam A B K L, quæ plano D E I H <lb/>per duo puncta A, K ſimili ſitu atque tranſverſim in parallelarum ellipſium am-<lb/>bitu ſibi mutuo reſpondentia incidetur: </s> <s xml:id="echoid-s3630" xml:space="preserve">ſed quęlibet columna cujus baſis eſt <lb/>planum regulare, ſectum plano per duo puncta in oppoſitis iſtis hedris tranſver-<lb/>ſim <anchor type="note" xlink:href="" symbol="*"/> ὁμοταγῆ ab ipſo in duas æquas partes dirimitur: </s> <s xml:id="echoid-s3631" xml:space="preserve">Quare ſegmentum co- <anchor type="note" xlink:label="note-527.01.124-01a" xlink:href="note-527.01.124-01"/> lumnæ hujus infra planum D E I H, eſt ſemiſſis columnæ A B K L in ellipſi <lb/>A B tanquam baſe inſidentis. </s> <s xml:id="echoid-s3632" xml:space="preserve">Columnam autem A B K L æquari columnæ <lb/>baſis A B, altitudinis A C, hinc palam eſt quia ipſius altitudo altitudini A C <lb/>æqualis ſit. </s> <s xml:id="echoid-s3633" xml:space="preserve">Pondus itaque ſubnixum ellipſi A B ęquatur dimidiæ columnæ <lb/>cujus baſis ellipſis A B, altitudo æqualis rectæ A C.</s> <s xml:id="echoid-s3634" xml:space="preserve"/> </p> <div xml:id="echoid-div513" type="float" level="2" n="1"> <note symbol="*" position="left" xlink:label="note-527.01.124-01" xlink:href="note-527.01.124-01a" xml:space="preserve">Similiter <lb/>fita.</note> </div> <pb o="125" file="527.01.125" n="125" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div515" type="section" level="1" n="369"> <head xml:id="echoid-head386" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s3635" xml:space="preserve">Quamvis tribus diagrammatis {γρ}αμμικῶς theorematis hujus veritatem evi-<lb/>cerimus atque iſta via rationes & </s> <s xml:id="echoid-s3636" xml:space="preserve">cauſæ plenius uberiuſque pateſcant, uberta-<lb/>tem tamen iſtam arithmetico calculo 4 hoc exemplo fœcundiorem efficere pla-<lb/>cuit. </s> <s xml:id="echoid-s3637" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3638" xml:space="preserve">Vaſis A B aqua pleni fundum A C D E quadratum hori-<lb/>zonti perpendiculare eſto, cujus ſupremum labrum A C pedalis longitudinis <lb/>ſit in ſummitate aquæ A C F G, ſitq́ue altitudo A E item pedalis; </s> <s xml:id="echoid-s3639" xml:space="preserve">reliqualon-<lb/>gitudo A B pro libitu exporrigatur.</s> <s xml:id="echoid-s3640" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3641" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3642" xml:space="preserve">Pondus aquæ fundo <lb/> <anchor type="figure" xlink:label="fig-527.01.125-01a" xlink:href="fig-527.01.125-01"/> A C D E annixum dimidię aqueæ columnæ, <lb/>baſe iſti fundo, altitudine perpendiculari A E <lb/>æquali æquari demonſtrator. </s> <s xml:id="echoid-s3643" xml:space="preserve">At cum columna <lb/>iſta hîc cubus ſit pedalis, demonſtrandum erit <lb/>fundo A C D E incumbere pondus cubici pe-<lb/>dis dimidium.</s> <s xml:id="echoid-s3644" xml:space="preserve"/> </p> <div xml:id="echoid-div515" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.125-01" xlink:href="fig-527.01.125-01a"> <image file="527.01.125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.125-01"/> </figure> </div> <p> <s xml:id="echoid-s3645" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3646" xml:space="preserve">Tres parallelę H I, K L, M N, contra A C æquali diſtan-<lb/>tia rectam A E quadripartitò dividant.</s> <s xml:id="echoid-s3647" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div517" type="section" level="1" n="370"> <head xml:id="echoid-head387" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3648" xml:space="preserve">Manife<unsure/>ſtum eſt fundo A I plus inſidere quam o, nam ſi iſtiuſmodi fundum <lb/>horizonti effet parallelum nihil ſeu o ipſi inſideret, at nunc cùm infra conſiſtat <lb/>plus nihilo ſeu o ipſi incumbit: </s> <s xml:id="echoid-s3649" xml:space="preserve">Et tamen pondus illud citra {1/16} pedis eſt, cum <lb/>enim ad horizontem parallelum agitur per H I tantò urgetur pondere: </s> <s xml:id="echoid-s3650" xml:space="preserve">quare <lb/>cum in hâc theſi ſuperiori loco conſiſtat, minus ſuſtinet quam pedis cubici {1/16}. <lb/></s> <s xml:id="echoid-s3651" xml:space="preserve">ſimili ratione efficitur in fundo H L plus inſidere quàm {1/16}, minusq́ue {2/16}: </s> <s xml:id="echoid-s3652" xml:space="preserve">item <lb/>fundo K N plus {2/16}, minus autem {3/16}: </s> <s xml:id="echoid-s3653" xml:space="preserve">denique fundo M D plus {3/16}, at mi-<lb/>nus {4/16}. </s> <s xml:id="echoid-s3654" xml:space="preserve">Addita igitur quatuor pondera (ſi o huc annumeres) in ſingulis termi-<lb/>nis priora & </s> <s xml:id="echoid-s3655" xml:space="preserve">minora, hoc eſt, o, {1/16}. </s> <s xml:id="echoid-s3656" xml:space="preserve">{2/16}. </s> <s xml:id="echoid-s3657" xml:space="preserve">{3/16}, danttotum {6/16}: </s> <s xml:id="echoid-s3658" xml:space="preserve">item quatuor poſte-<lb/>riora & </s> <s xml:id="echoid-s3659" xml:space="preserve">majora {1/16}, {2/16}, {3/16}, {4/16}, colligunt {10/16}. </s> <s xml:id="echoid-s3660" xml:space="preserve">Quamobrem fundo A C D E inſi-<lb/>detpondus quoddam majus quàm {6/16} pedis, at minus quàm {10/16}, interq́ue hos <lb/>terminos pes dimidius medius conſiſtit, quem fundo A C D E inſidere de-<lb/>monſtrare neceſſum eſt.</s> <s xml:id="echoid-s3661" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3662" xml:space="preserve">Cæterùm qua ratione tribus parallelis fundum quadrifariam diſſecuimus, <lb/>eadem omninò via in partes quotlibet optatas dirimetur. </s> <s xml:id="echoid-s3663" xml:space="preserve">ſit denarius ſegmen-<lb/>torum optatus numerus. </s> <s xml:id="echoid-s3664" xml:space="preserve">Iam ob cauſas ante expoſitas, decem priora in colla-<lb/>tione & </s> <s xml:id="echoid-s3665" xml:space="preserve">minora pondera quæſingulis inſident fundis, uto, {1/100}, {2/100}, {3/100}, {4/100}, <lb/>{5/100}, {6/100}, {7/100}, {8/100}, {9/100}, collecta efficiunt ſummam {45/100}: </s> <s xml:id="echoid-s3666" xml:space="preserve">ſimiliter poſteriora & </s> <s xml:id="echoid-s3667" xml:space="preserve"><lb/>graviora ut {1/100}, {2/100}, {3/100}, {4/100}, {5/100}, {6/100}, {7/100}, {8/100}, {9/100}, {10/100}. </s> <s xml:id="echoid-s3668" xml:space="preserve">dant ſumman; </s> <s xml:id="echoid-s3669" xml:space="preserve">{55/100}. <lb/></s> <s xml:id="echoid-s3670" xml:space="preserve">quamobrem fundo A C D E plus inſidet quàm {45/100} & </s> <s xml:id="echoid-s3671" xml:space="preserve">minus quàm {55/100}, qui <lb/>termini utrimque à dimidio pede, propter quem demonſtratio inſtituitur, ab-<lb/>ſunt pari intervallo. </s> <s xml:id="echoid-s3672" xml:space="preserve">Atqui quemadmodum hi proprius abſunt à pede dimidio <lb/>prioribus illis, nam {45/100} differentia ab {1/2} minor eſt quàm {6/16}; </s> <s xml:id="echoid-s3673" xml:space="preserve">& </s> <s xml:id="echoid-s3674" xml:space="preserve">{55/100}, minor item <lb/>differentia quam {10/16}: </s> <s xml:id="echoid-s3675" xml:space="preserve">ita in quo plura ſegmenta æqualia fundum A C D E par-<lb/>titus eris, continuò magis magisq́ue ad ipſum dimidium pedem accedes.</s> <s xml:id="echoid-s3676" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3677" xml:space="preserve">Quibus ritè intellectis, fingamus (ſi ſieri poſſit) fundo A C D E plus minusvé <lb/>{1/1000} pondere dimidii pedis inſidere: </s> <s xml:id="echoid-s3678" xml:space="preserve">veritatem igitur, fundo in 1000 partis lineis <lb/>parallelis ut ante diſtributo, ratione viaq́ue jam uſitata inquiramus. </s> <s xml:id="echoid-s3679" xml:space="preserve">Hîc propter <lb/>antecedentes cauſas mille ponduſcula priora mille particulis inſidentia erunt <lb/>O {1/1000000}, {2/1000000}, atque ita deinceps, ultimumq́; </s> <s xml:id="echoid-s3680" xml:space="preserve">{999/1000000}, horum omniũ ſumma (cujus <pb o="126" file="527.01.126" n="126" rhead="4 L*IBER* S*TATICÆ*"/> colligendæ compendium infra damus) erit {499500/1000000}: </s> <s xml:id="echoid-s3681" xml:space="preserve">Similiter mille ponduſcula <lb/>graviora {1/1000000}, {2/1000000}, {3/1000000}, &</s> <s xml:id="echoid-s3682" xml:space="preserve">c. </s> <s xml:id="echoid-s3683" xml:space="preserve">quorum noviſſimum {1000/100000}, in ſummam colle-<lb/>cta efficiunt {500500/1000000}. </s> <s xml:id="echoid-s3684" xml:space="preserve">Quamobrem fundo inſidet põdus majus quàm {499500/1000000}, minus <lb/>autem quàm {500500/1000000} unius pedis cubici, atqui {499500/1000000} abeſt duntaxat {1/1000} ab {1/27}, quare <lb/>pondus quod inſidet fundo ACDE deficit à dimidio pede defectu minore <lb/>quam fit {1/10000}; </s> <s xml:id="echoid-s3685" xml:space="preserve">ſic {500500/1000000} excedit {1/2} pedis ſemiſſem {1/1000}, itaq; </s> <s xml:id="echoid-s3686" xml:space="preserve">ipſi non inſidet pon-<lb/>dus {1/1000} dimidium pedem excedens. </s> <s xml:id="echoid-s3687" xml:space="preserve">Simile ratiocinium inſtitues in cæteris, <lb/>etiam poſitis quibuſliber quam minimis particulis. </s> <s xml:id="echoid-s3688" xml:space="preserve">Quare evidens eſt difſeten-<lb/>tiam (ſi quæ tamen eſſet) inter aquam fundo ACDE inſidentem, & </s> <s xml:id="echoid-s3689" xml:space="preserve">cubici <lb/>aquei pedis dimidium, minorem eſſe qualibet quæ animo concipi aut cogita-<lb/>tione comprehendi poſſit. </s> <s xml:id="echoid-s3690" xml:space="preserve">Vnde ſyllogiſmum inſtituo.</s> <s xml:id="echoid-s3691" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3692" xml:space="preserve">Pondere, quod ab aquæ dimidio pede abeſt, aliud minore ab eo differentia distans <lb/>exhiberi poteſt:</s> <s xml:id="echoid-s3693" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3694" xml:space="preserve">Sed pondere aqueo fundo A C D E inſidente, nullum ab aquæ pede dimidio minus <lb/>differens exhiberi poteſt:</s> <s xml:id="echoid-s3695" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3696" xml:space="preserve">Itaque pondus aquæ inſidens fundo A C D E, à dimidio aquæ pede nihil differt.</s> <s xml:id="echoid-s3697" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3698" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3699" xml:space="preserve">Itaque fundi regularis, cujus ſummum punctum in aquæ <lb/>conſiſtit, &</s> <s xml:id="echoid-s3700" xml:space="preserve">c.</s> <s xml:id="echoid-s3701" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3702" xml:space="preserve">CAuſa cur ſemiſſis iſte inter duos numeros perpetuò magis vicinos, nun-<lb/>quam tamen concurrentes conſiſtar, hujuſmodi theoremate exprimitur.</s> <s xml:id="echoid-s3703" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3704" xml:space="preserve">Numeris quotcunqueab unitate deinceps continuatis, <lb/>dimidius noviſsimi numeri quadratus cedat ſummæom-<lb/>nium, eandemq́ue noviſsimo numero multatam excedit.</s> <s xml:id="echoid-s3705" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3706" xml:space="preserve">SEd ut fidem exſolvam, & </s> <s xml:id="echoid-s3707" xml:space="preserve">compendium in tanta numerorum multitudine <lb/>addenda nunc explicem, ita habe. </s> <s xml:id="echoid-s3708" xml:space="preserve">primùm partium iſtarum nomen unum <lb/>eſſe & </s> <s xml:id="echoid-s3709" xml:space="preserve">commune, quare hoc poſthabito ipſarum numeris animũ intendamus, <lb/>ii igitur ab unitate continuò progreſſu unitate mutuò ſe ſuperant. </s> <s xml:id="echoid-s3710" xml:space="preserve">Itaque ad fa-<lb/>ctum à noviſſimo in ſui ſemiſſem multiplicato, is ipſe ſemiſſis additus dabit <lb/>optatam ſummam. </s> <s xml:id="echoid-s3711" xml:space="preserve">Exemplum hujuſmodi eſto; </s> <s xml:id="echoid-s3712" xml:space="preserve">quæritur ſumma numerorum <lb/>1, 2, 3, 4, 5, 6. </s> <s xml:id="echoid-s3713" xml:space="preserve">Factus à noviſſimo 6 in ſuũ ſemiſſem 3 ad eundem 3 additus dabit <lb/>21 optatam ſummam. </s> <s xml:id="echoid-s3714" xml:space="preserve">Vel ſi noviſſimus ſit impar, ut 1, 2, 3, 4, 5, 6, 7: </s> <s xml:id="echoid-s3715" xml:space="preserve">7 in ſuum ſe-<lb/>miſſem 2 {1/2} ductus facit 24 {1/2}, qui cum ſemiſſe 3 {1/2} compoſitus dat 28 optatũ ſum-<lb/>mæ totius numerum. </s> <s xml:id="echoid-s3716" xml:space="preserve">At cum noviſſimus iſle impar erit, quî partium numeratio <lb/>declinetur, unitatead noviſſimum addito eodemq́; </s> <s xml:id="echoid-s3717" xml:space="preserve">noviſſimo per hujus ſemiſ-<lb/>ſem multiplicato commodius abſolves. </s> <s xml:id="echoid-s3718" xml:space="preserve">utſi in eodem exemplo 1, 2, 3, 4, 5, 6, 7, <lb/>quæratur ſumma; </s> <s xml:id="echoid-s3719" xml:space="preserve">adde 1 ad 7 fit 8, cujus ſemiſsis 4 cum noviſsimo 7 multipli-<lb/>catus dabit 28 ut priùs. </s> <s xml:id="echoid-s3720" xml:space="preserve">atque ita in cæteris.</s> <s xml:id="echoid-s3721" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div518" type="section" level="1" n="371"> <head xml:id="echoid-head388" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s3722" xml:space="preserve">Luia ſupraſcriptus columnæ ſemißis, æquatur integræ item columnæ cujus baſis ſit <lb/>fundum datum, altitudo autem ſemißis perpendicularis à ſummo fundi puncto, in pla-<lb/>num per imum eius punctum borizonti parallelum demiſſæ; </s> <s xml:id="echoid-s3723" xml:space="preserve">11 pr opoſhoc modo quoque <lb/>enuntiari poterit.</s> <s xml:id="echoid-s3724" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3725" xml:space="preserve">Si fundi regularis ſupremum punctũ ſit in ſumma aquæ <pb o="127" file="527.01.127" n="127" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> ſuperficie, pondusipſi inſidens æquatur columnæaqueæ, <lb/>cujus baſis ſit huic fundo æqualis, altitudo ſemiſsi perpen-<lb/>dicularis à fundi ſummo in planum per imum ejus pun-<lb/>ctum horizonti æquidiſtanter eductum, demiſſæ.</s> <s xml:id="echoid-s3726" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s3727" xml:space="preserve">Lua formula poſtremam partem buius 12 propoſitionis efferemus.</s> <s xml:id="echoid-s3728" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div519" type="section" level="1" n="372"> <head xml:id="echoid-head389" xml:space="preserve">10 THEOREMA. 12 PROPOSITIO.</head> <p> <s xml:id="echoid-s3729" xml:space="preserve">Si fundi regularis ſupremum punctum infra ſummam <lb/>aquæ ſuperficiem deliteſcat, pondus ipſi inſidens æquatur <lb/>columnæaqueæ cujus baſis huicſundo, altitudo perpendi-<lb/>culariab aquæ ſummo in planum per ſummũ ſundi pun-<lb/>ctum horizonti parallelum, demiſſæ, atque inſuper ſemiſsi <lb/>perpendicularis indidem in alterum planũ perimum fun-<lb/>di punctum, horizonti parallelum, continuatæ.</s> <s xml:id="echoid-s3730" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div520" type="section" level="1" n="373"> <head xml:id="echoid-head390" xml:space="preserve">I Exemplum.</head> <p> <s xml:id="echoid-s3731" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3732" xml:space="preserve">Fundum regulare A B C D primùm quadrangulum parallelo-<lb/>grammum latere ſummo A B infra aquam abditum horizonti parallelum ſumi-<lb/>tor, perpendicularis E A per ſummum A utrimque continuata illic aquæ ſum-<lb/>mo, hic plano per D C horizonti parallelo occurrat in F, ſitq́ue AG ipſius in-<lb/>ferioris continuationis ſemiſsis.</s> <s xml:id="echoid-s3733" xml:space="preserve"/> </p> <figure> <image file="527.01.127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.127-01"/> </figure> <p> <s xml:id="echoid-s3734" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3735" xml:space="preserve">Pondus aquaæ <lb/>molis nixæ fundo A B C D colum-<lb/>næ cujus baſis dicto fundo, altitudo <lb/>rectæ E G æqualis ſit, æquari demõ-<lb/>ſtrato. </s> <s xml:id="echoid-s3736" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3737" xml:space="preserve">Latera <lb/>D A, C B uſque ſuperam aquæ ſu-<lb/>perficiem in H, I continuata conne-<lb/>ctantur recta H I; </s> <s xml:id="echoid-s3738" xml:space="preserve">hinc C K, D L æ-<lb/>quales lateri C I & </s> <s xml:id="echoid-s3739" xml:space="preserve">horizonti paral-<lb/>lelæ acta L K compleant parallelo-<lb/>grammum C D L K & </s> <s xml:id="echoid-s3740" xml:space="preserve">jungantur rectę I K, H L; </s> <s xml:id="echoid-s3741" xml:space="preserve">denique B M, A N lateri <lb/>C O, item M O, N P ipſi B C æquales & </s> <s xml:id="echoid-s3742" xml:space="preserve">parallelæ conſtituantur.</s> <s xml:id="echoid-s3743" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3744" xml:space="preserve">Tumq́ue altera figura huic aqueæ ſimilis, magnitudine autem & </s> <s xml:id="echoid-s3745" xml:space="preserve">pondere <lb/>æqualis deformetur C D H I K L, hac lege ut C K horizontiad perpendicu-<lb/>lum immineat. </s> <s xml:id="echoid-s3746" xml:space="preserve">ut hic.</s> <s xml:id="echoid-s3747" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div521" type="section" level="1" n="374"> <head xml:id="echoid-head391" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3748" xml:space="preserve">Eadem eſt corporis ſolidi C D H I K L ſecundi diagrammatis per 11 propoſ. <lb/></s> <s xml:id="echoid-s3749" xml:space="preserve">in C D H I fundum impreſsio, quæ humidi primæ figuræ in fundum ſuum <lb/>C D H I, & </s> <s xml:id="echoid-s3750" xml:space="preserve">conſequenter qualis preſſus eſt in illius parte A B C D, talis in hu-<lb/>jus parte A B C D quoque erit: </s> <s xml:id="echoid-s3751" xml:space="preserve">ſed impreſsio illius in A B C D eſt ſolidum <lb/>A B C D L K M N æquale columnę cujus baſis A B C D; </s> <s xml:id="echoid-s3752" xml:space="preserve">altitudo G E: </s> <s xml:id="echoid-s3753" xml:space="preserve">quare <lb/>aquæ põdus inſidens primæ figuræ fundo A B C D æquatur quoque columnę <lb/>baſis quidem H B C D, altitudinis verò G E. </s> <s xml:id="echoid-s3754" xml:space="preserve">Corpus autĕ A B C D L K N M <pb o="128" file="527.01.128" n="128" rhead="4 L*IBER* S*TATICÆ*"/> æquari columnæbaſis A B C D, altitudinis GE, <lb/> <anchor type="figure" xlink:label="fig-527.01.128-01a" xlink:href="fig-527.01.128-01"/> patebit demiſſa O Q perpendiculari in planum <lb/>A B C D: </s> <s xml:id="echoid-s3755" xml:space="preserve">nam priſma A B C D P O M N æqua-<lb/>le eſt ſolido cujus baſis A B C D altitudo O Q: <lb/></s> <s xml:id="echoid-s3756" xml:space="preserve">ſed quia rectæ A H, O C, itemq́ue anguli HAE, <lb/>C O Q ſunt æquales, & </s> <s xml:id="echoid-s3757" xml:space="preserve">AE plano per H, E, pun-<lb/>cta trajecto perpendicularis, item O Q ei quod <lb/>per C, Q, propterea A E & </s> <s xml:id="echoid-s3758" xml:space="preserve">æquatur ipſi O Q: </s> <s xml:id="echoid-s3759" xml:space="preserve"><lb/>ideoq́ue parallelepipedum A B C D P O M N, <lb/>parallelepipedo in baſin A B C D altitudine <lb/>A E inſiſtente erit æquale. </s> <s xml:id="echoid-s3760" xml:space="preserve">At (quemadmodum <lb/>jam 11 propoſ. </s> <s xml:id="echoid-s3761" xml:space="preserve">demonſtratum fuit) priſma <lb/>M N P O K L æquatur parallelepipedo baſis <lb/>A B C D altitudinis A G. </s> <s xml:id="echoid-s3762" xml:space="preserve">quare duo iſta ſolida <lb/>addita conſtituunt priſma A B C D L K N M æquale parallelepipedo dictæ <lb/>baſis A B C D, altitudinis autem G E.</s> <s xml:id="echoid-s3763" xml:space="preserve"/> </p> <div xml:id="echoid-div521" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.128-01" xlink:href="fig-527.01.128-01a"> <image file="527.01.128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.128-01"/> </figure> </div> </div> <div xml:id="echoid-div523" type="section" level="1" n="375"> <head xml:id="echoid-head392" xml:space="preserve">ALTERA DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3764" xml:space="preserve">Si per A B agas planum horizonti parallelum ipſi A B C D ſimile & </s> <s xml:id="echoid-s3765" xml:space="preserve">æquale; <lb/></s> <s xml:id="echoid-s3766" xml:space="preserve">huic incumbet per 10 prop. </s> <s xml:id="echoid-s3767" xml:space="preserve">põdus aquæ æquale columnæ baſis A B C D, altitu-<lb/>dinis AE: </s> <s xml:id="echoid-s3768" xml:space="preserve">atqui minimùm tantũ põderis inſidet cuilibet fundo humiliori ipſiq́; </s> <s xml:id="echoid-s3769" xml:space="preserve"><lb/>æquali: </s> <s xml:id="echoid-s3770" xml:space="preserve">primùm igitur fundo A B C D incumbit columna baſis dictæ A B C D, <lb/>altitudinis A E. </s> <s xml:id="echoid-s3771" xml:space="preserve">remota igitur aqua iſta quæ ſuperiori fundo inſidet quodque <lb/>ipſi A B C D formavimus æquale, erit A B in reliquę aqu@ ſummitate, atque <lb/>ideo per 11 prop. </s> <s xml:id="echoid-s3772" xml:space="preserve">dicto fundo A B C D inſidebit aquea columna baſis A B C D <lb/>altitudinis A B; </s> <s xml:id="echoid-s3773" xml:space="preserve">quæ ad ſuperiorem addita cõſtituet columnam baſis A B C D, <lb/>altitudinis autem E G, quæ quantitas eſt ponderis fundo A B C D inſidentis.</s> <s xml:id="echoid-s3774" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div524" type="section" level="1" n="376"> <head xml:id="echoid-head393" xml:space="preserve">2 Exemplum.</head> <figure> <image file="527.01.128-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.128-02"/> </figure> <p> <s xml:id="echoid-s3775" xml:space="preserve">Fundi regularis A B ſupremum punctum A in aquæ <lb/>ſummo, B ſit in imo; </s> <s xml:id="echoid-s3776" xml:space="preserve">perpendicularis A C ab A ſurſum <lb/>ad C aquæ ſuperficiem extimam, & </s> <s xml:id="echoid-s3777" xml:space="preserve">deorſum in D ad <lb/>planum per B imum punctum horizonti parallelũ con-<lb/>tinuata, continuationisq́ue inferioris ſemiſſis eſto A E. <lb/></s> <s xml:id="echoid-s3778" xml:space="preserve">Ajo tantum pondus fundo inſidere, quantum eſt colum-<lb/>næ baſis A B altitudinis C E. </s> <s xml:id="echoid-s3779" xml:space="preserve">cujus demonſtratio ante-<lb/>cedenti ſimilis eſt.</s> <s xml:id="echoid-s3780" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3781" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3782" xml:space="preserve">Itaqueſi fundi regularis ſupremum <lb/>punctum, &</s> <s xml:id="echoid-s3783" xml:space="preserve">c.</s> <s xml:id="echoid-s3784" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div525" type="section" level="1" n="377"> <head xml:id="echoid-head394" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s3785" xml:space="preserve">Hoc T heoremate, adhibita perpendiculari per ſummum fundipunctum educta, quan-<lb/>tum eſſet pondus regulari plano inſidens demonſtr avimus, ſed fundo non regulari pon-<lb/>dus hoc istiuſmodi perpendiculari non invenitur. </s> <s xml:id="echoid-s3786" xml:space="preserve">Certum eſt ipſi pondus inſidere æquale <lb/>aqueæ columnæ, cuius baſis iſtud ſit fundum, & </s> <s xml:id="echoid-s3787" xml:space="preserve">altituào perp endicularis à ſupremo cius <lb/>fundi puncto ad aquæ ſub qua deliteſcit ſummitatem educta, ſedpræterea jamreliguum <lb/>@llud pondus non æquatur alteri, columnæ cuius baſis ſit idem fundum altitudo dimidiæ <lb/>perpendicularis ab altiſsimo fundi puncto in planum per infimum punctum horizonti <pb o="129" file="527.01.129" n="129" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> parallelum demiſſa. </s> <s xml:id="echoid-s3788" xml:space="preserve">Cuius cauſa hæc eſt, quod columna baſis irregularis, planoper pun-<lb/>ctain oppoſitarum baſium ambitu tranſverſim ὸμο{τα}{γῆ} (ut in columnis baſis regularis) <lb/>neceſſariò bifariam non dividatur. </s> <s xml:id="echoid-s3789" xml:space="preserve">Cæterùm ut generaliter pondus, etiam cuicunqueir-<lb/>regulari fundo inſidens cognoſcatur, Problema huiuſmodi exigimus.</s> <s xml:id="echoid-s3790" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div526" type="section" level="1" n="378"> <head xml:id="echoid-head395" xml:space="preserve">3 PROBLEMA. 13 PROPOSITIO.</head> <p> <s xml:id="echoid-s3791" xml:space="preserve">Aqueam molem ponderi fundo plano, formæ contin-<lb/>gentis inſidenti æqualem invenire.</s> <s xml:id="echoid-s3792" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3793" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3794" xml:space="preserve">A B fundum planum ſub aqua regularené an irregulare ſit nihil <lb/>intereſt. </s> <s xml:id="echoid-s3795" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3796" xml:space="preserve">Corpus aqueum, quod ponderi fundo A B inſi-<lb/>denti æquetur invenire.</s> <s xml:id="echoid-s3797" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div527" type="section" level="1" n="379"> <head xml:id="echoid-head396" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3798" xml:space="preserve">Plani A B infinitè continuati & </s> <s xml:id="echoid-s3799" xml:space="preserve">ſupremæ aqueæ ſuperficiei communis ſe-<lb/>ctio eſto C, hinc fundi planique alterius & </s> <s xml:id="echoid-s3800" xml:space="preserve">horizonti & </s> <s xml:id="echoid-s3801" xml:space="preserve">fundo perpendicula-<lb/>ris communis ſectio per C, ſit C D, ipſiq́ue in plano per D horizonti paralle-<lb/>lo agatur æqualis D E quæ hujus & </s> <s xml:id="echoid-s3802" xml:space="preserve">plani per A B communi lectioni perpen-<lb/>dicularis ſit: </s> <s xml:id="echoid-s3803" xml:space="preserve">deinde plano C D E excitetur perpendiculare planũ per C & </s> <s xml:id="echoid-s3804" xml:space="preserve">E. <lb/></s> <s xml:id="echoid-s3805" xml:space="preserve">Hinc infinita A F circumagatur æquidiſtanter contra D E per ambitum fun-<lb/> <anchor type="figure" xlink:label="fig-527.01.129-01a" xlink:href="fig-527.01.129-01"/> di A B, qua converſione deformatur corpus <lb/>A G H B a duabus infinitorum planorũ par-<lb/>tibus A B, G H & </s> <s xml:id="echoid-s3806" xml:space="preserve">ſuperficiemotu lineæ de-<lb/>ſcriptâ comprehenſum. </s> <s xml:id="echoid-s3807" xml:space="preserve">Iam ajo molem aquæ <lb/>corpori A G H Bæqualem, gravitate æquari <lb/>ponderi fundo dato inſidenti.</s> <s xml:id="echoid-s3808" xml:space="preserve"/> </p> <div xml:id="echoid-div527" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.129-01" xlink:href="fig-527.01.129-01a"> <image file="527.01.129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.129-01"/> </figure> </div> <p> <s xml:id="echoid-s3809" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s3810" xml:space="preserve">Alteram figuram prio-<lb/>ri ſimilem, æqualem, & </s> <s xml:id="echoid-s3811" xml:space="preserve">iſtiaquæ æquipondiam <lb/>figurato, hac lege ut D E horizonti ad perpendiculum immineat.</s> <s xml:id="echoid-s3812" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div529" type="section" level="1" n="380"> <head xml:id="echoid-head397" xml:space="preserve">DEMONSTRATIO.</head> <figure> <image file="527.01.129-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.129-02"/> </figure> <p> <s xml:id="echoid-s3813" xml:space="preserve">Quale pondus incumbit ſecundo fundo A B tale inſi-<lb/>det primo fundo A B, ut ſupra demonſtratum fuit, ſed <lb/>ſecundo A B inſidet pondus corporis A G H B: </s> <s xml:id="echoid-s3814" xml:space="preserve">itaque <lb/>etiam primo A B incumbit pondus æquale aqueæ moli <lb/>A G H B. </s> <s xml:id="echoid-s3815" xml:space="preserve">Quod inveniſſe & </s> <s xml:id="echoid-s3816" xml:space="preserve">demonſtraſſe fuit propoſi-<lb/>tum. </s> <s xml:id="echoid-s3817" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3818" xml:space="preserve">Quamobrem aqueam molem, <lb/>ponderi fundo plano, formæ contingentis inſidenti, <lb/>æqualem invenimus. </s> <s xml:id="echoid-s3819" xml:space="preserve">Quod poſtulabatur.</s> <s xml:id="echoid-s3820" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div530" type="section" level="1" n="381"> <head xml:id="echoid-head398" xml:space="preserve">11 THEOREMA. 14 PROPOSITIO.</head> <p> <s xml:id="echoid-s3821" xml:space="preserve">Si duo parallelogramma æqualis latitudinis ab aquæ <lb/>ſumma ſuperficie deorſum æquali altitudine abdantur, <lb/>ipſorum longitudines preſsibus proportionales erunt.</s> <s xml:id="echoid-s3822" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3823" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3824" xml:space="preserve">In aqua A B C D duo parallelogramma E F, G H, æquali la-<lb/>titudine, & </s> <s xml:id="echoid-s3825" xml:space="preserve">infra aquam altitudine, hoc eſt ut perpendiculares FI, H K ſint <pb o="130" file="527.01.130" n="130" rhead="4 L*IBER* S*TATICÆ*"/> æquales, & </s> <s xml:id="echoid-s3826" xml:space="preserve">ſumma latera E, G in ſuperna aquæ ſuperficie collocentur.</s> <s xml:id="echoid-s3827" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3828" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3829" xml:space="preserve">Longitudines EF, G H preſſibus aquæ, quibus fundã <lb/>EF, GH afficiuntur æquales eſſe.</s> <s xml:id="echoid-s3830" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div531" type="section" level="1" n="382"> <head xml:id="echoid-head399" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3831" xml:space="preserve">Pondus aquæ fundo E F inſidentis æquatur per 11 propoſ. </s> <s xml:id="echoid-s3832" xml:space="preserve">aqueæ columnæ <lb/>cujus altitudo I F, baſis autem fundum E F. </s> <s xml:id="echoid-s3833" xml:space="preserve">ſimiliter pondus aquæ quod in-<lb/>ſidet fundo G H æquatur columnæ aqueæ altitudinis K H baſis G H fundo <lb/> <anchor type="note" xlink:label="note-527.01.130-01a" xlink:href="note-527.01.130-01"/> æqualis. </s> <s xml:id="echoid-s3834" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/> quarę ſunt ut baſes: </s> <s xml:id="echoid-s3835" xml:space="preserve">ſed baſis ſeu fun- dum E F eſt ad fundum G H, ut recta E F ad re-<lb/> <anchor type="figure" xlink:label="fig-527.01.130-01a" xlink:href="fig-527.01.130-01"/> ctam G H, nam perhypotheſin æqualem habent <lb/>latitudinem: </s> <s xml:id="echoid-s3836" xml:space="preserve">ex æquo itaque longitudo E F erit <lb/>ad longitudinem G H ut illius columna, ad co-<lb/>lumnam hujus, & </s> <s xml:id="echoid-s3837" xml:space="preserve">conſequenter ut pondus aquæ <lb/>illi inſidentis, ad pondus huic inſidentis. </s> <s xml:id="echoid-s3838" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3839" xml:space="preserve">Itaque ſi duo pa-<lb/>rallelogramma ęqualis latitudinis ab aquæ ſuperficie deorlum altitudine æqua-<lb/>li recedunt, ipſorum longitudines preſſibus aquæ ipſi inſidentis proportionales <lb/>erunt. </s> <s xml:id="echoid-s3840" xml:space="preserve">Quod demonſtrandum fuit.</s> <s xml:id="echoid-s3841" xml:space="preserve"/> </p> <div xml:id="echoid-div531" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-527.01.130-01" xlink:href="note-527.01.130-01a" xml:space="preserve">32. p. 11. <lb/>t. E.</note> <figure xlink:label="fig-527.01.130-01" xlink:href="fig-527.01.130-01a"> <image file="527.01.130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.130-01"/> </figure> </div> </div> <div xml:id="echoid-div533" type="section" level="1" n="383"> <head xml:id="echoid-head400" xml:space="preserve">4 THEOREMA. 15 PROPOSITIO.</head> <p> <s xml:id="echoid-s3842" xml:space="preserve">Si parallelogrammi ad horizontem inclinati, cujus <lb/>ſupremum latus in aquæ ſuperficie ſumma conſiſtat, duæ <lb/>perpendiculares altera in latus imum, altera in planum per <lb/>imum latus horizonti parallelum notæ ſint; </s> <s xml:id="echoid-s3843" xml:space="preserve">aquæ ipſi inſi-<lb/>dentis pondus invenire.</s> <s xml:id="echoid-s3844" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div534" type="section" level="1" n="384"> <head xml:id="echoid-head401" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s3845" xml:space="preserve">Parallelogrammum eſſe aut rectangulum aut obliquangulum, cum{q́ue} ſummo latere <lb/>in aquæ ſuperficie collocato ipſa ad horizontem inclinabuntur, id fiet in angulo recto, vel <lb/>obliquo. </s> <s xml:id="echoid-s3846" xml:space="preserve">unde quadruplex exemplorum ratio exiſtit, cuius varietatis tam in hoc, quam <lb/>duabus ſequentibus propoſitionibus quatuor dabimus exempla. </s> <s xml:id="echoid-s3847" xml:space="preserve">Primum rectanguli ad <lb/>horizontem recti, ubi alterum laterum horizonti annuentium & </s> <s xml:id="echoid-s3848" xml:space="preserve">perpendicularis duæ <lb/>altera in imum latus, altera in planũ per imum latus horizonti parallelum, una eodem{q́ue} <lb/>recta ſunt. </s> <s xml:id="echoid-s3849" xml:space="preserve">Secundum parallelogrammi obliquanguli itidem ad horizontirecti ubi duæ <lb/>perpendiculares altera à ſummo latere in imum, altera indidem in planum per imum <lb/>latus horizonti parallelum, eadem ſunt linea. </s> <s xml:id="echoid-s3850" xml:space="preserve">Tertium parallelogrammi rectanguli ad <lb/>horizontem obliquati ubi latus unam horizonti annuens & </s> <s xml:id="echoid-s3851" xml:space="preserve">perpendicularis a latere <lb/>ſummo in imum eadem ſunt recta. </s> <s xml:id="echoid-s3852" xml:space="preserve">Quartum denique parallelogrammi obliquanguli, <lb/>ubi dictæ tres lineæ inter ſe diverſae; </s> <s xml:id="echoid-s3853" xml:space="preserve">ſunt.</s> <s xml:id="echoid-s3854" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div535" type="section" level="1" n="385"> <head xml:id="echoid-head402" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s3855" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3856" xml:space="preserve">Rectanguli A B C D ad horizontem recti latus extimum A B <lb/>inaquæ ſuperficie 4 eſto pedum, A D 3.