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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:YS05QMU8.xml</dcterms:identifier> <dcterms:creator identifier="GND:118503863">Archimedes</dcterms:creator> <dcterms:contributor identifier="GND:11862086X">Tartaglia, Niccolò</dcterms:contributor> <dcterms:title xml:lang="la">De insidentibus aquae</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1565</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> <dcterms:description xml:id="echoid-dcterms:description">test</dcterms:description> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> <pb file="0002" n="2"/> <handwritten/> <pb file="0003" n="3"/> <handwritten/> <pb file="0004" n="4"/> <handwritten/> <handwritten/> <handwritten/> <pb file="0005" n="5"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">ARCHIMEDIS <lb/>DE INSIDENTIBVS <lb/>AQV AE.</head> <head xml:id="echoid-head2" xml:space="preserve">LIBER PRIMVS.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head3" style="it" xml:space="preserve">CVM PRIVILEGIO.</head> <head xml:id="echoid-head4" style="it" xml:space="preserve">TROIANO CVRTIO</head> <figure> <image file="0005-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0005-02"/> </figure> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head5" xml:space="preserve">VENETIIS, <lb/>APVD CVRTIVM TROIANVM.</head> <head xml:id="echoid-head6" xml:space="preserve">M D LXV►</head> <pb file="0006" n="6"/> <handwritten/> <figure> <image file="0006-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0006-01"/> </figure> <pb o="2" file="0007" n="7"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head7" xml:space="preserve">FABRITIO DENORES FILIO</head> <head xml:id="echoid-head8" xml:space="preserve">IACOBI COMITIS TRIPOLIS</head> <head xml:id="echoid-head9" xml:space="preserve">VCRTIVS TROIANVS S. P. D.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">SI omni laude digni habiti ſunt magni uiri, qui <lb/>omne ſuum ſtudium in id contulerunt, ut cæ-<lb/>teris hominibus quàm maxime prodeſſe poſ-<lb/>ſent; </s> <s xml:id="echoid-s2" xml:space="preserve">in magnum dedecus incurrent omnes, <lb/>qui illorum opera aut occultant, aut, ut in pu <lb/>blicum prodeant, nullam curam afferunt. </s> <s xml:id="echoid-s3" xml:space="preserve">Cum <lb/>uero labores maximi illi quidem Nicolai Tar-<lb/>taleæ quotidie magis, ac magis cognoſcantur profuiſſe litera-<lb/>tisuiris, non modica uidebor ego dignus reprehenſione, qui <lb/>reliquias habui eiuſdem laborum, & </s> <s xml:id="echoid-s4" xml:space="preserve">uigiliarum, ni illas quoque <lb/>in medium proferam, & </s> <s xml:id="echoid-s5" xml:space="preserve">communi utilitati conſulam. </s> <s xml:id="echoid-s6" xml:space="preserve">Quare <lb/>cum habeam adhuc apud me Archimedem de inſidentibus a quæ <lb/>ab ipſo Nicolao in lucem reuocatum, & </s> <s xml:id="echoid-s7" xml:space="preserve">quantum ab ipſo fieri <lb/>potuit, ab erroribus librarij emendatum, & </s> <s xml:id="echoid-s8" xml:space="preserve">ſuis lucubrationi-<lb/>bus illuſtratum; </s> <s xml:id="echoid-s9" xml:space="preserve">uideor fraudare omnes literatos ſua poſſeſsio-<lb/>ne, ni omnia, quæ huius ingenioſiſsimi uiri apud me reſtant, in <lb/>lucem emiſero, & </s> <s xml:id="echoid-s10" xml:space="preserve">omnibus ea communicauero. </s> <s xml:id="echoid-s11" xml:space="preserve">Ac cum noue-<lb/>rim te cum omnibus rectiſsimis ſtudijs mirifice deditum, tum to <lb/>tum ad imitationem tuorum maiorum, & </s> <s xml:id="echoid-s12" xml:space="preserve">ad rem gerendam in-<lb/>flammatum; </s> <s xml:id="echoid-s13" xml:space="preserve">putaui hoc opus tibi, & </s> <s xml:id="echoid-s14" xml:space="preserve">tui ſimilibus, qui indiſcipli <lb/>nis uerſantur, & </s> <s xml:id="echoid-s15" xml:space="preserve">res magnas gerunt, fore peropportunùm. </s> <s xml:id="echoid-s16" xml:space="preserve">nec <lb/>uero meæ facultatis eſt, nec breuitas huius ſcriptionis poſtulat, <lb/>ut dete, ac de tuis maioribus ego nunc plura dicam. </s> <s xml:id="echoid-s17" xml:space="preserve">nam fi re-<lb/>peterem clariſsimos uiros, qui literis, & </s> <s xml:id="echoid-s18" xml:space="preserve">armis in tua familia flo <lb/>ruerunt, eorumq́ue res geſtas enarrarem, atque quibus rebus, <lb/>tu, & </s> <s xml:id="echoid-s19" xml:space="preserve">optimus, ac clariſsimus pater tuus eorum gloriam adauge <lb/>tis; </s> <s xml:id="echoid-s20" xml:space="preserve">longe maius opus mihi extaret, quàm eſſet hic paruus libel-<lb/>lus, quamq́ue ego poſſem perficere. </s> <s xml:id="echoid-s21" xml:space="preserve">Itaque hæc alijs, qui poſ-<lb/>ſunt, relinquens, & </s> <s xml:id="echoid-s22" xml:space="preserve">in aliud tempus differens, ut nonnullum per <lb/>me adiumentum addatur tibi, & </s> <s xml:id="echoid-s23" xml:space="preserve">cæteris, qui rerum naturam con <lb/>templantur, & </s> <s xml:id="echoid-s24" xml:space="preserve">ijs artibus ſtudent quibus res maximæ geruntur; <lb/></s> <s xml:id="echoid-s25" xml:space="preserve">hoc opus in tuo nomine peruulgari, atque ediuolui, ut noſcant <lb/>omnes dum ſtudeo prodeſſe communi utilitati, ſeparatim ta-<lb/>men pro mea in te obſeruantia uoluiſſe tuis ſtudijs, magnitudi-<lb/>niq́ animi inſeruire.</s> <s xml:id="echoid-s26" xml:space="preserve"/> </p> <pb file="0008" n="8"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head10" xml:space="preserve">ARCHIMEDIS DE <lb/>INSIDENTIBVS AQV AE.</head> <head xml:id="echoid-head11" xml:space="preserve">LIBER PRIMVS.</head> <head xml:id="echoid-head12" xml:space="preserve">Suppoſitio prima.</head> <p> <s xml:id="echoid-s27" xml:space="preserve">Suppon atur humidum habens talem naturam, ut partibus ip-<lb/>ſius ex æquo iacentibus, & </s> <s xml:id="echoid-s28" xml:space="preserve">exiſtentibus continuis, expellatur mi-<lb/>nus pulſa a magis pulſa, & </s> <s xml:id="echoid-s29" xml:space="preserve">unaqueque autem partium ipſius pel <lb/>litur humido, quod ſupra ipſius ex iſtente ſecundum perpendicu <lb/>larem, ſi humidum ſit deſcendens in aliquo, & </s> <s xml:id="echoid-s30" xml:space="preserve">ab alio aliquo <lb/>preſſum.</s> <s xml:id="echoid-s31" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div7" type="section" level="1" n="7"> <head xml:id="echoid-head13" xml:space="preserve">Theorema primum. Propoſitio prima.</head> <p> <s xml:id="echoid-s32" xml:space="preserve">Si ſuperficies aliqua plane ſecta per aliquod ſignum ſemper <lb/>idem ſignum ſectionem facientem circuli periferiam centrum <lb/>habẽtem ſignũ, per quod planoſecatur ſphæræ, erit ſuperficies.</s> <s xml:id="echoid-s33" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s34" xml:space="preserve">SI enim ſuperſicies aliqua ſesta per ſignum K, plano ſuper ſestionem fa-<lb/>cientes circuli periferiam, centrum autem ipſius k, ſi igitur ipſa ſuperfi-<lb/>cies non est ſphæræ ſuperficies, non erunt omnes, quæ a centro ad ſuperfi <lb/>ciem, occurrentes lineæ æquales. </s> <s xml:id="echoid-s35" xml:space="preserve">Sit itaque a, b, g, d, ſigna in ſuperficie, & </s> <s xml:id="echoid-s36" xml:space="preserve"><lb/>inæquales, quæ K. </s> <s xml:id="echoid-s37" xml:space="preserve">a, K, b, per ipſas autem K, a, k, b, planum educatur, & </s> <s xml:id="echoid-s38" xml:space="preserve">fa-<lb/>ciat ſectionem in ſuperficie lineam d, a, b, g, circuli ergo eſt ipſa centrum au-<lb/>tem ipſius K. </s> <s xml:id="echoid-s39" xml:space="preserve">Q uoniam ſupponebatur ſuperficies talis non ſunt ergo inæqua <lb/>les lineæ K, a, K, b, neceſſarium igitur eſt ſuperficies eſſe ſphæræ ſuperficiem.</s> <s xml:id="echoid-s40" xml:space="preserve"/> </p> <figure> <image file="0008-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0008-01"/> </figure> <pb o="3" file="0009" n="9" rhead="LIBER I."/> </div> <div xml:id="echoid-div8" type="section" level="1" n="8"> <head xml:id="echoid-head14" xml:space="preserve">Theorema ij. Propoſitio ij.</head> <p> <s xml:id="echoid-s41" xml:space="preserve">Omnis humidi conſiſtentis ita ut maneat in motum ſuperfi <lb/>cies habebit figuram ſphęrę habentis cẽtruni idem cum terra.</s> <s xml:id="echoid-s42" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s43" xml:space="preserve">INtelligatur enim bumidum conſiſtensita, ut maneat non motum, & </s> <s xml:id="echoid-s44" xml:space="preserve"><lb/>ſecetur ipſius. </s> <s xml:id="echoid-s45" xml:space="preserve">Superſicies plano per centrum terræ. </s> <s xml:id="echoid-s46" xml:space="preserve">Sit autemterra <lb/>centrum K, ſuperficiei autem ſecta linea a, b, g, d. </s> <s xml:id="echoid-s47" xml:space="preserve">Dico itaque line a a, b, <lb/>g, d, circuli eſſe periferiam centrum autem ipſius K. </s> <s xml:id="echoid-s48" xml:space="preserve">Sienim non est recte <lb/>a, K, ad lineam a, b, g, d, occurrentes non erunt æquales. </s> <s xml:id="echoid-s49" xml:space="preserve">Sumatur itaque <lb/>aliqua recta quę eſt quarundam quidem a, k, occurrentium ad lineam a, b, <lb/>g, d, maior quarundam autem minor, & </s> <s xml:id="echoid-s50" xml:space="preserve">centro quidem K, distantia autem <lb/>ſumptæ lineæ circulus deſcribatur. </s> <s xml:id="echoid-s51" xml:space="preserve">Cadetigitur periferia circuli habẽs hoc <lb/>quidem extra lineam a, b, g, d, hoc autem intra, quoniam quæ ex centro <lb/>quorundam qu dem a, K, occurrentium ad lineam a b, g, d, eſt maior quo-<lb/>rundam autem minor. </s> <s xml:id="echoid-s52" xml:space="preserve">Sint igitur deſcripti circuli periferia quæ r, b, b, & </s> <s xml:id="echoid-s53" xml:space="preserve"><lb/>a, b, ad K, recta ducantur, & </s> <s xml:id="echoid-s54" xml:space="preserve">copulentur quæ b, K, b, e, l, æquales facientes <lb/>angulos. </s> <s xml:id="echoid-s55" xml:space="preserve">Deſcribatur autem, & </s> <s xml:id="echoid-s56" xml:space="preserve">centro K, periferia quidem quæ x, o, p, in <lb/>plano & </s> <s xml:id="echoid-s57" xml:space="preserve">in bumido partes itaque bumidi, quæ ſecundum x, o, p, periferiã <lb/>ex æquo ſunt poſitæ cõtinue inuicem premuntur quæ quidem ſecundũ x, o, <lb/> <anchor type="figure" xlink:label="fig-0009-01a" xlink:href="fig-0009-01"/> periferia p, o, b, e, humido quæ ſecundum 2, b, locum quæ autemſecundum <lb/>periferiam o, p, humido quod ſecundum b, e, locum æqualiter igitur premũ <lb/>tur partes bumidi, quod ſecundum periferiam x, o, ei quæſecundum o, p. <lb/></s> <s xml:id="echoid-s58" xml:space="preserve">Quare non expelletur minus preſſa a magis preſſis. </s> <s xml:id="echoid-s59" xml:space="preserve">Non ettam ergo con-<lb/>ſtare fecimus aliquod humidum. </s> <s xml:id="echoid-s60" xml:space="preserve">Supponebatur autem constans ita ut ma-<lb/>neret non motum neceſſarium, ergo linea a, b, g, d, eſt circuli periferiam, et <lb/>centrum ipſius K. </s> <s xml:id="echoid-s61" xml:space="preserve">Similiter autem demonſtrabitur, & </s> <s xml:id="echoid-s62" xml:space="preserve">ſuperficies humidi <lb/>plano ſecta fuerit per centrum terræ, quòd ſectio erit circuliperiferia, &</s> <s xml:id="echoid-s63" xml:space="preserve"> <pb file="0010" n="10" rhead="DEINSID ENTIBVS AQVAE"/> centrum ipſius erit quòd & </s> <s xml:id="echoid-s64" xml:space="preserve">terræ centrum. </s> <s xml:id="echoid-s65" xml:space="preserve">Palàm igitur quòd ſuperficies <lb/>bumidi conſtantis non motibabet figuram ſpbæræ habentis centrum idem <lb/>cum terra quaniam talis est, ut ſecta per idem ſignum ſectionem faciat cir-<lb/>culi periferiam habentis ſignum per quod ſecatur plano.</s> <s xml:id="echoid-s66" xml:space="preserve"/> </p> <div xml:id="echoid-div8" type="float" level="2" n="1"> <figure xlink:label="fig-0009-01" xlink:href="fig-0009-01a"> <image file="0009-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0009-01"/> </figure> </div> </div> <div xml:id="echoid-div10" type="section" level="1" n="9"> <head xml:id="echoid-head15" xml:space="preserve">Theorema iij. Propoſitio iij.</head> <p> <s xml:id="echoid-s67" xml:space="preserve">Solidarum magnitudinum quæ ęqualis molis & </s> <s xml:id="echoid-s68" xml:space="preserve">ęqualis pon <lb/>deris cum humido dimiſſe in humidum demergentur ita ut ſu <lb/>perficiem humidi non excedant nihil & </s> <s xml:id="echoid-s69" xml:space="preserve">non adhuc referentur <lb/>ad inferius.</s> <s xml:id="echoid-s70" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s71" xml:space="preserve">DEmonstratur enim aliqua magnitudo æque grauium cum bumido <lb/>in bumidum, & </s> <s xml:id="echoid-s72" xml:space="preserve">ſi poſſibile eſt excedat ipſa ſuperſiciem humidi conſi <lb/>ſtat autem bumidum ut maneat immotum. </s> <s xml:id="echoid-s73" xml:space="preserve">Intelligatur autem ali-<lb/>quod planum eductum per centrum terræ, & </s> <s xml:id="echoid-s74" xml:space="preserve">humidi, & </s> <s xml:id="echoid-s75" xml:space="preserve">per ſolidam ma-<lb/>gnitudinem. </s> <s xml:id="echoid-s76" xml:space="preserve">Sectio autem ſit ſuperficiei quidem bumidi quæ a, b, g, d. </s> <s xml:id="echoid-s77" xml:space="preserve">Solide <lb/>autem magnitudines quæ e, z, b, t, inſidentia centrum autem terræ. </s> <s xml:id="echoid-s78" xml:space="preserve">Sint au <lb/>tem ſolidæ quidem magnitudinis quod quidem b, g, b, t, in bumido quod au <lb/>tem b, e, z, g extra intelligatur, & </s> <s xml:id="echoid-s79" xml:space="preserve">ſolida figura cõpreſſa pyramide baſſem <lb/>quidem babentem par alelogrommum, quod in ſuperficie bumidi, uerticem <lb/>autem centrum terræ ſectio autem ſit plani in quo est quæ a, b, g, d, perife-<lb/>ria, & </s> <s xml:id="echoid-s80" xml:space="preserve">planorum pyramidis quæ K, l, K, m, deſcribatur autem quędam al-<lb/>terius ſphæræ, ſuperficies circa centrum K, in bumido ſub e, z, b, t, quæ x, o, <lb/>p, ſecetur hoc a ſuperficie plani. </s> <s xml:id="echoid-s81" xml:space="preserve">Sumatur autem, & </s> <s xml:id="echoid-s82" xml:space="preserve">qnædam alia pyramis <lb/>æqualis, & </s> <s xml:id="echoid-s83" xml:space="preserve">ſimilis comprebendenti ſolidim continua ipſi ſectio autem ſit <lb/>planorum ipſius quæ K, m, K, n, & </s> <s xml:id="echoid-s84" xml:space="preserve">in bumido intelligatur quædam magni-<lb/> <anchor type="figure" xlink:label="fig-0010-01a" xlink:href="fig-0010-01"/> tudo bumido aſſumpta quæ r, s, e, y, æqualis, & </s> <s xml:id="echoid-s85" xml:space="preserve">ſimilis ſolidæ, quæſecundũ <pb o="4" file="0011" n="11" rhead="LIBER I."/> b, h, e, g, quod eſt ipſius in humido partes autem humidi quæ ſ, in prima py-<lb/>ramide ſub ſuperficie in qua eſt quæ x, o, & </s> <s xml:id="echoid-s86" xml:space="preserve">quæ in altera in qua quæ p, o, <lb/>ex quo ſunt poſitæ, & </s> <s xml:id="echoid-s87" xml:space="preserve">non coutinuæ. </s> <s xml:id="echoid-s88" xml:space="preserve">Similiter autem premuntur quæ qui-<lb/>dem etiam ſesundum x, o, premitur a ſolidot, h, e, r, & </s> <s xml:id="echoid-s89" xml:space="preserve">humido intermedio <lb/>ſuperfi ie quæ ſecundum x, o, l, m, & </s> <s xml:id="echoid-s90" xml:space="preserve">planorum pyramidis quæ autem ſe-<lb/>cundum p, o, ſolido r, ſ, c, y, & </s> <s xml:id="echoid-s91" xml:space="preserve">humido intermedio ſuperficierum quæ ſecun-<lb/>dum p, o, m, n, & </s> <s xml:id="echoid-s92" xml:space="preserve">planorum pyramidis minor autem erit grauitas humidi <lb/>quod ſecundum m, n, o, p, eo quòd ſecunduml, m, x, o. </s> <s xml:id="echoid-s93" xml:space="preserve">Quod. </s> <s xml:id="echoid-s94" xml:space="preserve">n. </s> <s xml:id="echoid-s95" xml:space="preserve">ſecundum r, <lb/>s, c, y, eſt minus ſolido e, z, h, t, i<unsure/>pſius enim ei quod ſecundum h, b, g, t, eſt æ-<lb/>quale quia magnitudine æ quale, & </s> <s xml:id="echoid-s96" xml:space="preserve">æque graue ſupponitur ſolidum cũ hu-<lb/>mido reliquum autem reliquo inæquale eſt. </s> <s xml:id="echoid-s97" xml:space="preserve">Palam igitur quia expelletur <lb/>pars quæ ſecundum periferiam o, p, ab ea quæ ſecundum periferiam o, x, & </s> <s xml:id="echoid-s98" xml:space="preserve"><lb/>non erit humidum non motum Supponitur autem non motum exiſtens. </s> <s xml:id="echoid-s99" xml:space="preserve">nõ <lb/>ergo excedet ſuperficiem humidi aliquid ſolidæ magnitudinis Demerſum <lb/>autem ſolidum non fertur ad inferiora. </s> <s xml:id="echoid-s100" xml:space="preserve">Similiter enim prementur omnes <lb/>partes humidi ex quo poſitæ quia ſolidum eſt æque graue.</s> <s xml:id="echoid-s101" xml:space="preserve"/> </p> <div xml:id="echoid-div10" type="float" level="2" n="1"> <figure xlink:label="fig-0010-01" xlink:href="fig-0010-01a"> <image file="0010-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0010-01"/> </figure> </div> </div> <div xml:id="echoid-div12" type="section" level="1" n="10"> <head xml:id="echoid-head16" xml:space="preserve">Theorema iiij. Propoſitio iiij.</head> <p> <s xml:id="echoid-s102" xml:space="preserve">Solidarum magnitudinum quęcunque leuior fuerit humidi <lb/>miſſa in humidum non demergetur, tota ſed erit aliquid ip-<lb/>ſius extra ſuperſiciem humidi.