</s> <s xml:id="echoid-s3857" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3858" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3859" xml:space="preserve">Aquæ inſidentis fundo A B C D pondus invenire.</s> <s xml:id="echoid-s3860" xml:space="preserve"/> </p> <pb o="131" file="527.01.131" n="131" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div536" type="section" level="1" n="386"> <head xml:id="echoid-head403" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3861" xml:space="preserve">Latus A B 3 per A D multiplicatum efficit 12 quæ ſecundò <lb/> <anchor type="figure" xlink:label="fig-527.01.131-01a" xlink:href="fig-527.01.131-01"/> per A D 3 multiplicata dabunt 36 cubicos pedes, ejus ſemiſſis <lb/>18 numerus optatus. </s> <s xml:id="echoid-s3862" xml:space="preserve">Idem aliter. </s> <s xml:id="echoid-s3863" xml:space="preserve">quadratum ab A B 3 in dimi-<lb/>dium lateris A B 4 exhibet 18 pedes, ut ſupra. </s> <s xml:id="echoid-s3864" xml:space="preserve">ſingulis autem <lb/>pedibus æſtimatis 65 ℔, efficitur pondus 1170 ℔ iſti fundo in-<lb/>nixum.</s> <s xml:id="echoid-s3865" xml:space="preserve"/> </p> <div xml:id="echoid-div536" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.131-01" xlink:href="fig-527.01.131-01a"> <image file="527.01.131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.131-01"/> </figure> </div> </div> <div xml:id="echoid-div538" type="section" level="1" n="387"> <head xml:id="echoid-head404" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s3866" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3867" xml:space="preserve">ABCD parallelogrammum obliquangulum horizonti per-<lb/>pendiculare, cujus latus A B in aquæ ſuperficie quatuor ſit pedum, perpendi-<lb/>cularis à ſummo latere A B in imum D C eſto A E 3.</s> <s xml:id="echoid-s3868" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3869" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3870" xml:space="preserve">Aquæ ſundo A B C D ſubnixæ pondus invenire.</s> <s xml:id="echoid-s3871" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div539" type="section" level="1" n="388"> <head xml:id="echoid-head405" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3872" xml:space="preserve">A E 3 multiplicata per A B 4 faciunt 12 quæ rur-<lb/> <anchor type="figure" xlink:label="fig-527.01.131-02a" xlink:href="fig-527.01.131-02"/> ſum in A E 3 multiplicata dabunt 36 pedes cubicos, <lb/>ejus ſemiſſis 18 ſunt quæſitus pedum cubicorum nu-<lb/>merus. </s> <s xml:id="echoid-s3873" xml:space="preserve">Vel, quadrato à 3 in ſemiſſem lateris A B 4 <lb/>multiplicato, redit idem 18.</s> <s xml:id="echoid-s3874" xml:space="preserve"/> </p> <div xml:id="echoid-div539" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.131-02" xlink:href="fig-527.01.131-02a"> <image file="527.01.131-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.131-02"/> </figure> </div> </div> <div xml:id="echoid-div541" type="section" level="1" n="389"> <head xml:id="echoid-head406" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s3875" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3876" xml:space="preserve">ABCD rectangulum ad horizontem obliquatum, ejus latus <lb/>A B, in ſuperficie aquæ 6 pedum, A D 4, A E per-<lb/> <anchor type="figure" xlink:label="fig-527.01.131-03a" xlink:href="fig-527.01.131-03"/> pendicularis ab A in planum per D C horizonti pa-<lb/>rallelum 3 eſto pedum.</s> <s xml:id="echoid-s3877" xml:space="preserve"/> </p> <div xml:id="echoid-div541" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.131-03" xlink:href="fig-527.01.131-03a"> <image file="527.01.131-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.131-03"/> </figure> </div> <p> <s xml:id="echoid-s3878" xml:space="preserve">Q*VAE SITVM*. </s> <s xml:id="echoid-s3879" xml:space="preserve">Aquæ fundo A B C D innixæ <lb/>pondus invenire.</s> <s xml:id="echoid-s3880" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div543" type="section" level="1" n="390"> <head xml:id="echoid-head407" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3881" xml:space="preserve">Quater ſena ſunt 24 hæc per 3 multiplicata fa-<lb/>ciunt 72, ejus ſemiſſis 36 optati cubici pedes. </s> <s xml:id="echoid-s3882" xml:space="preserve">Velſic. <lb/></s> <s xml:id="echoid-s3883" xml:space="preserve">factus à ter quaternis in dimidium numeri 6 multiplicatus efficit 36 ut ſupra.</s> <s xml:id="echoid-s3884" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div544" type="section" level="1" n="391"> <head xml:id="echoid-head408" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s3885" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3886" xml:space="preserve">ABCD parallelogrammum obliquangulum ad horizontem <lb/>obliquum, cujus latus A B in ſuperficie aquęſit pedum 6, A E 4 perpendicu-<lb/>laris in latus C D, A F 6 perpendicularis plano ho-<lb/>rizonti per C D parallelo.</s> <s xml:id="echoid-s3887" xml:space="preserve"/> </p> <figure> <image file="527.01.131-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.131-04"/> </figure> <p> <s xml:id="echoid-s3888" xml:space="preserve">Q*VAESITUM*. </s> <s xml:id="echoid-s3889" xml:space="preserve">Aquæ fundo A B C D in-<lb/>cumbentis pondus invenire.</s> <s xml:id="echoid-s3890" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div545" type="section" level="1" n="392"> <head xml:id="echoid-head409" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3891" xml:space="preserve">A E 4 in A B 6 faciunt 24, quæ multiplicata <lb/>cum A F 3 efficiunt 72 cubicos pedes, ſemiſſis 36 <lb/>quæſitus pedum numerus. </s> <s xml:id="echoid-s3892" xml:space="preserve">Velſic. </s> <s xml:id="echoid-s3893" xml:space="preserve">factus à ter qua-<lb/>ternis per ſenarii ſemiſſem dabit eoſdem 36 pedes.</s> <s xml:id="echoid-s3894" xml:space="preserve"/> </p> <pb o="132" file="527.01.132" n="132" rhead="4 L*IBER* S*TATIC Æ*"/> </div> <div xml:id="echoid-div546" type="section" level="1" n="393"> <head xml:id="echoid-head410" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3895" xml:space="preserve">Columnæ baſis pedum 12, altitudinis 3, ſoliditas eſt 36 pedum ſemiſſis 18. <lb/></s> <s xml:id="echoid-s3896" xml:space="preserve">rale autem corpus inſidet, per 11 propoſ. </s> <s xml:id="echoid-s3897" xml:space="preserve">fundo A B C D primi exempli. </s> <s xml:id="echoid-s3898" xml:space="preserve">Itaque <lb/>ſuſtinet pondus 18 pedum. </s> <s xml:id="echoid-s3899" xml:space="preserve">Cæterorum exemplorum demonſtratio huic ger-<lb/>mana eſt.</s> <s xml:id="echoid-s3900" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div547" type="section" level="1" n="394"> <head xml:id="echoid-head411" xml:space="preserve">1 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s3901" xml:space="preserve">Ex quo perſpicitur latere parallelogrammi etiam infra aquam abdito, quo <lb/>ratiocinio pondus aqueum ipſi inſidens concludi poſſit. </s> <s xml:id="echoid-s3902" xml:space="preserve">nam additâ, ad pondus <lb/>ſupra inventum, columnâ cujus baſis ſit iſtud fundum, & </s> <s xml:id="echoid-s3903" xml:space="preserve">altitudo perpendicu-<lb/>laris à ſupremo fundi latere in ſummam aquę ſuperficiem totum hoc erit opta-<lb/>tum.</s> <s xml:id="echoid-s3904" xml:space="preserve"/> </p> <figure> <image file="527.01.132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.132-01"/> </figure> <p> <s xml:id="echoid-s3905" xml:space="preserve">Exemplum tale eſto. </s> <s xml:id="echoid-s3906" xml:space="preserve">Quadranguli A B C D latus <lb/>ſupremum A B infra aquæ ſummam ſuperficiem E F <lb/>conſiſtat, perpĕdicularis G A, ab A ad ſuperficiem E F, <lb/>pedum 3, area parallelogrammi A B C D 20. </s> <s xml:id="echoid-s3907" xml:space="preserve">Iam quod-<lb/>ſi A B in ſumma aquæ ſuperficie ſtatuatur, tum ipſi <lb/>40 cubicos aqueos pedes inſidere tanquam perantece-<lb/>dentem doctrinam concluſum aſſumo; </s> <s xml:id="echoid-s3908" xml:space="preserve">quæritur igitur <lb/>quot nunc ſuſtineat@ multiplicato plano A B C D 20 <lb/>pedum in altitudinem G A 3 fit columna 60 pedum, <lb/>quæ compoſita cum 40 exhibet 100 pedes, quorum pondere A B C D fun-<lb/>dum prematur.</s> <s xml:id="echoid-s3909" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div548" type="section" level="1" n="395"> <head xml:id="echoid-head412" xml:space="preserve">2 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s3910" xml:space="preserve">Atque cum fundum irregulare dabitur, invenito per 13 propoſ aqueam mo-<lb/>lem ponderi fundo illi inſidenti æqualem; </s> <s xml:id="echoid-s3911" xml:space="preserve">ex cujus dimenſione deinde quæſi-<lb/>tam gravitatem concludas.</s> <s xml:id="echoid-s3912" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div549" type="section" level="1" n="396"> <head xml:id="echoid-head413" xml:space="preserve">5 PROBLEMA. 16 PROPOSITIO.</head> <p> <s xml:id="echoid-s3913" xml:space="preserve">Si parallelogrammi ad horizontem inclinati, cujus ſu-<lb/>premum latus in aquæ ſuperficie ſumma conſiſtat, duæ <lb/>perpendiculares altera à ſummo in latus imum, altera in-<lb/>didem in planum per imum latus horizonti parallelum, <lb/>cum põdere quod ipſi inſidet nota ſint; </s> <s xml:id="echoid-s3914" xml:space="preserve">ſummum ejuſdem <lb/>latus invenire.</s> <s xml:id="echoid-s3915" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div550" type="section" level="1" n="397"> <head xml:id="echoid-head414" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s3916" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3917" xml:space="preserve">Quadrangulo rectangulo A B C D ad horizontem recto, cujus <lb/>ſummi lateris A B in aquæ ſuperficie ſuprema conſiſtentis lon-<lb/> <anchor type="figure" xlink:label="fig-527.01.132-02a" xlink:href="fig-527.01.132-02"/> gitudo ignoratur, incumbat moles aquea ponderis 18 pedum, <lb/>atque A D ſit 3. </s> <s xml:id="echoid-s3918" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3919" xml:space="preserve">Lateris A B longitudinem <lb/>invenire.</s> <s xml:id="echoid-s3920" xml:space="preserve"/> </p> <div xml:id="echoid-div550" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.132-02" xlink:href="fig-527.01.132-02a"> <image file="527.01.132-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.132-02"/> </figure> </div> </div> <div xml:id="echoid-div552" type="section" level="1" n="398"> <head xml:id="echoid-head415" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3921" xml:space="preserve">Diviſis 18 per quadratum A D 3, redibunt 2, cujus duplum <lb/>4 pedes, lateris A D longitudinem definiunt.</s> <s xml:id="echoid-s3922" xml:space="preserve"/> </p> <pb o="133" file="527.01.133" n="133" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div553" type="section" level="1" n="399"> <head xml:id="echoid-head416" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s3923" xml:space="preserve">D*ATUM*. </s> <s xml:id="echoid-s3924" xml:space="preserve">Parallelogrammum obliquangulum <lb/> <anchor type="figure" xlink:label="fig-527.01.133-01a" xlink:href="fig-527.01.133-01"/> A B C D horizonti perpendiculare, huic moles aquea <lb/>ponderis 18 pedum incumbit, ejusq́ue ſummi lateris <lb/>A B in aquæ extima ſuperficie longitudo ignoratur; </s> <s xml:id="echoid-s3925" xml:space="preserve">at-<lb/>qui perpendicularis A E in baſin imam D C datur <lb/>pedum. </s> <s xml:id="echoid-s3926" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3927" xml:space="preserve">Invenire latus A B.</s> <s xml:id="echoid-s3928" xml:space="preserve"/> </p> <div xml:id="echoid-div553" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.133-01" xlink:href="fig-527.01.133-01a"> <image file="527.01.133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.133-01"/> </figure> </div> </div> <div xml:id="echoid-div555" type="section" level="1" n="400"> <head xml:id="echoid-head417" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3929" xml:space="preserve">18 per quadratum A E 3 diviſis, quotique 2 duplum erit 4 pro A B.</s> <s xml:id="echoid-s3930" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div556" type="section" level="1" n="401"> <head xml:id="echoid-head418" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s3931" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3932" xml:space="preserve">Rectangulum parallelogrammum <lb/> <anchor type="figure" xlink:label="fig-527.01.133-02a" xlink:href="fig-527.01.133-02"/> A B C D horizonti obliquum, cui inſidet moles <lb/>aquea pondere pedum 36, ejusq́ue latus ſupremum <lb/>in aquæ ſuperficie ſumma longitudinis ſit ignotæ <lb/>ſed latus A D 4, & </s> <s xml:id="echoid-s3933" xml:space="preserve">perpendicularis A E à ſummo <lb/>latere in planum per latus imum horizonti paralle-<lb/>lum, 3 pedum dantur.</s> <s xml:id="echoid-s3934" xml:space="preserve"/> </p> <div xml:id="echoid-div556" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.133-02" xlink:href="fig-527.01.133-02a"> <image file="527.01.133-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.133-02"/> </figure> </div> <p> <s xml:id="echoid-s3935" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3936" xml:space="preserve">Invenire latus A B.</s> <s xml:id="echoid-s3937" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div558" type="section" level="1" n="402"> <head xml:id="echoid-head419" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3938" xml:space="preserve">Planus ab A E 3 in A D 4 eſt 12, per quem diviſis 36 quotoque 3 duplicato <lb/>ſit A B pedum 6.</s> <s xml:id="echoid-s3939" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div559" type="section" level="1" n="403"> <head xml:id="echoid-head420" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s3940" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3941" xml:space="preserve">A B C D parallelogrammum obli-<lb/> <anchor type="figure" xlink:label="fig-527.01.133-03a" xlink:href="fig-527.01.133-03"/> quangulum ad horizontem obliquatum, cui inſi-<lb/>det moles aquea ponderis 36 pedum, ejusq́ue ſu-<lb/>premum latus A B in aquæ ſumma ſuperficie, lon. <lb/></s> <s xml:id="echoid-s3942" xml:space="preserve">gitudinis ſit ignotæ; </s> <s xml:id="echoid-s3943" xml:space="preserve">ſed A E perpendicularis à <lb/>ſummo latere in imum pedum 4, & </s> <s xml:id="echoid-s3944" xml:space="preserve">altera indidem <lb/>in planum per imum latus horizonti parallelum <lb/>ſit pedum 3.</s> <s xml:id="echoid-s3945" xml:space="preserve"/> </p> <div xml:id="echoid-div559" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.133-03" xlink:href="fig-527.01.133-03a"> <image file="527.01.133-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.133-03"/> </figure> </div> <p> <s xml:id="echoid-s3946" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3947" xml:space="preserve">Invenire latus A B.</s> <s xml:id="echoid-s3948" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div561" type="section" level="1" n="404"> <head xml:id="echoid-head421" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3949" xml:space="preserve">Planus ab A F 3 in A E 4 eſt 12, qui dividens 36, dabit quotum 3, isq́ue <lb/>duplicatus facit A B 6 pedum.</s> <s xml:id="echoid-s3950" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div562" type="section" level="1" n="405"> <head xml:id="echoid-head422" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s3951" xml:space="preserve">Si in primo exemplo latus A B majus minusvé eſſet 4 pedibus, pondus item <lb/>aqueum ipſi inſidens majus minusvé eſſet pedibus 18, quod theſi repugnat. <lb/></s> <s xml:id="echoid-s3952" xml:space="preserve">quare A B eſt pedum 4. </s> <s xml:id="echoid-s3953" xml:space="preserve">REliquorum exemplorum demonſtratio huic eſt ger-<lb/>mana. </s> <s xml:id="echoid-s3954" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s3955" xml:space="preserve">Itaque ſi parallelogrammi ad horizontem incli-<lb/>nati, & </s> <s xml:id="echoid-s3956" xml:space="preserve">c.</s> <s xml:id="echoid-s3957" xml:space="preserve"/> </p> <pb o="134" file="527.01.134" n="134" rhead="4 L*IBER* S*TATIC Æ*"/> </div> <div xml:id="echoid-div563" type="section" level="1" n="406"> <head xml:id="echoid-head423" xml:space="preserve">1 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s3958" xml:space="preserve">Fx quo perſpicitur, quis modus inveniendi ſummilateris erit quando ipſum <lb/>infra aquam abſcondetur: </s> <s xml:id="echoid-s3959" xml:space="preserve">nam cum ſubduces à toto aqueæ molis pondere illi <lb/>inſidente columnam, cujus baſis ſit ipſum fundum, altitudo perpendicularis à <lb/>ſummo latere ad ſupernam aquæ ſuperficiem, reliquum hoc pondus tantum <lb/>erit quaſi ſummum iſtud latus in aquæ ſuperficie conſiſteret, unde anteceden-<lb/>tem factionem imitatus longitudinem ejus concludes.</s> <s xml:id="echoid-s3960" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s3961" xml:space="preserve">Ad inventionem autem iſtius columnæ quam diximus ſubducendam, hac <lb/>via inſiſtes. </s> <s xml:id="echoid-s3962" xml:space="preserve">Secato integrum illud datum pondus, integramvé columnam ra-<lb/>tione ea, quam habet perpendicularis ab aquæ ſuperficre ſumma in ſupremum <lb/>fundi latus demiſſa, ad perpendicularem candem auctam ſemiſſe perpendicu-<lb/>laris indidem continuatæ uſque in planum per imum latus horiz onti paralle-<lb/>lum. </s> <s xml:id="echoid-s3963" xml:space="preserve">Quod lucem accipiet à 12 propoſ. </s> <s xml:id="echoid-s3964" xml:space="preserve">1 exemplo. </s> <s xml:id="echoid-s3965" xml:space="preserve">nam ſi pars hæc ſubducen-<lb/>da iſtic quæreretur, ſic concluderes. </s> <s xml:id="echoid-s3966" xml:space="preserve">ut EG ad EA, ſic datum pondus ad ſui <lb/>partem ſubducendam.</s> <s xml:id="echoid-s3967" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div564" type="section" level="1" n="407"> <head xml:id="echoid-head424" xml:space="preserve">2 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s3968" xml:space="preserve">Et licet hinc invenire longitudinem ſupremi lateris, quando fundum linea <lb/>alteri laterum ad horizontem annuentium parallela interſecabitur. </s> <s xml:id="echoid-s3969" xml:space="preserve">Sit, dicis <lb/>gratia, in diagrammate 4 exempli agenda G H parallela contra A D, ut in <lb/>A G H D inſideat pondus aqueum 12 pedum, jam quę ratio eſt ponderis 12 ad <lb/>36 ea eſt ſegmenti A G ad totum latus A B. </s> <s xml:id="echoid-s3970" xml:space="preserve">quare A G erit 2 pedum.</s> <s xml:id="echoid-s3971" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div565" type="section" level="1" n="408"> <head xml:id="echoid-head425" xml:space="preserve">6 PROBLEMA. 17 PROPOSITIO.</head> <p> <s xml:id="echoid-s3972" xml:space="preserve">Si parallelogrammi ad horizontem inclinati, cujus ſu-<lb/>premum latus notum in ſupera aquæ ſuperficie conſiſtat, <lb/>perpendicularis à ſummo in planum per latus imum hori-<lb/>zonti parallelum, cum pondere ipſi inſidente nota ſint; </s> <s xml:id="echoid-s3973" xml:space="preserve">re. <lb/></s> <s xml:id="echoid-s3974" xml:space="preserve">liquam perpendicularem à latere ſummo in imum demiſ-<lb/>ſam invenire.</s> <s xml:id="echoid-s3975" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div566" type="section" level="1" n="409"> <head xml:id="echoid-head426" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s3976" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3977" xml:space="preserve">A B C D rectangulum ad horizontem perpendiculare, cui pon-<lb/>dus aqueum 18 pedum inſideat, latusq́ue A B in ſuprema <lb/> <anchor type="figure" xlink:label="fig-527.01.134-01a" xlink:href="fig-527.01.134-01"/> aquæ ſuperficie pedum 4.</s> <s xml:id="echoid-s3978" xml:space="preserve"/> </p> <div xml:id="echoid-div566" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.134-01" xlink:href="fig-527.01.134-01a"> <image file="527.01.134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.134-01"/> </figure> </div> <p> <s xml:id="echoid-s3979" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3980" xml:space="preserve">Invenire A D.</s> <s xml:id="echoid-s3981" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div568" type="section" level="1" n="410"> <head xml:id="echoid-head427" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3982" xml:space="preserve">Diviſis 18 per 2 ſemiſſem lateris A B quotus erit 9, cujus <lb/>quadrati latus 3 pro A D.</s> <s xml:id="echoid-s3983" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div569" type="section" level="1" n="411"> <head xml:id="echoid-head428" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s3984" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3985" xml:space="preserve">A B C D parallelogrammum obliquãgu-<lb/> <anchor type="figure" xlink:label="fig-527.01.134-02a" xlink:href="fig-527.01.134-02"/> lum horizonti perpendiculare, cui inſiſtit moles aquea <lb/>18 pedum ejusq́ue latus ſummum A B, quod in aquę <lb/>ſummitate conſiſtit, ſit pedum 4.</s> <s xml:id="echoid-s3986" xml:space="preserve"/> </p> <div xml:id="echoid-div569" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.134-02" xlink:href="fig-527.01.134-02a"> <image file="527.01.134-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.134-02"/> </figure> </div> <p> <s xml:id="echoid-s3987" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s3988" xml:space="preserve">Invenire rectam A E.</s> <s xml:id="echoid-s3989" xml:space="preserve"/> </p> <pb o="135" file="527.01.135" n="135" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div571" type="section" level="1" n="412"> <head xml:id="echoid-head429" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3990" xml:space="preserve">Diviſis 18 per 2 ſemiſſem lateris A B, quotus erit 9 hujus quadrati latus 3 <lb/>pro A E.</s> <s xml:id="echoid-s3991" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div572" type="section" level="1" n="413"> <head xml:id="echoid-head430" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s3992" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s3993" xml:space="preserve">A B C D rectangulum horizonti obliquum, cui pondusaquæ <lb/>36 pedum inſidet, ejusq́ue ſupremum latus A B <lb/> <anchor type="figure" xlink:label="fig-527.01.135-01a" xlink:href="fig-527.01.135-01"/> 6 pedum in aquæ ſummitate jaceat, unde per-<lb/>pendicularis A E in planum per imum latus ho-<lb/>rizonti parallelum ſit pedum 3.</s> <s xml:id="echoid-s3994" xml:space="preserve"/> </p> <div xml:id="echoid-div572" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.135-01" xlink:href="fig-527.01.135-01a"> <image file="527.01.135-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.135-01"/> </figure> </div> <p> <s xml:id="echoid-s3995" xml:space="preserve">Q*VÆSITVM*. </s> <s xml:id="echoid-s3996" xml:space="preserve">Invenire A D.</s> <s xml:id="echoid-s3997" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div574" type="section" level="1" n="414"> <head xml:id="echoid-head431" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s3998" xml:space="preserve">Diviſis 36 per 3 ſemiſſem lateris A B, quotus <lb/>eſt 12, qui per A E 3 diviſus exhibet A D 4 pedum.</s> <s xml:id="echoid-s3999" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div575" type="section" level="1" n="415"> <head xml:id="echoid-head432" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s4000" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s4001" xml:space="preserve">A B C D parallelogrammum obliquangulum ad horizontem <lb/>obliquatum, ſuſtineat pondus molis aqueæ 36 pedum, latusq́ue ſummum A B <lb/>in aquæ ſummitate conſtitutum ſit pedum 6, unde A E perpendicularis in la-<lb/>tus imum, at A F perpendicularis in planum per <lb/> <anchor type="figure" xlink:label="fig-527.01.135-02a" xlink:href="fig-527.01.135-02"/> imum latus horizonti parallelum ſit pedum 3.</s> <s xml:id="echoid-s4002" xml:space="preserve"/> </p> <div xml:id="echoid-div575" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.135-02" xlink:href="fig-527.01.135-02a"> <image file="527.01.135-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.135-02"/> </figure> </div> <p> <s xml:id="echoid-s4003" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s4004" xml:space="preserve">Invenire lineam A E.</s> <s xml:id="echoid-s4005" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div577" type="section" level="1" n="416"> <head xml:id="echoid-head433" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s4006" xml:space="preserve">Diviſis 36 per 3 ſemiſſem ſenarii lateris A B, <lb/>quotoq́ue deinceps per A F 3, exibit tandem <lb/>A E 4 pedum.</s> <s xml:id="echoid-s4007" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div578" type="section" level="1" n="417"> <head xml:id="echoid-head434" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4008" xml:space="preserve">Si A D primi exempli 3 pedibus cederet excederetvé, pondus quoque fun-<lb/>do iſti innixum 18 pedibus aut cederet aut præſtaret, quod contra theſin dixiſſe <lb/>abſurdum fuerit. </s> <s xml:id="echoid-s4009" xml:space="preserve">quare A D erit pedum 3. </s> <s xml:id="echoid-s4010" xml:space="preserve">Ratio & </s> <s xml:id="echoid-s4011" xml:space="preserve">demonſtratio reliquorum <lb/>huic ſimillimè inſtituetur. </s> <s xml:id="echoid-s4012" xml:space="preserve">Itaque ſi parallelogrammi ad horizontem inclina-<lb/>ti, & </s> <s xml:id="echoid-s4013" xml:space="preserve">c.</s> <s xml:id="echoid-s4014" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div579" type="section" level="1" n="418"> <head xml:id="echoid-head435" xml:space="preserve">1 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s4015" xml:space="preserve">Ex quo perſpicitur, quæ ratio ſit factionis ubi ſummum latus ſub aqua deli-<lb/>teſcit. </s> <s xml:id="echoid-s4016" xml:space="preserve">etenim cum de toto aquæ pondere fundo inſidentis deducetur columna, <lb/>cujus baſis ſit ipſum fundum, altitudo autem perpĕdicularis à ſummitate aquæ <lb/>in fundi ſupremum latus. </s> <s xml:id="echoid-s4017" xml:space="preserve">relinquetur pondus illud quod inſideret ſi ſummum <lb/>latus in aquæ ſuperſicere conſiſteret. </s> <s xml:id="echoid-s4018" xml:space="preserve">unde ſecundum demonſtrata dicta per-<lb/>pendicularis concludetur.</s> <s xml:id="echoid-s4019" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div580" type="section" level="1" n="419"> <head xml:id="echoid-head436" xml:space="preserve">2 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s4020" xml:space="preserve">Et licet hinc, cùm recta ſupremo lateri parallela deſecabit fundi partem datus <lb/>ſuſtinens pondus, invenire longitudinem perpendicularis à ſummo latere in <lb/>parallelam dictam demiſſæ. </s> <s xml:id="echoid-s4021" xml:space="preserve">Vt ſi in 4 exempli paradigmate G H parallela con- <pb o="136" file="527.01.136" n="136" rhead="4 L*IBER* S*TATICÆ*"/> tra A B interſecet A D in I, & </s> <s xml:id="echoid-s4022" xml:space="preserve">ſegmento A B H I incumbat pondus aquæ 24 <lb/>pedum cubicorum. </s> <s xml:id="echoid-s4023" xml:space="preserve">diviſis 24 per 3, quæ femiſſis eſt lateris A B, quotus erit 8: <lb/></s> <s xml:id="echoid-s4024" xml:space="preserve">tumq́ue invenito duos numeros in ratione A F 3 ad A E 4 quorum planus ſit <lb/>dictus octonarius; </s> <s xml:id="echoid-s4025" xml:space="preserve">numeriq́ue iſti erunt √ 6 & </s> <s xml:id="echoid-s4026" xml:space="preserve">√ 10 {2/3}, poſterior hic definiet <lb/>quantitatem A G: </s> <s xml:id="echoid-s4027" xml:space="preserve">namque G H parallela ex G contra A B auferet planum <lb/>A B H I cui per 1, propoſ. </s> <s xml:id="echoid-s4028" xml:space="preserve">molcsaquea 24 pedum innitetur.</s> <s xml:id="echoid-s4029" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div581" type="section" level="1" n="420"> <head xml:id="echoid-head437" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s4030" xml:space="preserve">Tribus deinceps continuis propoſitionibus, ex ſententia Breviarii, agendum nobis de <lb/>centris gravitatis preſſuum in fundis collectorum Vbi jure funda borizonti parallela <lb/>primum ſibi locum depoſcerent, ſed quia ipſorum gravitatis centra (quæ ex ſecundi li-<lb/>bri doctrina pateſcunt) à centris preſſuum diverſa non ſint, brevitatis ſtudio novum <lb/>nullum de iis theorema inſtituimus. </s> <s xml:id="echoid-s4031" xml:space="preserve">Quare initio ducto à fundis in clinatis, ita or dimur.</s> <s xml:id="echoid-s4032" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div582" type="section" level="1" n="421"> <head xml:id="echoid-head438" xml:space="preserve">12 THEOREMA. 18 PROPOSITIO.</head> <p> <s xml:id="echoid-s4033" xml:space="preserve">Si parallelogrammiad horizonrem inclina ti recta ſupre-<lb/>mum ejus latus, in ſumma aquæ ſuperficie conſiſtens, & </s> <s xml:id="echoid-s4034" xml:space="preserve"><lb/>imum ſibi oppoſitum biſecet, hæc à preſſus gravitatis cen-<lb/>tro ita tribuitur ut pars ſumma reliquæ ſit dupla.</s> <s xml:id="echoid-s4035" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div583" type="section" level="1" n="422"> <head xml:id="echoid-head439" style="it" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s4036" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s4037" xml:space="preserve">* Exponatur aqua A B, fundumq́ue ACDE figura parallelo-<lb/>gramma ad horizontem annuens, cujus ſuperũ <lb/> <anchor type="figure" xlink:label="fig-527.01.136-01a" xlink:href="fig-527.01.136-01"/> latus A C ſit in aquæ ſuperficie ſumma, tumq́; <lb/></s> <s xml:id="echoid-s4038" xml:space="preserve">ſuperum inferumq́ue latus A C, E D, biſecen-<lb/>tur in F & </s> <s xml:id="echoid-s4039" xml:space="preserve">G, quæ jungat F G in puncto H ita <lb/>diviſa ut ſegmentum F H reliqui H G ſit du-<lb/>plum.</s> <s xml:id="echoid-s4040" xml:space="preserve"/> </p> <div xml:id="echoid-div583" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.136-01" xlink:href="fig-527.01.136-01a"> <image file="527.01.136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.136-01"/> </figure> </div> <p> <s xml:id="echoid-s4041" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s4042" xml:space="preserve">* H preſſus quo fundum affi-<lb/>citur gravitatis eſſe centrum demonſtrator.</s> <s xml:id="echoid-s4043" xml:space="preserve"/> </p> <figure> <image file="527.01.136-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.136-02"/> </figure> <p> <s xml:id="echoid-s4044" xml:space="preserve">P*RAEPARATIO.</s> <s xml:id="echoid-s4045" xml:space="preserve">* Acta CI ſubtendat an-<lb/>gulum C D I æquicrurum; </s> <s xml:id="echoid-s4046" xml:space="preserve">ut priſi<unsure/>ma A C I D E <lb/>ſit dimidia columna, cujus baſis A C D E, alti-<lb/>tudo perpendicularis ab A uſque ad planum per <lb/>E D horizonti parallelum.</s> <s xml:id="echoid-s4047" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4048" xml:space="preserve">Figurato item alterum corpus K L M N O P <lb/>ſimile priori A C I D E, planumque K L M N <lb/>plano A C E, & </s> <s xml:id="echoid-s4049" xml:space="preserve">M O horizonti perpendi-<lb/>cularis lateri D I homologa ſunto, itemq́ue Q R <lb/>ipſi F G: </s> <s xml:id="echoid-s4050" xml:space="preserve">hinc ab S medio lateris O P agantur S Q, S R, atque trianguli Q S R <lb/>gravitatis centrum T, unde V X horizonti perpendieularis ſit.</s> <s xml:id="echoid-s4051" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div585" type="section" level="1" n="423"> <head xml:id="echoid-head440" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4052" xml:space="preserve">Iam per 11 propof. </s> <s xml:id="echoid-s4053" xml:space="preserve">qua to preſſu corpus K L M N O P afficit fundum <lb/>K L M N, tanto afficit aqua A B fundum A C D E; </s> <s xml:id="echoid-s4054" xml:space="preserve">quare preſſuum gravita-<lb/>tis centra in fundis K L M N, A C D E ſimili erunt ſitu. </s> <s xml:id="echoid-s4055" xml:space="preserve">Cæterùm T quod ex <pb o="137" file="527.01.137" n="137" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> fabrica centrum eſt trianguli Q S R, idem quoque per 15 propoſ. </s> <s xml:id="echoid-s4056" xml:space="preserve">lib. </s> <s xml:id="echoid-s4057" xml:space="preserve">2. </s> <s xml:id="echoid-s4058" xml:space="preserve">Elem. <lb/></s> <s xml:id="echoid-s4059" xml:space="preserve">Statie. </s> <s xml:id="echoid-s4060" xml:space="preserve">gravitatis centrum eſt corporis K L M N O P, quare V X, cum per cen-<lb/>trum T horizonti ad perpendiculum immineat, erit ejus pendula gravitas <lb/>diameter, quâ deorſum continuatâ in Y, corpus K L M N O P in puncto X <lb/>rectæ X Y innixum (quod mathematicè intelligatur) datum ſervabit ſitum. </s> <s xml:id="echoid-s4061" xml:space="preserve"><lb/>ideoq́ue X eſt dicti corporis in fundo K L M N preſſionis gravitatis centrum; </s> <s xml:id="echoid-s4062" xml:space="preserve"><lb/>cumque V X educta per centrum T horizonti perpendicularis ſit, etiam pa-<lb/>rallela erit contra S R, ideoq́ue per 5 propoſ. </s> <s xml:id="echoid-s4063" xml:space="preserve">2. </s> <s xml:id="echoid-s4064" xml:space="preserve">lib. </s> <s xml:id="echoid-s4065" xml:space="preserve">Ele. </s> <s xml:id="echoid-s4066" xml:space="preserve">Static. </s> <s xml:id="echoid-s4067" xml:space="preserve">fecat rectam Q R <lb/>ratione dupla, ut Q X dupla ſit reliquæ X R. </s> <s xml:id="echoid-s4068" xml:space="preserve">Atqui, ut ſupra jam expoſitum <lb/>eſt, centra gravitatis in fundis A C D E, K L M N ſimili ſitu reſpondent. </s> <s xml:id="echoid-s4069" xml:space="preserve">Itaq; </s> <s xml:id="echoid-s4070" xml:space="preserve"><lb/>F G ſecabitur ratione dupla ſcilicet in H, atque iſtic erit gravitatis centrum <lb/>aqueæ preſſionis collectæ in fundo A C D E.</s> <s xml:id="echoid-s4071" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div586" type="section" level="1" n="424"> <head xml:id="echoid-head441" style="it" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s4072" xml:space="preserve">Propter cauſas 4 exemplo 11 propoſ. </s> <s xml:id="echoid-s4073" xml:space="preserve">expoſitas, linearem hanc demonſtra-<lb/>tionem aritbmetico calculo comprobabimus, hoc modo.</s> <s xml:id="echoid-s4074" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4075" xml:space="preserve">Fundum A B C D ſecetur recta E F biſecante oppoſita latera A B, C D, <lb/>hinc fundum in aliquot æquas partes (quas menſuras appellabimus) lineis pa-<lb/>rallelis diſtribuatur, primumq́ue bipartitò rectâ G H, quæ ſecet E F in I, inq́ue <lb/>eâdem ſtatuitor punctum K ut E K reliquæ K F ſit dupla. </s> <s xml:id="echoid-s4076" xml:space="preserve">atque <lb/> <anchor type="figure" xlink:label="fig-527.01.137-01a" xlink:href="fig-527.01.137-01"/> id centrum eſſe preſſionis demonſtrandum eſto, hoc qui ſequi-<lb/>tur modo. </s> <s xml:id="echoid-s4077" xml:space="preserve">Si in A B H G una libra aquæ inſideat, in reliquo <lb/>G H C D 3 inſidebunt: </s> <s xml:id="echoid-s4078" xml:space="preserve">quæ cum ita ſint, fingo primùm preſ-<lb/>ſus gravitatis centrum A B H G conſiſtere in I, ipſiusq́ue <lb/>G H C D in F (quamvis certum ſit centra ſublimiora eſſe) tum <lb/>igitur I K jugum foret, qui in ſuos radios rationis triplæ divi-<lb/>ſus in puncto L, fiet F L {1/4} menſuræ, hoc eſt rectæ I F. </s> <s xml:id="echoid-s4079" xml:space="preserve">Secundò <lb/>fingo gravitatis centrum preſſionis A B H G eſſe in E, ipſiusq́; <lb/></s> <s xml:id="echoid-s4080" xml:space="preserve">G H C D in I (quamvis centra manifeſtò infra conſiſtant) itaque commune <lb/>ipſorum gravitatis centrum ſupra I cadet in M. </s> <s xml:id="echoid-s4081" xml:space="preserve">Quare verum ipſorum centrum <lb/>neceſſariò inter M & </s> <s xml:id="echoid-s4082" xml:space="preserve">L interjacet. </s> <s xml:id="echoid-s4083" xml:space="preserve">ſed quâ viâ fundum hic bipartitò diviſimus, <lb/>ita in ſegmenta infinita ipſum poterit diſtingui, inter quæ verum gravitatis cen-<lb/>trum perpetuò conſiſtat. </s> <s xml:id="echoid-s4084" xml:space="preserve">Simili inquam ſectione continuata infinitò propiùs <lb/>acceditur, & </s> <s xml:id="echoid-s4085" xml:space="preserve">cum experientia ipſa clamet, L punctũ nunquam congruere cum <lb/>K, ſed aliquantillum infra ſubſiſtere; </s> <s xml:id="echoid-s4086" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s4087" xml:space="preserve">altrinſecus punctum M nunquam <lb/>ad K deſcendere ſed ſupra conſiſtere, concludemus K verum eſſe centrum. </s> <s xml:id="echoid-s4088" xml:space="preserve">Ve-<lb/>rumenimverò quia iſta omnium fundorum communis centri inveſtigatio tæ-<lb/>dii moleſtiæq́ue plena eſt, aliam compĕdiariam deſcripſimus. </s> <s xml:id="echoid-s4089" xml:space="preserve">Formato ab uni-<lb/>tate arith meticam binarii intervallo progreſſionem continuam 1, 3, 5, 7, 9, & </s> <s xml:id="echoid-s4090" xml:space="preserve">c. </s> <s xml:id="echoid-s4091" xml:space="preserve"><lb/>nam per 15 propoſ. </s> <s xml:id="echoid-s4092" xml:space="preserve">ſegmentorum fundi A B C D æqualium preſſiones in iſtiuſ-<lb/>modi ſunt progreſſu, deinde {1/4} (quæ quantitas eſt ſegmenti F L ante inventa) <lb/>ſubjiciatur 3 ut hic vides.</s> <s xml:id="echoid-s4093" xml:space="preserve"/> </p> <div xml:id="echoid-div586" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.137-01" xlink:href="fig-527.01.137-01a"> <image file="527.01.137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.137-01"/> </figure> </div> <p> <s xml:id="echoid-s4094" xml:space="preserve">{1/4}</s> </p> <p> <s xml:id="echoid-s4095" xml:space="preserve">1. </s> <s xml:id="echoid-s4096" xml:space="preserve">3. </s> <s xml:id="echoid-s4097" xml:space="preserve">5. </s> <s xml:id="echoid-s4098" xml:space="preserve">7. </s> <s xml:id="echoid-s4099" xml:space="preserve">9. </s> <s xml:id="echoid-s4100" xml:space="preserve">11.</s> <s xml:id="echoid-s4101" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4102" xml:space="preserve">Tumad 4 nomen {1/4} addito tertium ordinis numerum 5, totusq́ue inſcriba-<lb/>tur tertio loco, ipſiq́ue pronumeratore ſuperſcribatur 5, qui totus eſt compo-<lb/>ſitus ex {1/4} nomine ad numeratorem ſuum addito. </s> <s xml:id="echoid-s4103" xml:space="preserve">ut hic:</s> <s xml:id="echoid-s4104" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4105" xml:space="preserve">{1/4} {3/5}</s> </p> <p> <s xml:id="echoid-s4106" xml:space="preserve">1. </s> <s xml:id="echoid-s4107" xml:space="preserve">3. </s> <s xml:id="echoid-s4108" xml:space="preserve">5. </s> <s xml:id="echoid-s4109" xml:space="preserve">7. </s> <s xml:id="echoid-s4110" xml:space="preserve">9. </s> <s xml:id="echoid-s4111" xml:space="preserve">11.</s> <s xml:id="echoid-s4112" xml:space="preserve"/> </p> <pb o="138" file="527.01.138" n="138" rhead="4 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s4113" xml:space="preserve">Similiter in cæteris, nam ad numerum qui ipſi 7 inſcribatur inveniendum, <lb/>addes nomen 9 ad 7, totus 16 eſt nomen novum, cuiſuperſcribes 14 à 9 & </s> <s xml:id="echoid-s4114" xml:space="preserve">5 (qui <lb/>ſunt numerus nomenq́ue {5/9}) compoſitum. </s> <s xml:id="echoid-s4115" xml:space="preserve">atque ita {14/16} erit numerus debitus ipſi <lb/>7, ut infra vides:</s> <s xml:id="echoid-s4116" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4117" xml:space="preserve">{1/4} {5/9} {14/36}</s> </p> <p> <s xml:id="echoid-s4118" xml:space="preserve">1. </s> <s xml:id="echoid-s4119" xml:space="preserve">3. </s> <s xml:id="echoid-s4120" xml:space="preserve">5. </s> <s xml:id="echoid-s4121" xml:space="preserve">7. </s> <s xml:id="echoid-s4122" xml:space="preserve">9. </s> <s xml:id="echoid-s4123" xml:space="preserve">11.</s> <s xml:id="echoid-s4124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4125" xml:space="preserve">Qua ratione in cæteris continuata, numeros ipſis 9 & </s> <s xml:id="echoid-s4126" xml:space="preserve">11 inſcribendos inve-<lb/>neris. </s> <s xml:id="echoid-s4127" xml:space="preserve">quales hic vides:</s> <s xml:id="echoid-s4128" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4129" xml:space="preserve">{1/4} {5/9} {14/36} {30/29} {55/36}.</s> <s xml:id="echoid-s4130" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4131" xml:space="preserve">1. </s> <s xml:id="echoid-s4132" xml:space="preserve">3. </s> <s xml:id="echoid-s4133" xml:space="preserve">5. </s> <s xml:id="echoid-s4134" xml:space="preserve">7. </s> <s xml:id="echoid-s4135" xml:space="preserve">9. </s> <s xml:id="echoid-s4136" xml:space="preserve">11.</s> <s xml:id="echoid-s4137" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4138" xml:space="preserve">Quibus intellectis, ſi quæratur quo punctum L aſcendat fundo in quinque <lb/>ęquas partes diſtributo. </s> <s xml:id="echoid-s4139" xml:space="preserve">Sumito numerum quinto loco, hoceſt ipſi 9 inſcriptum <lb/>is erit {30/25} ſeu in minimis terminis {6/5}, hic indicabit L F talis fundi quinque-partiti <lb/>fore {6/5} menſuræ cognominis partibus, in quas fundum tributũ erit. </s> <s xml:id="echoid-s4140" xml:space="preserve">Sed eam mi-<lb/>norem eſſe quam {1/3} E F, punctumq́ue ejus ſummũ L hærere infra K demõſtra-<lb/>bitur hoc modo. </s> <s xml:id="echoid-s4141" xml:space="preserve">{6/5} partis unius in quas fundũ ſecatur hoc eſt {6/5} {6/5} ſunt totius EF <lb/>{6/25} quas {1/3} excedit {7/75} ejuſdem. </s> <s xml:id="echoid-s4142" xml:space="preserve">tantoq́ue intervallo tunc L punctum in citra K <lb/>conſiſtet. </s> <s xml:id="echoid-s4143" xml:space="preserve">porro ut in eadem ſectionelocum ipſius M invenias, addito integram <lb/>menſuram ad ſui {6/5} ſumma erit {11<unsure/>/9}, quæ ſunt {11/25} totius E F & </s> <s xml:id="echoid-s4144" xml:space="preserve">majores quam {1/3} <lb/>ejuſdem, nam de {11/28<unsure/>} deducta {1/3} relinquitur {8/75}, tantumq́ue M punctum ſupra K <lb/>conſiſtet, punctumq́ue hoc ſupernate M cadet ab K {1<unsure/>/75} diſtantius quam in-<lb/>fernate L. </s> <s xml:id="echoid-s4145" xml:space="preserve">atque ita in cæteris omnibus. </s> <s xml:id="echoid-s4146" xml:space="preserve">ut cum A B C D ſecabitur in partes <lb/>40, F L deprehen detur {20550/1600} unius menſuræ hoc eſt unius quadrageſimæ ipſius <lb/>E F. </s> <s xml:id="echoid-s4147" xml:space="preserve">Quo ratiocinio infinitè continuato punctorum L, M acceſſio ad K infi-<lb/>nitè quoque vicinior invenietur, quæ tamen nunquam eo pertingat. </s> <s xml:id="echoid-s4148" xml:space="preserve">cujus <lb/>neceſſitas ſuperiore exemplo γραμμικως demonſtrata eſt. </s> <s xml:id="echoid-s4149" xml:space="preserve">Cauſam compendii <lb/>hujus noſtri, is facilè animadvertet, qui modum 2 propoſ. </s> <s xml:id="echoid-s4150" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4151" xml:space="preserve">Elem. </s> <s xml:id="echoid-s4152" xml:space="preserve">Static. <lb/></s> <s xml:id="echoid-s4153" xml:space="preserve">factione prolixa perſequetur. </s> <s xml:id="echoid-s4154" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s4155" xml:space="preserve">Itaque ſi parallelogrammi ad <lb/>horizontem inclinati, & </s> <s xml:id="echoid-s4156" xml:space="preserve">c.</s> <s xml:id="echoid-s4157" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div588" type="section" level="1" n="425"> <head xml:id="echoid-head442" xml:space="preserve">13 THEOREMA. 19 PROPOSITIO.</head> <p> <s xml:id="echoid-s4158" xml:space="preserve">Si parallelogrammi ad horizontem inclinati ſummum <lb/>latus horizonti parallelum intra aquam abditum recta & </s> <s xml:id="echoid-s4159" xml:space="preserve"><lb/>ipſum & </s> <s xml:id="echoid-s4160" xml:space="preserve">latus oppoſitũ biſecet; </s> <s xml:id="echoid-s4161" xml:space="preserve">preſſus gravitatis centrum <lb/>in iſto fundo collecti partem dictæ rectæ inter ſui ſemiſ-<lb/>ſem & </s> <s xml:id="echoid-s4162" xml:space="preserve">trientem inferiorem interjectam ita ſecat, ut pars <lb/>trienti inferiori vicina ad reliquam ſit, quemadmodum per-<lb/>pendicularis à ſupero fundi latere uſque ad aquæ ſuperfi-<lb/>ciem ſummam, ad ſemiſſem perpendicularis indidem de-<lb/>miſſæ in planum per imum latus horizonti parallelum.</s> <s xml:id="echoid-s4163" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4164" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s4165" xml:space="preserve">Fundum A B C D ad horizontem inclinatum, ejuſq́ue ſupe-<lb/>rum latus A B intra aquam E F deliteſcens horizonti parallela eſt, unde G A <lb/>perpendicularis eſt in ſuperam aquæ ſuperficiem, eademq́ue continuata deor-<lb/>ſum in ſuperficiem per D C horizonti parallelam ſit A H, ſemiſſis A I, hinc <pb o="139" file="527.01.139" n="139" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> K L biſecet latera A B, D C, cujus triens inferior L M, atque L N ſemiſſis, <lb/>ſeu quod idem eſt N ſit centrum fundi parallelogrammi A B C D: </s> <s xml:id="echoid-s4166" xml:space="preserve">Denique <lb/>intervallum M N itaſecetur in O, ut M O ad O N ſit quemadmodum G A <lb/>ad H I. </s> <s xml:id="echoid-s4167" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s4168" xml:space="preserve">Gravitatis centrum preſſus aquæ in fundo A B C D <lb/>in O conſiſtere demonſtrator. </s> <s xml:id="echoid-s4169" xml:space="preserve">P*RAEPARATIO*. </s> <s xml:id="echoid-s4170" xml:space="preserve">C B, D A uſqueadaquæ <lb/>ſuperficiem in P & </s> <s xml:id="echoid-s4171" xml:space="preserve">E, continuan-<lb/> <anchor type="figure" xlink:label="fig-527.01.139-01a" xlink:href="fig-527.01.139-01"/> tor; </s> <s xml:id="echoid-s4172" xml:space="preserve">ſit q́ue C Q horizonti paralle-<lb/>la lateri C D perpendicularis ipſiq́; <lb/></s> <s xml:id="echoid-s4173" xml:space="preserve">adeò C P æqualis; </s> <s xml:id="echoid-s4174" xml:space="preserve">denique B R, <lb/>A S, lateri C T, item R T, S V ipſi <lb/>B C æquales conſtituantur & </s> <s xml:id="echoid-s4175" xml:space="preserve">pa-<lb/>rallelæ.</s> <s xml:id="echoid-s4176" xml:space="preserve"/> </p> <div xml:id="echoid-div588" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.139-01" xlink:href="fig-527.01.139-01a"> <image file="527.01.139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.139-01"/> </figure> </div> <p> <s xml:id="echoid-s4177" xml:space="preserve">Hanc alteram figuram antece-<lb/>denti E P C D Q æqualem, ſimi-<lb/>lem, & </s> <s xml:id="echoid-s4178" xml:space="preserve">æquipondiam deformato, <lb/>cujus latus C D horizonti ad per-<lb/> <anchor type="figure" xlink:label="fig-527.01.139-02a" xlink:href="fig-527.01.139-02"/> pendiculũ immineat ſitq́; </s> <s xml:id="echoid-s4179" xml:space="preserve">X centrũ <lb/>gravitatis columnę ABCDRSVT, <lb/>atque Y centrum gravitatis priſma-<lb/>tis R S V T Q; </s> <s xml:id="echoid-s4180" xml:space="preserve">denique jungito <lb/>X N, Y M.</s> <s xml:id="echoid-s4181" xml:space="preserve"/> </p> <div xml:id="echoid-div589" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.139-02" xlink:href="fig-527.01.139-02a"> <image file="527.01.139-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.139-02"/> </figure> </div> </div> <div xml:id="echoid-div591" type="section" level="1" n="426"> <head xml:id="echoid-head443" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4182" xml:space="preserve">Cum in ſecundo hoc diagram-<lb/>mate X gravitatis centrum ſit pa-<lb/>rallelepipedi A B C D R S V T, & </s> <s xml:id="echoid-s4183" xml:space="preserve"><lb/>N baſis A B C D, itemq́ue C T <lb/>horizonti perpendicularis, etiam <lb/>X N horizonti perpendicularis ejusq́; </s> <s xml:id="echoid-s4184" xml:space="preserve">gravitatis pendula diameter erit. </s> <s xml:id="echoid-s4185" xml:space="preserve">ideoq́; <lb/></s> <s xml:id="echoid-s4186" xml:space="preserve">N eſt columnæ iſtius preſſionis centrum, quod autem M ſit preſſus corporis <lb/>S R T V Q gravitatis centrum è 18 propoſ. </s> <s xml:id="echoid-s4187" xml:space="preserve">perſpicitur, quamobrem M N erit <lb/>ipſorum jugum, iſtud autem in O ita eſt ſectum ut ratio ſegmentorum O M, <lb/>O N eadem ſit quæ A G ad A I, ſed ita quoq e eſt parallelepipedum <lb/>A B C D R S V T ad priſma S R T V Q: </s> <s xml:id="echoid-s4188" xml:space="preserve">itaq; </s> <s xml:id="echoid-s4189" xml:space="preserve">æqueordinatè ut ABCDRSVT <lb/>ad S R T V Q ſic O M ad O N. </s> <s xml:id="echoid-s4190" xml:space="preserve">quare per 1 propoſ. </s> <s xml:id="echoid-s4191" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4192" xml:space="preserve">Elem. </s> <s xml:id="echoid-s4193" xml:space="preserve">Static. </s> <s xml:id="echoid-s4194" xml:space="preserve">O cen-<lb/>trum erit preſſionis hujus ſecundæ figuræ. </s> <s xml:id="echoid-s4195" xml:space="preserve">Et cum, propter cauſas jam ſæpe di-<lb/>ctas, primæ ſecundæq́; </s> <s xml:id="echoid-s4196" xml:space="preserve">figuræ centra ſimili ſitu congruant, O quoque in prima <lb/>figura gravitatis erit centrum.</s> <s xml:id="echoid-s4197" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4198" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s4199" xml:space="preserve">Itaque ſi parallelogrammi ad horizontem inclinati, & </s> <s xml:id="echoid-s4200" xml:space="preserve">c.</s> <s xml:id="echoid-s4201" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div592" type="section" level="1" n="427"> <head xml:id="echoid-head444" xml:space="preserve">7 PROBLEMA. 20 PROPOSITIO.</head> <p> <s xml:id="echoid-s4202" xml:space="preserve">Dati fundi plani rectilinei, preſſus gravitatis centrum <lb/>invenire.</s> <s xml:id="echoid-s4203" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4204" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s4205" xml:space="preserve">Aquæ A B ſuperficies ſuperna A C, datumq́ue fundum recti-<lb/>lineum D E. </s> <s xml:id="echoid-s4206" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s4207" xml:space="preserve">Preſſus gravitatis centrum in fundo iſtoc col-<lb/>le<unsure/>cti invenire.</s> <s xml:id="echoid-s4208" xml:space="preserve"/> </p> <pb o="140" file="527.01.140" n="140" rhead="4 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div593" type="section" level="1" n="428"> <head xml:id="echoid-head445" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s4209" xml:space="preserve">Primùm corpus aqueum preſſu fundo <lb/> <anchor type="figure" xlink:label="fig-527.01.140-01a" xlink:href="fig-527.01.140-01"/> D E inſidenti æquepondium per 13 propoſ. <lb/></s> <s xml:id="echoid-s4210" xml:space="preserve">inveniatur, idque eſto D E F G, cujus gra-<lb/>vitatis centrum per 21 propoſ. </s> <s xml:id="echoid-s4211" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4212" xml:space="preserve">Static. </s> <s xml:id="echoid-s4213" xml:space="preserve"><lb/>inventum ſit H, unde H I parallela agatur <lb/>contra G E, ejus in fundo D E terminus I <lb/>optatũ erit preſſus gravitatis centrum; </s> <s xml:id="echoid-s4214" xml:space="preserve">cujus <lb/>demonſtratio antecedentium 18 & </s> <s xml:id="echoid-s4215" xml:space="preserve">19 propoſ. </s> <s xml:id="echoid-s4216" xml:space="preserve">ſimilis erit.</s> <s xml:id="echoid-s4217" xml:space="preserve"/> </p> <div xml:id="echoid-div593" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.140-01" xlink:href="fig-527.01.140-01a"> <image file="527.01.140-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.140-01"/> </figure> </div> <p> <s xml:id="echoid-s4218" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s4219" xml:space="preserve">Itaque dati fundi plani rectilinei, & </s> <s xml:id="echoid-s4220" xml:space="preserve">c.</s> <s xml:id="echoid-s4221" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div595" type="section" level="1" n="429"> <head xml:id="echoid-head446" xml:space="preserve">8 PROBLEMA. 21 PROPOSITIO.</head> <p> <s xml:id="echoid-s4222" xml:space="preserve">Data aqua magnitudinis ignotæ, gravitatis verò notæ; <lb/></s> <s xml:id="echoid-s4223" xml:space="preserve">magnitudinem ex ſua propriaq́ue ponderitate invenire.</s> <s xml:id="echoid-s4224" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div596" type="section" level="1" n="430"> <head xml:id="echoid-head447" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s4225" xml:space="preserve">Quamvis magnitudinis inventio Geometrica ratione inveniri & </s> <s xml:id="echoid-s4226" xml:space="preserve">explicari <lb/>poſſit, quia tamen Statica factio expeditior certiorq́ue ſit, & </s> <s xml:id="echoid-s4227" xml:space="preserve">in corporibus, præ-<lb/>ſertim inordinatis propius verum collimet, eam hic exponere ſtatui.</s> <s xml:id="echoid-s4228" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4229" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s4230" xml:space="preserve">Aquæ A magnitudo ignoretur, gravitas autem nota eſto, <lb/>hoc eſt per 1 defin. </s> <s xml:id="echoid-s4231" xml:space="preserve">hujus cujus nota magnitudo cognita ponderitate expri-<lb/>mitur. </s> <s xml:id="echoid-s4232" xml:space="preserve">Itaque pedem cubicum 65 ℔ pondere taxabo.</s> <s xml:id="echoid-s4233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4234" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s4235" xml:space="preserve">* Magnitudinem A ex ſua ponderitate concludere.</s> <s xml:id="echoid-s4236" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div597" type="section" level="1" n="431"> <head xml:id="echoid-head448" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s4237" xml:space="preserve">Ponderato aquam A, ſitq́ue 5 ℔, quæ per 65 ℔ diviſæ efficiunt 1 {1/3} pedis <lb/>cubici pro quæſita magnitudine corporis A.</s> <s xml:id="echoid-s4238" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div598" type="section" level="1" n="432"> <head xml:id="echoid-head449" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4239" xml:space="preserve">Cum enim A ſit 5 ℔, pes autem aquæ ejuſdem 65 ℔ pendeat, <lb/> <anchor type="figure" xlink:label="fig-527.01.140-02a" xlink:href="fig-527.01.140-02"/> ipſaq́ue per 2 poſtul. </s> <s xml:id="echoid-s4240" xml:space="preserve">ponderitatis ſit homogeneæ, ratio ponderi-<lb/>tatis ſuæ ad 65 ℔ eadem erit, quæ magnitudinis ad pedem cubi-<lb/>cum; </s> <s xml:id="echoid-s4241" xml:space="preserve">atqui 5 ℔ & </s> <s xml:id="echoid-s4242" xml:space="preserve">65 ℔ eſt ſubtredecupla, itaque etiam magnitu-<lb/>dine æquatur 1 {1/3} pedis. </s> <s xml:id="echoid-s4243" xml:space="preserve">quod demonſtraſſe oportebat.</s> <s xml:id="echoid-s4244" xml:space="preserve"/> </p> <div xml:id="echoid-div598" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.140-02" xlink:href="fig-527.01.140-02a"> <image file="527.01.140-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.140-02"/> </figure> </div> <p> <s xml:id="echoid-s4245" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s4246" xml:space="preserve">* Quare ex aquæ gravitate nota, licet ignota <lb/>magnitudine; </s> <s xml:id="echoid-s4247" xml:space="preserve">ipſam magnitudinem eruimus.</s> <s xml:id="echoid-s4248" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div600" type="section" level="1" n="433"> <head xml:id="echoid-head450" xml:space="preserve">9 PROBLEMA. 22 PROPOSITIO.</head> <p> <s xml:id="echoid-s4249" xml:space="preserve">Datis duorum corporum magnitudinis & </s> <s xml:id="echoid-s4250" xml:space="preserve">materiæ gra-<lb/>vitatis rationibus inter ſe, cum pondere alterius, reliqui <lb/>quoque pondus invenire.</s> <s xml:id="echoid-s4251" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4252" xml:space="preserve">D*ATVM*. </s> <s xml:id="echoid-s4253" xml:space="preserve">Exponantur corpora A B & </s> <s xml:id="echoid-s4254" xml:space="preserve">C, ſitq́ue A B ad C ut 3 ad 1, <lb/>materiæ autem gravitas ut 1 ad 2, atque A B librarum 6.</s> <s xml:id="echoid-s4255" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4256" xml:space="preserve">Q*VAESITVM*. </s> <s xml:id="echoid-s4257" xml:space="preserve">Corporis C pondus invenire.</s> <s xml:id="echoid-s4258" xml:space="preserve"/> </p> <pb o="141" file="527.01.141" n="141" rhead="*DE* H*YDROSTATICES ELEMENTIS*."/> </div> <div xml:id="echoid-div601" type="section" level="1" n="434"> <head xml:id="echoid-head451" xml:space="preserve">CONSTRVCTIO.</head> <p> <s xml:id="echoid-s4259" xml:space="preserve">Deſignato D B æquale ipſi C, cum igitur D B triens ſit totius A B 6 ℔, <lb/>ipſum D B erit 2 ℔: </s> <s xml:id="echoid-s4260" xml:space="preserve">ſed gravitas materiæ D B ad C, ut 1 ad 2; </s> <s xml:id="echoid-s4261" xml:space="preserve">quare C pen-<lb/>det 4 ℔.</s> <s xml:id="echoid-s4262" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div602" type="section" level="1" n="435"> <head xml:id="echoid-head452" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4263" xml:space="preserve">Etenim ſi C majoris eſſet ponderis quam 4 ℔, gravi-<lb/>tas ejus ad C quæ eſt 2 ℔ (nam C five D B æquantur <lb/> <anchor type="figure" xlink:label="fig-527.01.141-01a" xlink:href="fig-527.01.141-01"/> tertiæ parti A B) erit majore ratione quam dupla; </s> <s xml:id="echoid-s4264" xml:space="preserve">quod <lb/>tamen theſi repugnat. </s> <s xml:id="echoid-s4265" xml:space="preserve">quare C non eſt majus eſt 4 ℔. <lb/></s> <s xml:id="echoid-s4266" xml:space="preserve">Sed neque minus eſſe eadem ratione concludes. </s> <s xml:id="echoid-s4267" xml:space="preserve">Itaque <lb/>ipſis 4 ℔ æquale. </s> <s xml:id="echoid-s4268" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s4269" xml:space="preserve">Datis itaque duo-<lb/>rum corporum magnitudinis & </s> <s xml:id="echoid-s4270" xml:space="preserve">ſoliditatis rationibus, cum pondere alterius; </s> <s xml:id="echoid-s4271" xml:space="preserve"><lb/>reliqui corporis pondus, ut petcbatur, invenimus.</s> <s xml:id="echoid-s4272" xml:space="preserve"/> </p> <div xml:id="echoid-div602" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.141-01" xlink:href="fig-527.01.141-01a"> <image file="527.01.141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.141-01"/> </figure> </div> </div> <div xml:id="echoid-div604" type="section" level="1" n="436"> <head xml:id="echoid-head453" xml:space="preserve">C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s4273" xml:space="preserve">Ex his liquet,</s> </p> <p style="it"> <s xml:id="echoid-s4274" xml:space="preserve">Magnitudinis ratione ſublatâ à ratione ponderis, relinqui materiæ gravitatis ra-<lb/>tionem.</s> <s xml:id="echoid-s4275" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4276" xml:space="preserve">Et, Materiæ gravitatis ratione ſublatâ à ratione ponderis, relinqui magnitudinis <lb/>rationem.</s> <s xml:id="echoid-s4277" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4278" xml:space="preserve">Et, Materiæ gravitatis ratione addita ad rationem magnitudinis binc ponderis ra-<lb/>tionem existere.</s> <s xml:id="echoid-s4279" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4280" xml:space="preserve">EX quibus perſpicitur, datis quinque harum rationum terminis ſextum con-<lb/>ſtanti ratione inveniri. </s> <s xml:id="echoid-s4281" xml:space="preserve">In exemplo A 6 ℔ eſto, <lb/> <anchor type="figure" xlink:label="fig-527.01.141-02a" xlink:href="fig-527.01.141-02"/> magnitudine 5 pedũ; </s> <s xml:id="echoid-s4282" xml:space="preserve">ponduſq́ue alterius corporis B <lb/>ignoretur, aſt magnitudine eſto 2 pedũ, ponderitatis <lb/>autem materiæ A ad B ratio, ut 4 ad 7. </s> <s xml:id="echoid-s4283" xml:space="preserve">Iam ad in-<lb/>ventionem ignorati ponderis B, addes rationem materiæ ponderitatis, nempe <lb/>{4/7} ad rationem magnitudinis {5/2} unde oritur ratio ponderis {10/7}, pondus igitur A <lb/>ad B eſt ut 10 ad 7. </s> <s xml:id="echoid-s4284" xml:space="preserve">Itaque quia A pendet 6 ℔, concludes ut 10 ad 7 ſic 6 ℔ ad <lb/>pondus B 4 {1/5} ℔.</s> <s xml:id="echoid-s4285" xml:space="preserve"/> </p> <div xml:id="echoid-div604" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.141-02" xlink:href="fig-527.01.141-02a"> <image file="527.01.141-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.141-02"/> </figure> </div> <p> <s xml:id="echoid-s4286" xml:space="preserve">SEcundò ignoretur magnitudo B, cujus inventio è cæteris quinque terminis <lb/>inveſtiganda. </s> <s xml:id="echoid-s4287" xml:space="preserve">Deducito materiæ ponderitatis rationem {4/7}, de ponderis ra-<lb/>tione {10/7}, relinquetur magnitudinis ratio {5/2}. </s> <s xml:id="echoid-s4288" xml:space="preserve">Itaque magnitudo A eſt ad B ut 5 <lb/>ad 2, atqui A eſt 5 pedum, unde concludes etiam B 2 eſſe pedum.</s> <s xml:id="echoid-s4289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4290" xml:space="preserve">DEnique ignoretur materiæ gravitatis ratio, quę è cognitis reliquis duabus <lb/>rationibus ſit eruenda. </s> <s xml:id="echoid-s4291" xml:space="preserve">Subducito magnitudinis rationem {5/2} de ratione <lb/>ponderis {11<unsure/>/7}, reliqua erit materiæ gravitatis ratio 4 ad 7.</s> <s xml:id="echoid-s4292" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4293" xml:space="preserve">Quamvis propoſitio iſta & </s> <s xml:id="echoid-s4294" xml:space="preserve">antecedens omni materiæ homogeneæ genera-<lb/>lis ſint, maximus tamen uſus circa aquea Zetemata verſari videtur. </s> <s xml:id="echoid-s4295" xml:space="preserve">Atque ita <lb/>quarti libri</s> </p> </div> <div xml:id="echoid-div606" type="section" level="1" n="437"> <head xml:id="echoid-head454" xml:space="preserve">FINIS ESTO.</head> <pb file="527.01.142" n="142"/> <pb file="527.01.143" n="143"/> </div> <div xml:id="echoid-div607" type="section" level="1" n="438"> <head xml:id="echoid-head455" xml:space="preserve">LIBER QVINTVS</head> <head xml:id="echoid-head456" xml:space="preserve">STATICAE <lb/>DE <lb/>INITIIS PRAXIS <lb/>HYDROSTATICES.</head> <pb o="144" file="527.01.144" n="144" rhead="5 L*IBER* S*TATICÆ*"/> </div> <div xml:id="echoid-div608" type="section" level="1" n="439"> <head xml:id="echoid-head457" xml:space="preserve">AD LECTOREM.</head> <p style="it"> <s xml:id="echoid-s4296" xml:space="preserve">EX*PLICATIS* Hydrostatices Elementis, Praxis <lb/>ejus bunc ſibi depoſcit locum, vel ſaltem illa quæ <lb/>ſuper hâc nobis ſunt cognita; </s> <s xml:id="echoid-s4297" xml:space="preserve">quæ tamen nondum <lb/>ſcripto viſum fuit publicare, aſt pragmatice ſolùm <lb/>exercere: </s> <s xml:id="echoid-s4298" xml:space="preserve">tribus his propoſitionibus exceptis, quas <lb/>modò in lucem damus, cum{q́ue} ex antecedentibus tanquam con-<lb/>ſectaria quædam deducantur Hydrostatices praxis nomine <lb/>judicamus indignas, ſed quia ipſiaſsident istac parte opusboc <lb/>non perfectum, ſed tenui orſu aff ectum exbibemus, quem ami-<lb/>ce Lector æquiboni{q́ue} conſulas, cæteraſuo tempore cumfænere <lb/>recepturus.</s> <s xml:id="echoid-s4299" xml:space="preserve"/> </p> <pb o="145" file="527.01.145" n="145" rhead="5 L*IBER* S*TAT. DE INITIIS PRAXIS* H*YDROSTAT*."/> <p> <s xml:id="echoid-s4300" xml:space="preserve">QVomodo navis rerumq́ue quas vehit, aut cujuſlibet ſolidi aquæ innatan-<lb/>tis pondus, è data parte demerſa inveniri & </s> <s xml:id="echoid-s4301" xml:space="preserve">concludi queat, jam ſupra 6 <lb/>propoſitione apertè ſatis ediſſerui; </s> <s xml:id="echoid-s4302" xml:space="preserve">his igitur omiſſis paucula quædam 7 propo-<lb/>ſitioni cognata & </s> <s xml:id="echoid-s4303" xml:space="preserve">ab ea dependentia etiam hic proponemus.</s> <s xml:id="echoid-s4304" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div609" type="section" level="1" n="440"> <head xml:id="echoid-head458" xml:space="preserve">1 PROPOSITIO.</head> <p> <s xml:id="echoid-s4305" xml:space="preserve">Invenire quantò idem corpus materiæ levioris quàm <lb/>aqua, in hacaltiùs demergatur, quàm in illa.</s> <s xml:id="echoid-s4306" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4307" xml:space="preserve">Verbigratia, Quæritur quantò altiùs navis Lugodini Batavorum in Rheno <lb/>flumine immergatur quam in mari propè <anchor type="note" xlink:href="" symbol="*"/> Cattorum vicum haud lõgè iſtinc.</s> <s xml:id="echoid-s4308" xml:space="preserve"> <anchor type="note" xlink:label="note-527.01.145-01a" xlink:href="note-527.01.145-01"/> Primum aquæ utriuſque, Rhenanæ ſcilicet & </s> <s xml:id="echoid-s4309" xml:space="preserve">marinæ, materiæ gravitatem in-<lb/>quirito, ea erit ut 42 ad 43, ita enim menſe Sextili experientia approbante edo-<lb/>ctus ſum. </s> <s xml:id="echoid-s4310" xml:space="preserve">Sumptis enim duobus corporibus parili magnitudine, Rhenanum <lb/>pendebat ſcruptula 4260, marinum autem 4362, quorum ratio in minoribus <lb/>terminis, ut 42 ad 43 ſatis vicina eſt.</s> <s xml:id="echoid-s4311" xml:space="preserve"/> </p> <div xml:id="echoid-div609" type="float" level="2" n="1"> <note symbol="*" position="right" xlink:label="note-527.01.145-01" xlink:href="note-527.01.145-01a" xml:space="preserve">@ att<gap/></note> </div> <p> <s xml:id="echoid-s4312" xml:space="preserve">Vnde concludes partem in Rheno demerſam, ad eam quæ in mari juxta <lb/>Catto-vicum mergeretur eſſe, ut 43 ad 42, hinc Geometra pro datæ navis for-<lb/>mâ rationem altitudinis hujus ad illam judicabit. </s> <s xml:id="echoid-s4313" xml:space="preserve">Cujus concluſionis neceſſitas <lb/>è 7 propoſ. </s> <s xml:id="echoid-s4314" xml:space="preserve">hydroſtat. </s> <s xml:id="echoid-s4315" xml:space="preserve">perſpicitur.</s> <s xml:id="echoid-s4316" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div611" type="section" level="1" n="441"> <head xml:id="echoid-head459" xml:space="preserve">2 PROPOSITIO.</head> <p> <s xml:id="echoid-s4317" xml:space="preserve">Exemplis pragmaticis 10 propoſitionis Hydroſtatices <lb/>veritatem comprobare.</s> <s xml:id="echoid-s4318" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4319" xml:space="preserve">Quinto conſectario 10 propoſ. </s> <s xml:id="echoid-s4320" xml:space="preserve">Hydroſtat. </s> <s xml:id="echoid-s4321" xml:space="preserve">mathematicè demonſtratum no-<lb/>bis eſt, fundum aquæ characteribus E F iſtic inſignitum, aqua copioſiore, pau-<lb/>cioreq́ue in eadem altitudine perindeaffici. </s> <s xml:id="echoid-s4322" xml:space="preserve">Quia tamen non nemo hoc à vero <lb/>alienum & </s> <s xml:id="echoid-s4323" xml:space="preserve">naturæ contrarium ſuſpicari poſſet, Mathematicam illam demon-<lb/>ſtrationem quinque exemplis, cujus experientia cuilibet in procinctuſit, dein-<lb/>ceps ſumus illuſtraturi.</s> <s xml:id="echoid-s4324" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div612" type="section" level="1" n="442"> <head xml:id="echoid-head460" xml:space="preserve">1 Exemplum.</head> <p> <s xml:id="echoid-s4325" xml:space="preserve">Fundum A B fundo C D ſimilis eſto & </s> <s xml:id="echoid-s4326" xml:space="preserve">æqualis, itemq́ue altitudo E F al-<lb/>titudini G H; </s> <s xml:id="echoid-s4327" xml:space="preserve">ſed pars I E inſiſtens ſujectæ aquæ K L B A minor ſit, quam pars <lb/>ipſius G C D ſibi reſpondens, pendatq́ue aqua E A B <lb/> <anchor type="figure" xlink:label="fig-527.01.145-01a" xlink:href="fig-527.01.145-01"/> 1 ℔, G C D 10 ℔, ſitq́ue G C D cylindrus, is igitur <lb/>ipſius E A B erit decuplus, hujus tamen in fundum <lb/>A B impreſſum eſſe tantum aſſerimus, quantus ſit to-<lb/>tius G C D in fundum C D. </s> <s xml:id="echoid-s4328" xml:space="preserve">Quod reapſe pragma-<lb/>tica machinatione ita demonſtratur.