</s> <s xml:id="echoid-s103" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s104" xml:space="preserve">F It enim ſolida magnitudo leuior humido, & </s> <s xml:id="echoid-s105" xml:space="preserve">dimiſſa in humidum. </s> <s xml:id="echoid-s106" xml:space="preserve">de-<lb/>mergatur tota ſi poſſibile eſt, & </s> <s xml:id="echoid-s107" xml:space="preserve">nihil ipſius ſit extra ſuperficiem hu-<lb/>midi, Conſiſtat autem humidum ita, ut maneat non motum. </s> <s xml:id="echoid-s108" xml:space="preserve">Intelliga <lb/>tur etiam aliqu pl danum eductum per centrum terræ, & </s> <s xml:id="echoid-s109" xml:space="preserve">per humidum, <lb/> <anchor type="figure" xlink:label="fig-0011-01a" xlink:href="fig-0011-01"/> & </s> <s xml:id="echoid-s110" xml:space="preserve">per ſolidamw agnitudmum. </s> <s xml:id="echoid-s111" xml:space="preserve">Secetur autem a plano hoc ſuperficies qui-<lb/>dem humidi ſecundum ſuperficiem a, b, g, d. </s> <s xml:id="echoid-s112" xml:space="preserve">Solida autem magnitudo per fi <pb file="0012" n="12" rhead="DE IN SIDENTIBVS AQV AE"/> guram in r. </s> <s xml:id="echoid-s113" xml:space="preserve">Centrum autem terræ ſit K. </s> <s xml:id="echoid-s114" xml:space="preserve">Intelligatur autem quædam pyra-<lb/>mis comprendens figuram r, ſecundum quod & </s> <s xml:id="echoid-s115" xml:space="preserve">prius uerticem habens ſi-<lb/>gnum K. </s> <s xml:id="echoid-s116" xml:space="preserve">Secentur autem ipſius plana aſuperficie plani a, b, g, ſecundum a, <lb/>K, K, b. </s> <s xml:id="echoid-s117" xml:space="preserve">Accipiatur autem, & </s> <s xml:id="echoid-s118" xml:space="preserve">aliqua alia pyramis æqualis, & </s> <s xml:id="echoid-s119" xml:space="preserve">ſimilis huic. <lb/></s> <s xml:id="echoid-s120" xml:space="preserve">Secentur autem ipſius plana a plano a, b, g, ſecundum K, b, K, g, deſcribatur <lb/>autem & </s> <s xml:id="echoid-s121" xml:space="preserve">quædam alterius ſphæræ ſuperficies in humido circa centrum K. </s> <s xml:id="echoid-s122" xml:space="preserve"><lb/>Sub ſolida autem magnitudine ſecetur ipſa ab eodem plano ſecundũ x, o, p. </s> <s xml:id="echoid-s123" xml:space="preserve"><lb/>Intelligatur autem, & </s> <s xml:id="echoid-s124" xml:space="preserve">magnitudo abſumpta ab humido quæ ſecundumh, <lb/>in poſteriori pyramide æqualis ſolidæ quæ ſecundum r, partes aũt humidi, <lb/>quòd in prima pyramide quæ ſub ſuperficiebus, quæ ſecundum ſuperficiem <lb/>x, o, & </s> <s xml:id="echoid-s125" xml:space="preserve">quod in ſecunda quæ ſub ſuperficiebus quę ſuperficie o p, ex quo ſunt <lb/>poſitæ, & </s> <s xml:id="echoid-s126" xml:space="preserve">continuæ inuicem non ſimiliter autem premuntur quæ quidẽ in <lb/>prima pyramide premitur a ſolida magnitudine, quæ ſecundumr, & </s> <s xml:id="echoid-s127" xml:space="preserve">ab hu <lb/>mido continente ipſas, & </s> <s xml:id="echoid-s128" xml:space="preserve">exiſtente in loco pyramidis, quæ ſecundum a, b, o, <lb/>x. </s> <s xml:id="echoid-s129" xml:space="preserve">Quæ autem in altera pyramide præmittitur ab humido continent ipſam <lb/>exiſtente in loco pyramidis qui ſecundum p, o, b, g, eſt autem, & </s> <s xml:id="echoid-s130" xml:space="preserve">grauitas <lb/>quæ ſccundum r, minor grauitate humidi, quod ſecundum h, quoniam ma-<lb/>gnitudinem quidem eſt æqualis. </s> <s xml:id="echoid-s131" xml:space="preserve">Solida autem magnitudo ſupponitur eß le <lb/>uior humido humidi continentis magnitudines r, b, eritq́ pyramidum æ-<lb/>qualis. </s> <s xml:id="echoid-s132" xml:space="preserve">Magis igitur premitur pars humidi quòd ſub ſuperficiebus, quæſe-<lb/>cundum periferiam o, p, expellet ergo quod minus premitur, & </s> <s xml:id="echoid-s133" xml:space="preserve">non manet <lb/>humidum non motum. </s> <s xml:id="echoid-s134" xml:space="preserve">Supponebatur autem non motum n n ergo demerge <lb/>tur tota, ſed erit aliquid ipſius extra ſuperficiem humidi.</s> <s xml:id="echoid-s135" xml:space="preserve"/> </p> <div xml:id="echoid-div12" type="float" level="2" n="1"> <figure xlink:label="fig-0011-01" xlink:href="fig-0011-01a"> <image file="0011-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0011-01"/> </figure> </div> </div> <div xml:id="echoid-div14" type="section" level="1" n="11"> <head xml:id="echoid-head17" xml:space="preserve">Theorema v. Propoſitio v.</head> <p> <s xml:id="echoid-s136" xml:space="preserve">Solidarum magnitudinum quæcunque fuerit leuior dimiſſa <lb/>in humidũ in tanto demergetur ut tanta moles humidi quan-<lb/>ta eſt moles demerſæ habeat æ qualem grauitatem cumtota ma <lb/>gnitudine.</s> <s xml:id="echoid-s137" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s138" xml:space="preserve">D Iſponantur autem eandem prioribus, & </s> <s xml:id="echoid-s139" xml:space="preserve">ſit humidum nou motum. <lb/></s> <s xml:id="echoid-s140" xml:space="preserve">Sit autem ma nitudo e, z, b, t, leuior humido. </s> <s xml:id="echoid-s141" xml:space="preserve">Siigitur humidum eſt <lb/>non motum ſimiliter prementur partes ipſius ex æquo poſitæ, ſimi-<lb/>liter ergo premetur humidum quodſub ſuperficiebus, quæ ſecundum perife <lb/>rias x, o, & </s> <s xml:id="echoid-s142" xml:space="preserve">p, o. </s> <s xml:id="echoid-s143" xml:space="preserve">Quare æqualis eſt grauitas quæ premitur. </s> <s xml:id="echoid-s144" xml:space="preserve">eſt autem, & </s> <s xml:id="echoid-s145" xml:space="preserve"><lb/>bumidi grauitas, quod in prima pyramide ſine b, h, t, g, ſolido æqualis graui <lb/>tati bumidi, quod in altera pyramide ſiue r, s, c, y, humido palam igitur, ꝙ <lb/>grauitas magnitudinis e, Z, h, t, eſt æqualis grauitati humidir, s, c, y. </s> <s xml:id="echoid-s146" xml:space="preserve">Mani <lb/>feſtum igitur quòd tanta moles humidi quanta eſt demerſa pars ſolidæ ma <lb/>gnitudinis habet grauitatem æqualem toti magnitudini.</s> <s xml:id="echoid-s147" xml:space="preserve"/> </p> <pb o="5" file="0013" n="13" rhead="LIBER I."/> <figure> <image file="0013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0013-01"/> </figure> </div> <div xml:id="echoid-div15" type="section" level="1" n="12"> <head xml:id="echoid-head18" xml:space="preserve">Theorema vj. Propoſitio vj.</head> <p> <s xml:id="echoid-s148" xml:space="preserve">Solida leuiora humido ui preſſa in humidum ſurrexi feruntur <lb/>tanta ui ad ſuperius, quanto humidum habens molẽ æqualem cũ <lb/>magnitudine eſt grauius magnitudine.</s> <s xml:id="echoid-s149" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s150" xml:space="preserve">S It enim magnitudo a, leuior humido. </s> <s xml:id="echoid-s151" xml:space="preserve">Sit autem magnitudinis, quidem <lb/>in qua a, grauitas b, humidi autem habentis molẽ æqualem cum a, gra-<lb/>uitas b, g, demonſtrandum, quod magnitudo a, ubi preſſa in humidum <lb/>refertur ad ſuperius tanta ui quanta est, grauitas g. </s> <s xml:id="echoid-s152" xml:space="preserve">Accipiatur enim quæ <lb/>dam magnitudo, in qua d, habens grauitatem æqualem ipſi g. </s> <s xml:id="echoid-s153" xml:space="preserve">Magnitudo <lb/>autem ex utriſque magnitudinibus in quibus a, d, in eadem compoſita eſt le <lb/> <anchor type="figure" xlink:label="fig-0013-02a" xlink:href="fig-0013-02"/> uior humido, eſt enim magnitudinis quidem ex utriuſque, grauitas autem <lb/>humidi habentis molẽ æqualem cum a, grauitas eſt b, g, dimittatur igitur <lb/>in bumidem magnitudo ex utriſque a, d, compoſita ad tantum demergetur <lb/>donec tanta moles humidi, quantum eſt demerſum magnitudinis habeat <lb/>grauitatem æqualem cum tota magnitudine, demonſtratum eſt hoc. </s> <s xml:id="echoid-s154" xml:space="preserve">Sit au- <pb file="0014" n="14" rhead="DE INSIDENTIBVS AQV AE"/> tem ſuperficies quædam humidi alicuius quæ a, b, g, d, periferia. </s> <s xml:id="echoid-s155" xml:space="preserve">Quoniam <lb/>igitur tanta mole shumidi. </s> <s xml:id="echoid-s156" xml:space="preserve">quanta eſt magnitudo a, habet grauitatem æqua-<lb/>lem cum magnitudinibus a, d, palam quod demerſum ipſius erit magnitudo <lb/>a, reliquum' autem in quo d, erit totum deſuper ſupra ſuperficiem humidi. </s> <s xml:id="echoid-s157" xml:space="preserve">Si <lb/>enim. </s> <s xml:id="echoid-s158" xml:space="preserve">Palàm igitur quòd quanta uimagnitudo a, refertur ad ſuperius tan-<lb/>ta ab eo quod ſupraſ, d, premitur ad inferius quoniam neutra a neutra ex-<lb/>pellitur, ſed d, ad deorſum premit tanta grauitate quanta eſt g, ſupponeba-<lb/>tur enim grauitas eius, in quo g, d, eſſe æqualem ipſi g, palàm igitur quod <lb/>oportebat demonſtrare.</s> <s xml:id="echoid-s159" xml:space="preserve"/> </p> <div xml:id="echoid-div15" type="float" level="2" n="1"> <figure xlink:label="fig-0013-02" xlink:href="fig-0013-02a"> <image file="0013-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0013-02"/> </figure> </div> </div> <div xml:id="echoid-div17" type="section" level="1" n="13"> <head xml:id="echoid-head19" xml:space="preserve">Theorema vij. Propoſitio vij.</head> <p> <s xml:id="echoid-s160" xml:space="preserve">Grauiora humido demiſſa in humidum ferrentur deorſum <lb/>donec deſcendant, & </s> <s xml:id="echoid-s161" xml:space="preserve">erunt leuiora in humido tantum, quantum <lb/>habet grauitas humidi habentis tantam molẽ, quanta eſt moles <lb/>ſolidæ magnitudinis.</s> <s xml:id="echoid-s162" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s163" xml:space="preserve">QVod quidem feretur in deorſum donec deſcendat, palàm partes e-<lb/>nim humidi, quæ ſubipſius premuntur magis, quæ partes ex quo ipſas <lb/>iacentes, quoniam ſolida magnitudo ſupponitur grauior humido. <lb/></s> <s xml:id="echoid-s164" xml:space="preserve">Quod autem leuiora erunt, ut dictum est, demostrabitur. </s> <s xml:id="echoid-s165" xml:space="preserve">Sit enim aliqua ma <lb/>gnitudo, quæ a, quæ grauior humido, grauitas autem magnitudinis, quidem <lb/>in qua a, ſitq́ b, g, humidi autem habentis molẽ æqualem ipſi a, grauitas <lb/>b, demonſtrandum, quòd magnitudo a, in humido exiſtens habebit grauita-<lb/>tem æqualem ipſig, accipiatur enim aliqua alia magnitudo in quad, leuior <lb/>humido moli æqualis cum ipſo. </s> <s xml:id="echoid-s166" xml:space="preserve">Sit autcm magnitudinis quidem in quad, gra <lb/>uitas æqualis grauitatib, humidi autem habentis molẽ ęqualẽ magnitudini <lb/>d, grauitas ſit æqualis grauitatib, g. </s> <s xml:id="echoid-s167" xml:space="preserve">Compoſiti, autem <lb/> <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> magnitudmibus in quibus, a, d, magnitudo ſimul utra-<lb/>rumq́ erit ęque grauis humido, grauitas enim magnitu-<lb/>dinum ſimul utrarumq; </s> <s xml:id="echoid-s168" xml:space="preserve">est æqualis ambabus grauitati <lb/>bus, ſcilicet b, g, & </s> <s xml:id="echoid-s169" xml:space="preserve">b, grauitas humidi buius habentis <lb/>molẽ æqualem ambabus magnitudinibus, eſt æqualis eiſ-<lb/>dem grauitatibus. </s> <s xml:id="echoid-s170" xml:space="preserve">Dimißis igitur magnitudinibus, & </s> <s xml:id="echoid-s171" xml:space="preserve"><lb/>proiectis in humidum æquerepentes erunt humido & </s> <s xml:id="echoid-s172" xml:space="preserve"><lb/>nec ad ſurſum ferentur, neque ad deorſum: </s> <s xml:id="echoid-s173" xml:space="preserve">quoniam <lb/>magnitudo quidem in qua a, exiſtens grauior humido <lb/>feretur ad deorſum, & </s> <s xml:id="echoid-s174" xml:space="preserve">tanta uia magnitudine in qua <lb/>d, retrabitur. </s> <s xml:id="echoid-s175" xml:space="preserve">Magnitudo autem, in qua d, quoniam eſt <lb/>leuior humido, eleuabitur ſurſum tanta ui quanta eſt grauitas g. </s> <s xml:id="echoid-s176" xml:space="preserve">Demon- <pb o="6" file="0015" n="15" rhead="LIBER I."/> ſiratum eſt enim quòd magnitudines ſolidæ leuioris humido impreſſæ in bn<unsure/> <lb/>midum tanta ui referuntur ad ſurſum quanto humidum æque molis cum <lb/>magnitudine eſt grauius magnitudine. </s> <s xml:id="echoid-s177" xml:space="preserve">Eſt autem humidum habens molem <lb/>æqualem cum d. </s> <s xml:id="echoid-s178" xml:space="preserve">Palàm igitur quòd magnitudo in qua, a, fertur in deor-<lb/>ſum tanta grauitate quanta eſt g.</s> <s xml:id="echoid-s179" xml:space="preserve"/> </p> <div xml:id="echoid-div17" type="float" level="2" n="1"> <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a"> <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0014-01"/> </figure> </div> </div> <div xml:id="echoid-div19" type="section" level="1" n="14"> <head xml:id="echoid-head20" xml:space="preserve">Suppoſitio ſecunda.</head> <p> <s xml:id="echoid-s180" xml:space="preserve">Supponatur eorum quæ in humido ſurſum feruntur vnum-<lb/>quodque ſurſum feri ſecundum perpendicularem quę per cen <lb/>trum grauitatis ipſorum produccitur.</s> <s xml:id="echoid-s181" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div20" type="section" level="1" n="15"> <head xml:id="echoid-head21" xml:space="preserve">Theorema viij. Propoſitio viij.</head> <p> <s xml:id="echoid-s182" xml:space="preserve">Si aliqua ſolida magnitudo habens figuram portionis ſphæ-<lb/>ræ in humidum demittatur ita ut baſis portionis nõ tangat hu-<lb/>midum figura inſidebit recta ita, ut axis portionis ſecundum <lb/>perpendicularem ſit. </s> <s xml:id="echoid-s183" xml:space="preserve">& </s> <s xml:id="echoid-s184" xml:space="preserve">ſi ab aliquo trahitur figura ita, ut ba-<lb/>ſis portionis tangat humidum, non manet declinata ſecun-<lb/>dum dimittatur, ſed recta reſtituatur.</s> <s xml:id="echoid-s185" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s186" xml:space="preserve">E T igitur ſi figura leuior exiſtens humido dimittatur in humidum ita <lb/>ut baſis ipſius tota ſit in humido figura inſidebit recta ita ut axis ip-<lb/>ſius ſit ſecundum perpendicularem. </s> <s xml:id="echoid-s187" xml:space="preserve">Intelligatur enim aliqua ma-<lb/>gnitudo qualis dicta eſt in humidum demiſſa intelligatur etiam & </s> <s xml:id="echoid-s188" xml:space="preserve">planum <lb/>productum per axem portionis & </s> <s xml:id="echoid-s189" xml:space="preserve">per centrum terræ. </s> <s xml:id="echoid-s190" xml:space="preserve">Sectio autem ſit ſu-<lb/> <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> perficiei quidem humidi quæ a, b, g, d, periferia, figuræ autem e, z, b, <lb/>periferia & </s> <s xml:id="echoid-s191" xml:space="preserve">quæ a, b, recta axis autem portionis ſitq́ue z, t. </s> <s xml:id="echoid-s192" xml:space="preserve">Siigitur eſt <pb file="0016" n="16" rhead="DE INSIDENTIBVS AQVAE"/> poſſibile non ſecundum perpendicularem ſit quæ z, t. </s> <s xml:id="echoid-s193" xml:space="preserve">Demenſtrandum igi-<lb/>tur quòd non manet figura ſecundum in rectum ſtatuetur, eſt autem cen-<lb/>trum ſpbæræ uſquez, t. </s> <s xml:id="echoid-s194" xml:space="preserve">Rurſum enim ſit ſigura maior emiſperio, & </s> <s xml:id="echoid-s195" xml:space="preserve">ſit <lb/> <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> centrum ſpbæræ uſque ad emiſperium ſcilicet t, in minori autem p, in maio <lb/>ri autem K, per K autem, & </s> <s xml:id="echoid-s196" xml:space="preserve">per centrum terræ l. </s> <s xml:id="echoid-s197" xml:space="preserve">ducatur k l. </s> <s xml:id="echoid-s198" xml:space="preserve">figura au <lb/>tem extra bumidum aſſumpta a ſuperficie bumidi axem babet in perpen-<lb/>diculari quæ per k, propter eandem prioribus eſt centrum grauitatis ipſius <lb/>in linean, k. </s> <s xml:id="echoid-s199" xml:space="preserve">Sit enim r, totius autem portionis centrum grauitatis eſt in <lb/>linea z, t, inter k, & </s> <s xml:id="echoid-s200" xml:space="preserve">z, & </s> <s xml:id="echoid-s201" xml:space="preserve">ſit c. </s> <s xml:id="echoid-s202" xml:space="preserve">Reliquæ ergo figur æ eius quæ in bumido <lb/>centrum erit in recta c, r, inducta & </s> <s xml:id="echoid-s203" xml:space="preserve">aſſumpta quæ babebit ad c, r, ean-<lb/>dem proportionem quam babet grauitas portionis quæ extra bumidum ad <lb/>grauitatem figuræ quæ in bumido. </s> <s xml:id="echoid-s204" xml:space="preserve">Sit autem o, centrum dictæfiguræ, & </s> <s xml:id="echoid-s205" xml:space="preserve"><lb/>per o, perpendiculari feretur igitur grauitas portionis quidem quæ est ex-<lb/>tra bumidum ſecundum recta n, r, o, ad deorſum, figuræ autem quæ in bu <lb/>mido ſecundum rectam o, l, ad ſurſum non manet igitur figura ſed partes <lb/>quidem figuræ quæ uerſus b, ferrentur ad deorſum. </s> <s xml:id="echoid-s206" xml:space="preserve">Quæ autem uerſus e, <lb/>adſurſum & </s> <s xml:id="echoid-s207" xml:space="preserve">ſuper boc erit donec quęz, t, ſecundum perpẽdicularem fiat.</s> <s xml:id="echoid-s208" xml:space="preserve"/> </p> <div xml:id="echoid-div20" type="float" level="2" n="1"> <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-01"/> </figure> <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0016-01"/> </figure> </div> <figure> <image file="0016-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0016-02"/> </figure> <figure> <image file="0016-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0016-03"/> <caption xml:id="echoid-caption1" style="it" xml:space="preserve">Explicit de Inſidentibus Aquæ Liber Primus.</caption> </figure> <pb file="0017" n="17"/> </div> <div xml:id="echoid-div22" type="section" level="1" n="16"> <head xml:id="echoid-head22" xml:space="preserve">AR CHIM EDIS <lb/>DE INSIDENTIBVS <lb/>AQV AE.</head> <figure> <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0017-01"/> </figure> </div> <div xml:id="echoid-div23" type="section" level="1" n="17"> <head xml:id="echoid-head23" xml:space="preserve">LIBER SECVNDVS.