</s> <s xml:id="echoid-s4329" xml:space="preserve"/> </p> <div xml:id="echoid-div612" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.145-01" xlink:href="fig-527.01.145-01a"> <image file="527.01.145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.145-01"/> </figure> </div> <pb o="146" file="527.01.146" n="146" rhead="5 L*IBER* S*TATICÆ*"/> <p> <s xml:id="echoid-s4330" xml:space="preserve">M N O libra eſto, cujus lances M, O; <lb/></s> <s xml:id="echoid-s4331" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.146-01a" xlink:href="fig-527.01.146-01"/> atque M quidem figuræ cylindraceæ æ-<lb/>qualis expoſito G C D ideoq́ue 10 li-<lb/>brarum aquæ capax; </s> <s xml:id="echoid-s4332" xml:space="preserve">tum Pſolidum ſimi-<lb/>le lanci M & </s> <s xml:id="echoid-s4333" xml:space="preserve">minus, ſcapo affigatur uthic <lb/>vides.</s> <s xml:id="echoid-s4334" xml:space="preserve"/> </p> <div xml:id="echoid-div613" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.146-01" xlink:href="fig-527.01.146-01a"> <image file="527.01.146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.146-01"/> </figure> </div> <p> <s xml:id="echoid-s4335" xml:space="preserve">Inſeratur igitur ſolidum P in lancem <lb/>M, ut in ſecunda figura, lanciq́ue O im-<lb/>ponatur pondus Q 10 ℔; </s> <s xml:id="echoid-s4336" xml:space="preserve">jam fundum <lb/>M tam validè impingetin corpus P quàm <lb/>à 10 ℔ impelletur. </s> <s xml:id="echoid-s4337" xml:space="preserve">ſit autĕ corpus P deci-<lb/>ma parte minus quam M, ut vacuus inter <lb/>utrumque locus 1 ℔ aquæ expleatur, hoc <lb/>eſtaquea mole æquante corpus E A B. </s> <s xml:id="echoid-s4338" xml:space="preserve">Itaque 1 ℔ aquæ in lancem infuſa hanc <lb/>deprimet, reliquamq́ue attollet, id ipſum teſtante experientia, & </s> <s xml:id="echoid-s4339" xml:space="preserve">10 propoſi-<lb/>tionis demonſtratione approbante. </s> <s xml:id="echoid-s4340" xml:space="preserve">Quare 1 ℔ aquæ in lance M iſtic tantæ erit <lb/>potentiæ, quàm 10 ℔ plumbiferrivé aut alterius materiæ ſolidæ eidem lanci M <lb/>affixæ. </s> <s xml:id="echoid-s4341" xml:space="preserve">Atque eadem ratione 1 ℔ aquæ, hujuſmodi partium diſpoſitione ma-<lb/>joris erit efficaciæ, quam millæ libræ materiæ alterius. </s> <s xml:id="echoid-s4342" xml:space="preserve">Quæ cum ita ſint, aqua <lb/>quæ inter utriuſque fundum, corporis P lancisq́ue M interceſſit, fundum M <lb/>nunc tam validè preſſat ac prius fun dum corporis P, hoc eſt, ac 10 ℔; </s> <s xml:id="echoid-s4343" xml:space="preserve">cùm pon-<lb/> <anchor type="figure" xlink:label="fig-527.01.146-02a" xlink:href="fig-527.01.146-02"/> dus Q 10 ℔ in reliqua lance O immiſſum ſit. <lb/></s> <s xml:id="echoid-s4344" xml:space="preserve">Itemq́ue contra aqua tanta vehementia premit <lb/>fundum lancis M, quanta eſt efficientia 10 ℔ <lb/>Q. </s> <s xml:id="echoid-s4345" xml:space="preserve">ponamus autem aquam in fundo M æqua-<lb/>riipſi K L B A, reliquam autem ipſi P circum-<lb/>fuſam, reliquæ I E. </s> <s xml:id="echoid-s4346" xml:space="preserve">Quare aqua E A B tam <lb/>potenter preſiat fundum A B, quàm hæc aqua <lb/>fundum M, ideoq́ue E A B premit ſuum fun-<lb/>dum A B æquivalenter 10 ℔, ſed tantus eſt <lb/>item preſius aquæ G C D contra fundum <lb/>C D. </s> <s xml:id="echoid-s4347" xml:space="preserve">Quamobrem, quod pragmaticè con-<lb/>firmare ſtatueramus, aqua E A B pondere 1 ℔ <lb/>ſuum fundum A B æquè validè premit, atque <lb/>G C D 10 ℔ fundum C D. </s> <s xml:id="echoid-s4348" xml:space="preserve">Pari ratione evin-<lb/>ces, vel 1 ℔ preſſare potentiùs mille libris.</s> <s xml:id="echoid-s4349" xml:space="preserve"/> </p> <div xml:id="echoid-div614" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.146-02" xlink:href="fig-527.01.146-02a"> <image file="527.01.146-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.146-02"/> </figure> </div> </div> <div xml:id="echoid-div616" type="section" level="1" n="443"> <head xml:id="echoid-head461" xml:space="preserve">2 Exemplum.</head> <p> <s xml:id="echoid-s4350" xml:space="preserve">Tubulus eſto A B C D, & </s> <s xml:id="echoid-s4351" xml:space="preserve">C D E F vas amplum ac ſpiſſum, utraque aquæ <lb/>plena, ſuperficiebus in eadem mundana ſuperficie conſi-<lb/> <anchor type="figure" xlink:label="fig-527.01.146-03a" xlink:href="fig-527.01.146-03"/> ſtentibus, inſidentia communi fundo C D. </s> <s xml:id="echoid-s4352" xml:space="preserve">hîc fundum <lb/>C D ab amplo vaſe C D E F non validius premi quam à <lb/>tubulo A B C D, patebit ipſo ablato, ut aqua aquam <lb/>tangat; </s> <s xml:id="echoid-s4353" xml:space="preserve">nam ſi ante aqua C D E F fundum D C preſſit va-<lb/>lidiùs quàm A B C D, idem quoque nunc fiet, & </s> <s xml:id="echoid-s4354" xml:space="preserve">poten-<lb/>tior debiliorem loco pellet, quare aquam A B C D aſcen-<lb/>dere, & </s> <s xml:id="echoid-s4355" xml:space="preserve">C D E F deſcendere neceſſe fuerit: </s> <s xml:id="echoid-s4356" xml:space="preserve">atque ita ipſa-<lb/>rum ſupernæ ſuperficies inæquali altitudine ſupra horizontem extarent, quod <pb o="147" file="527.01.147" n="147" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> experientiæ manifeſtè repugnat. </s> <s xml:id="echoid-s4357" xml:space="preserve">Quamobrem minor aquæ copia A B C D <lb/>premit fundum C D æquivalenter majori C D E F.</s> <s xml:id="echoid-s4358" xml:space="preserve"/> </p> <div xml:id="echoid-div616" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.146-03" xlink:href="fig-527.01.146-03a"> <image file="527.01.146-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.146-03"/> </figure> </div> </div> <div xml:id="echoid-div618" type="section" level="1" n="444"> <head xml:id="echoid-head462" xml:space="preserve">3 Exemplum.</head> <p> <s xml:id="echoid-s4359" xml:space="preserve">Vas A B C D aquæ plenum, cujus fundum D C horizonti parallelum ro-<lb/>tundo foramine pertundatur, quod ligneus tegat orbis G H materie quàm <lb/>aqua levior. </s> <s xml:id="echoid-s4360" xml:space="preserve">Exponatur deinde vas alterum I K L ſuperiori quidem æquealtum, <lb/>ſed minus & </s> <s xml:id="echoid-s4361" xml:space="preserve">aquæ item plenum, cujus fundũ ad M N perforatum æqualiter an-<lb/>tecedenti E F, & </s> <s xml:id="echoid-s4362" xml:space="preserve">orbe quoque O P ipſi G H æquali obtegatur. </s> <s xml:id="echoid-s4363" xml:space="preserve">Quibus poſitis, <lb/>orbis G H contra communem ligni naturam vimq́ue ingenitam ex aqua non <lb/>emerget, ſed foramini E F incumbens tam valenter premet quàm columna <lb/>aquea E F Q R multata differentiâ ponde-<lb/> <anchor type="figure" xlink:label="fig-527.01.147-01a" xlink:href="fig-527.01.147-01"/> rũ lignei orbis G H & </s> <s xml:id="echoid-s4364" xml:space="preserve">aquæ ipſi æqualis. </s> <s xml:id="echoid-s4365" xml:space="preserve">Et <lb/>quî experimento hoc cognoſcas, orbi G H <lb/>libram affigito, cujus pondus S ponderi di-<lb/>cto æquale ſit, eritq́ue orbis G H ipſi æqui-<lb/>libris. </s> <s xml:id="echoid-s4366" xml:space="preserve">Similiter firmato ad orbem O P li-<lb/>bram, cujus pondus T ſuperiori S æque-<lb/>ponderet, orbisq́ue hic O P ponderi T ma-<lb/>nebit æquilibris. </s> <s xml:id="echoid-s4367" xml:space="preserve">auctis autem ponderibus <lb/>S, T, orbes G H, O P attollentur, atque <lb/>adeò hâc viâ deprehĕdes orbes iſtos in fun-<lb/>da ſubjecta æqualem impreſsionem facere; </s> <s xml:id="echoid-s4368" xml:space="preserve">unde propoſiti veritas perſpicitur <lb/>videlicet minorem aquæ molem I K L tam validè quàm majorem A B C D <lb/>premere fundum ſibi ſubjectum.</s> <s xml:id="echoid-s4369" xml:space="preserve"/> </p> <div xml:id="echoid-div618" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.147-01" xlink:href="fig-527.01.147-01a"> <image file="527.01.147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.147-01"/> </figure> </div> </div> <div xml:id="echoid-div620" type="section" level="1" n="445"> <head xml:id="echoid-head463" xml:space="preserve">NOTATO.</head> <p> <s xml:id="echoid-s4370" xml:space="preserve">Si differentia ponderis orbis G H à pondere ãquæ ſibi æqualis, excederet <lb/>columnam aqueam E F Q R, orbem G H iſtic non hæſurum ſed à foramine <lb/>ſurſum emerſurum.</s> <s xml:id="echoid-s4371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4372" xml:space="preserve">Præterea ſi diſcus G H eſſet è plumbo, fervovè, aut alia materiâ graviore <lb/>quam aqua formatus, ejus in ſubjectum foramen impreſsionem fore tantam, <lb/>quanta ſit columnæ aqueę E F Q R auctæ differentia ponderis, quę inter H or-<lb/>bem dictum & </s> <s xml:id="echoid-s4373" xml:space="preserve">aquæ molem ſibi æqualem intercedit.</s> <s xml:id="echoid-s4374" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4375" xml:space="preserve">Denique ſi G H materiâ eſſet aquæ æquilibri, impreſſionem ejus in fora-<lb/>men E F, columnæ aqueæ E F Q R, æqualem fore.</s> <s xml:id="echoid-s4376" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div621" type="section" level="1" n="446"> <head xml:id="echoid-head464" xml:space="preserve">4 Exemplum.</head> <p> <s xml:id="echoid-s4377" xml:space="preserve">Eſto A B C D vas aquę plenum, cujus fundum C D pertuſum ſit ſpatio E F <lb/>atque ipſi incumbat orbis diſcusve materię levioris quam <lb/> <anchor type="figure" xlink:label="fig-527.01.147-02a" xlink:href="fig-527.01.147-02"/> aqua, is tantam impreſſionem faciet in foramen E F quan-<lb/>tam ſupra oſten dimus: </s> <s xml:id="echoid-s4378" xml:space="preserve">exponatur item tubulus I K L cu-<lb/>jus ſummum foramen I eâdem ſit altitudine cum A B, <lb/>imum eſto E F. </s> <s xml:id="echoid-s4379" xml:space="preserve">Canaliculus hic aqua oppletus tam vali-<lb/>dè parte inferna preſſabit orbem G H quam univerſa aqua <lb/>A B C D ipſi inſidens parte oppoſita, quia orbis G H ad-<lb/>ſcendet. </s> <s xml:id="echoid-s4380" xml:space="preserve">Atque adeò 1 ℔ aquę (tantæ enim aquæ capacem <lb/>fingo canaliculum) in orbe G H majorem efficientiam exerere poterit quam <pb o="148" file="527.01.148" n="148" rhead="5 L*IBER* S*TATICÆ*"/> centies millenę libræ, cujuſmodi in ſuperiore figura S: </s> <s xml:id="echoid-s4381" xml:space="preserve">cujus ſi cauſa ignota <lb/>eſſet, naturæ arcanum meritiſſimò dici poſſet.</s> <s xml:id="echoid-s4382" xml:space="preserve"/> </p> <div xml:id="echoid-div621" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.147-02" xlink:href="fig-527.01.147-02a"> <image file="527.01.147-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.147-02"/> </figure> </div> </div> <div xml:id="echoid-div623" type="section" level="1" n="447"> <head xml:id="echoid-head465" xml:space="preserve">5 Exemplum.</head> <figure> <image file="527.01.148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.148-01"/> </figure> <p> <s xml:id="echoid-s4383" xml:space="preserve">Denique quî etiam pragmaticè exempla 3 conſ. </s> <s xml:id="echoid-s4384" xml:space="preserve">10 <lb/>propoſ. </s> <s xml:id="echoid-s4385" xml:space="preserve">ubi aqua fundum premit inſernâ parte, com-<lb/>probemus; </s> <s xml:id="echoid-s4386" xml:space="preserve">exponatur aqua A B C D, & </s> <s xml:id="echoid-s4387" xml:space="preserve">tubus E F, <lb/>G orbis plum beus materię põderioſioris quàm aqua, <lb/>ut vides in priore diagrammate.</s> <s xml:id="echoid-s4388" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4389" xml:space="preserve">Is orbis foramini F fubditus accuratè cõgruat, jam tubo <lb/> <anchor type="figure" xlink:label="fig-527.01.148-02a" xlink:href="fig-527.01.148-02"/> cum orbe in aquam immiſſo, orbis G ſecundum naturam <lb/>plumbo ingenitam non mergetur, ſed tubo affixus hære-<lb/>bit, ac tam valide aquam premet, quàm pondus aqueæ co-<lb/>lumnæ cujus fundum F, altitudo H I, multatum differentia <lb/>ponderum orbis G atque aquæ ſibi æqualis. </s> <s xml:id="echoid-s4390" xml:space="preserve">Sin verò orbis <lb/>G non ſatis arctè claudat tubum ut aqua ingrediatur, non <lb/>ante G à tubo recedet, quam dictum pondus ab aqua in tu-<lb/>bum admiſſa ſuperetur.</s> <s xml:id="echoid-s4391" xml:space="preserve"/> </p> <div xml:id="echoid-div623" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.148-02" xlink:href="fig-527.01.148-02a"> <image file="527.01.148-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.148-02"/> </figure> </div> <p> <s xml:id="echoid-s4392" xml:space="preserve">Et ne quis arbitretur magnam aquę molem quâ cir-<lb/> <anchor type="figure" xlink:label="fig-527.01.148-03a" xlink:href="fig-527.01.148-03"/> cumquaque tubus cingitur, majori preſſu tubum af-<lb/>ficere, quàm minorem ejuſdem altitudinis. </s> <s xml:id="echoid-s4393" xml:space="preserve">tollatur <lb/>omnis aqua circumfuſa ut reliqua duntaxat ſit quan-<lb/>tulam in ſubjecto diagrammate vides, nihilominus ta-<lb/>men cognoſces aquam minorem (exploratâ preſſio-<lb/>nis potentiâ tubo G modo in hoc, modo in illud vas <lb/>impoſito) tam potenter tubum premere, quàm iſtam <lb/>majorem. </s> <s xml:id="echoid-s4394" xml:space="preserve">Cujus cauſa ſupra accuratè expoſita eſt.</s> <s xml:id="echoid-s4395" xml:space="preserve"/> </p> <div xml:id="echoid-div624" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.148-03" xlink:href="fig-527.01.148-03a"> <image file="527.01.148-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.148-03"/> </figure> </div> <p> <s xml:id="echoid-s4396" xml:space="preserve">C*ONCLVSIO.</s> <s xml:id="echoid-s4397" xml:space="preserve">* Itaque exemplis 10 propoſitionem Elementorum Hy-<lb/>droſtatices pragmaticè illuſtravimus.</s> <s xml:id="echoid-s4398" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div626" type="section" level="1" n="448"> <head xml:id="echoid-head466" xml:space="preserve">NOTATO.</head> <p style="it"> <s xml:id="echoid-s4399" xml:space="preserve">Ex II propoſitione præter cætera etiam hoc cognoſci poteſt quantum aquæ pondus <lb/>emiſſariorum cataractis, & </s> <s xml:id="echoid-s4400" xml:space="preserve">ſimilibus incumbat. </s> <s xml:id="echoid-s4401" xml:space="preserve">Præterea aquam vel unguis unius la-<lb/>titudine ab una parte æquè potenter premere, atque altrinſecus alteram cuius latitudo <lb/>vel ipſi Oceano æqualis ſit, modo in eâdem altitudine conſistant. </s> <s xml:id="echoid-s4402" xml:space="preserve">Quæ cum per ſe ſint <lb/>ſatis manifeſta nullis exemplis ea declaramus.</s> <s xml:id="echoid-s4403" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div627" type="section" level="1" n="449"> <head xml:id="echoid-head467" xml:space="preserve">3 PROPOSITIO.</head> <p> <s xml:id="echoid-s4404" xml:space="preserve">Cauſam reddere cur homo altè infra aquam natans <lb/>maximo ejus pondere non opprimatur.</s> <s xml:id="echoid-s4405" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4406" xml:space="preserve">Humani corporis planum occupet pedes 10, is infra aquam 20 pedes de-<lb/>merſus, aquę pede cubico 65 ℔ æſtimato, ſuſtinebit per 10 & </s> <s xml:id="echoid-s4407" xml:space="preserve">11 prop. </s> <s xml:id="echoid-s4408" xml:space="preserve">de Elem. <lb/></s> <s xml:id="echoid-s4409" xml:space="preserve">Hydroſtat. </s> <s xml:id="echoid-s4410" xml:space="preserve">13000 ℔. </s> <s xml:id="echoid-s4411" xml:space="preserve">Quamobrem qui poteſt, ut tanto pondere preſſus non <lb/>opprimatur;</s> <s xml:id="echoid-s4412" xml:space="preserve"><unsure/> Cauſa autem hæceſt:</s> <s xml:id="echoid-s4413" xml:space="preserve"/> </p> <pb o="149" file="527.01.149" n="149" rhead="*DE* H*YDROSTATICES ELEMENTIS.*"/> <p style="it"> <s xml:id="echoid-s4414" xml:space="preserve">Omni preſſu quo corpus dolore afficitur, pars aliqua corporis luxatur.</s> <s xml:id="echoid-s4415" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4416" xml:space="preserve">Sed isto preſſu nulla corporis pars luxatur.</s> <s xml:id="echoid-s4417" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4418" xml:space="preserve">Iſto igitur preſſu corpus dolore nullo afficitur.</s> <s xml:id="echoid-s4419" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4420" xml:space="preserve">Aſſumptio ſyllogiſmi reipſa manifeſta eſt, nam ſi pars aliqua ut caro, ſanguis, <lb/>humor aut quodlibet denique membrum luxaretur, in alium locum concedat <lb/>neceſſe eſſet; </s> <s xml:id="echoid-s4421" xml:space="preserve">atqui locus ille non eſt extra corpus, cum aqua undique æquali <lb/>preſſu circumfuſa ſit (quod verò pars ima per 11 propoſ. </s> <s xml:id="echoid-s4422" xml:space="preserve">Hydroſtat. </s> <s xml:id="echoid-s4423" xml:space="preserve">paulò vali-<lb/>diùs prematur ſuperiore, id hoc caſu nullius momenti eſt, quia tantula differen-<lb/>tia partem nullam ſede ſua dimovere poteſt) neque item intra ipſum corpus <lb/>concedit, cum iſtic corpore omnia oppleta ſint, unde ſingulæ partes ſingulis <lb/>partibus æqualiter reſiſtunt, namq́ue aqua undiquaque eadem ratione corpus <lb/>totum circumſtat. </s> <s xml:id="echoid-s4424" xml:space="preserve">Quare cum locus is nec intra neque extra corpus ſit, abſur-<lb/>dum imò impoſsibile fuerit partem ullam ſuo loco emoveri, ideoq́; </s> <s xml:id="echoid-s4425" xml:space="preserve">nec corpus <lb/>hinc afficietur ullo dolore.</s> <s xml:id="echoid-s4426" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4427" xml:space="preserve">Sed in exemplo clarius ita intelliges, eſto A B C D aqua cujus fundũ D C, <lb/>in quo foramen E habeat epiſtomium ſibi inſertum, cui dorſo incumbat ho-<lb/>mo F. </s> <s xml:id="echoid-s4428" xml:space="preserve">Quæ cum ita ſint, ab aquæ pondere ipſi inſidente nulla pars corporis <lb/>luxari poterit, cùm aqua ut dictum eſt, undique urgeat æqualiter.</s> <s xml:id="echoid-s4429" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4430" xml:space="preserve">Si vero ejus veritatem explorare libeat, eximito epiſto <lb/> <anchor type="figure" xlink:label="fig-527.01.149-01a" xlink:href="fig-527.01.149-01"/> mium E, tumq́ue tergum nulla re fultum ſuſtinebitur, ut in <lb/>locis cæteris; </s> <s xml:id="echoid-s4431" xml:space="preserve">ideoq́ue iſtic tanto preſſu afficietur, quantus <lb/>3 exemplo 2 propoſitionis hujus demonſtratus eſt, vide-<lb/>licet quantam efficit columna aquea cujus baſis ſit foramen <lb/>E altitudo autem eadem quæ aquæ ipſi inſidentis. </s> <s xml:id="echoid-s4432" xml:space="preserve">Quo <lb/>exemplo propoſiti veritas manifeſtò declaratur. </s> <s xml:id="echoid-s4433" xml:space="preserve">Itaque hic <lb/>Quinti Libri</s> </p> <div xml:id="echoid-div627" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.149-01" xlink:href="fig-527.01.149-01a"> <image file="527.01.149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.149-01"/> </figure> </div> </div> <div xml:id="echoid-div629" type="section" level="1" n="450"> <head xml:id="echoid-head468" xml:space="preserve">FINIS ESTO.</head> <pb o="150" file="527.01.150" n="150"/> </div> <div xml:id="echoid-div630" type="section" level="1" n="451"> <head xml:id="echoid-head469" xml:space="preserve">*APPENDIX*</head> <head xml:id="echoid-head470" xml:space="preserve">STATICES, <lb/>VBI INTER ALIA ERRORES <lb/>quidam Στατπκών I’ {δι}ω{μά}των refelluntur.</head> <head xml:id="echoid-head471" xml:space="preserve">AD LECTOREM.</head> <p style="it"> <s xml:id="echoid-s4434" xml:space="preserve">COgitanti mihi & </s> <s xml:id="echoid-s4435" xml:space="preserve">memoria repetenti quantùm <lb/>ſæpenumero contentiones autorum concer ta-<lb/>tiones{q́ue} in diſputando pertinaces diſplicuiſſent, <lb/>cùm animi pervicacia impulſi aliorum erratis <lb/>adeò contemptim inſultarent, ut morum ſuo-<lb/>rum vitia longè deteriora proderent; </s> <s xml:id="echoid-s4436" xml:space="preserve">quamvis amplum cam-<lb/>pum & </s> <s xml:id="echoid-s4437" xml:space="preserve">tanquam Mar at honium ad errores in Staticis idio-<lb/>matis à ſcriptoribus commiſſos refellendum, apertum cerne-<lb/>rem: </s> <s xml:id="echoid-s4438" xml:space="preserve">ſcrupulus tamen ille injectus mihi eſt ne iis detegendis Le-<lb/>ctori in idem vitium, quod in aliis reprehendo, ipſemet impin-<lb/>gere viderer. </s> <s xml:id="echoid-s4439" xml:space="preserve">Contra autem conſider anti taciturnitate nostra <lb/>(nam ſuperioribus libris ſtudiosè cavimus, ne doctrina iſtiuſ-<lb/>modi argumentorum velit atione obſcur aretur) nonnullos in <lb/>errores falſas{q́ue} opiniones incidere poſſe; </s> <s xml:id="echoid-s4440" xml:space="preserve">medium quoddam ge-<lb/>nus ſecutus ſum, ut pro ſingulis variorum erroribus, ſumma <lb/>duntaxat genera, & </s> <s xml:id="echoid-s4441" xml:space="preserve">communes cauſas duobus proximis capi-<lb/>tibus explicarem: </s> <s xml:id="echoid-s4442" xml:space="preserve">non quì adeò celebrium ſcriptorum exiſtima-<lb/>tioni aut famæ quidquam derogem; </s> <s xml:id="echoid-s4443" xml:space="preserve">ſed potius ut eam gratare-<lb/>cordatione augeam, utpote qui ſectatores ſuos ad horum inve-<lb/>ſtigationem commoverint, ſine quibus eſſet, multa egregia & </s> <s xml:id="echoid-s4444" xml:space="preserve"><lb/>ſcitu neceſſaria in occulto abdita laterent.</s> <s xml:id="echoid-s4445" xml:space="preserve"/> </p> <pb o="151" file="527.01.151" n="151" rhead="S*TATICES.*"/> </div> <div xml:id="echoid-div631" type="section" level="1" n="452"> <head xml:id="echoid-head472" xml:space="preserve">C*APVT I.*</head> <head xml:id="echoid-head473" style="it" xml:space="preserve">Cauſam æquilibritatis ſitus non eſſe in circulis ab extre= <lb/>mitatibus radiorum deſcriptis.</head> <p> <s xml:id="echoid-s4446" xml:space="preserve">CVr pondera æqualia in æqualibus radiis ſitu æquiponderent, communi <lb/>notioneſcitur: </s> <s xml:id="echoid-s4447" xml:space="preserve">at non perinde patet cauſa æquamenti ponderum inæ-<lb/>qualium in radiis diſparibus, quique ponderibus ſuis reciprocè propor-<lb/>tionales ſint. </s> <s xml:id="echoid-s4448" xml:space="preserve">hanc veteres circulis decircinatis à radiorum extremitatibus ineſſe <lb/>crediderunt, quemadmodum apud Ariſtotelem in Mechanicis ejusq́ueſectato-<lb/>res videre licet, quod falſum eſſehoc pacto redarguimus.</s> <s xml:id="echoid-s4449" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4450" xml:space="preserve">Quieſcens nullum deſcribit circulum:</s> <s xml:id="echoid-s4451" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4452" xml:space="preserve">Duo ſitu æquilibria quieſcunt:</s> <s xml:id="echoid-s4453" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s4454" xml:space="preserve">Itaque duo ſitu æquilibria nullum deſcribunt circulum.</s> <s xml:id="echoid-s4455" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4456" xml:space="preserve">Et conſequenter nullus erit circulus; </s> <s xml:id="echoid-s4457" xml:space="preserve">atqui ſublato circulo etiam cauſa tolli-<lb/>tur quæ ipſi ſubeſt, quare cauſa æquilibritatis ſitus in circulis hîc non latet. </s> <s xml:id="echoid-s4458" xml:space="preserve">Mo-<lb/>tus autem iſte (ut aſſumptionem noſtri ſyllogiſmi confirmemus) ſivè circulo-<lb/>rum deſcriptio qui hic cernitur, æquiponderantibus propriè non ineſt; </s> <s xml:id="echoid-s4459" xml:space="preserve">ſed <lb/>cafu contingit, ut vento aliove impulſu quo non hæc ſolùm, ſed ἀνιοὀῤῥοπα <lb/>pondera quælibet etiam circulos deſcribent. </s> <s xml:id="echoid-s4460" xml:space="preserve">Quamobrem in his circulis cauſa <lb/>æquilibritatis nulla eſt: </s> <s xml:id="echoid-s4461" xml:space="preserve">ſed iis quæ 1 propoſ. </s> <s xml:id="echoid-s4462" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4463" xml:space="preserve">Elem. </s> <s xml:id="echoid-s4464" xml:space="preserve">Staticor. </s> <s xml:id="echoid-s4465" xml:space="preserve">Mathematicè <lb/>demonſtravimus: </s> <s xml:id="echoid-s4466" xml:space="preserve">minimeq́ue mirum ſi ii, qui errores iſtiuſmodi pro veritate <lb/>uſurpabant, ad cauſarum cognitionem non penetrarint, aut nulla Statices for-<lb/>mâ inventa, à verò adeò diverſi abierint, multis falſis propoſitionibus ſeſe exer-<lb/>centes, quas ſigillatim, propter cauſas ſupra expoſitas, nuncnõ refellimus; </s> <s xml:id="echoid-s4467" xml:space="preserve">atque <lb/>eo magis quod à contraria veritatis norma facillimè coarguantur.</s> <s xml:id="echoid-s4468" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4469" xml:space="preserve">Propoſitiones item Cardani nonnullas lib. </s> <s xml:id="echoid-s4470" xml:space="preserve">5. </s> <s xml:id="echoid-s4471" xml:space="preserve">Proportion. </s> <s xml:id="echoid-s4472" xml:space="preserve">de ponderibus <lb/>obliquatis, quarum judicium è certis quibuſdam angulis, lineis, planisq́ue in-<lb/>ſtituit, falſas & </s> <s xml:id="echoid-s4473" xml:space="preserve">à vero alienas eſſe, ne longior hic ſim, è 19 propoſ. </s> <s xml:id="echoid-s4474" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4475" xml:space="preserve">Static. <lb/></s> <s xml:id="echoid-s4476" xml:space="preserve">facillime perſpicitur.</s> <s xml:id="echoid-s4477" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div632" type="section" level="1" n="453"> <head xml:id="echoid-head474" xml:space="preserve">C*APVT II.*</head> <head xml:id="echoid-head475" style="it" xml:space="preserve">Res motas impedimentis ſuis non eſſe propor-<lb/>tionales.</head> <p> <s xml:id="echoid-s4478" xml:space="preserve">IN Praxis Statices ad Lectorem præfatione diximus res motas ſuis impedi-<lb/>mentis non eſſe proportionales, ejusq́ue demonſtrationi hunc locum de-<lb/>ſtinavimus, ut argumenta aliter ſentientium refutemus. </s> <s xml:id="echoid-s4479" xml:space="preserve">Principiò Ariſtoteles <lb/>ejusq́ue ſectatores 4 Phyſic. </s> <s xml:id="echoid-s4480" xml:space="preserve">cap. </s> <s xml:id="echoid-s4481" xml:space="preserve">de inani exiſtimat corporibus duobus ſimili-<lb/>bus, & </s> <s xml:id="echoid-s4482" xml:space="preserve">materiâ æquipondiis per aërem delapſis eandem eſſe rationem ponde-<lb/>ris ad pondus quæ velocitatis illius ad velocitatem hujus, id eſt quæ ſit impe-<lb/>dimenti ad impedimentum. </s> <s xml:id="echoid-s4483" xml:space="preserve">Quam ſententiam variis locis clarius proponit, ut <lb/>6 Phyſic. </s> <s xml:id="echoid-s4484" xml:space="preserve">item 1, 2, 3 4 de Cælo, aliisq́ue compluribus. </s> <s xml:id="echoid-s4485" xml:space="preserve">ſententiam hanc Ioannes <lb/>Taiſnerus Hannonius oppugnavit, proportionem quidem hactenus admittens <lb/>ut corpora iſta æquali temporis ſpatio æqualia permeent intervallo. </s> <s xml:id="echoid-s4486" xml:space="preserve">Cui opi-<lb/>nioni Cardanus lib. </s> <s xml:id="echoid-s4487" xml:space="preserve">5. </s> <s xml:id="echoid-s4488" xml:space="preserve">Proportion. </s> <s xml:id="echoid-s4489" xml:space="preserve">propoſ. </s> <s xml:id="echoid-s4490" xml:space="preserve">110. </s> <s xml:id="echoid-s4491" xml:space="preserve">conſentit. </s> <s xml:id="echoid-s4492" xml:space="preserve">Sed utroſque hallucinari <lb/>ipſa experientia demonſtrabimus, ac deinde ejus cauſam declarabimus. </s> <s xml:id="echoid-s4493" xml:space="preserve">Expe- <pb o="152" file="527.01.152" n="152" rhead="A*PPENDIX*"/> rientia verò contra Ariſtotelem iſtiuſmodi eſt; </s> <s xml:id="echoid-s4494" xml:space="preserve">ſumito duos plumbeos globos <lb/>(quod Cl. </s> <s xml:id="echoid-s4495" xml:space="preserve">vir I*OANNES* G*ROTIVS* ſedulus naturæ indagator, & </s> <s xml:id="echoid-s4496" xml:space="preserve">ego <lb/>quondam experti ſumus) ponderis ratione decupla, eos altitudine 30 pedum <lb/>pariter demittito in ſubjectum aſſerem, aliudve ſolidum unde ſonus clarè red-<lb/>datur; </s> <s xml:id="echoid-s4497" xml:space="preserve">manifeſtè cognoſces leviorem non decuplo tardius graviore, ſed pariter <lb/>in aſſerem incidere ut ſonitus utriuſque illiſu redditus unus idemq́ue videatur. <lb/></s> <s xml:id="echoid-s4498" xml:space="preserve">Idem contingit in corporibus magnitudinis æqualis, gravitatis verò decuplæ: </s> <s xml:id="echoid-s4499" xml:space="preserve"><lb/>Quare dicta iſta Ariſtotelis proportio à vero aliena eſt. </s> <s xml:id="echoid-s4500" xml:space="preserve">Sed alterum experimen-<lb/>tum hujuſmodi cõtra T aiſnerum facit: </s> <s xml:id="echoid-s4501" xml:space="preserve">Sumito è goſſipio lanavè tenue quoddam <lb/>& </s> <s xml:id="echoid-s4502" xml:space="preserve">exile filum, atq; </s> <s xml:id="echoid-s4503" xml:space="preserve">ſarcinulam ex eadem materia pondere unius libræ densè fir-<lb/>miterq́ue colligatam, & </s> <s xml:id="echoid-s4504" xml:space="preserve">formâ filo ſimili, hęc pariter quinque aut ſex pedum al-<lb/>titudine demittito, re ipſa cognoſces filum longe diutius in aëre morari, quàm <lb/>ſarcinulam, etſi fili materia longe compactior denſiorq́; </s> <s xml:id="echoid-s4505" xml:space="preserve">ſit ſarcinulâ quæ mul-<lb/>tum aëris admittit. </s> <s xml:id="echoid-s4506" xml:space="preserve">Quare æquale ſpacium ab ipſis pari velocitate nõ tranſitur. </s> <s xml:id="echoid-s4507" xml:space="preserve"><lb/>Altera item experientia T aiſnerum redarguit in pondere adſcĕ dente ſive emer-<lb/>gente, in phialâ enim vitreâ aquæ plenâ agitatâ, ut multæ excitentur bullulæ <lb/>ſimulac quievit videbis majores bullas citiſſime atque unico momento, mino-<lb/>res verò emergere tardiùs, minimas autem bullulas inſtar tenuiſſimarum arenu-<lb/>larum lentiſſimè, & </s> <s xml:id="echoid-s4508" xml:space="preserve">tanquam teſtudineo gradu ſurſum prorepere, quarum om-<lb/>nium motus ab æquali velocitate vel tarditate longè abeſt. </s> <s xml:id="echoid-s4509" xml:space="preserve">Atque hactenus de <lb/>experientia. </s> <s xml:id="echoid-s4510" xml:space="preserve">Supereſt ut dicamus cur hîc nulla ſit proportio, hoc modo. </s> <s xml:id="echoid-s4511" xml:space="preserve">Quod-<lb/>libet corpus movens habet quoddam motus ſui impedimentum, quod in cor-<lb/>pore per aërem delato eſt aëris & </s> <s xml:id="echoid-s4512" xml:space="preserve">ſuperficiei ſuæ contactus; </s> <s xml:id="echoid-s4513" xml:space="preserve">ideoq́ue ſimilium <lb/>corporum majus, majore quoque afficitur impedimento, ſed quia ſimilia ſolida <lb/>ſuperficiebus ſuis non ſunt proportionalia (nam cubi in ratione octupla, ha-<lb/>bent ſuperficies ratione quadrupla) nec impedimentis proportionalia eſſe poſ-<lb/>ſunt: </s> <s xml:id="echoid-s4514" xml:space="preserve">atque hinc eſt quod minora corpora majus impedimentum patiantur, ra-<lb/>tione proportionis, & </s> <s xml:id="echoid-s4515" xml:space="preserve">propterea tardiùs deſcendant quam majora.</s> <s xml:id="echoid-s4516" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4517" xml:space="preserve">Imò quamvis ſuperficies corporibus ſuis eſſent proportionales, medium ta-<lb/>men per quod cadunt, quodammodo proportionem hanc evertit, ut in duobus <lb/>corpo ribus altero in aqua innatante, altero mergente animad erti facilè poteſt, <lb/>in quibus impedimenta ſuperficierum quandam inter ſe habent rationem, tem-<lb/>pora verò nullam, ideoq́ue proportionalia non ſunt. </s> <s xml:id="echoid-s4518" xml:space="preserve">ſed dicat aliquis id intelli-<lb/>gi ſolum cæteris paribus, videlicet quando utrumque corpus mergetur. </s> <s xml:id="echoid-s4519" xml:space="preserve">Nego <lb/>tamen in his ullam proportionem conſiſtere. </s> <s xml:id="echoid-s4520" xml:space="preserve">Sumptis enim duobus corporibus <lb/>A, B quarum utrumque in aqua mergatur, ſintq́ue in dicta proportione. </s> <s xml:id="echoid-s4521" xml:space="preserve">his po-<lb/>ſitis, manifeſtum eſt infinita poſſe inveniri corpora inæquali gravitate minori <lb/>quàm B, & </s> <s xml:id="echoid-s4522" xml:space="preserve">quæ in aqua demergantur, paulatimq́ue ita propius accedetur ad <lb/>corpus immerſabile, cujus nulla cum corpore quod mergitur ſit proportio. </s> <s xml:id="echoid-s4523" xml:space="preserve">Sed <lb/>illis eò contimè accedentibus, atque A, B in data proportione conſiſtentibus, <lb/>certè nullum infinitorũ illorum corporum ipſi A comparatum proportionem <lb/>iſtam habebit; </s> <s xml:id="echoid-s4524" xml:space="preserve">quia ſi in his eſſet, certè ad alterum non accederent quod theſi <lb/>cõceſſæ repugnat. </s> <s xml:id="echoid-s4525" xml:space="preserve">Quamobrem medium quoque per quod corpora permeant <lb/>dictam proportionem prohibet.</s> <s xml:id="echoid-s4526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4527" xml:space="preserve">Cumq́ue in mediis ordinatiſſimis & </s> <s xml:id="echoid-s4528" xml:space="preserve">ubique homogeneis nullam motus & </s> <s xml:id="echoid-s4529" xml:space="preserve"><lb/>impedimentorum proportionem ineſſe demonſtraverimus, ubi ſimplex ſuper-<lb/>ficiei cum aëre vel aqua ſit contactus. </s> <s xml:id="echoid-s4530" xml:space="preserve">longè firmiori ratione nulla proportio <lb/>inerit exemplis magis inordinatis materiæq́ue non unius generis ſed variæ, ut <lb/>in machinis partim ligneis partim ferreis cæterisq́ue ſimilibus, namq́ue ibi hoc <pb o="153" file="527.01.153" n="153" rhead="S*TATICES*."/> axungia illud oleo perungitur; </s> <s xml:id="echoid-s4531" xml:space="preserve">aliud humido aëre turgeſcit, aliud erugine cor-<lb/>ripitur, quæ omnia (ut multa alia omittam) machinarum motus modò expe-<lb/>diunt, modò impediunt. </s> <s xml:id="echoid-s4532" xml:space="preserve">Itaque ut in Statices Praxis præfatione dictum eſt, hu-<lb/>juſmodi proportioni, quæ probabilis videtur, nullo modo fidendum eſt: </s> <s xml:id="echoid-s4533" xml:space="preserve">quin <lb/>omnia iſta quæ Cardanus 5 Proport. </s> <s xml:id="echoid-s4534" xml:space="preserve">lib. </s> <s xml:id="echoid-s4535" xml:space="preserve">in variis propoſitionibus, aliiq́ue quam <lb/>plurimi hinc deducunt, falſa & </s> <s xml:id="echoid-s4536" xml:space="preserve">veri vana habenda ſunt. </s> <s xml:id="echoid-s4537" xml:space="preserve">Eo<unsure/>ſohis motus & </s> <s xml:id="echoid-s4538" xml:space="preserve">mo-<lb/>vendi æquilibrio contenti ſimus, quippe quæ his abundè ſatis faciat.</s> <s xml:id="echoid-s4539" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div633" type="section" level="1" n="454"> <head xml:id="echoid-head476" xml:space="preserve">C*AP. III.*</head> <head xml:id="echoid-head477" style="it" xml:space="preserve">Staticam eſſe Mathematicarum Liberalium <lb/>artium unam.</head> <p> <s xml:id="echoid-s4540" xml:space="preserve">QVamvis de iſtarum rerum nominibus quibus doctrina nihil obſcuratur <lb/>controvertere, aut in diſceptationem vocare ſupervacaneum ſit; </s> <s xml:id="echoid-s4541" xml:space="preserve">non eſt <lb/>tamen cur hæc eò pertinere quis arbitretur; </s> <s xml:id="echoid-s4542" xml:space="preserve">nam nos cum uſui erit Staticam li-<lb/>beralem artem appellabimus, atq; </s> <s xml:id="echoid-s4543" xml:space="preserve">ideo ejus appellationis rationem redderene-<lb/>ceſſe fuerit. </s> <s xml:id="echoid-s4544" xml:space="preserve">Quemadmodum igitur numeri & </s> <s xml:id="echoid-s4545" xml:space="preserve">magnitudinis materia diverſa, <lb/>meritiſſimò quoque artium ſuarum limites & </s> <s xml:id="echoid-s4546" xml:space="preserve">confinia ſecernenda diſtinguit; <lb/></s> <s xml:id="echoid-s4547" xml:space="preserve">illaq́ue Arithmeticæ hæc Geometriæ terminis circumſcribitur, ut ſingulæ con-<lb/>venientiori ordine, magis propriè, magisq́; </s> <s xml:id="echoid-s4548" xml:space="preserve">perſpicuè deſcribantur & </s> <s xml:id="echoid-s4549" xml:space="preserve">docean-<lb/>tur: </s> <s xml:id="echoid-s4550" xml:space="preserve">cumq́ue ſubtilis materies iſtis artibus ſubjecta à natura nobis ingenita aut <lb/>cognita non ſit; </s> <s xml:id="echoid-s4551" xml:space="preserve">ſed è variorum ſcriptis qui ſummo ſtudio maximaq́ue diligen-<lb/>tia in his ſeſe exercuere, imò non rarò caſu notitiam excellentium rerum ſunt <lb/>adepti, perdiſcenda ſit; </s> <s xml:id="echoid-s4552" xml:space="preserve">atque illarum cognitio humano uſui perquam neceſſa-<lb/>ria, ideoq́ue homine ingenuo digna, unde ipſæ liberales artes appellantur, & </s> <s xml:id="echoid-s4553" xml:space="preserve"><lb/>ſua certitudine reliquas artes longè antecellant; </s> <s xml:id="echoid-s4554" xml:space="preserve">meritò Mathematicæ appellan-<lb/>tur, cum ita eſſe non perſuadeant, ſed demonſtrando cogant, doceantq́ue. </s> <s xml:id="echoid-s4555" xml:space="preserve">Pari <lb/>ratione etiam Statica iſtis connumeranda, partim quia hujus materia gravitas, <lb/>ſit ab illarum utraque, numero ſcilicet & </s> <s xml:id="echoid-s4556" xml:space="preserve">magnitudine, diverſa; </s> <s xml:id="echoid-s4557" xml:space="preserve">partim etiam <lb/>quia proprietates hujus ſubtilitate quoque illis non cedant, cujus vel hoc ar-<lb/>gumentum ſit, quod omnium tardiſſimè è tenebris eruta, in lucem ſit edita. </s> <s xml:id="echoid-s4558" xml:space="preserve">De-<lb/>nique cum ab ultimis uſque initiis tantæ certitudinis ſit, quantæ illæ pari jure <lb/>peculiaris quædam ars Liberalis Mathematica, & </s> <s xml:id="echoid-s4559" xml:space="preserve">propria quoque cenſebitur.</s> <s xml:id="echoid-s4560" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4561" xml:space="preserve">Objiciat autem quis, Geometricas figurationes in ipſius demonſtrationibus <lb/>non rarò adhiberi, ideoq́; </s> <s xml:id="echoid-s4562" xml:space="preserve">ejus eſſe quandam ſpeciem. </s> <s xml:id="echoid-s4563" xml:space="preserve">Reſpondeo idem Arith-<lb/>meticæ accidere; </s> <s xml:id="echoid-s4564" xml:space="preserve">etenim quæ habet theoremata quarum cognitio non penitus <lb/>ex ipſa Geometria ſit repetenda? </s> <s xml:id="echoid-s4565" xml:space="preserve">Imo ne Geometria quidem ipſa ſe abſq; </s> <s xml:id="echoid-s4566" xml:space="preserve">nu-<lb/>meris tuebitur, aut defendet. </s> <s xml:id="echoid-s4567" xml:space="preserve">Inſpice ſodes elementa Geometrica, quoties quæ-<lb/>ſo figura figuræ dupla, item tria plana duobus æquari dicuntur? </s> <s xml:id="echoid-s4568" xml:space="preserve">unde conſtat <lb/>propoſitiones Geometricas abſque numeris demonſtrari non poſſe, quamvis <lb/>artes inter ſe ſint diverſæ & </s> <s xml:id="echoid-s4569" xml:space="preserve">diſtinctæ; </s> <s xml:id="echoid-s4570" xml:space="preserve">atque eadem Staticæ doctrinæ ratio erit.</s> <s xml:id="echoid-s4571" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4572" xml:space="preserve">Præterea Opticæ & </s> <s xml:id="echoid-s4573" xml:space="preserve">Catoptricæ, quæ non peculiares artes Mathematicæ ſed <lb/>Geometrici generis omnino cenſentur, alia eſt ratio, quam Staticæ, & </s> <s xml:id="echoid-s4574" xml:space="preserve">multùm <lb/>diſſimilis. </s> <s xml:id="echoid-s4575" xml:space="preserve">nam Staticæ materia ſeu ſubjectum quæ eſt gravitas, non ſecus quàm <lb/>magnitudo, & </s> <s xml:id="echoid-s4576" xml:space="preserve">numerus (quia omnia, ut eſt in veteri proverbio conſtant pon-<lb/>dere, numero, & </s> <s xml:id="echoid-s4577" xml:space="preserve">menſura) in qualibet ſubſtantia magno hominum commodo <lb/>deprehenduntur; </s> <s xml:id="echoid-s4578" xml:space="preserve">ſed ſuperiores iſtæ non item. </s> <s xml:id="echoid-s4579" xml:space="preserve">Quare ut diximus Statica me-<lb/>ritò liberalium Mathematicarum artium unafuerit.</s> <s xml:id="echoid-s4580" xml:space="preserve"/> </p> <pb o="154" file="527.01.154" n="154" rhead="A*PPENDIX*"/> </div> <div xml:id="echoid-div634" type="section" level="1" n="455"> <head xml:id="echoid-head478" xml:space="preserve">C*APVT IV.*</head> <head xml:id="echoid-head479" style="it" xml:space="preserve">Demonstrationum ſupraſcriptarum nonnullas per nume@ <lb/>rosinstitutas, Mathematicas eſſe.</head> <p> <s xml:id="echoid-s4581" xml:space="preserve">MAthematicæ & </s> <s xml:id="echoid-s4582" xml:space="preserve">Mechanicæ demonſtrationis à doctis annotatur differen-<lb/>tia, neque injuria. </s> <s xml:id="echoid-s4583" xml:space="preserve">Nam illa omnibus generalis eſt, & </s> <s xml:id="echoid-s4584" xml:space="preserve">rationem cur ita ſit <lb/>penitus demonſtrat, hæc verò in ſubjecto duntaxat paradigmate numeris de-<lb/>clarat. </s> <s xml:id="echoid-s4585" xml:space="preserve">Vt ſi demonſtraturus in rectangulo triangulo baſin recti æquè poſſe cru-<lb/>ribus, aſſumat triangulum cujus minimum latus ſit 3, ſecundum 4, tertium 5 pe-<lb/>dum, hocq́ue rectangulum eſſe deprehendatur; </s> <s xml:id="echoid-s4586" xml:space="preserve">tumq́ue oſtendat maximi la-<lb/>teris quadratum 25, æquari reliquorum laterum quadratis 16 & </s> <s xml:id="echoid-s4587" xml:space="preserve">9. </s> <s xml:id="echoid-s4588" xml:space="preserve">Sed demon-<lb/>ſtratio hujuſmodi tantum eſt propoſiti exempli, unde non concluſeris omnibus <lb/>rectangulis triangulis idem contingere, neque hinc cur id fiat evidens eſt; </s> <s xml:id="echoid-s4589" xml:space="preserve">& </s> <s xml:id="echoid-s4590" xml:space="preserve"><lb/>quia opus hujuſmodi machinationein ſpeciali exemplo inſtituitur, mechanica <lb/>demonſtratio appellatur: </s> <s xml:id="echoid-s4591" xml:space="preserve">ſed illa quam Euclides 47 propoſ. </s> <s xml:id="echoid-s4592" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4593" xml:space="preserve">uſurpat ca-<lb/>tholica eſt & </s> <s xml:id="echoid-s4594" xml:space="preserve">univerſalis, cauſam repetens ab ipſis elementis cur ita neceſſariò, & </s> <s xml:id="echoid-s4595" xml:space="preserve"><lb/>non aliter ſe habere poſſit: </s> <s xml:id="echoid-s4596" xml:space="preserve">hæc propter certitudinem in demonſtrando, & </s> <s xml:id="echoid-s4597" xml:space="preserve">do-<lb/>cendo infallibilem Mathematica dicitur; </s> <s xml:id="echoid-s4598" xml:space="preserve">ideoq́ue etiam ab ipſis Mathematicis <lb/>potior cenſetur & </s> <s xml:id="echoid-s4599" xml:space="preserve">frequentiùs uſurpatur, quam illa per numeros mechanica. <lb/></s> <s xml:id="echoid-s4600" xml:space="preserve">Vnde objectionem mihi paratam intelligo, cur 4, 11, 12, 18 propoſitiones 2. </s> <s xml:id="echoid-s4601" xml:space="preserve">lib. </s> <s xml:id="echoid-s4602" xml:space="preserve"><lb/>Elem. </s> <s xml:id="echoid-s4603" xml:space="preserve">Statices numeris adhibitis explicarim & </s> <s xml:id="echoid-s4604" xml:space="preserve">demonſtrarim. </s> <s xml:id="echoid-s4605" xml:space="preserve">Cui occurritur, <lb/>demonſtrationem in numeris dupliciter inſtitui; </s> <s xml:id="echoid-s4606" xml:space="preserve">alteram ubi tanquam termini <lb/>rationem, proportionemq́ue partium expoſitæ figuræ declarant; </s> <s xml:id="echoid-s4607" xml:space="preserve">alteram ubi <lb/>quantitatem. </s> <s xml:id="echoid-s4608" xml:space="preserve">Illa Mathematica eſt quia univerſim ſpeciei datæ figuræ conve-<lb/>niat, & </s> <s xml:id="echoid-s4609" xml:space="preserve">in ipſis cauſam declaret; </s> <s xml:id="echoid-s4610" xml:space="preserve">hæc autem non item, ob rationes iſtis contra-<lb/>rias. </s> <s xml:id="echoid-s4611" xml:space="preserve">Quain re Eutochius in ſuis in Apollonium cõmentariis 11 prop. </s> <s xml:id="echoid-s4612" xml:space="preserve">lib. </s> <s xml:id="echoid-s4613" xml:space="preserve">1. </s> <s xml:id="echoid-s4614" xml:space="preserve">mecum <lb/>facit, dum ait: </s> <s xml:id="echoid-s4615" xml:space="preserve">Non perturbentur qui in hæc inciderint, quodillud ex Arithmeticis de-<lb/>monſtretur: </s> <s xml:id="echoid-s4616" xml:space="preserve">antiqui enim hujuſmodi demonſtrationibus ſæpe uti conſueverunt; </s> <s xml:id="echoid-s4617" xml:space="preserve">quæ ta-<lb/>men Mathematicæ potius ſunt, quam Arithmeticæpropter analogias. </s> <s xml:id="echoid-s4618" xml:space="preserve">adde quod quæſi-<lb/>tum Arithmeticam ſit; </s> <s xml:id="echoid-s4619" xml:space="preserve">nam rationes & </s> <s xml:id="echoid-s4620" xml:space="preserve">rationum quantitates, & </s> <s xml:id="echoid-s4621" xml:space="preserve">multiplicationes pri-<lb/>mò numeris, ſecundo loco per numeros & </s> <s xml:id="echoid-s4622" xml:space="preserve">magnitudinibus inſunt, ex illius ſententia <lb/>qui ita ſcripſit: </s> <s xml:id="echoid-s4623" xml:space="preserve">Ταῦ{τα} {γὰρ} τὰ μα{θή}μα{τα} δοκ{οῦ}ν{τι} {εἶ}μεν ἀδελφά. </s> <s xml:id="echoid-s4624" xml:space="preserve">hoc eſt, hæ enim Ma-<lb/>thematicæ diſciplinæ germanæ eſſe videntur. </s> <s xml:id="echoid-s4625" xml:space="preserve">Inſuperautĕ objiciatur in Archimedis, <lb/>Ptolomæi, Apollonii, & </s> <s xml:id="echoid-s4626" xml:space="preserve">inter recentiores Comandini, Regiomontani, aliorumq́ue <lb/>ſimilium propoſitionibus ipſam rationem, non autem rationis terminos in nu-<lb/>meris nominatim proponi, quod à nobis factitatum ſit; </s> <s xml:id="echoid-s4627" xml:space="preserve">cui reſponſio expedita <lb/>eſt & </s> <s xml:id="echoid-s4628" xml:space="preserve">in promptu; </s> <s xml:id="echoid-s4629" xml:space="preserve">eodem jure atque ab illis citatur ratio dupla, tripla, quadru-<lb/>pla, eodem in quam jure, citari etiam rationem duodecuplam, quale illud in di-<lb/>cta 23 propoſ. </s> <s xml:id="echoid-s4630" xml:space="preserve">A D ad R D; </s> <s xml:id="echoid-s4631" xml:space="preserve">item rationem 37 ad 23 ſive ſuperquatuordecu-<lb/>partientem vigeſimastertias ipſius A R ad R D in ſupraſcripta propoſitione 11, <lb/>cum idem ſit rationem, & </s> <s xml:id="echoid-s4632" xml:space="preserve">rationum terminos proponere. </s> <s xml:id="echoid-s4633" xml:space="preserve">nam iſtarum li-<lb/>nearum in expoſitis figurarum illarum generibus alia ratio nulla eſt. </s> <s xml:id="echoid-s4634" xml:space="preserve">Cum au-<lb/>tem numerorum uſus ſit in perveſtigandis iſtiuſmodi figurarum proprietati-<lb/>bus, ut his ducibus & </s> <s xml:id="echoid-s4635" xml:space="preserve">commõſtratoribus facilè & </s> <s xml:id="echoid-s4636" xml:space="preserve">perſpicuè res ipſas pernoſca-<lb/>mus, etiam neceſſe fuit in illarum deſcriptione numeros eoſdem adſcribere, ne <lb/>aliis obſcurum ſit, quod earum autoribus & </s> <s xml:id="echoid-s4637" xml:space="preserve">inventoribus clarum fuerit, namq́; </s> <s xml:id="echoid-s4638" xml:space="preserve"><lb/>hæc ipſa eſt vera & </s> <s xml:id="echoid-s4639" xml:space="preserve">Mathematica demonſtratio, propoſiti veritatem ab ipſis <lb/>cauſis repetere.</s> <s xml:id="echoid-s4640" xml:space="preserve"/> </p> <pb o="155" file="527.01.155" n="155" rhead="S*TATICES*."/> <p> <s xml:id="echoid-s4641" xml:space="preserve">Notandum autem nonnullas demonſtrationes 1 lib. </s> <s xml:id="echoid-s4642" xml:space="preserve">Static. </s> <s xml:id="echoid-s4643" xml:space="preserve">itemq́ue Hydro-<lb/>ſtat. </s> <s xml:id="echoid-s4644" xml:space="preserve">ubi gravitas numero, notaq́ue librarum menſura exprimitur, ut mechanicis <lb/>demonſtrationibus accenſeri debere videantur, geminas à nobis exhibitas eſſe; <lb/></s> <s xml:id="echoid-s4645" xml:space="preserve">alteras Arithmeticas ut 1 exemplo 1 propoſ. </s> <s xml:id="echoid-s4646" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s4647" xml:space="preserve">ubi propoſitionis ſententia <lb/>Arithmetico calculo oſtenditur, quam Mathematica demonſtratio ſecundo <lb/>exemplo ſtatim ſubſequitur. </s> <s xml:id="echoid-s4648" xml:space="preserve">Vt Mechanica demonſtratio Mathematicæ non-<lb/>nunquam tanquam miniſtra facem alluceat.</s> <s xml:id="echoid-s4649" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div635" type="section" level="1" n="456"> <head xml:id="echoid-head480" xml:space="preserve">C*APVT V.*</head> <head xml:id="echoid-head481" style="it" xml:space="preserve">Vbi Propoſitio 8 Hydrostatices illustratur, <lb/>& clariùs exponitur.</head> <p> <s xml:id="echoid-s4650" xml:space="preserve">OCtava propoſitio Hydroſt. </s> <s xml:id="echoid-s4651" xml:space="preserve">docet: </s> <s xml:id="echoid-s4652" xml:space="preserve">Solidum in aqua levius eſſe quam in aëre <lb/>pondere aquæ magnitudine ſibi æqualis. </s> <s xml:id="echoid-s4653" xml:space="preserve">Vnde quis conſectaria huj<unsure/>uſmodi de-<lb/>duceret: </s> <s xml:id="echoid-s4654" xml:space="preserve">Solidum quodlibet in hydrargyro levius eſt quàm in aqua magnitudine hy-<lb/>drargyriſibi æqualis. </s> <s xml:id="echoid-s4655" xml:space="preserve">Velaliud hoc modo: </s> <s xml:id="echoid-s4656" xml:space="preserve">Solidum quodlibet in aqua levius eſt quam <lb/>in oleo magnitudine aquæ ſibi æqualis; </s> <s xml:id="echoid-s4657" xml:space="preserve">ſimiliq́ue analogia in cæteris. </s> <s xml:id="echoid-s4658" xml:space="preserve">Quæ vera de-<lb/>ductio, re ſimpliciter conſiderata experiĕtiæ contraria videtur; </s> <s xml:id="echoid-s4659" xml:space="preserve">nam libra plum-<lb/>binõ erit in aqua levior quam in oleo, põdere aquę ſibi æqualis, ſed tantò dun-<lb/>taxat levior quanta erit differentia aquæ & </s> <s xml:id="echoid-s4660" xml:space="preserve">olei dictæ plumbeæ libræ magnitu-<lb/>dine æqualium. </s> <s xml:id="echoid-s4661" xml:space="preserve">Sed tamen re penſiculatius expenſa theorema noſtrum omni-<lb/>bus numeris perfectum animadvertet, ſiquidem 1 poſtul. </s> <s xml:id="echoid-s4662" xml:space="preserve">Elem. </s> <s xml:id="echoid-s4663" xml:space="preserve">Hydroſt. </s> <s xml:id="echoid-s4664" xml:space="preserve">petie-<lb/>rim concedi, Ponderitatem corporum in aëre appellari propriè, item 5 poſt. </s> <s xml:id="echoid-s4665" xml:space="preserve">Vas ſu-<lb/>perficiarium effuſa aqua vacuum eſſe, hoc eſt per 11 defin. </s> <s xml:id="echoid-s4666" xml:space="preserve">aëris duntaxat plenum. <lb/></s> <s xml:id="echoid-s4667" xml:space="preserve">ſi igitur media, in quibus gravitas æſtimatur, hydrargyrum & </s> <s xml:id="echoid-s4668" xml:space="preserve">aqua ponantur, <lb/>ac tum poſtuletur, Corporum gravitatem in aqua dici propriè. </s> <s xml:id="echoid-s4669" xml:space="preserve">Item, Vas ſuperſicia-<lb/>rium effuſo hydrargyro aquæplenam eſſe, certè his ita conſtitutis dicta propoſitio <lb/>(Solidum quodlibet in hydrargyro levius eſſe quam in aqua, pondere aquæ magnitudine <lb/>ſibi æqualis) omninò vera fuerit. </s> <s xml:id="echoid-s4670" xml:space="preserve">Vtres hæc magis fiat perſpicua, cogitatione <lb/>fingito hominem ſub aqua conſtitutum ſecum habere hydrargyrum & </s> <s xml:id="echoid-s4671" xml:space="preserve">aurum, <lb/>ſitq́ue aqua vice aëris, Ajo aurum iſtictantò fore levius, quam in hydrargyro <lb/>quantum erit pondus hydrargyri aurum magnitudine æquantis: </s> <s xml:id="echoid-s4672" xml:space="preserve">quod ſanè <lb/>manifeſtum eſt. </s> <s xml:id="echoid-s4673" xml:space="preserve">At verò ſi Corporum pondus in inani verè dici ſumatur, ut revera <lb/>ſe res habet, ſecundum hac inquam affectionem ita enuntiari poſſet. </s> <s xml:id="echoid-s4674" xml:space="preserve">Omne ſoli-<lb/>dum in aqua gravius eſt, quàm in inani pondere aquæ ſibi æqualis. </s> <s xml:id="echoid-s4675" xml:space="preserve">Verùm cum uſus <lb/>& </s> <s xml:id="echoid-s4676" xml:space="preserve">effectio (quò theoriam perpetuò dirigere decet) non in vacuo ſed in aëre <lb/>fiant, ſatiùs eſt ſecun dum modum nobis ſupra uſitatum, pondus rei proprium <lb/>in aëre ſupponi, cujus ratione & </s> <s xml:id="echoid-s4677" xml:space="preserve">reſpectu 8 noſtra propoſitio cæteræq́ue inde <lb/>derivatæ omnibus numeris perfectæ ſunt. </s> <s xml:id="echoid-s4678" xml:space="preserve">Quod annotaſſe fuit operæ pre-<lb/>tium.</s> <s xml:id="echoid-s4679" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div636" type="section" level="1" n="457"> <head xml:id="echoid-head482" xml:space="preserve">A*PPENDICIS* F*INIS*.</head> <pb file="527.01.156" n="156"/> <pb file="527.01.157" n="157"/> </div> <div xml:id="echoid-div637" type="section" level="1" n="458"> <head xml:id="echoid-head483" xml:space="preserve">ADDITAMENTVM <lb/>STATICÆ.</head> <pb file="527.01.158" n="158"/> </div> <div xml:id="echoid-div638" type="section" level="1" n="459"> <head xml:id="echoid-head484" xml:space="preserve">BREVIARIVM <lb/>ADDITAMENTI.</head> <p> <s xml:id="echoid-s4680" xml:space="preserve">APrima Staticæ editione varia cùm in Praxi, tum <lb/>etiam in Theoria mihi occurrerunt, quæ ſingula <lb/>ſecunda hâc editione ſuo quæque loco diſponi, <lb/>inq́ue unum corpus digeſta ordinari potuere: </s> <s xml:id="echoid-s4681" xml:space="preserve">ſed cùm vero <lb/>conſentaneum videatur, etiam plura hujuſmodi uſum & </s> <s xml:id="echoid-s4682" xml:space="preserve"><lb/>tempus nobis paritura, quorum ordinatio iterum ite-<lb/>rumq́ novanda foret, idq; </s> <s xml:id="echoid-s4683" xml:space="preserve">ſine fine, quamvis fortaſſe illud <lb/>ſatius eſſet; </s> <s xml:id="echoid-s4684" xml:space="preserve">nunc tamen magis neceſlaria diſpoſitioni huic <lb/>me vacare nõ ſinunt. </s> <s xml:id="echoid-s4685" xml:space="preserve">Ideoq́; </s> <s xml:id="echoid-s4686" xml:space="preserve">de priore forma Staticæ omni-<lb/>nò nihil (iis exceptis quæ mutare neceſſe erat) detraxi, aut <lb/>immutavi. </s> <s xml:id="echoid-s4687" xml:space="preserve">reliquas appendiculas, quarum acceſsione au-<lb/>cta eſt, hîc uno A*DDITAMENTI* titulo complexus ſum. <lb/></s> <s xml:id="echoid-s4688" xml:space="preserve">Cujus hoc eſt argumentum:</s> <s xml:id="echoid-s4689" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4690" xml:space="preserve">Primò de Spartoſtatica.</s> <s xml:id="echoid-s4691" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Funium Sta-<lb/>t<unsure/>ica.</note> <p> <s xml:id="echoid-s4692" xml:space="preserve">Secundò de Trochleoſtatica.</s> <s xml:id="echoid-s4693" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Trochlearum <lb/>Statica.</note> <p> <s xml:id="echoid-s4694" xml:space="preserve">Tertiò de Fluitantibus Acrobaricis.</s> <s xml:id="echoid-s4695" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Ponderis in <lb/>ſummo verti-<lb/>ce gravitate.</note> <p> <s xml:id="echoid-s4696" xml:space="preserve">Quartò de Chalinothlipſi.</s> <s xml:id="echoid-s4697" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Freni preſſura <lb/>&<unsure/> tenacia.</note> <p> <s xml:id="echoid-s4698" xml:space="preserve">Quintò de Hydatholcia.</s> <s xml:id="echoid-s4699" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Aque attra<unsure/>-<lb/>c<unsure/>tu.</note> <p> <s xml:id="echoid-s4700" xml:space="preserve">Sextò de Aëroſtatica.</s> <s xml:id="echoid-s4701" xml:space="preserve"/> </p> <note position="left" xml:space="preserve">Aëris Stati-<lb/>ca.</note> <pb file="527.01.159" n="159"/> </div> <div xml:id="echoid-div639" type="section" level="1" n="460"> <head xml:id="echoid-head485" xml:space="preserve">ADDITAMENTI</head> <head xml:id="echoid-head486" xml:space="preserve">STATICÆ <lb/>PARS PRIMA <lb/>DE <lb/>SPARTOSTATICA.</head> <pb file="527.01.160" n="160"/> </div> <div xml:id="echoid-div640" type="section" level="1" n="461"> <head xml:id="echoid-head487" xml:space="preserve">BREVIARIVM</head> <head xml:id="echoid-head488" xml:space="preserve">SPARTOSTATICES.</head> <p style="it"> <s xml:id="echoid-s4702" xml:space="preserve">NOviſsimis tribus 1 libri Staticæ propoſitionibus ex-<lb/>poſuimus affectiones ponderum de rectis duabus, <lb/>quæ duobus diverſis locis fixæ ſunt, dependentium: <lb/></s> <s xml:id="echoid-s4703" xml:space="preserve">ſed quia ponder a pluribus modis èrectis ſuſpendi poſsint, qua-<lb/>rum quantũ quæ{q́ue} ferat ſcire expetas, istud ad ſpecialem bunc <lb/>tractatum retulimus. </s> <s xml:id="echoid-s4704" xml:space="preserve">Cum autem linearum vicem funes ple-<lb/>rumque ſubeant, ab uſu vulg atiſsimo S*PARTO STATI CEN* <lb/>appellamus, qua appellatione intellig as doctrinam declar antem <lb/>quantum ponderis diverſorum funium, ex quibus idem cogni-<lb/>tum pondus dependet, quilibet ſufferat. </s> <s xml:id="echoid-s4705" xml:space="preserve">Totius autem argu-<lb/>menti ſumma hujuſmodi eſt. </s> <s xml:id="echoid-s4706" xml:space="preserve">Cum propoſitione 27 lib. </s> <s xml:id="echoid-s4707" xml:space="preserve">1 Staticæ <lb/>demonstratum ſit, columnâ contra duo obliqua pondera in <lb/>æquilibrio conſtitutâ, eſſe ut linea obliquata ad lineam rectam, <lb/>ſic pondus obliquè ducens ad pondus ducens rectà, hinc prima <lb/>istaparte varia deducemus conſectaria, quorum loco propoſi-<lb/>tiones formare licuit; </s> <s xml:id="echoid-s4708" xml:space="preserve">ſed partim brevitatis gratiâ, partim <lb/>quiaistinc manifeſtiſsime deducuntur, id negotii omiſſum eſt.</s> <s xml:id="echoid-s4709" xml:space="preserve"/> </p> <pb o="161" file="527.01.161" n="161"/> </div> <div xml:id="echoid-div641" type="section" level="1" n="462"> <head xml:id="echoid-head489" xml:space="preserve">PRIMVM CONSECTARIVM</head> <head xml:id="echoid-head490" xml:space="preserve">è 27 propoſitione 1 Libri S*TATICÆ*.</head> <p> <s xml:id="echoid-s4710" xml:space="preserve">SI in figura 27 propoſitionis 1 lib. </s> <s xml:id="echoid-s4711" xml:space="preserve">in E <lb/>loco ponderis obliquè attollentis ſub-<lb/>ſtituatur firmitudinis punctum quale <lb/>hic vides, perſpicuum eſt hoc affici <lb/>preſſu ponderi G æquali, atque iſtiuſmodi obliqui-<lb/>tate niti, qualem oſtendit obliqua linea L E.</s> <s xml:id="echoid-s4712" xml:space="preserve"/> </p> <figure> <image file="527.01.161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.161-01"/> </figure> </div> <div xml:id="echoid-div642" type="section" level="1" n="463"> <head xml:id="echoid-head491" xml:space="preserve">2 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4713" xml:space="preserve">Item ſi in eadem figura 27 propoſ. </s> <s xml:id="echoid-s4714" xml:space="preserve">LE, MF continuatæ concurrant,' pun <lb/>ctum concurſus per 25 propoſ. </s> <s xml:id="echoid-s4715" xml:space="preserve">incidet in pendulam <lb/> <anchor type="figure" xlink:label="fig-527.01.161-02a" xlink:href="fig-527.01.161-02"/> gravitatis ejus diametrum. </s> <s xml:id="echoid-s4716" xml:space="preserve">Quamobrem ut cognoſ-<lb/>catur quanta obliqua preſſio puncto E infideat; </s> <s xml:id="echoid-s4717" xml:space="preserve">du-<lb/>cito pendulam diametrum à centro P quæ occur-<lb/>rat continuatæ M F in Q, hinc ab Q per E rectam <lb/>Q R ut R ſit in A M. </s> <s xml:id="echoid-s4718" xml:space="preserve">quæ cum ita ſint, preſſio erit <lb/>ab R verſus E. </s> <s xml:id="echoid-s4719" xml:space="preserve">Atqui ut etiam quanta ea ſit cognoſ-<lb/>cas, uſurpato E R tanquam lineam obliquè tollen-<lb/>tem, & </s> <s xml:id="echoid-s4720" xml:space="preserve">ES tanquam tollentem rectè, unde reliqua <lb/>erunt in proclivi.</s> <s xml:id="echoid-s4721" xml:space="preserve"/> </p> <div xml:id="echoid-div642" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.161-02" xlink:href="fig-527.01.161-02a"> <image file="527.01.161-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.161-02"/> </figure> </div> </div> <div xml:id="echoid-div644" type="section" level="1" n="464"> <head xml:id="echoid-head492" xml:space="preserve">3 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4722" xml:space="preserve">Sed ut rationem ponderum è funibus dependentium explicemus, eſto co-<lb/>lumna AB, cujus centrum C, eq̀ue duobus firmi-<lb/> <anchor type="figure" xlink:label="fig-527.01.161-03a" xlink:href="fig-527.01.161-03"/> tudinis punctis D, E ſuſpenſum, eductis ex cen-<lb/>tro C duabus lineis C D, CE, quare iſtæ per 5 defin. <lb/></s> <s xml:id="echoid-s4723" xml:space="preserve">ſunt columnæ gravitatis diametri, ideoq́ue H I paral-<lb/>lela contra C E inter C D, C F educta erit C I per 13 <lb/>defin. </s> <s xml:id="echoid-s4724" xml:space="preserve">linea rectà attollens, C H autem obliquè, unde <lb/>efficitur ut C I ad C H ſic pondus illius recta attol-<lb/>lens ad pondus hujus attollens obliquè. </s> <s xml:id="echoid-s4725" xml:space="preserve">Sed pondus re-<lb/>ctà tollens quod pertinetad C I, totius columnæ pon-<lb/>deri æquatur; </s> <s xml:id="echoid-s4726" xml:space="preserve">itaque ut C I ad C H, ſic totius columnæ <lb/>pondus, ad pondus quod pertinetad D. </s> <s xml:id="echoid-s4727" xml:space="preserve">Eademq́ue via <lb/>concludetur pondus pertinens ad E ductâ ab I in C E <lb/>rectâ IK contra D C parallelâ; </s> <s xml:id="echoid-s4728" xml:space="preserve">atque tum erit ut rectà <lb/>tollens C I ad tollentem obliquè C K, ſic totius columnæ pondus, ad pon-<lb/>dus ſubnixum ipſi E.</s> <s xml:id="echoid-s4729" xml:space="preserve"/> </p> <div xml:id="echoid-div644" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.161-03" xlink:href="fig-527.01.161-03a"> <image file="527.01.161-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.161-03"/> </figure> </div> <p> <s xml:id="echoid-s4730" xml:space="preserve">Verùm quia C K perpetuò eſt æqualis HI, nihil eſt neceſſe ducere hanc <lb/>poſtremam I K, omnesq́ue neceſſarii cogniti termini inſunt tribus trianguli <lb/>H I C lateribus: </s> <s xml:id="echoid-s4731" xml:space="preserve">unde ita fari licet.</s> <s xml:id="echoid-s4732" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4733" xml:space="preserve">Vt CI ad C H, ſic pondus columnæ ad pondus pertingens ad D. </s> <s xml:id="echoid-s4734" xml:space="preserve">Item <lb/>ut C I ad I H, ſic pondus columnæ ad id quod pertinet ad E. </s> <s xml:id="echoid-s4735" xml:space="preserve">Et denique ut <lb/>CH ad HI, fic pondus quod ab D ad pondus quod ab E ſuſtinetur.</s> <s xml:id="echoid-s4736" xml:space="preserve"/> </p> <pb o="162" file="527.01.162" n="162" rhead="A*DDITAMENTI* S*TATICÆ PARS PRIMA*"/> </div> <div xml:id="echoid-div646" type="section" level="1" n="465"> <head xml:id="echoid-head493" xml:space="preserve">4 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4737" xml:space="preserve">Verumenimverò ut propiùs ad ra-<lb/>tionem ponderum è funibus depen-<lb/> <anchor type="figure" xlink:label="fig-527.01.162-01a" xlink:href="fig-527.01.162-01"/> dentium accedamus; </s> <s xml:id="echoid-s4738" xml:space="preserve">columna A B <lb/>paulum infra deſcĕdat utin hâc figu-<lb/>râ, & </s> <s xml:id="echoid-s4739" xml:space="preserve">per 3 poſtulatum hoc loco non <lb/>erit ponderis diverſi ab antecedente, <lb/>ubi ſublimiùs pendebat. </s> <s xml:id="echoid-s4740" xml:space="preserve">Itaque etiam <lb/>proportio 3 conſectario expoſita in <lb/>hoc 4 ſine ulla varietate etiamnum <lb/>permanet.</s> <s xml:id="echoid-s4741" xml:space="preserve"/> </p> <div xml:id="echoid-div646" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.162-01" xlink:href="fig-527.01.162-01a"> <image file="527.01.162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.162-01"/> </figure> </div> </div> <div xml:id="echoid-div648" type="section" level="1" n="466"> <head xml:id="echoid-head494" xml:space="preserve">5 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4742" xml:space="preserve">Tandem in locum columnæ 4 conſectarii aliud pondus ipſi æquale ſubſtitui-<lb/>tor, ſed formæ & </s> <s xml:id="echoid-s4743" xml:space="preserve">gravitatis materiæ cujuſliber, ut hîc AB. </s> <s xml:id="echoid-s4744" xml:space="preserve">atque etiamnum <lb/>ratum eſt, & </s> <s xml:id="echoid-s4745" xml:space="preserve">perſpicuum C I eſſe ad <lb/> <anchor type="figure" xlink:label="fig-527.01.162-02a" xlink:href="fig-527.01.162-02"/> C H, ut pondus A B ad partem quæ <lb/>pertinet ad D. </s> <s xml:id="echoid-s4746" xml:space="preserve">Item ut C I ad I H, <lb/>ſic pondus A B ad id quod ex E <lb/>ſuſtinetur, denique ut C H ad HI <lb/>ſic pondus ex D ad id quod ex E.</s> <s xml:id="echoid-s4747" xml:space="preserve"/> </p> <div xml:id="echoid-div648" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.162-02" xlink:href="fig-527.01.162-02a"> <image file="527.01.162-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.162-02"/> </figure> </div> <p> <s xml:id="echoid-s4748" xml:space="preserve">Vnde in promptu erit, ſi ex D C E <lb/>tanquam fune dependeat notũ pon-<lb/>dus AB, notiq́ue ſint anguli F C D, <lb/>F C E, concludere quantum ponderis quilibet iſtorum DC, CE perferat.</s> <s xml:id="echoid-s4749" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div650" type="section" level="1" n="467"> <head xml:id="echoid-head495" xml:space="preserve">6 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4750" xml:space="preserve">Si verò eodem modo è lineis duo pluravé pondera dependeant, ut in ſubje-<lb/>ctâ figurâ A B C D E F, cujus extima firmitudinis puncta ſint A, F, è qua li-<lb/>nea quatuor pondera G, H, I, K ſuſpenſa ſint, etiam ponderis potentiam ab il-<lb/> <anchor type="figure" xlink:label="fig-527.01.162-03a" xlink:href="fig-527.01.162-03"/> larum quinque linearum ſingulis AB, BC, CD, DE, EF dependentem <lb/>inveniri poſſe manifeſtum eſt: </s> <s xml:id="echoid-s4751" xml:space="preserve">namq́ue cõtinuata ſurſum dicis gratiâ, G B in L, <pb o="163" file="527.01.163" n="163" rhead="*DE* S*PARTOSTATICA*."/> ductaq́ue M N parallela contra B C, concludes ut B N ad B M, ſic Gadpon-<lb/>dus ſuſtentatum ab A B.</s> <s xml:id="echoid-s4752" xml:space="preserve"/> </p> <div xml:id="echoid-div650" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.162-03" xlink:href="fig-527.01.162-03a"> <image file="527.01.162-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.162-03"/> </figure> </div> <p> <s xml:id="echoid-s4753" xml:space="preserve">Item ut B M ad B N, ſic pondus G ad pondus quod ſuſtinetur à B C.</s> <s xml:id="echoid-s4754" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4755" xml:space="preserve">Secundò continuata etiam H C ſurſum vorſum in O, & </s> <s xml:id="echoid-s4756" xml:space="preserve">B P parallela du-<lb/>cta contra C D: </s> <s xml:id="echoid-s4757" xml:space="preserve">concludes ſimiliter ſuperiori, ut C P ad C B, ſic H ad pondus <lb/>ſui partem quod pertiner ad C B. </s> <s xml:id="echoid-s4758" xml:space="preserve">Ex quo perſpicitur idem quod ſupra pro <lb/>B C concluſum eſt nunc redire. </s> <s xml:id="echoid-s4759" xml:space="preserve">Factio cæterarum concluſionum his ſimiliter <lb/>inſtituetur. </s> <s xml:id="echoid-s4760" xml:space="preserve">In his & </s> <s xml:id="echoid-s4761" xml:space="preserve">aliis ſimilibus I*LLVSTRISSIMVS* P*RINCEPS* cer-<lb/>tiſſimis experimentis cognovit, Praxin Theoriæ exactiſſimè conſentire.