</head> <head xml:id="echoid-head24" style="it" xml:space="preserve">TROIANO CVRTIO</head> <head xml:id="echoid-head25" xml:space="preserve">VENETIIS, <lb/>APVD TROIANVM CVRTIVM. <lb/>M D L X V</head> <pb file="0018" n="18"/> <pb o="2" file="0019" n="19"/> </div> <div xml:id="echoid-div24" type="section" level="1" n="18"> <head xml:id="echoid-head26" xml:space="preserve">FABRITIO DENORES <lb/>FILIO IACOBI COMITIS <lb/>TRIPOLIS</head> <head xml:id="echoid-head27" style="it" xml:space="preserve">CVRTIVS TROIANVS S. P. D.</head> <p> <s xml:id="echoid-s209" xml:space="preserve">SI omni laude digni habiti ſunt <lb/>magniuiri, qui omne ſuum ſtu-<lb/>dium in id contulerunt, ut cæte-<lb/>ris hominibus quàm maxime pro <lb/>deſſe poſſent; </s> <s xml:id="echoid-s210" xml:space="preserve">in magnum dede-<lb/>cus incurrent omnes, qui illo-<lb/>rum opera aut occultant, aut, ut in publicum pro-<lb/>deant, nullam curam afferunt. </s> <s xml:id="echoid-s211" xml:space="preserve">Cum uero labores <lb/>maximi illi quidem Nicolai Tartaleæ quotidie ma-<lb/>gis, ac magis cognoſcantur profuiſſe literatis uiris; <lb/></s> <s xml:id="echoid-s212" xml:space="preserve">non modica uidebor ego dignus reprehenſione, qui <lb/>reliquias habui eiuſdem laborum, & </s> <s xml:id="echoid-s213" xml:space="preserve">uigiliarum, ni <lb/>illas quoque in medium proferam, & </s> <s xml:id="echoid-s214" xml:space="preserve">communi uti-<lb/>litati conſulã. </s> <s xml:id="echoid-s215" xml:space="preserve">Quare cũ habeam adhuc apud me Ar <lb/>chimedem de inſidentibus aquæ ab ipſo Nicolao in <lb/>lucem reuocatum, & </s> <s xml:id="echoid-s216" xml:space="preserve">quantum ab ipſo fieri potuit, <lb/>ab erroribus librarij emendatum, & </s> <s xml:id="echoid-s217" xml:space="preserve">ſuis locubratio= <lb/>nibus illuſtratum; </s> <s xml:id="echoid-s218" xml:space="preserve">uideor fraudare omnes literatos <lb/>ſua poſſeſsione, niomnia, quæ huius ingenioſiſsimi <pb file="0020" n="20"/> uiriapud me reſtant, in lucem emiſero, & </s> <s xml:id="echoid-s219" xml:space="preserve">omnibus <lb/>ea communicauero. </s> <s xml:id="echoid-s220" xml:space="preserve">Ac cum nouerim te cum om-<lb/>nibus rectiſsimis ſtudijs mirifice deditum, tum to-<lb/>tum ad imitationem tuorum maiorum, & </s> <s xml:id="echoid-s221" xml:space="preserve">ad rem <lb/>gerendam inflammatum, putaui hoc opus tibi, & </s> <s xml:id="echoid-s222" xml:space="preserve"><lb/>tui ſimilibus, qui in diſciplinis uerſantur, & </s> <s xml:id="echoid-s223" xml:space="preserve">res ma-<lb/>gnas gerunt, fore peropportunûm. </s> <s xml:id="echoid-s224" xml:space="preserve">nec uero meæ fa-<lb/>cultatis eſt, nec breuitas huius ſcriptionis poſtulat, ut <lb/>de te, ac de tuis maioribus ego nunc plura dicam. <lb/></s> <s xml:id="echoid-s225" xml:space="preserve">nam ſirepeterem clariſsimos uiros, qui literis, & </s> <s xml:id="echoid-s226" xml:space="preserve">ar-<lb/>mis in tua familia floruerunt, eorumq́ue res geſtas <lb/>enarrarem, atque quibus rebus tu, & </s> <s xml:id="echoid-s227" xml:space="preserve">optimus, ac cla <lb/>riſsimus pater tuus eorum gloriam adaugetis; </s> <s xml:id="echoid-s228" xml:space="preserve">lon-<lb/>ge maius opus mihi extaret, quàm eſſet hic paruus li <lb/>bellus, & </s> <s xml:id="echoid-s229" xml:space="preserve">quamq́ue ego poſſem perficere. </s> <s xml:id="echoid-s230" xml:space="preserve">Itaque <lb/>hæc alijs, qui poſſunt, relinquens, & </s> <s xml:id="echoid-s231" xml:space="preserve">in aliud tem-<lb/>pus differens, ut nonullum per me adiumentum ad-<lb/>datur tibi, & </s> <s xml:id="echoid-s232" xml:space="preserve">cæteris, qui rerum naturam contem-<lb/>plantur, & </s> <s xml:id="echoid-s233" xml:space="preserve">ijs artibus ſtudent, quibus res maximæ <lb/>gerunt; </s> <s xml:id="echoid-s234" xml:space="preserve">hoc opus in tuo nomine peruulgari, atque e-<lb/>diuolui. </s> <s xml:id="echoid-s235" xml:space="preserve">ut noſcant omnes, dum ſtudeo piodeſſe com <lb/>muni utilitati, ſeparatim tamen pro mea in te ob-<lb/>ſeruantia uoluiſſe tuis ſtudijs, & </s> <s xml:id="echoid-s236" xml:space="preserve">magnitudini ani-<lb/>mi inſeruire.</s> <s xml:id="echoid-s237" xml:space="preserve"/> </p> <pb o="3" file="0021" n="21" rhead="ARCHIMEDIS DE"/> </div> <div xml:id="echoid-div25" type="section" level="1" n="19"> <head xml:id="echoid-head28" xml:space="preserve">INSIDENTIBVS <lb/>AQV AE. LIB. II.</head> <head xml:id="echoid-head29" xml:space="preserve">PRIMVS.</head> <p> <s xml:id="echoid-s238" xml:space="preserve">SI a liqua magnitudo exiſtens leuior humi-<lb/>do, dimittatur in humidum; </s> <s xml:id="echoid-s239" xml:space="preserve">hanc habebit <lb/>proportionem in grauitate ad humidum <lb/>mobilis æqualis ſibi, quam habct demerſa <lb/>magnitudo ad totam magnitudinem.</s> <s xml:id="echoid-s240" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s241" xml:space="preserve">_D_Emittatur enim in bumidum aliqua magnitu-<lb/> <anchor type="figure" xlink:label="fig-0021-01a" xlink:href="fig-0021-01"/> do ſolida, quàm ſit f, a, leuior bumido. </s> <s xml:id="echoid-s242" xml:space="preserve">Sit au-<lb/>tem quod quidem demerſum ipſum a, quod au-<lb/>tem extra humidum f, demonſtrandum quòd magni-<lb/>tudo f, a, ad humidum æqualis molis in grauitate, hanc <lb/>babet proportionem; </s> <s xml:id="echoid-s243" xml:space="preserve">quama, ad f, a. </s> <s xml:id="echoid-s244" xml:space="preserve">Accipiatur enim <lb/>aliqua humida magnitudo quàm ſit n, i, molis æqualis <lb/>cum f, a, & </s> <s xml:id="echoid-s245" xml:space="preserve">ipſi quidem f, ſit æqualen, ipſi autem a, i, <lb/>& </s> <s xml:id="echoid-s246" xml:space="preserve">adbuc grauitas quidem magnitudinis f, a, ſit b, ip-<lb/>ſius autem n, i, quær, o, ipſius autem, ir, magnitudo igi <lb/>turf, a, ad n, i, hanc habet proportionem quam graui-<lb/>tas b, ad grauit atem r, o, ſed quoniam magnitudof, a, <lb/>in humidum dimiſſa eſt leuior exiſtens bumido. </s> <s xml:id="echoid-s247" xml:space="preserve">Palam <lb/>quòd demeræſ magnitudinis moles humidi babet gra-<lb/>uitatem: </s> <s xml:id="echoid-s248" xml:space="preserve">æqualem cum magnitudine f, a, demonſtra-<lb/>tum eſt enim boc, & </s> <s xml:id="echoid-s249" xml:space="preserve">quoniam quod ſecundum a, humi-<lb/>dum est i, ipſius autemi, grauitas eſtr, ipſius autem f, <lb/>a, grauitas eſt b, grauitas b, quæ eſt babentis <lb/> <anchor type="figure" xlink:label="fig-0021-02a" xlink:href="fig-0021-02"/> æqualitate mole totius magnitudinis f, a, eſt <lb/>aqualis grauitati humidi i, ſcilicet ipſi r, & </s> <s xml:id="echoid-s250" xml:space="preserve"><lb/>quoniam eſt, ut magnitudo f, a, ad humidum <lb/>quod ſecundum ipſam ſcilicet n, i, itab, o, ad <lb/>r, o, æquale autem eſt b, ipſt r, ut autemr, ad <lb/>ro, ita i, ad n, i, & </s> <s xml:id="echoid-s251" xml:space="preserve">a, adf, a, ut ergof, a, ad bu- <pb file="0022" n="22" rhead="DE INS IDENTIBVS AQV AE"/> <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> midum quod ſecundum ipſa in grauitate ma <lb/>gnitudo a, ad f, a, factum eſt æquale demer-<lb/>ſæ magnitudinis, ſcilicet a, habet ergo magni <lb/>tudo f, a, in grauitate ad n, i, ita b, ad r, o. <lb/></s> <s xml:id="echoid-s252" xml:space="preserve">Quam autem proportionem habet r, ad r, o, <lb/>hanc habet proportionem adr, <lb/>& </s> <s xml:id="echoid-s253" xml:space="preserve">a, ad f, a, _demonstratum_ eſt enim.</s> <s xml:id="echoid-s254" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="1"> <figure xlink:label="fig-0021-01" xlink:href="fig-0021-01a"> <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0021-01"/> </figure> <figure xlink:label="fig-0021-02" xlink:href="fig-0021-02a"> <image file="0021-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0021-02"/> </figure> <figure xlink:label="fig-0022-01" xlink:href="fig-0022-01a"> <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0022-01"/> </figure> </div> </div> <div xml:id="echoid-div27" type="section" level="1" n="20"> <head xml:id="echoid-head30" xml:space="preserve">SECVNDVS.</head> <p> <s xml:id="echoid-s255" xml:space="preserve">Recta portio rectanguli conoidalis quando axem habue-<lb/>rit maiorem, quàm emiolium eius, quæ uſque axem omnem <lb/>proportionem habens ad humidum in grauitate dimiſia in <lb/>humido ita, ut baſis ipſius non tangat humidum, poſita incli <lb/>nata non manet inclin ata, ſed reſtituetur recta.