</s> <s xml:id="echoid-s4762" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4763" xml:space="preserve">Proportionem 17 propoſitione à nobis deſcriptam, aliter quoq; </s> <s xml:id="echoid-s4764" xml:space="preserve">efferre & </s> <s xml:id="echoid-s4765" xml:space="preserve">effa-<lb/>ri poſſumus, unde uſus paulo facilior emanet Cujus explicationi diagramma id <lb/>oculis hic ſubjeci. </s> <s xml:id="echoid-s4766" xml:space="preserve">ubi pro eo quodita enuntia-<lb/>tur, ut põdus oblique attollĕs ad pondus attol-<lb/> <anchor type="figure" xlink:label="fig-527.01.163-01a" xlink:href="fig-527.01.163-01"/> lens rectè, ſic propriũ cujuſq́; </s> <s xml:id="echoid-s4767" xml:space="preserve">pondus obliquè <lb/>tollĕs ad pondus tollĕs rectè ut aliter efferam, <lb/>unde factio expeditior derivetur: </s> <s xml:id="echoid-s4768" xml:space="preserve">agatur LP, <lb/>inter lineas rectè & </s> <s xml:id="echoid-s4769" xml:space="preserve">obliquè attollentes, paral <lb/>lela contra F M, his poſitis, dico ut linea rectè <lb/>attollens, ad tollentem oblique, ſic totius co-<lb/>lumnæ pondus ad pondus ipſum tollens obli-<lb/>què, hoc eſt, ut EP ad EL, ſic pondus columnę <lb/>totius ad G. </s> <s xml:id="echoid-s4770" xml:space="preserve">& </s> <s xml:id="echoid-s4771" xml:space="preserve">rurſum ut E P ad P L, ſic pon-<lb/>dus columnæ ad H. </s> <s xml:id="echoid-s4772" xml:space="preserve">qua via ignotorum ter-<lb/>minorum inventio multò fit brevior & </s> <s xml:id="echoid-s4773" xml:space="preserve">ſuccin-<lb/>ctior. </s> <s xml:id="echoid-s4774" xml:space="preserve">Animadvertas item pro LP potuiſſe duci M Q, inter alteras rectè & </s> <s xml:id="echoid-s4775" xml:space="preserve">obli-<lb/>què extollentes lineas, parallelam contra EL, quâ ratiocinium, ut ſupra cum <lb/>L E, inire liceat. </s> <s xml:id="echoid-s4776" xml:space="preserve">namq́ue ut P E ad EL, ſic Q F ad FM, cùm triangula <lb/>FMQ & </s> <s xml:id="echoid-s4777" xml:space="preserve">L P E ſimilia ſint, ob parallelas Q F PE, MF LP.</s> <s xml:id="echoid-s4778" xml:space="preserve"/> </p> <div xml:id="echoid-div651" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.163-01" xlink:href="fig-527.01.163-01a"> <image file="527.01.163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.163-01"/> </figure> </div> </div> <div xml:id="echoid-div653" type="section" level="1" n="468"> <head xml:id="echoid-head496" xml:space="preserve">7 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s4779" xml:space="preserve">Hactenus pondera è duabus lineis de-<lb/>pendĕtia expoſita ſunt, ſequuntur deinceps <lb/> <anchor type="figure" xlink:label="fig-527.01.163-02a" xlink:href="fig-527.01.163-02"/> quæ pluribus lineis ſuſpenduntur. </s> <s xml:id="echoid-s4780" xml:space="preserve">Cui fini <lb/>quinti conſectarii diagramma aſſumamus, <lb/>hâc tantùm difſerentiâ, utrecta C G troch-<lb/>leam K hîc ſtrictim tangat, utrecta K C F <lb/>horizonti ſit obliqua, cæterum pondus AB <lb/>idem eſto, iidemq́ue anguli aſſumantur <lb/>D C F, F CE. </s> <s xml:id="echoid-s4781" xml:space="preserve">jam per 5 conſectarium patet <lb/>C I ad C H eſſe, ut pondus A B ad id quod <lb/>ſuſtinetur à D. </s> <s xml:id="echoid-s4782" xml:space="preserve">porro ut CI ad I H, ſic A B <lb/>ad id quod pertinet ad E. </s> <s xml:id="echoid-s4783" xml:space="preserve">Denique ut C H <lb/>ad H I, ſic id quod ab D ad id quod ab E <lb/>ſuſtinetur.</s> <s xml:id="echoid-s4784" xml:space="preserve"/> </p> <div xml:id="echoid-div653" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.163-02" xlink:href="fig-527.01.163-02a"> <image file="527.01.163-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.163-02"/> </figure> </div> <p> <s xml:id="echoid-s4785" xml:space="preserve">Ex quo efficitur ſi ab D C E, tanquam <lb/>fune, dependeat pondus A B manifeſtum <lb/>eſſe quantum pars quæq́ue D C, C E ſuffe-<lb/>rant.</s> <s xml:id="echoid-s4786" xml:space="preserve"/> </p> <pb o="164" file="527.01.164" n="164" rhead="A*DDITAMENTI* S*TATICÆ PARS PRIMA*"/> </div> <div xml:id="echoid-div655" type="section" level="1" n="469"> <head xml:id="echoid-head497" xml:space="preserve">8 C*ONSECTARIVM*.</head> <p> <s xml:id="echoid-s4787" xml:space="preserve">Si pondus tribus lineis ſuſpenſum ſit, ut hîc, ubi A B ſuſtinetur duabus C D, <lb/>C E, tumq́ue C E ab alteris dua-<lb/>bus EF, E G, ut univerſim totum <lb/> <anchor type="figure" xlink:label="fig-527.01.164-01a" xlink:href="fig-527.01.164-01"/> pondus A B è tribus lineis C D, <lb/>E F, E G dependeat, etiam tum <lb/>ſciri poterit quantum quæq́ue <lb/>ferat namq́ue per 5 conſ. </s> <s xml:id="echoid-s4788" xml:space="preserve">conclu-<lb/>detur quid ad C D & </s> <s xml:id="echoid-s4789" xml:space="preserve">C E perti-<lb/>neat: </s> <s xml:id="echoid-s4790" xml:space="preserve">deinde per 7 cõſectarium ſin-<lb/>gulis EF, EG ratam partem pon-<lb/>deris quod ad C E pertinet di-<lb/>ſtribues.</s> <s xml:id="echoid-s4791" xml:space="preserve"/> </p> <div xml:id="echoid-div655" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.164-01" xlink:href="fig-527.01.164-01a"> <image file="527.01.164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.164-01"/> </figure> </div> <p> <s xml:id="echoid-s4792" xml:space="preserve">Præterea etiam C D in duo alia <lb/>retinacula D H, D I diviſa, quid <lb/>illorum cujuſque propriumſit, eo-<lb/>dem quoque modo concludes. </s> <s xml:id="echoid-s4793" xml:space="preserve">quare quantum ponderis ſingulis lineis E F, <lb/>E G, DH, DI cedat ſiverectæ iſtæ in eodem ſint plano, ſeu in diverſis, co-<lb/>gnoſcere licebit.</s> <s xml:id="echoid-s4794" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4795" xml:space="preserve">Notato autem lineas C E G, C E F ac cæteras ſimiles non porrigi in di-<lb/>rectum, ſed ab ipſis ad E angulum neceſſariò comprehendi, cum E F ex <lb/> <anchor type="figure" xlink:label="fig-527.01.164-02a" xlink:href="fig-527.01.164-02"/> hypotheſi alicujus efficientiæ ſit, unde angulus exiſtit ad E, eadem mode <lb/>quoque recta E G aget in rectam C E F.</s> <s xml:id="echoid-s4796" xml:space="preserve"/> </p> <div xml:id="echoid-div656" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.164-02" xlink:href="fig-527.01.164-02a"> <image file="527.01.164-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.164-02"/> </figure> </div> <p> <s xml:id="echoid-s4797" xml:space="preserve">Præterea ſi ab F alia duo retinacula adjungantur F K, F L, etiam hic quan-<lb/>@m ponderis ad utramlibet ipſarum pertingat invenire in promptu eſt. </s> <s xml:id="echoid-s4798" xml:space="preserve">Itaque <pb o="165" file="527.01.165" n="165" rhead="*DE* S*PARTOSTATIOA*."/> <anchor type="figure" xlink:label="fig-527.01.165-01a" xlink:href="fig-527.01.165-01"/> quantum cuique harum quinque linearum D H, D I, F K, F L, E G cedat, <lb/>cognoſces. </s> <s xml:id="echoid-s4799" xml:space="preserve">quæ ratio infinitè continuari poteſt.</s> <s xml:id="echoid-s4800" xml:space="preserve"/> </p> <div xml:id="echoid-div657" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.165-01" xlink:href="fig-527.01.165-01a"> <image file="527.01.165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.165-01"/> </figure> </div> </div> <div xml:id="echoid-div659" type="section" level="1" n="470"> <head xml:id="echoid-head498" xml:space="preserve">9 C*ONSECTARIUM*.</head> <p> <s xml:id="echoid-s4801" xml:space="preserve">Dictum fuit hactenus de pondere, ut A B, quæ ab una continua linea uſque <lb/>ad C dependeat, unde deinde duæ alteræ exiſtant. </s> <s xml:id="echoid-s4802" xml:space="preserve">Sed quando ex eo puncto <lb/>tres lineæ educentur, conſideratio paulo diverſa erit; </s> <s xml:id="echoid-s4803" xml:space="preserve">utque hoc diſtinctè pro-<lb/>ponam, ſic habe, tres iſtas lineas vel eſſein uno plano, vel in diverſis: </s> <s xml:id="echoid-s4804" xml:space="preserve">atque cum <lb/>in eodem erunt plano, nihil certò concludi poſſe. </s> <s xml:id="echoid-s4805" xml:space="preserve">Sit enim pondus A B, tresq́; <lb/></s> <s xml:id="echoid-s4806" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.165-02a" xlink:href="fig-527.01.165-02"/> lineæ ex quibus pendet C D, C E, C F, à quarum communi termino C ſit <lb/>CG, unde pondus ſuſpendatur. </s> <s xml:id="echoid-s4807" xml:space="preserve">I am ſublato medio retinaculo F C, à duabus <lb/>duntaxat lineis C D, C E pondus A B ſuſtineatur. </s> <s xml:id="echoid-s4808" xml:space="preserve">quibus poſitis, pondus A B <lb/>tamen loco non movetur, anguliq́ue D C G, E C G, iidem permanebunt, <lb/>quamvis plus ponderis ab iſtis duabus C D, C E, quàm prius addita C F ſu-<lb/>ſtineatur; </s> <s xml:id="echoid-s4809" xml:space="preserve">cum hæc ſua potentia iſtas allevet: </s> <s xml:id="echoid-s4810" xml:space="preserve">ſed ad C F potentiæ variæ & </s> <s xml:id="echoid-s4811" xml:space="preserve">mul- <pb o="166" file="527.01.166" n="166" rhead="A*DDITAMENTI* S*TATICÆ PARS PRIMA*"/> tùm diverſe collocari poſſunt, unde, ut dixi, palam eſt unam certamq́ue con-<lb/>cluſionem non admittere.</s> <s xml:id="echoid-s4812" xml:space="preserve"/> </p> <div xml:id="echoid-div659" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.165-02" xlink:href="fig-527.01.165-02a"> <image file="527.01.165-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.165-02"/> </figure> </div> </div> <div xml:id="echoid-div661" type="section" level="1" n="471"> <head xml:id="echoid-head499" xml:space="preserve">10 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s4813" xml:space="preserve">Cumautem tresiſtæ retinaculorum lineæ in duobus conſtituentur planis, <lb/>concluſio duntaxat unica, certaq́; </s> <s xml:id="echoid-s4814" xml:space="preserve">erit. </s> <s xml:id="echoid-s4815" xml:space="preserve">Vtſi pondus A B è tribus lineis C D, <lb/>C E, C F dependeat, quæ omnes tamen in eodem plano non ſint, & </s> <s xml:id="echoid-s4816" xml:space="preserve">ab harum <lb/>concurſu unicalinea C G porrigatur, unde pondusex G dependeat. </s> <s xml:id="echoid-s4817" xml:space="preserve">Iam quî <lb/>invenias, quantum quæq́ue ponderis ferat, ut v. </s> <s xml:id="echoid-s4818" xml:space="preserve">g. </s> <s xml:id="echoid-s4819" xml:space="preserve">CF. </s> <s xml:id="echoid-s4820" xml:space="preserve">fingas cogitatione, ſo-<lb/>lummodo à recta C F & </s> <s xml:id="echoid-s4821" xml:space="preserve">à H C cõmuni ſectione planorũ D C E. </s> <s xml:id="echoid-s4822" xml:space="preserve">G C F pon-<lb/>dus A B eſſeíuſpenſum. </s> <s xml:id="echoid-s4823" xml:space="preserve">reliquis lineis ſublatis, patet igitur angulum G C F <lb/>omninò non mutari, ſed eundem eſſe qui priùs fuerat, itemq́ue C F pondus <lb/> <anchor type="figure" xlink:label="fig-527.01.166-01a" xlink:href="fig-527.01.166-01"/> idem ferre quod ante. </s> <s xml:id="echoid-s4824" xml:space="preserve">fiinquam, fingas pondus A B è duabus lineis, C F, C H <lb/>dependere, tumq́; </s> <s xml:id="echoid-s4825" xml:space="preserve">pers conſectarium, quanto in ſuſtinendo pondere recta C F <pb o="167" file="527.01.167" n="167" rhead="*DE* S*PARTOSTATICA.*"/> diſtendatur concludes. </s> <s xml:id="echoid-s4826" xml:space="preserve">ac pari ratione quo pondere reliquæ C D, C E diſti-<lb/>neantur cognoſces.</s> <s xml:id="echoid-s4827" xml:space="preserve"/> </p> <div xml:id="echoid-div661" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.166-01" xlink:href="fig-527.01.166-01a"> <image file="527.01.166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.166-01"/> </figure> </div> <p> <s xml:id="echoid-s4828" xml:space="preserve">Præterea ſi hæ retinaculorum lineæ in alias inſuper lineas diducantur, pon-<lb/>dus quo ipſarum unaquæque diſtenditur, conſimiliter 9 conſectario conclu-<lb/>detur.</s> <s xml:id="echoid-s4829" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div663" type="section" level="1" n="472"> <head xml:id="echoid-head500" xml:space="preserve">11 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s4830" xml:space="preserve">Si quatuor lineæ adidem punctum cohæreſcant, cujuſmodi antecedenti cõ-<lb/>ſectario tres, propoſitio hæc unam certamq́ue determinationem non habet. <lb/></s> <s xml:id="echoid-s4831" xml:space="preserve">Sic A, B, C, D tanquam ſuprema linearum, è quibus pondus dependet, <lb/>puncta intelligantor. </s> <s xml:id="echoid-s4832" xml:space="preserve">lam pendula ejus diameter vel incidet in rectam A D, vel <lb/>extra ipſam intra trian gulum A D B, vel intra A D C (fieri enim non poteſt ut in <lb/>ambitu quadranguli A B C D cadat, nedum extra) ſi incidat in A D, conſtat <lb/>rectas ad B & </s> <s xml:id="echoid-s4833" xml:space="preserve">C pertingentes, potentiam quâ iſtæ ſub A & </s> <s xml:id="echoid-s4834" xml:space="preserve">D diſtenduntur <lb/>quiddam allevare; </s> <s xml:id="echoid-s4835" xml:space="preserve">ſed cùm triangulũ dua-<lb/>rum iſtarum linearum quæ ſub A, B, con-<lb/> <anchor type="figure" xlink:label="fig-527.01.167-01a" xlink:href="fig-527.01.167-01"/> ſiſtunt & </s> <s xml:id="echoid-s4836" xml:space="preserve">tertiæ A D, nullam admittat mu-<lb/>tationem varietatemvé, diverſę & </s> <s xml:id="echoid-s4837" xml:space="preserve">multipli-<lb/>ces potĕtiæ, ad C & </s> <s xml:id="echoid-s4838" xml:space="preserve">B adjungi poſſunt quę <lb/>linearũ ſub A, D diſtenſionem immutent, <lb/>manente tamen eâdĕ datæ formæ diſpoſi-<lb/>tione, adeò ut certa ſingularũ partium de-<lb/>terminatio nulla hic inveniri poſſit Cumautem pendula gravitatis diameter in <lb/>E intra triangulum A D B cadet, tum quarta C à mutatione ſitus ponderum <lb/>quæ ad A, B, C pertinent planè erit immunis. </s> <s xml:id="echoid-s4839" xml:space="preserve">ex quibus efficitur propoſitio-<lb/>nem hujuſmodinullam partium ratam determinationem admittere.</s> <s xml:id="echoid-s4840" xml:space="preserve"/> </p> <div xml:id="echoid-div663" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.167-01" xlink:href="fig-527.01.167-01a"> <image file="527.01.167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.167-01"/> </figure> </div> <p> <s xml:id="echoid-s4841" xml:space="preserve">Advertendum autem inſuper, cùm quatuor lineæ certam aliquam determi-<lb/>nationem reſpuant, multò firmioriratione complurium linearum concluſio-<lb/>nem incertam eſſe. </s> <s xml:id="echoid-s4842" xml:space="preserve">Similiq́ue ratione, cum 9 conſectario demonſtratum ſit, <lb/>tres lineas in eodem plano nullam certam cõcluſionem habere, etiam quatuor, <lb/>aliasq́ue plures longè minùs determinatione aliqua certa circumſcribi.</s> <s xml:id="echoid-s4843" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div665" type="section" level="1" n="473"> <head xml:id="echoid-head501" xml:space="preserve">NOTATO</head> <p> <s xml:id="echoid-s4844" xml:space="preserve">Corpus etiam modo ab 11 conſectario diverſo è tribus lineis depĕdere poſſe, <lb/>cùm ſcilicet tribus diſtantibus locis ipſæ corpori affigentur, ut continuatæ ta-<lb/>men in eodem puncto non concurrant; </s> <s xml:id="echoid-s4845" xml:space="preserve">quod per 25 propoſ. </s> <s xml:id="echoid-s4846" xml:space="preserve">lib. </s> <s xml:id="echoid-s4847" xml:space="preserve">1. </s> <s xml:id="echoid-s4848" xml:space="preserve">ſide duabus <lb/>tantum lineis ſuſpendantur neceſſariò cõtingit. </s> <s xml:id="echoid-s4849" xml:space="preserve">Sed quâ viâ inveniatur pondus <lb/>iſtarum unicuique debitum, nondum etiam, dum hæctypis excuderentur, aſſe-<lb/>cutus eram: </s> <s xml:id="echoid-s4850" xml:space="preserve">ſi quid vel à meipſo, vel ab alio quopiam hoc problema juvabi-<lb/>tur, id temporis proceſſu fiet palam.</s> <s xml:id="echoid-s4851" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div666" type="section" level="1" n="474"> <head xml:id="echoid-head502" xml:space="preserve">12 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s4852" xml:space="preserve">Ponderis igitur ab unâlineâ ſuſpenſi, ex qua deinde duæ treſvé aliæ in partes <lb/>diverſas diſtractæ exiſtant, ratio hujuſmodi fuit; </s> <s xml:id="echoid-s4853" xml:space="preserve">unde affectiones Staticæ dua-<lb/>rum triumvé itidem linearum, eidem ponderi affixarum & </s> <s xml:id="echoid-s4854" xml:space="preserve">ſurſum tendentium, <lb/>inq́ue idem pendulæ gravitatis diametri punctum incurrentium, in procinctu <lb/>erunt. </s> <s xml:id="echoid-s4855" xml:space="preserve">enimverò A B pondus eſto, de duabus lineis D C, E C ſuſpenſum, inq́ <lb/>C puncto concurrunto, pendulaq́ue diameter F C. </s> <s xml:id="echoid-s4856" xml:space="preserve">quantum igitur harum <pb o="168" file="527.01.168" n="168" rhead="A*DDIT.* S*TAT. PARS PRIMA DE* S*PARTOSTAT.*"/> D C, E C utrique cedat. </s> <s xml:id="echoid-s4857" xml:space="preserve">Cognoſces continuatâ F C deorſum in G, actâque <lb/>H I, interipſam & </s> <s xml:id="echoid-s4858" xml:space="preserve">D C, parallelâ contra C E. </s> <s xml:id="echoid-s4859" xml:space="preserve">Iam enim per 5 conſectarium erit <lb/>C I ad C H, ut pondus A B ad id quod ſuſtinetur ex D: </s> <s xml:id="echoid-s4860" xml:space="preserve">item C I ad I H, ut <lb/> <anchor type="figure" xlink:label="fig-527.01.168-01a" xlink:href="fig-527.01.168-01"/> pondus A B ad id quod ex E. </s> <s xml:id="echoid-s4861" xml:space="preserve">Denique etiam C H ad H I, ut pondus quodâ <lb/>D, adid quod ſuſtinetur ab E.</s> <s xml:id="echoid-s4862" xml:space="preserve"/> </p> <div xml:id="echoid-div666" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.168-01" xlink:href="fig-527.01.168-01a"> <image file="527.01.168-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.168-01"/> </figure> </div> <p> <s xml:id="echoid-s4863" xml:space="preserve">Inſuper autem liquet proprietates, cujuſmodi 8, 9, 10, & </s> <s xml:id="echoid-s4864" xml:space="preserve">11 conſectario ac-<lb/>cidunt, etiam in ſimiles 13 hujus conſectarii figuras incidere.</s> <s xml:id="echoid-s4865" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div668" type="section" level="1" n="475"> <head xml:id="echoid-head503" xml:space="preserve">S*PARTOSTATICES*</head> <head xml:id="echoid-head504" xml:space="preserve">FINIS.</head> <pb file="527.01.169" n="169"/> </div> <div xml:id="echoid-div669" type="section" level="1" n="476"> <head xml:id="echoid-head505" xml:space="preserve">ADDIT AMENTI <lb/>STATICÆ <lb/>PARS SECVNDA. <lb/>DE <lb/>TROCHLEOSTATICA.</head> <pb file="527.01.170" n="170"/> </div> <div xml:id="echoid-div670" type="section" level="1" n="477"> <head xml:id="echoid-head506" xml:space="preserve">BREVIARIVM</head> <head xml:id="echoid-head507" xml:space="preserve">TROCHLEOSTATICES.</head> <p style="it"> <s xml:id="echoid-s4866" xml:space="preserve">CV*M* I*LLVSTR ISSIMVS* P*RIN CEPS* <lb/>perſpexiſſet librum delle Fortificationi di Buo-<lb/>najuto Lorini, inter cæter a etiam trochlearum <lb/>doctrinam perlegit, ubitantummodo ponderum <lb/>rectà adſcendentium, à potentiis rectà ad per-<lb/>pendiculum deſcendentibus explicatur: </s> <s xml:id="echoid-s4867" xml:space="preserve">quia verò non ſemper <lb/>iſtarectà ſurſum, deorſumvé, ſed in obliquum nonunquam <lb/>moventur, harum quoque potentiarum, rationum, cauſa-<lb/>rumq́ue cognoſcendi cupiditas ipſum inceſsit, ut doctrinam <lb/>hanc omnibus numeris haberet perfectam. </s> <s xml:id="echoid-s4868" xml:space="preserve">Et ſane cognitionis <lb/>& </s> <s xml:id="echoid-s4869" xml:space="preserve">ſcientiæ hujus cupiditas juſtis rationibus innixa videtur, <lb/>cùm trochlearum uſus in majorum ponderum ſubvectationi-<lb/>bus permagnus ſit; </s> <s xml:id="echoid-s4870" xml:space="preserve">& </s> <s xml:id="echoid-s4871" xml:space="preserve">utile nonunquam fuerit præſciri, quæ <lb/>potentia dato ponderi attollendo ſit congrua. </s> <s xml:id="echoid-s4872" xml:space="preserve">Quare poſtquam <lb/>in Statica, & </s> <s xml:id="echoid-s4873" xml:space="preserve">prima hujus additamĕtiparte ſeſe I*LLUSTRISS*. <lb/></s> <s xml:id="echoid-s4874" xml:space="preserve">P*RINCEPS* exercuiſſet, quibus cognitis trochlearum proprie-<lb/>tates & </s> <s xml:id="echoid-s4875" xml:space="preserve">affectiones deinceps eſſent in promptu, at ſe jam huic <lb/>negotio daret; </s> <s xml:id="echoid-s4876" xml:space="preserve">ista quoque, quæ ſuper his tractata nobis ſunt, <lb/>inter Mathematica ejus Hypomnemata referenda cenſui.</s> <s xml:id="echoid-s4877" xml:space="preserve"/> </p> <pb o="171" file="527.01.171" n="171"/> </div> <div xml:id="echoid-div671" type="section" level="1" n="478"> <head xml:id="echoid-head508" xml:space="preserve">PROPOSITIO.</head> <p> <s xml:id="echoid-s4878" xml:space="preserve">Ponderum trochleis ſublimè tractorum formas inqui-<lb/>rere.</s> <s xml:id="echoid-s4879" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4880" xml:space="preserve">PRiuſquam rem ipſam exordimur generaliter intelligito, & </s> <s xml:id="echoid-s4881" xml:space="preserve">cogitatione <lb/>concipito, datum pondus hic conſtitui à trochlea infima cum pon-<lb/>dere ipſi alligato: </s> <s xml:id="echoid-s4882" xml:space="preserve">præterea differentiam gravitatis quæ à funibus exiſtit, <lb/>nullius momentià nobis nunc æſtimari.</s> <s xml:id="echoid-s4883" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div672" type="section" level="1" n="479"> <head xml:id="echoid-head509" xml:space="preserve">I Exemplum ponderum quærectà attolluntur.</head> <p> <s xml:id="echoid-s4884" xml:space="preserve">Eſto in primo hoc diagrammate trochlea A, ex qua dependet pondus B, fu-<lb/>nis C D E F, cujus duæ partes C D, F E parallelæ ſint, & </s> <s xml:id="echoid-s4885" xml:space="preserve">utraque horizonti <lb/>perpendicularis, Quibus poſitis, totoq́ue pondere B ita è duabus iſtis partibus <lb/>C D, F E ſuſpenſo, ututraque pars pari potentia afficiatur, etiam ſingulis pro-<lb/> <anchor type="figure" xlink:label="fig-527.01.171-01a" xlink:href="fig-527.01.171-01"/> pter orbiculi volubilitatem cedet ſemiſſis ponderis B. </s> <s xml:id="echoid-s4886" xml:space="preserve">quamobrem ſi quis ma-<lb/>nu ſua funem in F ſuſtineat, is ferret gravitatem dimidii ponderis B, ex quo li- <pb o="172" file="527.01.172" n="172" rhead="A*DDITAMENTI* S*TATICÆ PARS SECUNDA*"/> quet, cur etiam unica trochlea facilius, quam ſine ea pondus attollatur. </s> <s xml:id="echoid-s4887" xml:space="preserve">Notato <lb/>autem hic illud Staticum axioma etiam locum habere:</s> <s xml:id="echoid-s4888" xml:space="preserve"/> </p> <div xml:id="echoid-div672" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.171-01" xlink:href="fig-527.01.171-01a"> <image file="527.01.171-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.171-01"/> </figure> </div> <p> <s xml:id="echoid-s4889" xml:space="preserve">Vt ſpatium agentis, ad ſpatium patients:</s> <s xml:id="echoid-s4890" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4891" xml:space="preserve">Sic potentia patientis, ad potentiam agentis.</s> <s xml:id="echoid-s4892" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4893" xml:space="preserve">Nam manu F, quæ hîc agit; </s> <s xml:id="echoid-s4894" xml:space="preserve">duos pedes promota, pondus, quod patitur uni-<lb/>cum duntaxat pedem procedet: </s> <s xml:id="echoid-s4895" xml:space="preserve">cujus cauſa manifeſta eſt.</s> <s xml:id="echoid-s4896" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4897" xml:space="preserve">Ex his ubi unici orbiculi rotatu pondus attollitur, facilè cognoſcetur ſimi-<lb/>lium formarum rauo in trochlea geminata, ut hic. </s> <s xml:id="echoid-s4898" xml:space="preserve">ubirurſum C alterum funis <lb/> <anchor type="figure" xlink:label="fig-527.01.172-01a" xlink:href="fig-527.01.172-01"/> terminum denotat. </s> <s xml:id="echoid-s4899" xml:space="preserve">Et cùm pondus B de tribus funibus dependeat ſingulis <lb/>duntaxat cedit ponderis una tertia, quare manus F tertiam tantum partem <lb/>ſuſtinebit ponderis B.</s> <s xml:id="echoid-s4900" xml:space="preserve"/> </p> <div xml:id="echoid-div673" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.172-01" xlink:href="fig-527.01.172-01a"> <image file="527.01.172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.172-01"/> </figure> </div> <pb o="173" file="527.01.173" n="173" rhead="*DE* T*ROCHLEOSTATICA*."/> <p> <s xml:id="echoid-s4901" xml:space="preserve">Similiter cum tribus orbiculis fanis ductorius obvolvetur, ſingulis partibus <lb/>cedet ponderis B pars quarta, ideoq́ue F manus feret partem quartam pon-<lb/> <anchor type="figure" xlink:label="fig-527.01.173-01a" xlink:href="fig-527.01.173-01"/> deris B. </s> <s xml:id="echoid-s4902" xml:space="preserve">Vnde ſimili proceſſu generale axioma ponderum pluribus etiam <lb/>trochleis tractorum inſtitui & </s> <s xml:id="echoid-s4903" xml:space="preserve">efformari poteſt.</s> <s xml:id="echoid-s4904" xml:space="preserve"/> </p> <div xml:id="echoid-div674" type="float" level="2" n="3"> <figure xlink:label="fig-527.01.173-01" xlink:href="fig-527.01.173-01a"> <image file="527.01.173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.173-01"/> </figure> </div> <p> <s xml:id="echoid-s4905" xml:space="preserve">Advertendum autem rariſſimè F hoc modo ſurſum duci, quod nos in tribus <lb/>expoſitis diagrammatis clarioris demonſtrationis gratia fecimus, ſed plerumq; <lb/></s> <s xml:id="echoid-s4906" xml:space="preserve">additur inſuper orbiculus unus, ut per ejus ambitum ductus funis deorſum tra-<lb/>hatur, ut quarto hoc diagrammate ſpectandum exhibemus. </s> <s xml:id="echoid-s4907" xml:space="preserve">notandum tamen <lb/>quartum iſtum adventitium orbiculum manui nullam ponderis allevationem <lb/>mutationem vé in ducere; </s> <s xml:id="echoid-s4908" xml:space="preserve">quia pondus B è quatuor tantùm funibus, perinde <lb/>atque in tertiâ diagraphâ ſuſtinetur; </s> <s xml:id="echoid-s4909" xml:space="preserve">nam noviſſimus iſte funis qui quintus vi-<lb/>deri poſſet, ipſe unus idemq́ue eſt cum quarto. </s> <s xml:id="echoid-s4910" xml:space="preserve">Ex quo intelligitur etiamſi fu-<lb/>nis ductorius per centum iſtiuſmodi trochleas traducatur, trahentem planè ni-<lb/>hil juvari.</s> <s xml:id="echoid-s4911" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4912" xml:space="preserve">Verumenimverò ſi hujus mechanicam veritatem deſideres, in F loco ma-<lb/>nus, quartâ hâc figurâ ſubſtituito pondus æquale quartæ parti ponderis attol-<lb/>lendi, & </s> <s xml:id="echoid-s4913" xml:space="preserve">hæc (ſi opus rite inſtitutum ſit) inter ſe ſitu erunt æquilibria. </s> <s xml:id="echoid-s4914" xml:space="preserve">pondus <lb/>autem tollendum, ut accuratè definiatur, notato id hic conſtitui à dato ponde-<lb/>re B, trochleâ imâ A, atque inſuper a gravitate funis. </s> <s xml:id="echoid-s4915" xml:space="preserve">atqui ut funis gravitatem <pb o="174" file="527.01.174" n="174" rhead="A*DDITAMENTI* S*TATICÆ PARS SECUNDA*"/> fuſius explicem, ſunto D, E extimi ejus contactus contra orbiculum A; </s> <s xml:id="echoid-s4916" xml:space="preserve">Ttem <lb/>G, H funis novifſimi contactus contra trochleam ſupremam, itemq́ue L, M, <lb/>contactus trochleæ ultimæ atque N ſit loco contactuum G, H, interme-<lb/> <anchor type="figure" xlink:label="fig-527.01.174-01a" xlink:href="fig-527.01.174-01"/> dio, ſic O inter I K. </s> <s xml:id="echoid-s4917" xml:space="preserve">ſitq́ue C finis ipſius funis. </s> <s xml:id="echoid-s4918" xml:space="preserve">Hinc P ſtatuatur in G ut G P <lb/>ipſi H F æquetur; </s> <s xml:id="echoid-s4919" xml:space="preserve">ſic Q in K D, ut K Q, I L æquales ſint. </s> <s xml:id="echoid-s4920" xml:space="preserve">Quibus poſitis, <lb/>N G P pondere & </s> <s xml:id="echoid-s4921" xml:space="preserve">magnitudine ęquatur ipſi N H F; </s> <s xml:id="echoid-s4922" xml:space="preserve">itemq́; </s> <s xml:id="echoid-s4923" xml:space="preserve">O I L ipſi O K Q. <lb/></s> <s xml:id="echoid-s4924" xml:space="preserve">ſed C M omninò nec degravat nec allevat; </s> <s xml:id="echoid-s4925" xml:space="preserve">adeò ut ad pondus datum & </s> <s xml:id="echoid-s4926" xml:space="preserve">gravita-<lb/>tem trochleæ accedant inſuper ſemicircularia funis ſegmenta L M, D E, cum <lb/>recto ſegmento D Q.</s> <s xml:id="echoid-s4927" xml:space="preserve"/> </p> <div xml:id="echoid-div675" type="float" level="2" n="4"> <figure xlink:label="fig-527.01.174-01" xlink:href="fig-527.01.174-01a"> <image file="527.01.174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.174-01"/> </figure> </div> <p> <s xml:id="echoid-s4928" xml:space="preserve">Denique animadvertito cum trochlearum ope quid attollitur, ut pars com-<lb/>moti funis in aëre ſuſpenſus à ſolo abſit: </s> <s xml:id="echoid-s4929" xml:space="preserve">quanta erit iſtius ductorii funis gravi-<lb/>tas, tantò minori potentia ductori opus eſſe.</s> <s xml:id="echoid-s4930" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div677" type="section" level="1" n="480"> <head xml:id="echoid-head510" xml:space="preserve">2 Exemplum <anchor type="note" xlink:href="" symbol="*"/> {λο}ξοβα{ρί}ας</head> <note symbol="*" position="right" xml:space="preserve">nbi pondera <lb/>in obliquum <lb/>@mmovisht</note> <p> <s xml:id="echoid-s4931" xml:space="preserve">Primum hoc diagramma cætera ſimile eſto primo primi exempli diagram-<lb/>mati, ſolummodo hîc manus F non recta ſurſum ſed obliquè & </s> <s xml:id="echoid-s4932" xml:space="preserve">in latus non <lb/>nihil commoveatur; </s> <s xml:id="echoid-s4933" xml:space="preserve">quibus poſitis, pondus quod à funium unoquoque ſuffer-<lb/>tur, per 5 conſectariũ primæ partis hujus ad Staticam additamenti pateſcet. </s> <s xml:id="echoid-s4934" xml:space="preserve">Sed <lb/>@t explicatione fiat illuſtrius; </s> <s xml:id="echoid-s4935" xml:space="preserve">continuato rectam, ex quâ pondus B dependet, <pb o="175" file="527.01.175" n="175" rhead="*DE* T*ROCHLEOSTATICA*."/> ſurſum in G; </s> <s xml:id="echoid-s4936" xml:space="preserve">itemq́ue B G, F E donec ſeſe interſecent in H; </s> <s xml:id="echoid-s4937" xml:space="preserve">atqueipſas in-<lb/>terſecet I K parallela contra D C. </s> <s xml:id="echoid-s4938" xml:space="preserve">His ita diſpoſitis, ajo eſſe I K ad K H, ut <lb/>pondus ab F manu ductum ad datum B; </s> <s xml:id="echoid-s4939" xml:space="preserve">itemq́ue ut H I ad I K, (quæ in exem-<lb/>plis unius trochleæ, quale hoc eſt, perpetuò æquantur, quia continuatam C D <lb/>in H occurrere neceſſe eſt; </s> <s xml:id="echoid-s4940" xml:space="preserve">angulumq́ue G H I angulo G H C æquari) ſic <lb/>pondus quod à manu F ad id quod ſuſtinetur à C. </s> <s xml:id="echoid-s4941" xml:space="preserve">has potentias ob cauſas <lb/>jam expoſitas itidem in unius trochleæ exemplo æquari manifeſtum eſt, ſingu-<lb/> <anchor type="figure" xlink:label="fig-527.01.175-01a" xlink:href="fig-527.01.175-01"/> lis quippe ponderis ſemiſſem perferentibus; </s> <s xml:id="echoid-s4942" xml:space="preserve">ponderis inquam, cujus ad datum <lb/>pondus ratio ſit, per 5 conſectarium 1 partis additamenti ad Staticam, quæ H K <lb/>ad H I.</s> <s xml:id="echoid-s4943" xml:space="preserve"/> </p> <div xml:id="echoid-div677" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.175-01" xlink:href="fig-527.01.175-01a"> <image file="527.01.175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.175-01"/> </figure> </div> <p> <s xml:id="echoid-s4944" xml:space="preserve">Sed fune ductorio hoc obliquo circa duas pluresvé trochleas voluto, uni-<lb/>verſa item ponderum ratio cognoſcetur. </s> <s xml:id="echoid-s4945" xml:space="preserve">Etenim, dicis gratia, ſecunda figura <lb/>omninò ſimilis effingatur ſecundæ figuræ primi exempli, tantum hoc uno di-<lb/>verſæ ſint, quod manus F hîc obliquè & </s> <s xml:id="echoid-s4946" xml:space="preserve">in latus ſurſum trahat. </s> <s xml:id="echoid-s4947" xml:space="preserve">Iam igitur per <lb/>5 conſectarium 1 partis hujus Additamenti quantum ponderis ſingulis funibus <lb/>cedat manifeſtè liquet. </s> <s xml:id="echoid-s4948" xml:space="preserve">Cujus declarationi exemplum tale eſto. </s> <s xml:id="echoid-s4949" xml:space="preserve">Recta ex qua <lb/>pondus dependet ſurſum educitor in G ut B G, tumq́ue F E continuata, ſe-<lb/>cet infinitam B G in H. </s> <s xml:id="echoid-s4950" xml:space="preserve">& </s> <s xml:id="echoid-s4951" xml:space="preserve">à puncto I ſuprema trochlea dependeat, unde <lb/>ad H adjungatur recta H I, cui inter F H & </s> <s xml:id="echoid-s4952" xml:space="preserve">G H parallela agatur K L. <lb/></s> <s xml:id="echoid-s4953" xml:space="preserve">His poſitis, ajo ut K H ad L H, ſic pondus à manu ſuſtentatum ad pondus da- <pb o="176" file="527.01.176" n="176" rhead="A*DDITAMENTI* S*TATICÆ PARS SECUNDA*"/> tum: </s> <s xml:id="echoid-s4954" xml:space="preserve">ſed K H in exemplo geminarum trochlearum, quale hoc eſt, æquatur di-<lb/>midiæ K L; </s> <s xml:id="echoid-s4955" xml:space="preserve">quare potentia pertinens ad F, eſt dimidia ejus quæ ſuſtinetur ab <lb/>I, quare ſingulis funibus par cedit onus, ſcilicettriens ponderis, cujus ad da-<lb/> <anchor type="figure" xlink:label="fig-527.01.176-01a" xlink:href="fig-527.01.176-01"/> tum ratio eſt quæ L H ad H K. </s> <s xml:id="echoid-s4956" xml:space="preserve">quamobrem in ſimilibus exemplis concludes <lb/>hoc modo, ut K H ad H L, ſic datum pondusad quod aliud? </s> <s xml:id="echoid-s4957" xml:space="preserve">ejus, quod hinc <lb/>efficitur, pars tertia dabit vim ponderis ab F ſuſtentati; </s> <s xml:id="echoid-s4958" xml:space="preserve">itemq́ue quantum reli-<lb/>quorum funium ſingulis cedat. </s> <s xml:id="echoid-s4959" xml:space="preserve">Cum autem tres proponenturtrochleæ, ma-<lb/>nifeſtum eſt quartam partem ponderis, hujuſmodi proportione concluſi, aſſu-<lb/>mendam. </s> <s xml:id="echoid-s4960" xml:space="preserve">atque eo deinceps in cæteris progreſſu continuo.