</s> <s xml:id="echoid-s256" xml:space="preserve"/> </p> <figure> <image file="0022-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0022-02"/> </figure> <p style="it"> <s xml:id="echoid-s257" xml:space="preserve">_R_Ectam dico conſiſtere talem portionem, quando quod ſecuit ip-<lb/>ſam fuerit æquidistanter ſuperficiei humidi. </s> <s xml:id="echoid-s258" xml:space="preserve">Sit portio rectanguli <lb/>conoidalis, qualis dicta est: </s> <s xml:id="echoid-s259" xml:space="preserve">& </s> <s xml:id="echoid-s260" xml:space="preserve">iaceat inclinata, demonstrandum <lb/>quòd non manet, ſed reſtituetur recta. </s> <s xml:id="echoid-s261" xml:space="preserve">Secta autem ipſa plano per axem <lb/>recte ad planum, quod in ſuperficie humidi portionis ſectio ſitq́ue apol. <lb/></s> <s xml:id="echoid-s262" xml:space="preserve">rectanguli coni fectio, axis autem pertionis, & </s> <s xml:id="echoid-s263" xml:space="preserve">diameter ſestionis quæn, <lb/>o. </s> <s xml:id="echoid-s264" xml:space="preserve">Superficiei autem hnmidi, quàm K. </s> <s xml:id="echoid-s265" xml:space="preserve">Siigitur portio non eſt recta, non <lb/>ntique erit quæ a, l, æquidiſtans ipſi i, s, K. </s> <s xml:id="echoid-s266" xml:space="preserve">Quare non faciet angulum <lb/>rectum quæ n, o, ad i, s, ducatur ergo quæ K, ***, contingens ſectionem <lb/>coni penes, p.</s> <s xml:id="echoid-s267" xml:space="preserve"/> </p> <pb o="4" file="0023" n="23" rhead="LBERI II."/> <figure> <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0023-01"/> </figure> </div> <div xml:id="echoid-div28" type="section" level="1" n="21"> <head xml:id="echoid-head31" xml:space="preserve">TERTIVS.</head> <p> <s xml:id="echoid-s268" xml:space="preserve">Recta portio rectanguli conoydalis, quando axem habue <lb/>rit maiorem, quam emiolium eius, quæ uſq; </s> <s xml:id="echoid-s269" xml:space="preserve">ad axem omnẽ <lb/>proportionem habens ad humidum in grauitate dimiſſa in <lb/>humido ita, ut baſis ipſius tota ſit in humido poſita inclina-<lb/>ta: </s> <s xml:id="echoid-s270" xml:space="preserve">non manet in clinata, ſed reſtituetur ita, ut axis ipfius ſe-<lb/>cundum perpendicularem ſit.</s> <s xml:id="echoid-s271" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s272" xml:space="preserve">_D_Imittatur e<unsure/>nim aliqua portio in humidum æqualis dicta eſt: </s> <s xml:id="echoid-s273" xml:space="preserve">& </s> <s xml:id="echoid-s274" xml:space="preserve"><lb/>ſit ipſius baſis in humido. </s> <s xml:id="echoid-s275" xml:space="preserve">Secta autem plano per axem recto ad <lb/>ſuperficiem humidi ſectio ſit quę apol. </s> <s xml:id="echoid-s276" xml:space="preserve">rectanguli coni ſectio a-<lb/>xis autem portionis, & </s> <s xml:id="echoid-s277" xml:space="preserve">dyameter ſectio m, ſ, quæ p, f, ſuperficiei autem <lb/>humidi ſectio ſitq́ue i, ſ, & </s> <s xml:id="echoid-s278" xml:space="preserve">ſi inclinata iacet portio non erit ſecundum <lb/>perpendicularem axis, non ergo faciet quæ p, f, angulos æquales ad i,s, <lb/>ducatur autem quædam quæ K, ***, æquediſtanter ipſi i, s, contingens ſe-<lb/>ctionem apol penes o, & </s> <s xml:id="echoid-s279" xml:space="preserve">ſolidæ quidem magnitudinis apol centrum <lb/>grauitatis ſit r, ipſius autem i, p, o, s, ſolidi centrum b, & </s> <s xml:id="echoid-s280" xml:space="preserve">eopulata, quæ <lb/>b, r, educatur, & </s> <s xml:id="echoid-s281" xml:space="preserve">centrum grauitatis reliquæ figuræ ſcilicet i, s, l, a, ſit g. <lb/></s> <s xml:id="echoid-s282" xml:space="preserve">Similiter demonſtrabitur angulus quidem qui ſubr, ***, K, acutus per-<lb/>pendicularis quæ a, b, r, t, r, ad K, o, producitur cadens inter K, & </s> <s xml:id="echoid-s283" xml:space="preserve">o, ſitq́ <lb/>r, t. </s> <s xml:id="echoid-s284" xml:space="preserve">Si autem ab ipſis g, b, ducantur æquedistanter ipſir, t, quod quidem <lb/>in humido abſum ptum ferret ſurſum ſecundum productam per g. </s> <s xml:id="echoid-s285" xml:space="preserve">Quod <lb/>autem extra humidum ſecundum producta per b, feretur deorſum, & </s> <s xml:id="echoid-s286" xml:space="preserve"><lb/>non manet ſolidum. </s> <s xml:id="echoid-s287" xml:space="preserve">apol ſic ſe habens in humido, ſed quod quidem ſe-<lb/>cundum a, habcbit lationem ſurſum. </s> <s xml:id="echoid-s288" xml:space="preserve">Quod autem ſecundum l, deorſum <lb/>donec fiat, quæp, f, ſecundum perpendicularem.</s> <s xml:id="echoid-s289" xml:space="preserve"/> </p> <pb file="0024" n="24" rhead="DE INSIDENTIBVS AQV AE"/> </div> <div xml:id="echoid-div29" type="section" level="1" n="22"> <head xml:id="echoid-head32" xml:space="preserve">QVARTVS.</head> <p> <s xml:id="echoid-s290" xml:space="preserve">Recta portio rectanguli conoydalis quando fuerit leuior <lb/>humido, & </s> <s xml:id="echoid-s291" xml:space="preserve">axẽ habuerit maiorem, quàm emiolium eius, quę <lb/>uſque ad axem: </s> <s xml:id="echoid-s292" xml:space="preserve">ſi in grauitate ad humidum æque molis non <lb/>minorem proportionem habeat illa. </s> <s xml:id="echoid-s293" xml:space="preserve">quàm habet tetragonũ <lb/>quod ab exceſſu, quo maior eſt axis, quàm emiolius eius, quę <lb/>uſque ad axem dimiſſa in humido ita ut baſis ipſius non tan-<lb/>gat humidum poſita, inclinata, non manet inclinata, ſed re-<lb/>ſtituetur in rectum.</s> <s xml:id="echoid-s294" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s295" xml:space="preserve">_E_Sto portio rectangula conoydalis, qualis dicta est: </s> <s xml:id="echoid-s296" xml:space="preserve">& </s> <s xml:id="echoid-s297" xml:space="preserve">dimiſſa in <lb/>bumidum, ſi eſt poſſibile, ſit nõ recta, ſed ſit inclinata. </s> <s xml:id="echoid-s298" xml:space="preserve">Secta autem <lb/>ipſa per axem plano recto ad ſnperficiem humidi, portionis quidẽ <lb/>ſe<unsure/>ctio ſit rectanguli coni: </s> <s xml:id="echoid-s299" xml:space="preserve">ſectio quæ apol. </s> <s xml:id="echoid-s300" xml:space="preserve">axis autem portionis, & </s> <s xml:id="echoid-s301" xml:space="preserve">dyæ <lb/>meter, quæn, o, ſuperficiei autem humidi ſectio ſit i, s. </s> <s xml:id="echoid-s302" xml:space="preserve">Siigitur portio non <lb/>eſt recta, non faciet quæ n, o, ad is angulos æquales: </s> <s xml:id="echoid-s303" xml:space="preserve">ducatur autẽ quæ <lb/>K, ***, contingens ſectionem rectanguli coni penes, p, æquidiſtans autem <lb/>ipſi i s. </s> <s xml:id="echoid-s304" xml:space="preserve">A, p, autem æquedistanter ipſi o, n, ducaturq́ue p, f, & </s> <s xml:id="echoid-s305" xml:space="preserve">accipian <lb/>tur contra grauitum, & </s> <s xml:id="echoid-s306" xml:space="preserve">erit ſolidi quidem apol. </s> <s xml:id="echoid-s307" xml:space="preserve">centrumr, eius autem <lb/>quod inter humidum centrum b, & </s> <s xml:id="echoid-s308" xml:space="preserve">copuletur g, t, r, & </s> <s xml:id="echoid-s309" xml:space="preserve">educatur ad g, <lb/>& </s> <s xml:id="echoid-s310" xml:space="preserve">ſit ſolidi, quod ſupra humidi centrum grauitatis g, & </s> <s xml:id="echoid-s311" xml:space="preserve">quoniam quæ <lb/>n, o, ipſius quidem r, o, eſt emiolia eius autem, quæ uſque ad axem eſt ma <lb/>ior, quàm emiolia, palam quòd quær, o, eſt maior, quàm quæ uſque ad a-<lb/>xem. </s> <s xml:id="echoid-s312" xml:space="preserve">Sit igitur quæ r, m, æqualis ei, quæ uſque ad axem, quæ autem o, <lb/>n, dupla ipſius r, m. </s> <s xml:id="echoid-s313" xml:space="preserve">Quoniam igitur ſit quæ quidem n, o, ipſius r, o, emio-<lb/>lia, quæ autem m, o, ipſius o, b, & </s> <s xml:id="echoid-s314" xml:space="preserve">reliqua, quæm, n, reliqua ſcilicet r, b, <lb/>æmiolia eſt ipſi m, o, eſt maior, quàm emiolius eſt axis eius, quæ uſque ad <lb/>axem, ſcilicet r, m, & </s> <s xml:id="echoid-s315" xml:space="preserve">quoniam ſupponebatur portio ad humidum in gra <lb/>uitate non minuerem proportionem habens illa, quam habet tetragonũ <lb/>quod ab exceſſu, quo. </s> <s xml:id="echoid-s316" xml:space="preserve">axis eſt maior, quàm æmiolius eius, quæ uſq; </s> <s xml:id="echoid-s317" xml:space="preserve">ad a-<lb/>xem ad tetragonum quod ab axe. </s> <s xml:id="echoid-s318" xml:space="preserve">palam quòd non minorem proportio <lb/>nem babet portio ad humidum in grauitate illa proportionem quam ha <lb/>bet tetragonum, quod ab m, o, ad id, quod ab n, o. </s> <s xml:id="echoid-s319" xml:space="preserve">Q uam autem propor-<lb/>tìonem habet portio ad humidum in grauitate, hanc habet demerſa ip-<lb/>ſius portio adtotam ſolidam portionem, demonſtratum eſt enim hoc, ſed <lb/>quam habet proportionem demerſa, proportio adtotam hanc habet te-<lb/>iragonum quod _Demonstratum_ eſt enim in ijs <lb/>quæ de conoydalibus quòd ſi a rectangulo conoydaliduæ portiones qua- <pb o="5" file="0025" n="25" rhead="LIBER II."