</s> <s xml:id="echoid-s4961" xml:space="preserve"/> </p> <div xml:id="echoid-div678" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.176-01" xlink:href="fig-527.01.176-01a"> <image file="527.01.176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.176-01"/> </figure> </div> <p> <s xml:id="echoid-s4962" xml:space="preserve">Curautem K L potiùs parallela nobis ſit conſtituta contra H I, quàm con-<lb/>tra funes ductorios, id ex his quæ deſimilibus, adſecundum tertiumq́ue con-<lb/>ſectarium primæ partis hujus additamenti dicta nobis ſunt, in promptu erit. <lb/></s> <s xml:id="echoid-s4963" xml:space="preserve">nam totius machinæ & </s> <s xml:id="echoid-s4964" xml:space="preserve">ponderis pendula diametros tendit per punctum H, à <lb/>quo manifeſtò duæ lineæ, ex quibus ratiocinium inſtituendum, ſunt educen-<lb/>dæ. </s> <s xml:id="echoid-s4965" xml:space="preserve">C*ONCLVSIO*. </s> <s xml:id="echoid-s4966" xml:space="preserve">Itaque ponderum trochleis ſublatorum tormas, ut po-<lb/>ſtulabatur, inquiſivimus.</s> <s xml:id="echoid-s4967" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div680" type="section" level="1" n="481"> <head xml:id="echoid-head511" xml:space="preserve">T*ROCHLEOSTATICÆ*</head> <head xml:id="echoid-head512" xml:space="preserve">FINIS.</head> <pb file="527.01.177" n="177"/> </div> <div xml:id="echoid-div681" type="section" level="1" n="482"> <head xml:id="echoid-head513" xml:space="preserve">ADDITAMENTI <lb/>STATICÆ <lb/>PARS TERTIA <lb/>DE <lb/>FLVITANTIBVS <lb/>ACROBARICIS.</head> <pb file="527.01.178" n="178"/> </div> <div xml:id="echoid-div682" type="section" level="1" n="483"> <head xml:id="echoid-head514" xml:space="preserve">BREVIARIVM <lb/>Fluitantium Acrobaricorum.</head> <p style="it"> <s xml:id="echoid-s4968" xml:space="preserve">ACcidit aliquando, ut in navibus ſcalæ 20 pedes altæ <lb/>erigendæ eſſent, per quas milites ar matiadſcenderĕt; <lb/></s> <s xml:id="echoid-s4969" xml:space="preserve">quia autem verendum erat ne in ſummo prægraves <lb/>naviculæ everterentur, milites ſimul in aquam præcipita-<lb/>rentur, unius fabrica, unde experimentum tutò ſumipoſſet <lb/>primùm fuit instituta. </s> <s xml:id="echoid-s4970" xml:space="preserve">Quæ res mihi cogitationis hujus ini-<lb/>tium injecit, utrum id Statico rationcinio data figura, data{q́ue} <lb/>gravitate demonſtrari & </s> <s xml:id="echoid-s4971" xml:space="preserve">concludi poſſet. </s> <s xml:id="echoid-s4972" xml:space="preserve">Cuifini hoc Theore-<lb/>ma invenimus deſcripſimus{q́ue}: </s> <s xml:id="echoid-s4973" xml:space="preserve">Et ſipeculiari appellatione pla-<lb/>ceret inſigniri, à finè & </s> <s xml:id="echoid-s4974" xml:space="preserve">uſu potiſsimo Theorema de Fluctuan. </s> <s xml:id="echoid-s4975" xml:space="preserve"><lb/>tibus Acrobaricis indigitari poſſet, ideſt de Acrobaricis, ſive in <lb/>ſummo prægravibus, quæ aquæ inſident innatant{q́ue}: </s> <s xml:id="echoid-s4976" xml:space="preserve">nam de cæ-<lb/>teris Acrobaricis, quæ in ſolido ſolo ponuntur, quæque tum ca-<lb/>dunt, cùm pendula gravitatis diameter in latus incidit atque <lb/>extra baſin propendet, diſſerere nunc noninstitui.</s> <s xml:id="echoid-s4977" xml:space="preserve"/> </p> <pb o="179" file="527.01.179" n="179"/> </div> <div xml:id="echoid-div683" type="section" level="1" n="484"> <head xml:id="echoid-head515" xml:space="preserve">THEOREMA.</head> <p> <s xml:id="echoid-s4978" xml:space="preserve">Corpus fluitans hunc ſibi vindicat ſitum, ut ſuæ gravi-<lb/>tatis centrum ſit in ſegmenti in aqua demerſi pendula gra-<lb/>vitatis diametro.</s> <s xml:id="echoid-s4979" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s4980" xml:space="preserve">DA*tum.</s> <s xml:id="echoid-s4981" xml:space="preserve">* Corpus ABCD innatetaquæ EFGH, ſuperna ſuper-<lb/>ficies EF, in quam corpus uſquead IK demerſum ſit, arque adeò <lb/>ICK cavitatem aqueam denotet, & </s> <s xml:id="echoid-s4982" xml:space="preserve">hujus gravitatis centrum L, <lb/>pendula autem diameter MLN, denique totius corporis ABCD <lb/>centrum O. </s> <s xml:id="echoid-s4983" xml:space="preserve">Q*VAESITVM.</s> <s xml:id="echoid-s4984" xml:space="preserve">* Corporis ABCD gravitatis centrum O in <lb/>cavitatis aqueæ ICK pendula gravitatis diametro MN conſiſtere demon-<lb/>ſtrato.</s> <s xml:id="echoid-s4985" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div684" type="section" level="1" n="485"> <head xml:id="echoid-head516" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s4986" xml:space="preserve">Exempto exaqua ſolido ABCD, animo concipito iſtam cavitatem aqueam <lb/>ICK, quaſi fixam immotamq́ueſubſiſtere; </s> <s xml:id="echoid-s4987" xml:space="preserve">atque inſuper majoris perſpicuitatis <lb/> <anchor type="figure" xlink:label="fig-527.01.179-01a" xlink:href="fig-527.01.179-01"/> gratiâ eam cavitatem vas ſuperficiarium eſſe fingito, ſecundum 7 definitionem <lb/>hydroſtatices. </s> <s xml:id="echoid-s4988" xml:space="preserve">hocipſum corporeſuo orbum, aqua infuſâ compleatur. </s> <s xml:id="echoid-s4989" xml:space="preserve">& </s> <s xml:id="echoid-s4990" xml:space="preserve">quia <pb o="180" file="527.01.180" n="180" rhead="A*DDIT.* S*TAT. PARS TERTIA DE* F*LUIT.* A*CROB.*"/> per 1 propoſ. </s> <s xml:id="echoid-s4991" xml:space="preserve">hydroſtat. </s> <s xml:id="echoid-s4992" xml:space="preserve">aqua quemlibet ſibi datum in aqua locum ſervat, vas <lb/>iſtud ſuperficiarium in iſto ſitu permanebit; </s> <s xml:id="echoid-s4993" xml:space="preserve">atqueadeò, ſiveaqua, ſeu corpore <lb/>ABCD oppletum ſit, retinebit ſitum. </s> <s xml:id="echoid-s4994" xml:space="preserve">ſed infuſæ aquæ centrum, itemq́ue va-<lb/>ſis ſuperficiarii unum idemq́ue eſt, utpote L; </s> <s xml:id="echoid-s4995" xml:space="preserve">quare corporis ABCD gravita-<lb/>tis centrum, erit neceſſariò in vaſis ſuperficiarli pendula diametro M N. </s> <s xml:id="echoid-s4996" xml:space="preserve">Enim-<lb/>verò ſi fieri poſſit ſumatur extra, utin P; </s> <s xml:id="echoid-s4997" xml:space="preserve">id fieri plané nequit abſque mutatione <lb/>figuræ cavitatis aqueæ ICK, cum enim hunc habeat ſitum totius corporis cen-<lb/>tro conſtituto in O ex hypotheſi, tranſpoſita corporis materia, ut gravitatis cen-<lb/>trum deveniret ad P neceſſum eſſet B deſcendere, D autem adſcendere, & </s> <s xml:id="echoid-s4998" xml:space="preserve">C <lb/>converti verſus K, quod omnino theſi repugnat, atque alia eſſet aquæ cavitas, <lb/>quàm ea de qua quæſtio inſtituitur. </s> <s xml:id="echoid-s4999" xml:space="preserve">Quare corporis gravitatis centrum erit in <lb/>MN, videlicet infra cavitatis aqueæ gravitatis centrum L, vel ſupra, vel in ipſo.</s> <s xml:id="echoid-s5000" xml:space="preserve"/> </p> <div xml:id="echoid-div684" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.179-01" xlink:href="fig-527.01.179-01a"> <image file="527.01.179-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.179-01"/> </figure> </div> <p> <s xml:id="echoid-s5001" xml:space="preserve">C*ONCLUSIO.</s> <s xml:id="echoid-s5002" xml:space="preserve">* Itaque corpus fluitans iſtum ſibi vendicabit ſitum ut gra-<lb/>vitatis ſuæ centrum ſit in ſegmenti ſui in aquâ demerſi pendula gravitatis dia-<lb/>metro.</s> <s xml:id="echoid-s5003" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div686" type="section" level="1" n="486"> <head xml:id="echoid-head517" xml:space="preserve">1 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s5004" xml:space="preserve">Cum corporis gravitatis centrum ſupra aqueæ cavitatis gravitatis centrum <lb/>cõſiſtet, palam eſt ita ſumma prægravari ut omnia invertantur (ni ſuſtineantur) <lb/>donec corporis gravitatis diameter ſubeat cavitatis aqueæ gravitatis centrum <lb/>in pendula ejuſdem diametro. </s> <s xml:id="echoid-s5005" xml:space="preserve">Nam, verbi gratiâ, baculum incurvum aquæ in-<lb/>natans certum ſervabit ſitum, ut quamvis partem inferiorem vertas ſurſum, non <lb/>tamen ita permanebit, ſed ad priorem, quem ab initio obtinebat, ſitum redibit, <lb/>quia baculi gravitatis centrum tum non conſiſtit in aqueæ cavitatis gravitatis <lb/>diametro infra ejuſdem gravitatis centrum.</s> <s xml:id="echoid-s5006" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div687" type="section" level="1" n="487"> <head xml:id="echoid-head518" xml:space="preserve">2 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s5007" xml:space="preserve">Et pondere in navi aliove vaſe tranſpoſito ut aqueæ cavitatis figura mutetur, <lb/>aqueæ cavitatis centrilocum quoque mutari manifeſtum eſt.</s> <s xml:id="echoid-s5008" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div688" type="section" level="1" n="488"> <head xml:id="echoid-head519" xml:space="preserve">3 C*ONSECTARIUM.*</head> <p> <s xml:id="echoid-s5009" xml:space="preserve">Liquetitem omne pondus, quod infra aqueæ cavitatis planum diametrale <lb/>horizonti parallelum collocatur, navem ſtabiliorem, quoque minus in ſummo <lb/>gravis, ſit efficere. </s> <s xml:id="echoid-s5010" xml:space="preserve">Contra autem, quod ſupra collocabitur ſummitatem aggra-<lb/>vat, ut facilius ſubvertatur.</s> <s xml:id="echoid-s5011" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div689" type="section" level="1" n="489"> <head xml:id="echoid-head520" xml:space="preserve">NOTATO.</head> <p> <s xml:id="echoid-s5012" xml:space="preserve">Siduo gravitatis centra, videlicet altera aqueæ cavitatis, altera navis cætera-<lb/>rumq́ue rerum quas fert, ſint cognita, inveniri poſſe factione Staticâ nulla ad-<lb/>hibitâ experientiâ, quis oneratæ navis ſitus, quævé obliquitas futura ſit; </s> <s xml:id="echoid-s5013" xml:space="preserve">atque <lb/>utrum margines ejus infra aquam abdentur, nec ne, cui fini theorema iftud à <lb/>nobis inſtitutum fuit. </s> <s xml:id="echoid-s5014" xml:space="preserve">Sed quia inveſtigatio centrorum gravitatis adeò diverſa-<lb/>rum rerum, quibus navis ut plerumque oneratur, moleſta tædiiq́ue maximi <lb/>plena eſt, iſti negotio fuerit inutile Attamen quia Acrobaricorum in aqua flui-<lb/>tantium corporum cognitio alterubi forté uſui eſſe poſſit, autſi cui in inveſti-<lb/>gando ſubſidio eſſe poſſit, hoc pacto commentationem noſtram deſcribere <lb/>placuit.</s> <s xml:id="echoid-s5015" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div690" type="section" level="1" n="490"> <head xml:id="echoid-head521" xml:space="preserve">F*LVITANTIVM* A*CROBARICORVM*</head> <head xml:id="echoid-head522" xml:space="preserve">FINIS.</head> <pb file="527.01.181" n="181"/> </div> <div xml:id="echoid-div691" type="section" level="1" n="491"> <head xml:id="echoid-head523" xml:space="preserve">ADDITAMENTI <lb/>STATICÆ <lb/>PARS QVARTA, <lb/>DE <lb/> <anchor type="note" xlink:href="" symbol="*"/> CHALINOTHLIPSI.</head> <note symbol="*" position="right" xml:space="preserve">Frenorum <lb/>preſſu<unsure/>.</note> <pb file="527.01.182" n="182"/> </div> <div xml:id="echoid-div692" type="section" level="1" n="492"> <head xml:id="echoid-head524" xml:space="preserve">BREVIARIVM <lb/>CHALINOTHLIPSIS.</head> <p style="it"> <s xml:id="echoid-s5016" xml:space="preserve">CV *M* ab ineunte ætate P*RINCEPS* I*LLV*-<lb/>*STRISSIMVS* ἱ{πω}ικ{η\‘ν} non minus ſtudiosè, <lb/>quam aſsidue tractarit exercuerit{q́ue}, ut non ſo-<lb/>lum cum peritiſsimo{q́ue} quoque ſuper eâ ſermones <lb/>conferret: </s> <s xml:id="echoid-s5017" xml:space="preserve">ſed etiam, ſi quidquam literarum <lb/>monumentis à quoquam conſignatum eſſet diligentiſsimèlege-<lb/>ret, veteres iuxta & </s> <s xml:id="echoid-s5018" xml:space="preserve">novos. </s> <s xml:id="echoid-s5019" xml:space="preserve">nunquam tamen χαλινοθλίψεως <lb/>preſſuræ frenorum cauſas cognoſcere penitus, vel aſſequi po-<lb/>tuit. </s> <s xml:id="echoid-s5020" xml:space="preserve">quod freni partium aliquâ imminutâ, auctâ, inflexâ, <lb/>ſubitò magnæ & </s> <s xml:id="echoid-s5021" xml:space="preserve">incertæ mutationes in equi duct u exiſterent. <lb/></s> <s xml:id="echoid-s5022" xml:space="preserve">Vt, cum ob alias, tum potiſsimum quoque banc ipſam ob cau-<lb/>ſam non parva ſtaticæ cognoſcendæ cupiditate flagraret. </s> <s xml:id="echoid-s5023" xml:space="preserve">ſpera-<lb/>bat enim istâ ſibi viâ ad cognitionem cauſarum planiorem <lb/>aditum fore, quæ ſpes eum baud fefellit. </s> <s xml:id="echoid-s5024" xml:space="preserve">ut nunc frenos fabri-<lb/>cet, non incertis (ut olim) conjecturis ſolum ductus, ſedcauſis <lb/>ante perſpectis & </s> <s xml:id="echoid-s5025" xml:space="preserve">cognitis. </s> <s xml:id="echoid-s5026" xml:space="preserve">Et quia istæ matbematicæ demon-<lb/>ſtrationis ratiocinio nituntur, banc de frenorum preſſudoctri-<lb/>nam, quam voce græca χαλινοθλί{ψι}ν dicìmus, inter mathe-<lb/>matica I*ILVST*. </s> <s xml:id="echoid-s5027" xml:space="preserve">P*RINCIPIS* hypomnemata referen-<lb/>dam cenſui. </s> <s xml:id="echoid-s5028" xml:space="preserve">Idque eò magis, ut aliibis initiis incitati ulteri{us} <lb/>banc artem promoveant explicentque.</s> <s xml:id="echoid-s5029" xml:space="preserve"/> </p> <pb o="183" file="527.01.183" n="183"/> </div> <div xml:id="echoid-div693" type="section" level="1" n="493"> <head xml:id="echoid-head525" xml:space="preserve">DEFINITIONES.</head> <p> <s xml:id="echoid-s5030" xml:space="preserve">FReni partes & </s> <s xml:id="echoid-s5031" xml:space="preserve">earum appellationem characteribus & </s> <s xml:id="echoid-s5032" xml:space="preserve">notis tantum indi-<lb/>camus.</s> <s xml:id="echoid-s5033" xml:space="preserve"/> </p> <figure> <image file="527.01.183-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.183-01"/> </figure> </div> <div xml:id="echoid-div694" type="section" level="1" n="494"> <head xml:id="echoid-head526" xml:space="preserve">1 DEFINITIO.</head> <p> <s xml:id="echoid-s5034" xml:space="preserve">AB Scapus.</s> <s xml:id="echoid-s5035" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div695" type="section" level="1" n="495"> <head xml:id="echoid-head527" xml:space="preserve">2 DEFINITIO.</head> <p> <s xml:id="echoid-s5036" xml:space="preserve">C Axiculus anſatus.</s> <s xml:id="echoid-s5037" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div696" type="section" level="1" n="496"> <head xml:id="echoid-head528" xml:space="preserve">3 DEFINITIO.</head> <p> <s xml:id="echoid-s5038" xml:space="preserve">D Annellus.</s> <s xml:id="echoid-s5039" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5040" xml:space="preserve">Pollux. </s> <s xml:id="echoid-s5041" xml:space="preserve">Οἱ δὲ {σι}δηροὶ κύκλοι δἰ ὧν {δι}{εί}ροντ{αι} {αἱ} ἡνί{αι}, δακτύλιοι.</s> <s xml:id="echoid-s5042" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div697" type="section" level="1" n="497"> <head xml:id="echoid-head529" xml:space="preserve">4 DEFINITIO.</head> <p> <s xml:id="echoid-s5043" xml:space="preserve">EF Scapipars ſumma, ſeu capitellum.</s> <s xml:id="echoid-s5044" xml:space="preserve"/> </p> <pb o="184" file="527.01.184" n="184" rhead="A*DDITAMENTI* S*TATICÆ PARS QVARTA*"/> </div> <div xml:id="echoid-div698" type="section" level="1" n="498"> <head xml:id="echoid-head530" xml:space="preserve">5 DEFINITIO.</head> <p> <s xml:id="echoid-s5045" xml:space="preserve">G Ocellus.</s> <s xml:id="echoid-s5046" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div699" type="section" level="1" n="499"> <head xml:id="echoid-head531" xml:space="preserve">6 DEFINITIO.</head> <p> <s xml:id="echoid-s5047" xml:space="preserve">HI Lupus.</s> <s xml:id="echoid-s5048" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div700" type="section" level="1" n="500"> <head xml:id="echoid-head532" xml:space="preserve">7 DEFINITIO.</head> <p> <s xml:id="echoid-s5049" xml:space="preserve">KL Ψέ{λλ}ιον.</s> <s xml:id="echoid-s5050" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5051" xml:space="preserve">Catena eſt. </s> <s xml:id="echoid-s5052" xml:space="preserve">Poll. </s> <s xml:id="echoid-s5053" xml:space="preserve">{τὸ} ὶ {πξ<unsure/>ὶ} τὸ {γρν}{εί}ον {δι}{ει}ρόμενον, ψέ{λλ}ιον.</s> <s xml:id="echoid-s5054" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div701" type="section" level="1" n="501"> <head xml:id="echoid-head533" xml:space="preserve">8 DEFINITIO.</head> <p> <s xml:id="echoid-s5055" xml:space="preserve">KM Sigma.</s> <s xml:id="echoid-s5056" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5057" xml:space="preserve">Nihil aptius adhuc quidem nobis occurrit. </s> <s xml:id="echoid-s5058" xml:space="preserve">Sigma autem helicem <lb/>iſtam dicimus, non quod veteris, ſed quodnoſtri ævilitteræ ſimili-<lb/>tudine reſpondeat.</s> <s xml:id="echoid-s5059" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div702" type="section" level="1" n="502"> <head xml:id="echoid-head534" xml:space="preserve">9 DEFINITIO.</head> <p> <s xml:id="echoid-s5060" xml:space="preserve">NO Pſellii uncinus.</s> <s xml:id="echoid-s5061" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div703" type="section" level="1" n="503"> <head xml:id="echoid-head535" xml:space="preserve">10 DEFINITIO.</head> <p> <s xml:id="echoid-s5062" xml:space="preserve">P, Q Catellæ duæ intermediæ.</s> <s xml:id="echoid-s5063" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div704" type="section" level="1" n="504"> <head xml:id="echoid-head536" xml:space="preserve">11 DEFINITIO.</head> <p> <s xml:id="echoid-s5064" xml:space="preserve">Aſperum frenum, frenivè partes aſperæ ſunt, quæ lupum <lb/>vehementius inferiori mandibulæ, mentoq́ue in primunt. <lb/></s> <s xml:id="echoid-s5065" xml:space="preserve">remiſſæ quæ lenius.</s> <s xml:id="echoid-s5066" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div705" type="section" level="1" n="505"> <head xml:id="echoid-head537" xml:space="preserve">INTER PRETAMENTVM.</head> <p> <s xml:id="echoid-s5067" xml:space="preserve">Freni ab habena adducti preſſuræ variis partibus variæ ſunt, nam præter <lb/>iſtas in mandubulis mentoq́ue, catenæ intermediæ pectus, annelli ſcapos <lb/>preſſant. </s> <s xml:id="echoid-s5068" xml:space="preserve">nobis tamen aſperitas hîc haud aliam ſibi habet notionem ſubjectam, <lb/>quam vehementiam preſſus illius, quo mandibula inferior à lupo, & </s> <s xml:id="echoid-s5069" xml:space="preserve">men-<lb/>tum à ψελλίω afficiuntur, nam his equus ducitur, maloq́; </s> <s xml:id="echoid-s5070" xml:space="preserve">ſuo cogitur: </s> <s xml:id="echoid-s5071" xml:space="preserve">eamq́ <lb/>ob cauſam</s> </p> <p> <s xml:id="echoid-s5072" xml:space="preserve">-- equi frenato eſt auris in ore.</s> <s xml:id="echoid-s5073" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5074" xml:space="preserve">nam quî dolorem iſtum declinet mentum pectori adducit collumq́ue flectit. <lb/></s> <s xml:id="echoid-s5075" xml:space="preserve">ponamus enim equi mentum ab habena palmum adduci, cervicis flexu vehe-<lb/>mentiorem preſſum eludet. </s> <s xml:id="echoid-s5076" xml:space="preserve">indidem quoq; </s> <s xml:id="echoid-s5077" xml:space="preserve">retro agetur, ut ita dolori ſeſe ſub-<lb/>ducat, dum veretur ſi prorſum gradus promoveat, ne is ingraveſcat. </s> <s xml:id="echoid-s5078" xml:space="preserve">Hæciſta <lb/>aſperitas eſt quam deſinivimus. </s> <s xml:id="echoid-s5079" xml:space="preserve">unde & </s> <s xml:id="echoid-s5080" xml:space="preserve">freni, frenorumq́; </s> <s xml:id="echoid-s5081" xml:space="preserve">partes aſperę appel-<lb/>lantur, quæ lupum mandibulæ inferiori mentoq́ue vehementius inprimunt: </s> <s xml:id="echoid-s5082" xml:space="preserve"><lb/>remiſſæ quæ lenius. </s> <s xml:id="echoid-s5083" xml:space="preserve">ut frenum aſperius ſeu lupatum, frenum lenius: </s> <s xml:id="echoid-s5084" xml:space="preserve">ſcapus <lb/>aſperior, lenior: </s> <s xml:id="echoid-s5085" xml:space="preserve">& </s> <s xml:id="echoid-s5086" xml:space="preserve">ejuſdem pars ſumma aſperior leniórve.</s> <s xml:id="echoid-s5087" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div706" type="section" level="1" n="506"> <head xml:id="echoid-head538" xml:space="preserve">12 DEFINITIO.</head> <p> <s xml:id="echoid-s5088" xml:space="preserve">Inverſuræ ſunt ſcaporum curvaturæ</s> </p> <pb o="185" file="527.01.185" n="185" rhead="*DE* C*HALINOTHLIPSI.*"/> </div> <div xml:id="echoid-div707" type="section" level="1" n="507"> <head xml:id="echoid-head539" xml:space="preserve">INTERPRETAMENTVM.</head> <p> <s xml:id="echoid-s5089" xml:space="preserve">Frenorum ſcapos & </s> <s xml:id="echoid-s5090" xml:space="preserve">rectos & </s> <s xml:id="echoid-s5091" xml:space="preserve">curvamine inflexos fabricant. </s> <s xml:id="echoid-s5092" xml:space="preserve">rectorum de-<lb/>formationem ſupra expreſſimus. </s> <s xml:id="echoid-s5093" xml:space="preserve">inflexorum hîc ob <lb/>oculos ponimus; </s> <s xml:id="echoid-s5094" xml:space="preserve">ſic enim deformantur ut ab X verſus <lb/> <anchor type="figure" xlink:label="fig-527.01.185-01a" xlink:href="fig-527.01.185-01"/> Y curvamine aſſurgant, deinde ab Y in Z, & </s> <s xml:id="echoid-s5095" xml:space="preserve">ab Z in a <lb/>ſinuamine inflectantur, unde inverſuræ illis nomen à <lb/>nobis inditum.</s> <s xml:id="echoid-s5096" xml:space="preserve"/> </p> <div xml:id="echoid-div707" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.185-01" xlink:href="fig-527.01.185-01a"> <image file="527.01.185-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.185-01"/> </figure> </div> </div> <div xml:id="echoid-div709" type="section" level="1" n="508"> <head xml:id="echoid-head540" xml:space="preserve">SEQVENTES DEFINITIO-<lb/>NES NOVÆ, ET THEORIÆ QVASI <lb/>PROPRIÆ SVNT.</head> <head xml:id="echoid-head541" xml:space="preserve">13 DEFINITIO.</head> <p> <s xml:id="echoid-s5097" xml:space="preserve">Mediumid punctum R, quo frenato <lb/>equo adductis habenis, annellus D tan-<lb/>git axiculum anſatum C, annelli conta-<lb/>ctus dicatur.</s> <s xml:id="echoid-s5098" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div710" type="section" level="1" n="509"> <head xml:id="echoid-head542" xml:space="preserve">14 DEFINITIO.</head> <p> <s xml:id="echoid-s5099" xml:space="preserve">Medium punctum S, quo frenato <lb/>equo adductis habenis ſigma ocellum, <lb/>& </s> <s xml:id="echoid-s5100" xml:space="preserve">T medium punctum uncini ſuum <lb/>ocellum contingit, ocellicontactus di-<lb/>catur.</s> <s xml:id="echoid-s5101" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div711" type="section" level="1" n="510"> <head xml:id="echoid-head543" xml:space="preserve">15 DEFINITIO.</head> <p> <s xml:id="echoid-s5102" xml:space="preserve">H axiculum olivæ medio infixum cir-<lb/>ca quem lupus verſatur, lupi axiculum <lb/>dicimus.</s> <s xml:id="echoid-s5103" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div712" type="section" level="1" n="511"> <head xml:id="echoid-head544" xml:space="preserve">16 DEFINITIO.</head> <p> <s xml:id="echoid-s5104" xml:space="preserve">Angulum R H S, ab contactu annelli R ad lupiaxicu-<lb/>lum H & </s> <s xml:id="echoid-s5105" xml:space="preserve">inde ad ocelli contactum S eductis lineis inter-<lb/>ceptum, angulum ſub eductis contingentibus dicemus.</s> <s xml:id="echoid-s5106" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div713" type="section" level="1" n="512"> <head xml:id="echoid-head545" xml:space="preserve">17 DEFINITIO.</head> <p> <s xml:id="echoid-s5107" xml:space="preserve">Frenum {προ}πιραςικὸν eſt omnibus equis explorandis <lb/>oportunum quale nam frenum illorum tenaciæ infre-<lb/>nandæ commodum ſit, utinde certa ratione frenum ido-<lb/>neum confieri poſsit.</s> <s xml:id="echoid-s5108" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5109" xml:space="preserve">Freni propeiraſtici deformationĕ & </s> <s xml:id="echoid-s5110" xml:space="preserve">qualitates infra oportuniore loco dicemus.</s> <s xml:id="echoid-s5111" xml:space="preserve"/> </p> <pb o="186" file="527.01.186" n="186" rhead="A*DDITAMENTI* S*TATICÆ* P*ARS QVARTA*"/> </div> <div xml:id="echoid-div714" type="section" level="1" n="513"> <head xml:id="echoid-head546" xml:space="preserve">1 PROPOSITIO.</head> <p> <s xml:id="echoid-s5112" xml:space="preserve">Scaporum inverſuris aſperitatem neque augeri neque <lb/>leniri.</s> <s xml:id="echoid-s5113" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5114" xml:space="preserve">I*LLVSTRISSIMVS* P*RINCEPS* haud ignarus hos, qui ſcaporum in-<lb/>verſuras, tribus punctis R, H, S eodem loco manentibus, frenorum aſperita-<lb/>ti lenitudinive quidquam momenti afferre credunt, falſos opinionis & </s> <s xml:id="echoid-s5115" xml:space="preserve">veri va-<lb/>nos eſſe. </s> <s xml:id="echoid-s5116" xml:space="preserve">eoſdem quoque his rationibus ſolet refellere. </s> <s xml:id="echoid-s5117" xml:space="preserve">Affigas enim, ajebat, <lb/>ſcapo directo, cujus figuram ſupra expreſſimus, laminam aliquam ferream, <lb/>quæ ſcapum tanquam majuſculis inverſuris deformatum exhibeat. </s> <s xml:id="echoid-s5118" xml:space="preserve">jam ſi quis <lb/>eſt ab hac mente, ut inde aſperitatem augeri ſuſpicetur: </s> <s xml:id="echoid-s5119" xml:space="preserve">id perinde foret ac ſi <lb/>credat inductam illam lamellam occulta quadam vi (magnetis inſtar) vel preſ-<lb/>ſim vel punctim aliquid agere. </s> <s xml:id="echoid-s5120" xml:space="preserve">porrò ſi qui pertinacius affirmĕt id re ipſa uſuq́; <lb/></s> <s xml:id="echoid-s5121" xml:space="preserve">comprobari, iſtos negando facile eſt refutare: </s> <s xml:id="echoid-s5122" xml:space="preserve">quid enim habent dicere@ Cæ-<lb/>terum ſi equiſones, χαλινέρ{γα}{πο<unsure/>}, quique ὶ{πω}ικ{ὴν} ſolo uſu tractant, tritum il-<lb/>lud, tanquam ex Apollinis tripode nobis occinant. </s> <s xml:id="echoid-s5123" xml:space="preserve">Luam quiſque novit ar-<lb/>tem, in hac fidem ei habendam. </s> <s xml:id="echoid-s5124" xml:space="preserve">illi ſciant hoc in ſe ipſos quadrare maximè. </s> <s xml:id="echoid-s5125" xml:space="preserve"><lb/>nam de ſtaticis effectionibus ſentĕtiam ferunt, quarum tamen ſint ignariſſimi. </s> <s xml:id="echoid-s5126" xml:space="preserve"><lb/>ex hac enim facile docerentur mutationem omnem & </s> <s xml:id="echoid-s5127" xml:space="preserve">varietatem exiſtere ab <lb/>tribus punctis R H S. </s> <s xml:id="echoid-s5128" xml:space="preserve">quibus eodem loco manentibus, & </s> <s xml:id="echoid-s5129" xml:space="preserve">propter eas lineis <lb/>imaginariis R H, H S ſede ſua haud emotis, anguloq́ue R H S invariato, ea-<lb/>dem & </s> <s xml:id="echoid-s5130" xml:space="preserve">par ſemper eritab ſcapo aſperitas. </s> <s xml:id="echoid-s5131" xml:space="preserve">niſi fortè quid momenti afferat la-<lb/>mellæ affixæ ponduſculum. </s> <s xml:id="echoid-s5132" xml:space="preserve">verum id à propoſita quæſtione alienum eſt. </s> <s xml:id="echoid-s5133" xml:space="preserve">& </s> <s xml:id="echoid-s5134" xml:space="preserve">ſi <lb/>hoc urgeant, eares forte non magis eorum ſententiæ patrocinabitur, quam <lb/>adverſabitur.</s> <s xml:id="echoid-s5135" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5136" xml:space="preserve">Præterea animadverſione dignum eſt, non latus V W in ſcapi capitello, ſed <lb/>rectam imaginariam H S ad generalis cujuſdam theorematis ordinationem, <lb/>unde inflexurarum modus capiatur, aliquid momenti afferre. </s> <s xml:id="echoid-s5137" xml:space="preserve">nam latitudo <lb/>ocelli diverſa, diverſos quoque effectus inducit.</s> <s xml:id="echoid-s5138" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div715" type="section" level="1" n="514"> <head xml:id="echoid-head547" xml:space="preserve">2 PROPOSITIO.</head> <p> <s xml:id="echoid-s5139" xml:space="preserve">Scapi quo breviores tanto ſuntaſperiores.</s> <s xml:id="echoid-s5140" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5141" xml:space="preserve">Cauſa ejus rei ferè gemina. </s> <s xml:id="echoid-s5142" xml:space="preserve">prima quod adductis tantundem habenis ſcapi <lb/>breviores plus commoveantur quam longiores. </s> <s xml:id="echoid-s5143" xml:space="preserve">cujus veritatem diagramma-<lb/>te demonſtrabo. </s> <s xml:id="echoid-s5144" xml:space="preserve">A B ſcapus longior, A C brevior, quo-<lb/>rum ſit idem capitellum A D, ejuſque ocelli contactus D: <lb/></s> <s xml:id="echoid-s5145" xml:space="preserve"> <anchor type="figure" xlink:label="fig-527.01.186-01a" xlink:href="fig-527.01.186-01"/> adductis habenis in ſcapo breviore aſſurgat annelli conta-<lb/>ctus in E per peripheriam C E, & </s> <s xml:id="echoid-s5146" xml:space="preserve">ocelli contactus D de-<lb/>ſcendat in F per peripheriam D F. </s> <s xml:id="echoid-s5147" xml:space="preserve">Conſimiliter in ſcapo <lb/>longiore annelli contactus adducatur ad G, ut peripheriæ <lb/>C E, B G æquales ſint, hîc D deſcendet ſolummodo in H <lb/>per peripheriam D H. </s> <s xml:id="echoid-s5148" xml:space="preserve">ſed D F major eſt quàm D H, ſunt <lb/>enim inter ſe rationeſcaporum inverſa, id eſt, quemadmo-<lb/>dum longior ſcapus A B ad breviorem A C. </s> <s xml:id="echoid-s5149" xml:space="preserve">Itaque pſellia <lb/>ocellis affixa minoribus ſcapis moventur validius. </s> <s xml:id="echoid-s5150" xml:space="preserve">verum <lb/>quantum pſellia moventur magis, hoc mandibula à lupo, <lb/>mentumq́ue à pſellio preſsius premitur conſtringiturq́ue. </s> <s xml:id="echoid-s5151" xml:space="preserve">unde eſſicitur ut <lb/>ſcapi breviores longioribus ſint aſperiores.</s> <s xml:id="echoid-s5152" xml:space="preserve"/> </p> <div xml:id="echoid-div715" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.186-01" xlink:href="fig-527.01.186-01a"> <image file="527.01.186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.186-01"/> </figure> </div> <pb o="187" file="527.01.187" n="187" rhead="DE C*HALINOTHLIPSI:*"/> <p> <s xml:id="echoid-s5153" xml:space="preserve">Alrera cauſa pendet ab equi cervice inflexa, nam hinc eſt quod catellæ in-<lb/>@ermediæ in ſcapo breviore lõgius ab equi pectore diſtent, quam in longiore. <lb/></s> <s xml:id="echoid-s5154" xml:space="preserve">ideoque habenæ ſcapi brevioris magis adducendę ſunt ut catellæ intermediæ <lb/>pectus tangant, quam in ſcapo longiore. </s> <s xml:id="echoid-s5155" xml:space="preserve">ideoque etiam cum tangent ſua aſpe-<lb/>ritate moleſtiores erunt.</s> <s xml:id="echoid-s5156" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div717" type="section" level="1" n="515"> <head xml:id="echoid-head548" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s5157" xml:space="preserve">Neceſt quod hic aliquis ſuſpicetur placita hæc cum ſtaticis theorematis <lb/>pugnare, quæ longioribus jugis plus virium tribuebant, hoc eſt, ſi B D libri-<lb/>le fingatur, cujus brevior radius ſit A D, longior, unde & </s> <s xml:id="echoid-s5158" xml:space="preserve">motus principium <lb/>exiſtat, A B. </s> <s xml:id="echoid-s5159" xml:space="preserve">tum ſupra dictis contrarium effici. </s> <s xml:id="echoid-s5160" xml:space="preserve">Nam ſcrupulus iſte facile <lb/>tolletur, ſi in mentem revocemus, quod hic in conſiderationem non veniat <lb/>vis ea, quam eques adducendis ſua manu habenis præſtat (nam ut in brevio-<lb/>re ſcapo ocelli contactum tantundem commoveat, magis allaborandum erit, <lb/>quam in longiore) ſed hoc, quod ſi ſatis intentæ ſint habenæ utri ſcapi ſuo <lb/>preſſu plus aſperitatis moleſtiæ q́ue faceſſant.</s> <s xml:id="echoid-s5161" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div718" type="section" level="1" n="516"> <head xml:id="echoid-head549" xml:space="preserve">3 PROPOSITIO.</head> <p> <s xml:id="echoid-s5162" xml:space="preserve">Capitella quo longiora ſunt tanto plus aſperitatis <lb/>habent.</s> <s xml:id="echoid-s5163" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5164" xml:space="preserve">Nam habenis tantundem adductis pſellium longiore capitello plus com-<lb/>movetur quam breviore. </s> <s xml:id="echoid-s5165" xml:space="preserve">diagrammate res illuſtrior fiet. </s> <s xml:id="echoid-s5166" xml:space="preserve">A B ſit capitellum <lb/>longius, ejuſque ocelli contactus B, A C vero ſit capitellum brevius, ejuſque <lb/>ocelli contactus C, communis utrique hic ſcapus intel-<lb/> <anchor type="figure" xlink:label="fig-527.01.187-01a" xlink:href="fig-527.01.187-01"/> ligatur A B. </s> <s xml:id="echoid-s5167" xml:space="preserve">jam adductis habenis annelli contactus ad-<lb/>ſcendat in E. </s> <s xml:id="echoid-s5168" xml:space="preserve">& </s> <s xml:id="echoid-s5169" xml:space="preserve">ocelli contactus B deſcendatad F per <lb/>peripheriam B F. </s> <s xml:id="echoid-s5170" xml:space="preserve">Sed ocelli contactus ab C uſque G <lb/>peragrata peripheriá C G minor eſt quam B F; </s> <s xml:id="echoid-s5171" xml:space="preserve">nam ut <lb/>A C ad A B, ſic G C ad B F. </s> <s xml:id="echoid-s5172" xml:space="preserve">Itaque pſellium ocello <lb/>B longioris capitelliaffixum, æquali habenarum ductu <lb/>plus movetur, quam ſi ocello C minoris capitelli fixum <lb/>hæreret. </s> <s xml:id="echoid-s5173" xml:space="preserve">quanto autem pſellrum magis ſurſum commo-<lb/>vetur, eò arctius mento hæret, & </s> <s xml:id="echoid-s5174" xml:space="preserve">ipſius lupi in man-<lb/>dibilam impreſſionem facit validiorem. </s> <s xml:id="echoid-s5175" xml:space="preserve">Ideoq́ue lon-<lb/>giora capitella plus aſperitatis habent. </s> <s xml:id="echoid-s5176" xml:space="preserve">quod ſi quiſpiam <lb/>quęrat cauſam, cur motus initio poſito in D longior ra-<lb/>dius A B, cõtra ſtatica principia vehementiores effectus <lb/>producat, quam brevior A C. </s> <s xml:id="echoid-s5177" xml:space="preserve">eum monemus videat <lb/>annotatiunculam 2 propoſitionis, ubi haud abſimilem <lb/>quæſtionis nodum diſſolvimus.</s> <s xml:id="echoid-s5178" xml:space="preserve"/> </p> <div xml:id="echoid-div718" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.187-01" xlink:href="fig-527.01.187-01a"> <image file="527.01.187-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.187-01"/> </figure> </div> </div> <div xml:id="echoid-div720" type="section" level="1" n="517"> <head xml:id="echoid-head550" xml:space="preserve">4 PROPOSITIO.</head> <p> <s xml:id="echoid-s5179" xml:space="preserve">Quanto contactus annelli à pectore longius diſtat tan-<lb/>to plus aſperitatis habet.</s> <s xml:id="echoid-s5180" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5181" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s5182" xml:space="preserve">* A lupi eſto axiculus, A B ſcapus, B C habena, Bannelli con-<lb/>tactus. </s> <s xml:id="echoid-s5183" xml:space="preserve">A D ſcapus alter æqualis priori A B, D C habena, D annelli conta- <pb o="188" file="527.01.188" n="188" rhead="A*DDITAMENTI* S*TATICÆ* P*ARS QVARTA*"/> ctus: </s> <s xml:id="echoid-s5184" xml:space="preserve">ſitq́ue prioris annelli contactus B ab equi pectore diſtantior, quam po-<lb/> <anchor type="figure" xlink:label="fig-527.01.188-01a" xlink:href="fig-527.01.188-01"/> ſterioris D. </s> <s xml:id="echoid-s5185" xml:space="preserve">Q*VÆSITVM.</s> <s xml:id="echoid-s5186" xml:space="preserve">* Demonſtrandum eſt contactum annelli B plus <lb/>habere aſperitatis, quam D. </s> <s xml:id="echoid-s5187" xml:space="preserve">P*RÆPARATIO.</s> <s xml:id="echoid-s5188" xml:space="preserve">* Centro A, intervallo A B de-<lb/>circinetur peripheria B D E: </s> <s xml:id="echoid-s5189" xml:space="preserve">Iam annelli contactus B adducatur in F, & </s> <s xml:id="echoid-s5190" xml:space="preserve">D <lb/>in E, peripheriæq́ue D E B F ſunto æquales.</s> <s xml:id="echoid-s5191" xml:space="preserve"/> </p> <div xml:id="echoid-div720" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.188-01" xlink:href="fig-527.01.188-01a"> <image file="527.01.188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.188-01"/> </figure> </div> </div> <div xml:id="echoid-div722" type="section" level="1" n="518"> <head xml:id="echoid-head551" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s5192" xml:space="preserve">Non dubium videtur, quin quanto B C linea longioreſt quam F C, tan-<lb/>to altius retrorſum adducta fuerit manus cum annelli cõtactus eſſet in loco F, <lb/>quam cum eſlet in B. </s> <s xml:id="echoid-s5193" xml:space="preserve">Haud aliter quanto D C major quam E C, tanto ma-<lb/>nus C elatius adducenda ſuit cum annelli contactus eſſet in E, quam cum in <lb/>D. </s> <s xml:id="echoid-s5194" xml:space="preserve">ſed E C, D C rectarum differentia major eſt, quam F C, B C. </s> <s xml:id="echoid-s5195" xml:space="preserve">Et quanta <lb/>ipſarum differentiarum differentia eſt, tanto elatius manus attollitur adducto <lb/>annelli contactu ab D in E, quam ab B in F. </s> <s xml:id="echoid-s5196" xml:space="preserve">Motus autem, ſeu peripheria <lb/>D E per conſtructionem æqualis eſt peripheriæ B F. </s> <s xml:id="echoid-s5197" xml:space="preserve">quare manus C, æqua-<lb/>libus ductuum peripheriis ab punctis B & </s> <s xml:id="echoid-s5198" xml:space="preserve">D altius tollitur ab D, quam à B. <lb/></s> <s xml:id="echoid-s5199" xml:space="preserve">ideoque ſi utrobique manus D æquali altitudine attollenda eſſet, major eſſet <lb/>motus peripheria ab B in F, quam ab D in E. </s> <s xml:id="echoid-s5200" xml:space="preserve">Sed majorem motum ab B <lb/>verſus F major ocelli motus ſequitur: </s> <s xml:id="echoid-s5201" xml:space="preserve">atque hunc quoq; </s> <s xml:id="echoid-s5202" xml:space="preserve">ipſius pſellii. </s> <s xml:id="echoid-s5203" xml:space="preserve">Quam-<lb/>obrem ſi utroque caſu manus pari altitudine attollatur, motus pſellii ex B, F <lb/>verſum, major erit motu pſellii ab D verſus E. </s> <s xml:id="echoid-s5204" xml:space="preserve">ſed majore motu vel elatio-<lb/>ne pſellii plaga in equi mentum fit vehementior, unde lupi quoque in mandi-<lb/>bulam impreſſio exiſtit violentior. </s> <s xml:id="echoid-s5205" xml:space="preserve">Itaque æquali manus C elatione, ſi annelli <lb/>contactus obtineat locum B in ſcapo A B plus aſperitatis habet, quam ſi eſſet <lb/>in D in ſcapo A D. </s> <s xml:id="echoid-s5206" xml:space="preserve">atque adeò aunelli contactus B, quia diſtantior eſt ab <lb/>equi pectore, aſperior eſt viciniore D.</s> <s xml:id="echoid-s5207" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div723" type="section" level="1" n="519"> <head xml:id="echoid-head552" xml:space="preserve">1 NOTA.</head> <p> <s xml:id="echoid-s5208" xml:space="preserve">Quia angulus A D C minus à recto abeſt, quam A B C, qui longeacutior <pb o="189" file="527.01.189" n="189" rhead="DE C*HALINOTHLIPSI.*"/> eſt, vis manus C clariores effectus habet in ſcapo A D, quam in ſcapo A B <lb/>per converſam 24 prop. </s> <s xml:id="echoid-s5209" xml:space="preserve">1 lib. </s> <s xml:id="echoid-s5210" xml:space="preserve">Staticæ. </s> <s xml:id="echoid-s5211" xml:space="preserve">& </s> <s xml:id="echoid-s5212" xml:space="preserve">quia hæc minus attentum fallere <lb/>poſſet tanquam expoſitæ propoſitioni contraria ſciſceret, monemus ut anno-<lb/>tatiunculam 2 propoſitionis adeat, indeq́; </s> <s xml:id="echoid-s5213" xml:space="preserve">lucem petat. </s> <s xml:id="echoid-s5214" xml:space="preserve">Nõ enim in hoc theo-<lb/>remate quæritur quantanam ſit potentia manus C, ſed ſi manus utroque caſu <lb/>æqualiter attollatur (quamvis ſub angulo A B C ei magis ſit allaborandum, <lb/>quam ſub angulo A D C) uter ductus tum plus aſperitatis habeat.</s> <s xml:id="echoid-s5215" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div724" type="section" level="1" n="520"> <head xml:id="echoid-head553" xml:space="preserve">2 NOTA.</head> <p> <s xml:id="echoid-s5216" xml:space="preserve">Ad eam quam rettulimus aſperitatis cauſam nonnunquam & </s> <s xml:id="echoid-s5217" xml:space="preserve">alia accedit. <lb/></s> <s xml:id="echoid-s5218" xml:space="preserve">iſta videlicet. </s> <s xml:id="echoid-s5219" xml:space="preserve">ut quanto annelli contactus ad equi pectus propius adducitur, <lb/>tanto in frenis, quales vulgo fabricant, intermediæ catellæ pectori quoq; </s> <s xml:id="echoid-s5220" xml:space="preserve">fiant <lb/>viciniores; </s> <s xml:id="echoid-s5221" xml:space="preserve">Sed cum jam eo adductæ ſunt ut pectus tangant, poſl<unsure/>ea freni du-<lb/>ctus aſperitatis amplius nihil habet. </s> <s xml:id="echoid-s5222" xml:space="preserve">quamvis enim deinceps plus annitaris <lb/>omnis plaga equi pectore avertitur, neque mento mandibulæve impreffio va-<lb/>lidior ulla infertur. </s> <s xml:id="echoid-s5223" xml:space="preserve">Sed ſiannelli contactus, atque ideo quoque intermediæ <lb/>catellæ longius à pectore abſint, conſequens eſt ſcapos longius retrorſum pe-<lb/>ctus verſum adduci poſſe ut catellæ intermediæ pectus non tangant, unde <lb/>major aſperitas exiſtit. </s> <s xml:id="echoid-s5224" xml:space="preserve">Veruntamen annotatiuncula hæc locum non habet ſi <lb/>catellæ intermediæ neutro modo pectus contingant.</s> <s xml:id="echoid-s5225" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div725" type="section" level="1" n="521"> <head xml:id="echoid-head554" xml:space="preserve">3 NOTA.</head> <p> <s xml:id="echoid-s5226" xml:space="preserve">Nonnulli equi (quorũ deformationem hic vides) caput ſurſum jactãdo ſeſe <lb/>frenorum preſſuræ eripiunt, neque tum illorum tenacia infrenari poteſt, ſed <lb/>equitem invitum rapiunt: </s> <s xml:id="echoid-s5227" xml:space="preserve">cum tamen annelli contactus eo ſitu longius à pe-<lb/>ctore abſit, ideoq́ue frenum eo caſu aſperitate ſua plus moleſtiæ faceſſere de-<lb/>beat. </s> <s xml:id="echoid-s5228" xml:space="preserve">Verum objectiunculam illam facile eſt diſſolvere, nam cum tenſa habe-<lb/> <anchor type="figure" xlink:label="fig-527.01.189-01a" xlink:href="fig-527.01.189-01"/> na A B parallela erit lineæ imaginariæ ab annelli cõtactu A ad lupiaxiculum <lb/>C eductæ, quemadmodum in hoc diagrammate notare potes, tum quan-<lb/>topere annitaris ductu habenæ capitellum ne hilum quidem commoveris, <lb/>neque pſellium elevâris, ideoq́ue omnis aſperitas conciderit nam quamvis <lb/>lupus retrorſum adducatur, inde tamen ſupraſcriptæ aſperitatis cauſa non de-<lb/>pendet. </s> <s xml:id="echoid-s5229" xml:space="preserve">Imo ſi habena ſupra ſitum jam delineatum adſcendat, quanto vehe-<lb/>mentius habena tum adducetur, tanto pſellium magis remittet. </s> <s xml:id="echoid-s5230" xml:space="preserve">Vt hæc à præ-<lb/>miſſo theoremate exceptio haud dubiis, aut obſcuris rationibus nitatur.</s> <s xml:id="echoid-s5231" xml:space="preserve"/> </p> <div xml:id="echoid-div725" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.189-01" xlink:href="fig-527.01.189-01a"> <image file="527.01.189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.189-01"/> </figure> </div> <pb o="190" file="527.01.190" n="190" rhead="A*DDITAMENTI* S*TATICÆ* P*ARS QVARTA*"/> </div> <div xml:id="echoid-div727" type="section" level="1" n="522"> <head xml:id="echoid-head555" xml:space="preserve">5 PROPOSITIO.</head> <p> <s xml:id="echoid-s5232" xml:space="preserve">Pſellia quo breviora eo ſunt aſperiora.</s> <s xml:id="echoid-s5233" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5234" xml:space="preserve">Vero conſentaneum eſt lupi in mandibulam impreſſionem tum demum <lb/>incipere cum pſellium mento primulum admovetur, ſed ut pſellium longiuſ. <lb/></s> <s xml:id="echoid-s5235" xml:space="preserve">culum mentum tangat manus magis adducenda, quam ſi ſit brevius. </s> <s xml:id="echoid-s5236" xml:space="preserve">quare æ-<lb/>qualiter mota manu preſſus brevioris pſellii arctior eſt, quam longioris: </s> <s xml:id="echoid-s5237" xml:space="preserve">ideoq́; </s> <s xml:id="echoid-s5238" xml:space="preserve"><lb/>pſellium brevius plus habet aſperitatis.</s> <s xml:id="echoid-s5239" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div728" type="section" level="1" n="523"> <head xml:id="echoid-head556" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s5240" xml:space="preserve">Diximus vero ſimile videri lupi preſſuram illic initium habere, ubi pſellium <lb/>mento primum affligitur. </s> <s xml:id="echoid-s5241" xml:space="preserve">nonnulli tamen equi ante afflictionem iſtam preſſu <lb/>quopiam afficiuntur, etiam ſolo freno abſque pſellio, alius alio citius, prout <lb/>ore eſt teneriore durioréve, vel ut frenum materiæ eſt mollioris duriorisve, <lb/>laxius ſtrictiúsve. </s> <s xml:id="echoid-s5242" xml:space="preserve">Verum hæc tam exigua tamq́ue inæqualis preſſura, non eſt <lb/>tanti ut accuratioribus qualitatum & </s> <s xml:id="echoid-s5243" xml:space="preserve">affectionum ſuarum rationibus inveſti-<lb/>gandis & </s> <s xml:id="echoid-s5244" xml:space="preserve">demonſtrandis opus ſit.</s> <s xml:id="echoid-s5245" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div729" type="section" level="1" n="524"> <head xml:id="echoid-head557" xml:space="preserve">6 PROPOSITIO.</head> <p> <s xml:id="echoid-s5246" xml:space="preserve">Frenum <anchor type="note" xlink:href="" symbol="*"/> {προ}πςραςιχὸν fabricari.</s> <s xml:id="echoid-s5247" xml:space="preserve"/> </p> <note symbol="*" position="left" xml:space="preserve">pr@tentati-<lb/>vum.</note> <p> <s xml:id="echoid-s5248" xml:space="preserve">Definitionem freni {προ}πειραςιχ{οῦ} ſupra ſuo loco 17 def. </s> <s xml:id="echoid-s5249" xml:space="preserve">rettulimus. </s> <s xml:id="echoid-s5250" xml:space="preserve">nunc <lb/>fabricam ejus docemus. </s> <s xml:id="echoid-s5251" xml:space="preserve">deformatio quidem eſt qualem ante oculos expreſ-<lb/>ſam vides, ubi A B ſcapos deſignant, qui vel augeri vel etiam imminui ob <lb/>partes C B immiſſas poterunt, quæq́; </s> <s xml:id="echoid-s5252" xml:space="preserve">cumopus erit cochleis ad D ſiſtuntur. <lb/></s> <s xml:id="echoid-s5253" xml:space="preserve">porro ſcapi ſuos habent axes ad E circa quos verſentur, quorum ope ad an-<lb/>gulum optatum cum capitello inflectentur, deinde cochleis E ſiſtuntur. </s> <s xml:id="echoid-s5254" xml:space="preserve">Ca-<lb/>pitella G H pari craſſitudine & </s> <s xml:id="echoid-s5255" xml:space="preserve">mole ſunt deformata, longitudinis porro tan-<lb/>tæ, quanta maxima opus erit: </s> <s xml:id="echoid-s5256" xml:space="preserve">Ocelli I ſurſum deorſumve adduci poſſunt, & </s> <s xml:id="echoid-s5257" xml:space="preserve"><lb/>ubi commodum videtur cochleis ad K quoque ſiſtuntur. </s> <s xml:id="echoid-s5258" xml:space="preserve">Vt hac deformatio-<lb/>ne & </s> <s xml:id="echoid-s5259" xml:space="preserve">ſuprema & </s> <s xml:id="echoid-s5260" xml:space="preserve">ima pars ad juſtam longitudinem augeri vel contrahi com-<lb/>mode poſſint.</s> <s xml:id="echoid-s5261" xml:space="preserve"/> </p> <pb o="191" file="527.01.191" n="191" rhead="DE C*HALINO THLIPSI*."/> <figure> <image file="527.01.191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.191-01"/> </figure> <p> <s xml:id="echoid-s5262" xml:space="preserve">Hactenus freni {προ}πςραςιχ{οῦ} deſcriptio & </s> <s xml:id="echoid-s5263" xml:space="preserve">deformatio univerſo quaſi cor-<lb/>pore, & </s> <s xml:id="echoid-s5264" xml:space="preserve">junctis jam partibus nobis eſt conſiderata. </s> <s xml:id="echoid-s5265" xml:space="preserve">Sed ut ipſas partes quoque <lb/>examinemus, eas ſigillatim hic ante oculos ponimus: </s> <s xml:id="echoid-s5266" xml:space="preserve">ubi iidem characteres <lb/>eaſdem notiones ſibi habent ſubjectas quas prius.</s> <s xml:id="echoid-s5267" xml:space="preserve"/> </p> <pb o="192" file="527.01.192" n="192" rhead="A*DDITAMENTI* S*TATIC Æ* P*ARS* *QVARTA*"/> <figure> <image file="527.01.192-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.192-01"/> </figure> <p> <s xml:id="echoid-s5268" xml:space="preserve">Hæc eſt freni {προ}πς-<lb/>ραςιχ{οῦ} deformatio, quã <lb/>I*LLVSTRISSIMVS* <lb/>P*RINCEPS* effinxit, & </s> <s xml:id="echoid-s5269" xml:space="preserve"><lb/>re ipſa atque uſu com-<lb/>modiſsimam deprehĕ-<lb/>dit. </s> <s xml:id="echoid-s5270" xml:space="preserve">ſi verò deincepsali-<lb/>quid accedat aut inve-<lb/>niatur commodius, æ-<lb/>quiſsimũ fuerit id ſuo <lb/>ſibi commodo uſur-<lb/>pare.</s> <s xml:id="echoid-s5271" xml:space="preserve"/> </p> <pb o="193" file="527.01.193" n="193" rhead="DE C*HALINO THLIPSI.*"/> </div> <div xml:id="echoid-div730" type="section" level="1" n="525"> <head xml:id="echoid-head558" xml:space="preserve">Quomodo ope {προ}πςραςιχ{οῦ} frenum uſuifiat <lb/>accommodatum.</head> <p> <s xml:id="echoid-s5272" xml:space="preserve">Freno {προ}πςραςιχ{\~ο} lupum, qualem equo idoneum judicabis, adaptato, <lb/>deinde ope ſcapi per cavum mobilis, reliquiſque partibus {προ}πςραςιχ{οῦ} mo-<lb/>bilibus, longitudo & </s> <s xml:id="echoid-s5273" xml:space="preserve">ſcapi, & </s> <s xml:id="echoid-s5274" xml:space="preserve">capitelli, & </s> <s xml:id="echoid-s5275" xml:space="preserve">angulus ſub rectis à punctis conta-<lb/>ctuum eductis comprehĕſus inſtituantur primò tanta, quemadmodum equo <lb/>commodiſſimum judicabis: </s> <s xml:id="echoid-s5276" xml:space="preserve">poſt autem cum indito ei freno edoctus eris ali-<lb/>quam harum quatuor partium, aut inſimul omnes mutatione opus habere, ut <lb/>vel ſcapi, vel capitella producenda ſint minuendave, vel angulus ſub lineis à <lb/>contactuum punctis eductis amplior arctiórve inſtituendus, vel pſellium au-<lb/>ctius contractiusve fabricandum. </s> <s xml:id="echoid-s5277" xml:space="preserve">iſta in ſingulis labore minimo nec fallenti-<lb/>bus indiciis novari poterunt. </s> <s xml:id="echoid-s5278" xml:space="preserve">ut neque frenum equo dematur, nec eques ab <lb/>equo deſcendat. </s> <s xml:id="echoid-s5279" xml:space="preserve">Cum igitur προπςραςιχὸν ita formaris prout equo erit com-<lb/>modiſſimum, ſimili partium præcipuarum poſitu & </s> <s xml:id="echoid-s5280" xml:space="preserve">reſponſu, hoc eſt, ut an-<lb/>gulus ſub rectis à tactuum punctis eductis comprehenſus, rectæq̀ue imagina-<lb/>riæ eum comprehendentes illis æquales ſint. </s> <s xml:id="echoid-s5281" xml:space="preserve">reliqua pro libitu adornentur <lb/>phaleræ enim ſcaporumq́ue inverſuræ cæteraq́; </s> <s xml:id="echoid-s5282" xml:space="preserve">varietatem nullam inducunt, <lb/>modo catellæ intermedię, pſellium, lupuſq́ue ſimiliter convenienti ſitu & </s> <s xml:id="echoid-s5283" xml:space="preserve">for-<lb/>ma confiant. </s> <s xml:id="echoid-s5284" xml:space="preserve">Frenum ita deformatum equo non minus congruum erit quam <lb/>προπδυςιχὸν fuerat, eodemq́ue plané malo equum coget <lb/>Ire viam quam docet eques ---<lb/>quod I*LLVSTRISSIMVS* P*RINCEPS* reapſe expertus comprobavit.</s> <s xml:id="echoid-s5285" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5286" xml:space="preserve">Nonnulli eorum, qui hancipſam artem libris editis tractarunt, frenos fa-<lb/>bricant in quos varii ſcapi inæqualium verſurarum inſeri poſsint. </s> <s xml:id="echoid-s5287" xml:space="preserve">ſed ſi illic an-<lb/>nelli contactus eodem maneat loco, majores minorésve inverſuræ ad rem ni-<lb/>hil afferent momenti, quemadmodum prima propoſitione docuimus. </s> <s xml:id="echoid-s5288" xml:space="preserve">vel ut <lb/>aliis verbis eandem ſententiam exponam, ſi annelli contactus alio ſit loco, <lb/>major minórve ſcapi inflexio non eſt cauſa effectus varietatis ejus, quam in <lb/>equi ductu ſentimus, qui inde exiſtit quod annelli contactus alio loco cõ ſiſtat <lb/>unde inveſtigatio iſtiuſmodi latentibus & </s> <s xml:id="echoid-s5289" xml:space="preserve">ignoratis cauſis incertiſsima eſt & </s> <s xml:id="echoid-s5290" xml:space="preserve"><lb/>obſcura, ut perquam rarò hac via freni idonei congruiq́ue confiant.</s> <s xml:id="echoid-s5291" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div731" type="section" level="1" n="526"> <head xml:id="echoid-head559" xml:space="preserve">NOTA.</head> <p> <s xml:id="echoid-s5292" xml:space="preserve">Qui catholicum ſtatices theorema de machinarum effectionibus conſide. <lb/></s> <s xml:id="echoid-s5293" xml:space="preserve">rabit, inde ita inferet. </s> <s xml:id="echoid-s5294" xml:space="preserve">quia pari manus ductu, etiamſi capitella ſint inæqualia, <lb/>par pſelliorum motus exiſtat, hinc quoq; </s> <s xml:id="echoid-s5295" xml:space="preserve">æqualem preſſus effectionem con-<lb/>ſequi. </s> <s xml:id="echoid-s5296" xml:space="preserve">exemplo res erit clarior. </s> <s xml:id="echoid-s5297" xml:space="preserve">ſi enim duo freni conſtruantur, quorum an-<lb/>guli ſub rectis à punctis contactuum eductis comprehenſi inter ſe æquentur, <lb/>pſelliaq́ue utrobique parem habeant laxitatem, ſcapiq́ue capitellis proportio-<lb/>nales quidem ſed inæquales ſint, in illis utrobique pſellium æquali manus du-<lb/>ctu tantundem commovebitur, unde par preſſus eſſicientia exiſtet. </s> <s xml:id="echoid-s5298" xml:space="preserve">quod po-<lb/>ſuo diagrammate illuſtiius erit.</s> <s xml:id="echoid-s5299" xml:space="preserve"/> </p> <pb o="194" file="527.01.194" n="194" rhead="A*DDITAMENTI* S*TATICÆ* P*ARS QUARTA*"/> <p> <s xml:id="echoid-s5300" xml:space="preserve">D*ATVM.</s> <s xml:id="echoid-s5301" xml:space="preserve">* A B ſcapus ſit præ-<lb/> <anchor type="figure" xlink:label="fig-527.01.194-01a" xlink:href="fig-527.01.194-01"/> longus, A C prælongum ejuſdem <lb/>capitellum, in cõtinuata A B: </s> <s xml:id="echoid-s5302" xml:space="preserve">con-<lb/>tra A D ſcapus breviuſculus, & </s> <s xml:id="echoid-s5303" xml:space="preserve"><lb/>A E ejus capitellum breviuſculum; <lb/></s> <s xml:id="echoid-s5304" xml:space="preserve">ſed ita, ut A E A D, A C A B. </s> <s xml:id="echoid-s5305" xml:space="preserve"><lb/>proportionales ſint. </s> <s xml:id="echoid-s5306" xml:space="preserve">agatur deinde <lb/>ipſi A B æqualis A F, & </s> <s xml:id="echoid-s5307" xml:space="preserve">A G ipſi <lb/>A C: </s> <s xml:id="echoid-s5308" xml:space="preserve">ſintq́ue ab F & </s> <s xml:id="echoid-s5309" xml:space="preserve">G in B C <lb/>perpendiculares, F H, G C. </s> <s xml:id="echoid-s5310" xml:space="preserve"><lb/>deinde A K æqualis rectæ A D <lb/>tantum attollatur ut K L perpen-<lb/>diculatis ſit æqualis ipſi H F. </s> <s xml:id="echoid-s5311" xml:space="preserve">ſimi-<lb/>liter ut altrinſecus A M, in con-<lb/>tinuata K A, æquetur ipſi A E, <lb/>& </s> <s xml:id="echoid-s5312" xml:space="preserve">ab M perpendicularis cadat <lb/>M N. </s> <s xml:id="echoid-s5313" xml:space="preserve">his conſtitutis, ponamus <lb/>annellum B ſcapi longioris ab B <lb/>adductum in F ut diſtantia à prio-<lb/>ri ſcapi ſitu ſit F A: </s> <s xml:id="echoid-s5314" xml:space="preserve">minoris au-<lb/>tem ſcapi annellum ab D didu-<lb/>ctum in K, ut diſtantia ſit L K. </s> <s xml:id="echoid-s5315" xml:space="preserve">his <lb/>ductibus ocellus majoris ſcapi mi-<lb/>grabit ab C in G, eritq́ue diſtan-<lb/>tia à priori ſitu I G: </s> <s xml:id="echoid-s5316" xml:space="preserve">ocellus verò E <lb/>capitelli minoris diſcedet in M, cu-<lb/>jus à capitelli primo ſitu diſtantia <lb/>eſt M N. </s> <s xml:id="echoid-s5317" xml:space="preserve">Sed motus intervalla H F <lb/>L K pro manus ductibus (quiaillis <lb/>æquantur) cenſenda ſunt: </s> <s xml:id="echoid-s5318" xml:space="preserve">item <lb/>IG M N pro pſelliorum motibus-<lb/>quod illis æquentur. </s> <s xml:id="echoid-s5319" xml:space="preserve">Quæ cum ita <lb/>ſint demonſtrandum eſt N M æ, <lb/>quari ipſi I G. </s> <s xml:id="echoid-s5320" xml:space="preserve">unde, quod propo-<lb/>ſitum nobis fuerat, par preſſus vio-<lb/>lentia neceſſariò concluditur.</s> <s xml:id="echoid-s5321" xml:space="preserve"/> </p> <div xml:id="echoid-div731" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.194-01" xlink:href="fig-527.01.194-01a"> <image file="527.01.194-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.194-01"/> </figure> </div> </div> <div xml:id="echoid-div733" type="section" level="1" n="527"> <head xml:id="echoid-head560" xml:space="preserve">DEMONSTRATIO.</head> <p> <s xml:id="echoid-s5322" xml:space="preserve">Triangulum A K L ſimile eſt triangulo A M N; </s> <s xml:id="echoid-s5323" xml:space="preserve">ideoq́ue latera quæ ſimili <lb/>ſitu reſpondent proportionalia, <lb/>Sic A K ad A M eſt, ut K L ad M N.</s> <s xml:id="echoid-s5324" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5325" xml:space="preserve">Triangulum A F H ſimile eſt triangulo A I G, ideoq́; </s> <s xml:id="echoid-s5326" xml:space="preserve">latera ſimiliter ſita <lb/>erunt proportionalia.</s> <s xml:id="echoid-s5327" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5328" xml:space="preserve">hoc eſt, A F ad A G, ut FH ad GI, <lb/>ſed ut A F ad A G, ſic A K ad A M. </s> <s xml:id="echoid-s5329" xml:space="preserve">itaque <lb/>ut A K ad A M, ſic F H ad G I.</s> <s xml:id="echoid-s5330" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5331" xml:space="preserve">atqui F H & </s> <s xml:id="echoid-s5332" xml:space="preserve">K L per hypotheſin æquantur. </s> <s xml:id="echoid-s5333" xml:space="preserve">erit igitur <lb/>ut A K ad A M, ſic K L ad GI.</s> <s xml:id="echoid-s5334" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5335" xml:space="preserve">Vt G I & </s> <s xml:id="echoid-s5336" xml:space="preserve">M N quartæ proportionales ſint, poſitis iiſdem tribus antece- <pb o="195" file="527.01.195" n="195" rhead="DE C*HALINO THLIPSI.*"/> dentibus terminis. </s> <s xml:id="echoid-s5337" xml:space="preserve">videlicet M N in prima proportione, & </s> <s xml:id="echoid-s5338" xml:space="preserve">G I in noviſsima. <lb/></s> <s xml:id="echoid-s5339" xml:space="preserve">ideoq́ue ipſæ inter ſe æquales.</s> <s xml:id="echoid-s5340" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5341" xml:space="preserve">Quamobrem, cum pſellium utroq; </s> <s xml:id="echoid-s5342" xml:space="preserve">freno tantundem impellatur, inde par <lb/>preſſus alperitas exiſtere videri poſſet, cum tamen hæc res ſe longe ſecus ha-<lb/>beat, quare non alienum videtur iſta paulo fuſius explicare.</s> <s xml:id="echoid-s5343" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5344" xml:space="preserve">Experientia teſtatur, quod à quibuſdam quoque annotatum memini, lon-<lb/>gioribus capitellis quoſdam æquos cervicem erectiorem geſtare quam bre-<lb/>vioribus. </s> <s xml:id="echoid-s5345" xml:space="preserve">cujus cauſam P*RINCEPS* I*LLVSTRISSIMVS* hanc eſſe arbi-<lb/>tratur. </s> <s xml:id="echoid-s5346" xml:space="preserve">A B deſignet capitellum longius, A C brevius; </s> <s xml:id="echoid-s5347" xml:space="preserve">longioris pſellium <lb/>BD, brevioris autem C D. </s> <s xml:id="echoid-s5348" xml:space="preserve">jam longius pſellium B D angulum cum capi-<lb/>tello conſtituit acutiorem, quam C D. </s> <s xml:id="echoid-s5349" xml:space="preserve">nam A B D angulus minor eſt angu-<lb/> <anchor type="figure" xlink:label="fig-527.01.195-01a" xlink:href="fig-527.01.195-01"/> lo A C D. </s> <s xml:id="echoid-s5350" xml:space="preserve">conſtat itaque quod (cum ductu habenæ E F capitellum A B <lb/>commovebitur) pſelliũ C D directius, at B D obliquius & </s> <s xml:id="echoid-s5351" xml:space="preserve">ſurſum verſus equi <lb/>mentũ preſſet: </s> <s xml:id="echoid-s5352" xml:space="preserve">quamobrem ut preſſum hunc evitet cervicem fert erectiorem.</s> <s xml:id="echoid-s5353" xml:space="preserve"/> </p> <div xml:id="echoid-div733" type="float" level="2" n="1"> <figure xlink:label="fig-527.01.195-01" xlink:href="fig-527.01.195-01a"> <image file="527.01.195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.195-01"/> </figure> </div> <p> <s xml:id="echoid-s5354" xml:space="preserve">Occurrat aliquis, ſi hæclongioris capitelli foret affectio id non ſolum in <lb/>quibuſdam equis (quemadmodum nos monuimus) ſed in omnibus omnino <lb/>re ipſa comprobatum iri, quod tamen ſecus eſſe multi teſtantur, atque inter <lb/>alios quoque le Sieur de la Brouë libro 3 peculiari tactatu cuititulum fecit <lb/>Occaſions pour leſquelles on doit faire l'æil de la branche plus baut ou plus ba@ <lb/>que la meſure ordinaire.</s> <s xml:id="echoid-s5355" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5356" xml:space="preserve">Ipſum quoque I*LLVSTRISSIMVM* P*RINCIPEM* teſtantem audivi <lb/>ſeſe experientia edoctum capitello productiore quoſdam equos cervicem at-<lb/>tollere, nonnullos contra demittere, quodq́; </s> <s xml:id="echoid-s5357" xml:space="preserve">ipſum tum nõ leviter ſit miratus. <lb/></s> <s xml:id="echoid-s5358" xml:space="preserve">ſed cum poſtmodum ſtaticen edoctus idem conſideraret diligentius, hanc eſſe <lb/>tam diverſi effectus cauſam ſibi perſuaſit. </s> <s xml:id="echoid-s5359" xml:space="preserve">quod productio capitelli, quæ plus <lb/>aſperitatis mento mandibulæq́ue per 3 propoſ. </s> <s xml:id="echoid-s5360" xml:space="preserve">infligit, cõtrarios ſimul habeat <lb/>effectus. </s> <s xml:id="echoid-s5361" xml:space="preserve">nam validiore lupi in mandibulã impreſſione equus caput demittit, <lb/>quî aſperitatem illam declinet; </s> <s xml:id="echoid-s5362" xml:space="preserve">at cõtra validiore pſelli contra mentum preſſu <lb/>caput arrigit, ut dolori huic ſeſe quoque ſubducat, quod jam ſupra demõſtra-<lb/>vimus. </s> <s xml:id="echoid-s5363" xml:space="preserve">cum autem utraque aſperitas concurrat dolori graviſsimo celerrimè <lb/>medicinam parat. </s> <s xml:id="echoid-s5364" xml:space="preserve">verum nonnulli equi ſunt mandibulæ gingivæq́ue tenerio- <pb o="196" file="527.01.196" n="196" rhead="A*DDITA.* S*TAT*. P*ARS QVARTA DE* C*HALIN*."/> ris, & </s> <s xml:id="echoid-s5365" xml:space="preserve">menti durioris; </s> <s xml:id="echoid-s5366" xml:space="preserve">at contra alii duriore ſunt gingiva, teneriore mento. <lb/></s> <s xml:id="echoid-s5367" xml:space="preserve">conſequens igitur eſt quoſdam equos caput erigere, quoſdam demittere. </s> <s xml:id="echoid-s5368" xml:space="preserve">Ve-<lb/>runtamen ut theorema iſtud generaliter definiamus: </s> <s xml:id="echoid-s5369" xml:space="preserve">longius capitellum (ſi <lb/>pſellium, quâ ſurſum ſuo preſlu tendit tantum conſideremus) non parum af-<lb/>fert momenti ut equus caput erigat, quamvisalio graviore malo coactus con-<lb/>tra nonnunquam ſe demittat.</s> <s xml:id="echoid-s5370" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5371" xml:space="preserve">Quamobrem conſequens videtur equis, qui capite ſatis arrecto incedunt, <lb/>quorumq́ue gingivæ non ſit nimia teneritudo, breviora capitella & </s> <s xml:id="echoid-s5372" xml:space="preserve">pſellia <lb/>ſtrictiora eſſe idonea, nam longiora capitella cum laxiore pſellio ſubito fortè <lb/>adducta quaſi plagam faciunt, & </s> <s xml:id="echoid-s5373" xml:space="preserve">equorum ora magis corrumpunt quam <lb/>breviora ſtrictiori pſellio. </s> <s xml:id="echoid-s5374" xml:space="preserve">quæminore quidem impetu eorum pectoriaddu-<lb/>cuntur & </s> <s xml:id="echoid-s5375" xml:space="preserve">tamen parem preſſionis habent efſecientiam. </s> <s xml:id="echoid-s5376" xml:space="preserve">Huc adde, longiora <lb/>pſellia ut B D facileab mento delabi, ut tum equus ſeſſoris imperio non ob-<lb/>temperet, quod brevioribus pſelliis C D non contingit.</s> <s xml:id="echoid-s5377" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5378" xml:space="preserve">Notato & </s> <s xml:id="echoid-s5379" xml:space="preserve">hoc. </s> <s xml:id="echoid-s5380" xml:space="preserve">cum non erit opus longioribus capitellis quîequus erectior <lb/>incedat (quod evenitubi mandibulæ teneritudo eadem erit quæ menti) per-<lb/>beve capitellum uſurpandum, ſcaporum autem longitudinem, quanta com-<lb/>modiſſi<unsure/>ma videbitur: </s> <s xml:id="echoid-s5381" xml:space="preserve">aſperitatem deinde amplitudine, vel brevitate pſellio-<lb/>rum pro placito tem perandam.</s> <s xml:id="echoid-s5382" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5383" xml:space="preserve">Verumenimverò, quia I*LLVSTRISSIMVS* P*RINCEPS* affectiones <lb/>has diligentiſſime indagavit, coronidis loco aliam quandam anomaliam de <lb/>frenis quorum ſcapi & </s> <s xml:id="echoid-s5384" xml:space="preserve">capitella proportionalia ſunt annotabo.</s> <s xml:id="echoid-s5385" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s5386" xml:space="preserve">Cui fini ponatur A B <lb/> <anchor type="figure" xlink:label="fig-527.01.196-01a" xlink:href="fig-527.01.196-01"/> longior ſcapus ejusq́ue <lb/>habena B C, A D bre-<lb/>vior cujus habena D E. <lb/></s> <s xml:id="echoid-s5387" xml:space="preserve">partes ſupremæ inferio-<lb/>ribus ſcapis ſtatuantur <lb/>proportionales. </s> <s xml:id="echoid-s5388" xml:space="preserve">quam-<lb/>vis jam duo iſti ſcapi lõ-<lb/>giores ob analogiam pa-<lb/>rem efficiant preſſum, <lb/>tamen manifeſtum eſt <lb/>manumnõ eodem utro-<lb/>bique loco verſari: </s> <s xml:id="echoid-s5389" xml:space="preserve">ſed <lb/>ſi, dum B adducet, ea in <lb/>C conſiſtat, cum D du-<lb/>cet oportebit eam ſta-<lb/>tui in E, ut D E paral-<lb/>lela ſit contra B C. </s> <s xml:id="echoid-s5390" xml:space="preserve">Nam adducta habena ſecundum D C eaalium compre-<lb/>henderet angulum cum A D quam D E, quæres mutationem aliquamne-<lb/>ceſſariò inducet.</s> <s xml:id="echoid-s5391" xml:space="preserve"/> </p> <div xml:id="echoid-div734" type="float" level="2" n="2"> <figure xlink:label="fig-527.01.196-01" xlink:href="fig-527.01.196-01a"> <image file="527.01.196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/527.01.196-01"/> </figure> </div> </div> <div xml:id="echoid-div736" type="section" level="1" n="528"> <head xml:id="echoid-head561" xml:space="preserve">C*HALINOTHLIPSIS*</head> <head xml:id="echoid-head562" xml:space="preserve">FINIS.</head> <pb file="527.01.197" n="197"/> </div> <div xml:id="echoid-div737" type="section" level="1" n="529"> <head xml:id="echoid-head563" xml:space="preserve">TOMVS <lb/>QVINTVS <lb/>MATHEMATICORVM <lb/>HYPOMNEMATVM, <lb/>DE <lb/>MISCELLANEIS.</head> <head xml:id="echoid-head564" xml:space="preserve">Quo comprehenduntur ea in quibus ſe exercuit</head> <head xml:id="echoid-head565" xml:space="preserve">ILLVSTRISSIMVS, ILLVSTRIS-</head> <p> <s xml:id="echoid-s5392" xml:space="preserve">ſimo & </s> <s xml:id="echoid-s5393" xml:space="preserve">antiquiſſimo ſtemmateortus Princeps, ac Dominus <lb/>M*AURITIUS*, Princeps Auraïcus, Comes Naſſoviæ, <lb/>Cattimelibocorum, Viandæ, Moerſii, & </s> <s xml:id="echoid-s5394" xml:space="preserve">c. </s> <s xml:id="echoid-s5395" xml:space="preserve">Marchio Veræ, & </s> <s xml:id="echoid-s5396" xml:space="preserve">Vliſſingę, &</s> <s xml:id="echoid-s5397" xml:space="preserve">c. <lb/></s> <s xml:id="echoid-s5398" xml:space="preserve">Dominus civitatis Gravæ, & </s> <s xml:id="echoid-s5399" xml:space="preserve">ditonis Cuyc, civitatum Uyt, <lb/>Daeſburch, &</s> <s xml:id="echoid-s5400" xml:space="preserve">c. </s> <s xml:id="echoid-s5401" xml:space="preserve">Gubernator Geldriæ, Hollandiæ, Zelandiæ, <lb/>Weſtfriſiæ, Zutphaniæ, Vltrajecti, Tranſiſalanę, &</s> <s xml:id="echoid-s5402" xml:space="preserve">c. </s> <s xml:id="echoid-s5403" xml:space="preserve"><lb/>Imperator exercitus Provinciarum fœdere <lb/>conſociatarum Belgii, Architalaſſus <lb/>Generalis, &</s> <s xml:id="echoid-s5404" xml:space="preserve">c.</s> <s xml:id="echoid-s5405" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div738" type="section" level="1" n="530"> <head xml:id="echoid-head566" xml:space="preserve">Conſcriptus à S*IMONE* S*TEVINO*.</head> <head xml:id="echoid-head567" xml:space="preserve">L*VGDVNI* B*ATAVORVM*, <lb/>Ex Officinâ Ioannis Patii, Academiæ Typographi. <lb/>Anno cIↄ<unsure/>. Iↄ<unsure/>. c. *VIII*.</head> </div></text> </echo>