/> litercunque productis planis abſcindantur portiones adinuicem eandens <lb/>habebunt proportionem quam tetragona quæ ab axibus ipſorum</s> </p> <p style="it"> <s xml:id="echoid-s320" xml:space="preserve">non minorem ergo proportionem: </s> <s xml:id="echoid-s321" xml:space="preserve">habet tetragonum quòd a, p, f, <lb/>ad tetragonum quod a, b, n, o, quam tetragonum quòd ab m, o, ad tetra-<lb/>gonum quod ab n, o, quare quæ p, f, non _est_ minor quàm m, o, neque quæ <lb/>b, p, quàm n, o. </s> <s xml:id="echoid-s322" xml:space="preserve">Si igitur ab m, ipſi n, o, recta ducatur, cadent intrab, & </s> <s xml:id="echoid-s323" xml:space="preserve">p. <lb/></s> <s xml:id="echoid-s324" xml:space="preserve">Quoniam igitur quæ quidem p, f, eſt æquediſtanter dyametro quæ au-<lb/>tem m, t, eſt perpendicularis ad dyametrum, & </s> <s xml:id="echoid-s325" xml:space="preserve">quæ r, m, æqualis ei quæ <lb/>uſque ad axem a, b, r, ad t, copulata, & </s> <s xml:id="echoid-s326" xml:space="preserve">educta facit angulos rectos ad <lb/>contingentem ſecundum p. </s> <s xml:id="echoid-s327" xml:space="preserve">Quare & </s> <s xml:id="echoid-s328" xml:space="preserve">ad i, s. </s> <s xml:id="echoid-s329" xml:space="preserve">& </s> <s xml:id="echoid-s330" xml:space="preserve">ad eam quæper i, s. </s> <s xml:id="echoid-s331" xml:space="preserve">ſu-<lb/>perficiem humidi faciet æquales angulos, ſi autem per b, g, ipſi r, t, æque-<lb/>diſtantes ducantur anguli recti erunt facti ad, ſuperficiem humidi, & </s> <s xml:id="echoid-s332" xml:space="preserve"><lb/>quod quidem in humido aſſumitur ſolidum conoydalis ſurſum fertur ſe-<lb/>cundum ea, quæ per b, æquediſtantem ipſir, t, quod autem extra humi-<lb/>dum aſſumpta deorſum fertur in humidum ſecundum productam per g, <lb/>æquediſtantem ipſir, t, & </s> <s xml:id="echoid-s333" xml:space="preserve">per totum idem erit, donec utique conoydale <lb/>rectum reſti tuatur.</s> <s xml:id="echoid-s334" xml:space="preserve"/> </p> <figure> <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0025-01"/> </figure> </div> <div xml:id="echoid-div30" type="section" level="1" n="23"> <head xml:id="echoid-head33" xml:space="preserve">QVINTVS.</head> <p> <s xml:id="echoid-s335" xml:space="preserve">Recta portio rectanguli conoydalis quando leuior exi-<lb/>ſtens humido habuerit axem maiorem, quàm emyolium e-<lb/>iusq́ue uſque ad axem ſi ad humidum in grauitate non ma-<lb/>iorem proportionem habeat illa, quam habet exceſſus, quo <lb/>maius eſt tetragonam quod ab axe tetragono quod ab ex-<lb/>ceſſu quo axis eſt maior, quàm emyolius eius, quæ uſque ad <lb/>axem ad tetragonum quod ab axe dimiſſa in humidum ita <pb file="0026" n="26" rhead="DE INSIDENTIBVS AQV AE"/> ut baſis ipſius tota ſit in humido poſita, inclinata, non manet <lb/>inclinata, ſed reſtituetur ita ut axis ipſius ſecundum perpen <lb/>dicularem ſit.</s> <s xml:id="echoid-s336" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s337" xml:space="preserve">_D_Emittatur enim in humidum aliqua portio qualis dicta eſt, & </s> <s xml:id="echoid-s338" xml:space="preserve">ſit <lb/>baſis ipſius tota in bumido. </s> <s xml:id="echoid-s339" xml:space="preserve">Secta autem ipſa plano per axem re-<lb/>cto ad ſuperficiem humidi erit ſectio rectanguli, coni ſectio, & </s> <s xml:id="echoid-s340" xml:space="preserve">ſit <lb/>quæ apol, axis autem, & </s> <s xml:id="echoid-s341" xml:space="preserve">dyameter ſectionis quàm n, o, ſuperficiei autem <lb/>bumidi ſectio, quæ i, s, & </s> <s xml:id="echoid-s342" xml:space="preserve">quoniam non eſt axis ſecundum perpendicula-<lb/>rem non faciet, quæ n, o, ad i, s, angulos æquales: </s> <s xml:id="echoid-s343" xml:space="preserve">ducatur autem quæ K, <lb/>***, contingens ſectionem apol ſecundum p, æquidiſtans ipſi i, s, & </s> <s xml:id="echoid-s344" xml:space="preserve">per <lb/>p, ipſi n, o, æquediſtans quæ p, f, & </s> <s xml:id="echoid-s345" xml:space="preserve">accipiantur centra grauitatem: </s> <s xml:id="echoid-s346" xml:space="preserve">& </s> <s xml:id="echoid-s347" xml:space="preserve"><lb/>ſit ipſius quidem apol. </s> <s xml:id="echoid-s348" xml:space="preserve">centrum r. </s> <s xml:id="echoid-s349" xml:space="preserve">eius autem quod extra humidum b, <lb/>& </s> <s xml:id="echoid-s350" xml:space="preserve">copulata quæ b, r, educatur ad g, & </s> <s xml:id="echoid-s351" xml:space="preserve">ſit g, centrum grauitatis ſolidi <lb/>aſſumpti in humido : </s> <s xml:id="echoid-s352" xml:space="preserve">& </s> <s xml:id="echoid-s353" xml:space="preserve">accipiatur quæ r, m, æqualis ei quæ uſque ad <lb/>axe Quæ autem o, h, dupla ipſius h, m, & </s> <s xml:id="echoid-s354" xml:space="preserve">alia fiant conſimili-<lb/>ter ſuperiori. </s> <s xml:id="echoid-s355" xml:space="preserve">Quoniam igitur ſupponitur portio ad humidum in graui-<lb/>tate non maiorem proportionem habens proportione, quam habet exceſ <lb/>ſus, quo maius eſt tetragonũ, quod ab n, o, tetragono, quod ab m, o, tetra <lb/>gonum, quod ab n, o, ſed quam proportionem habet in grauitate porti*** <lb/> <anchor type="figure" xlink:label="fig-0026-01a" xlink:href="fig-0026-01"/> ad humidum æqualis molis, hanc proportionẽ habet demerſa ipſius por <lb/>tio ad totum ſolidum : </s> <s xml:id="echoid-s356" xml:space="preserve">demonstratum est enim hoc in primo theorema-<lb/>te. </s> <s xml:id="echoid-s357" xml:space="preserve">Non maiorem ergo proportionem habet demerſa magnitudo por-<lb/>tionis ad totam portionem, quàm ſit dicta portio. </s> <s xml:id="echoid-s358" xml:space="preserve">Quare non maiorem <lb/>proportionem habet tota portio ad eam, quæ e xtra humidum proportio <lb/>nem, quam habet tetragonũ, quod ab n, o, ad tetragonum, quod ab m, t, <lb/>habet autem tota portio ad portionem, quàm extra bumidum eandeni <pb o="6" file="0027" n="27" rhead="LIBER II."/> proportionem quam habet tetragonum, quod ab n, o, ad id quod a, p, f, <lb/>non maiorem ergo proportionem habet, quæ ab n, o, ad id a, p, f, quàm <lb/>quòd ab n, o, ad id, quod ab m, o, non minor ergo fit, quæ p, f, quàm quæ <lb/>o, m, quare nec quæ p, b, quàm n, o. </s> <s xml:id="echoid-s359" xml:space="preserve">Quæ ergo ab m, producitur ipſi r, o, <lb/>æquidiſtans concidet ipſi b, p, intra p, & </s> <s xml:id="echoid-s360" xml:space="preserve">b, concidat ſecundum t, & </s> <s xml:id="echoid-s361" xml:space="preserve">quo <lb/>niam in rectanguli coni, Sectione quæ p, f, eſt æquidiſtanter dyametto r, <lb/>o. </s> <s xml:id="echoid-s362" xml:space="preserve">Quæ autem n, t, perpendicularis ſuper dyametrum, quæ autem r, m, <lb/>æqualis ei quæ uſque ad axem. </s> <s xml:id="echoid-s363" xml:space="preserve">Palam quòd quæ r, t, educta facit angu-<lb/>los rectos ad K, p, ***, quare & </s> <s xml:id="echoid-s364" xml:space="preserve">ad i, s. </s> <s xml:id="echoid-s365" xml:space="preserve">Quæ ergo r, t, eſt perpendicula-<lb/>ris ad ſuperficiem humidi, & </s> <s xml:id="echoid-s366" xml:space="preserve">per ſigna b, g, æquediſtanter ipſi r, t, produ <lb/>ctæ erunt perpendiculares ad ſuperficiem bumidi: </s> <s xml:id="echoid-s367" xml:space="preserve">quæ quidem igitur ex <lb/>tra humidum portio deorſum ferretur in humidum ſecundum producta <lb/>per b, perpendicularem. </s> <s xml:id="echoid-s368" xml:space="preserve">Quæ autem intra humidum ſurſum ferretur ſe <lb/>cundum perpendicularem, quæ per g, & </s> <s xml:id="echoid-s369" xml:space="preserve">non manet ſolida portio apol. <lb/></s> <s xml:id="echoid-s370" xml:space="preserve">ſed intra humidum erit in motum, donec utique quæ n, o, fiat ſecundum <lb/>perpendicularem.</s> <s xml:id="echoid-s371" xml:space="preserve"/> </p> <div xml:id="echoid-div30" type="float" level="2" n="1"> <figure xlink:label="fig-0026-01" xlink:href="fig-0026-01a"> <image file="0026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0026-01"/> </figure> </div> </div> <div xml:id="echoid-div32" type="section" level="1" n="24"> <head xml:id="echoid-head34" xml:space="preserve">SEXTVS.</head> <p> <s xml:id="echoid-s372" xml:space="preserve">Recta portio rectanguli conoydalis quando humido le-<lb/>uior exiſtens axem habuerit maiorem quidem quam hemio <lb/>lium minorem autem quam ut habet hãc proportionem ad <lb/>eam, quæ uſque ad axem quam habent quindecim ad quat-<lb/>tuor dimiſſa in humidum ita, ut baſis ipſius contingat humi-<lb/>dum, nunquam ſtabit inclinata ita, ut baſis ipſius ſecundum <lb/>vnum ſignum conting at humidum</s> </p> <p style="it"> <s xml:id="echoid-s373" xml:space="preserve">_S_It portio qualis dicta eſt, & </s> <s xml:id="echoid-s374" xml:space="preserve">dimiſſa in humidum conſiſtat, ſicut ſtẽ <lb/>ſum eſt : </s> <s xml:id="echoid-s375" xml:space="preserve">ita ut baſis ipſius ſecundum vnum ſignum contingat hu-<lb/>midum. </s> <s xml:id="echoid-s376" xml:space="preserve">Secta autem ipſa per axem plano recto ad ſuperficiem hu <lb/>midi: </s> <s xml:id="echoid-s377" xml:space="preserve">ſectio ſuperficiei portionis ſit, quæ apol. </s> <s xml:id="echoid-s378" xml:space="preserve">rectanguli coni ſectio : </s> <s xml:id="echoid-s379" xml:space="preserve">ſu <lb/>perficiei autem humidi quæ a, s, axis autem portionis, & </s> <s xml:id="echoid-s380" xml:space="preserve">dyameter ſit <lb/>quæ n, o, & </s> <s xml:id="echoid-s381" xml:space="preserve">ſecetur ſecundum f, quidem ita quæ o, f, ſit quæ dupla ipſius <lb/>f, n, ſecundum ***, autem ita, ut quæ n, o, ad f, ***, habe at proportionem <lb/>quam quindecim ad quattuor, & </s> <s xml:id="echoid-s382" xml:space="preserve">ipſi n, o, adducatur quæ ***, K. </s> <s xml:id="echoid-s383" xml:space="preserve">Quæ <lb/>autem n, o, maiorem proportionẽ habet ad f, ***, quàm ad ea, ꝗ̃ uſq; </s> <s xml:id="echoid-s384" xml:space="preserve">ad <lb/>axem. </s> <s xml:id="echoid-s385" xml:space="preserve">Sit quæ f, b, æqualis ei, quæ uſque ad axem, & </s> <s xml:id="echoid-s386" xml:space="preserve">ducatur quæ qui-<lb/>dem p, c, æquediſtanter ipſi a s contingens, ſectionem. </s> <s xml:id="echoid-s387" xml:space="preserve">apol. </s> <s xml:id="echoid-s388" xml:space="preserve">ſecundum p. <lb/></s> <s xml:id="echoid-s389" xml:space="preserve">Quæ autem p, i, æquediſtanter ipſi n, o, Secet autem quæ p, i, prius ipſam <lb/>K, ***. </s> <s xml:id="echoid-s390" xml:space="preserve">Quoniam igitur in portione apol contenta a recta, & </s> <s xml:id="echoid-s391" xml:space="preserve">aſectione <pb file="0028" n="28" rhead="DE INSIDENTIBVS AQV AE"/> rectanguli coni quæ quidem K, h, æquediſtanter ipſi a, l, quo autem p, i, <lb/>æquediſtanter dy imetro ſecta ipſa K, ***. </s> <s xml:id="echoid-s392" xml:space="preserve">Quæ autem a s, æquediſtanter <lb/>contingenti ſecundum p. </s> <s xml:id="echoid-s393" xml:space="preserve">neceſſarium eſt ipſam p, i, autem <lb/>eandem proportionem habere ad p, h, quam habet quæ n, ***, ad ***, o, <lb/>maiorem proportionem demonſtratum eſt enim hoc perſumpta. </s> <s xml:id="echoid-s394" xml:space="preserve">Quæ au <lb/>tem ***, h, _est_ æmyolia ipſius ***, o, & </s> <s xml:id="echoid-s395" xml:space="preserve">quæ i, h. </s> <s xml:id="echoid-s396" xml:space="preserve">Ergo aut æmyolia eſt ip-<lb/>ſius h, p, aut maior quàm æmyolia quæ ergo p, h, ipſius h, i, aut dupla _est_, <lb/>aut minor quàm dupla. </s> <s xml:id="echoid-s397" xml:space="preserve">Sit autem quæ p, t, ipſius t, i, dupla. </s> <s xml:id="echoid-s398" xml:space="preserve">Centrum er <lb/>go grauitatis eius quod in humido _est_ ſignum t, & </s> <s xml:id="echoid-s399" xml:space="preserve">copulata quæ t, f, edu <lb/>catur, & </s> <s xml:id="echoid-s400" xml:space="preserve">ſit centrum grauitatis eius quòd extra humidum g, & </s> <s xml:id="echoid-s401" xml:space="preserve">a, b, ip <lb/>ſi n, o, recta quę b, r. </s> <s xml:id="echoid-s402" xml:space="preserve">Quoniam igitur est quæ quidem p, i, æ quedistanter <lb/>dyametro n, o, quæ autem b, r, perpendicularis ſuper dyametrum. </s> <s xml:id="echoid-s403" xml:space="preserve">Quæ <lb/>autem f, b, æqualis ei quæ uſque ad axem palam quòd quę t, r, educta <lb/>æquales angulos ad contingentem ſectionem apol. </s> <s xml:id="echoid-s404" xml:space="preserve">ſecundum p, quare <lb/>& </s> <s xml:id="echoid-s405" xml:space="preserve">ad a, s, & </s> <s xml:id="echoid-s406" xml:space="preserve">ad ſuperficiem aquæ ductis autem per t, g, ęquediſtanter ip <lb/>ſif, b, erunt & </s> <s xml:id="echoid-s407" xml:space="preserve">ipſe perpendiculares ad ſuperficiem aquæ, & </s> <s xml:id="echoid-s408" xml:space="preserve">magnitudo<unsure/> <lb/> <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> <pb o="7" file="0029" n="29" rhead="LIBER II."/> quidem inter humidum aſſumpta ex ſolido apol ſurſum ferretur ſecun-<lb/>dum eam quæ per t, perpendicularem. </s> <s xml:id="echoid-s409" xml:space="preserve">Quæ autem extra humidum de-<lb/>orſum ferret in humidum ſecundum eam, quæ per g, perpendicularem. <lb/></s> <s xml:id="echoid-s410" xml:space="preserve">Reuoluetur ergo ſolidum apol. </s> <s xml:id="echoid-s411" xml:space="preserve">& </s> <s xml:id="echoid-s412" xml:space="preserve">baſis ipſius non tanget ſuperficiem <lb/>bumidi ſecundum vnum ſignum. </s> <s xml:id="echoid-s413" xml:space="preserve">Si autem quæ p, i, non ſecuerit lineam <lb/>K, ***, ſicut in ſolida ſigura deſcriptum eſt. </s> <s xml:id="echoid-s414" xml:space="preserve">Manifestum quod ſignum t, <lb/>quod eſt centrum grauitatis demerſę portionis cadet inter p, & </s> <s xml:id="echoid-s415" xml:space="preserve">i, & </s> <s xml:id="echoid-s416" xml:space="preserve">re-<lb/>liqua ſimiliter demonſtrabuntur.</s> <s xml:id="echoid-s417" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="1"> <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0028-01"/> </figure> </div> </div> <div xml:id="echoid-div34" type="section" level="1" n="25"> <head xml:id="echoid-head35" xml:space="preserve">SEPTIMVS.</head> <p> <s xml:id="echoid-s418" xml:space="preserve">Recta portio rectanguli conoy dalis quando humido le-<lb/>uior fuerit, & </s> <s xml:id="echoid-s419" xml:space="preserve">axem habuerit maiorem quidem quàm æmyo <lb/>lium eiusq́ uſque ad axem minorem, aut ut proportionem <lb/>habeat ad eam, quæ uſque ad axem quàm quindecim ad quat <lb/>tuor dimiſſa in humidum, ita ut baſis ipſius tota ſit in humi-<lb/>do, nunquam ſtabit ita ut baſis ipſius tangat ſuperficiem hu-<lb/>midi, ſed ut tota ſit in humido, necſecundum vnum ſignum <lb/>tangens ſuperficiem.</s> <s xml:id="echoid-s420" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s421" xml:space="preserve">_S_It portio qualis dicta eſt, & </s> <s xml:id="echoid-s422" xml:space="preserve">_dimißa in humidum, ſicut dictum eſt cõ <lb/>ſiſt at ita ut baſis ipſius tangat ſuperficiem humidi, demonstrandum <lb/>quòd non manet, ſed reuoluetur ita, ut baſis ipſius tangat ſuperficiẽ <lb/>humidi non ſecundum vnum ſignum : </s> <s xml:id="echoid-s423" xml:space="preserve">ſecta enim ipſa plano, recta ad ſu <lb/>perficiem humidi : </s> <s xml:id="echoid-s424" xml:space="preserve">Sectio ſit quæ apol rectanguli coni ſectio. </s> <s xml:id="echoid-s425" xml:space="preserve">Sit autem <lb/>& </s> <s xml:id="echoid-s426" xml:space="preserve">ſuperficiei humida ſectio quæ s, a, axis autem portionis, & </s> <s xml:id="echoid-s427" xml:space="preserve">dyameter <lb/>quę p, f, ſit i. </s> <s xml:id="echoid-s428" xml:space="preserve">Rurſum autem ſecetur quæ p, f, ſecundum r, quidem ita <lb/>ut quæ r, p, ſit dupla ipſius r, f, ſecundum ***. </s> <s xml:id="echoid-s429" xml:space="preserve">autem ita ut quæ p, ***, ad <lb/>r, ***, proportionem habeat quam quindecim ad quattuor. </s> <s xml:id="echoid-s430" xml:space="preserve">& </s> <s xml:id="echoid-s431" xml:space="preserve">quæ ***. <lb/></s> <s xml:id="echoid-s432" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> <pb file="0030" n="30" rhead="DE INSIDENTIBVS AQVAE"/> K, recta ducatur ſuper p, f, erit autem minor, quæ r, ***, quàm e, a, quæ <lb/>uſque ad axem. </s> <s xml:id="echoid-s433" xml:space="preserve">Accipiatur igitur ei quæ uſque ad axem æqũas quæ <lb/>r, h, & </s> <s xml:id="echoid-s434" xml:space="preserve">quæ quidem c, o, ducatur contingens ſectiones penes o, exiſtens <lb/>_æquedistans_ ipſi a, s, & </s> <s xml:id="echoid-s435" xml:space="preserve">quæ n, o, & </s> <s xml:id="echoid-s436" xml:space="preserve">_æquedistans_ ipſi p, f. </s> <s xml:id="echoid-s437" xml:space="preserve">Secet autem <lb/>quæ n, o, ipſam K, ***, prius ſecundum i. </s> <s xml:id="echoid-s438" xml:space="preserve">Conſimiliter autem præcedenti <lb/>demõſtrabitur, quòd quæn, o, aut hemiolia eſt ipſius o, i, aut maior quàm <lb/>bemiolia, ſit autẽ, quæ o, t, ipſi t, n, minor, quàm dupla. </s> <s xml:id="echoid-s439" xml:space="preserve">Sit igitur quæ o, <lb/>b, dupla ipſius b, n, & </s> <s xml:id="echoid-s440" xml:space="preserve">diſponantur tandem prioribus. </s> <s xml:id="echoid-s441" xml:space="preserve">Similiter igitur de <lb/>monstrabitur, quæ r, f, faciens angulos rectos ad c, o, & </s> <s xml:id="echoid-s442" xml:space="preserve">ad ſuperficiem <lb/>humidi, & </s> <s xml:id="echoid-s443" xml:space="preserve">ab ipſis b, g, productæ æquedistanter ipſi r, f, erunt perpendi <lb/>culares ſuper ſuperficiem humidi. </s> <s xml:id="echoid-s444" xml:space="preserve">Portio igitur, quę quidem extra hu-<lb/>midum deorſum ferretur in humidum, ſecundum eam, quæ per b, perpen <lb/>dicularem. </s> <s xml:id="echoid-s445" xml:space="preserve">Quæ autem inter humidum ſurſum ferretur, ſecũdum eam, <lb/>quàm per g. </s> <s xml:id="echoid-s446" xml:space="preserve">Maximum igitur, quod ad uoluit ſolidum ita, ut baſis ip-<lb/>ſius, necſecundum vnum contingat ſuperficiem humidi, quoniam nunc, <lb/>ſecundum vnum tangens ad deorſum, ferret ex parte a. </s> <s xml:id="echoid-s447" xml:space="preserve">Manifestum <lb/>autem quòd & </s> <s xml:id="echoid-s448" xml:space="preserve">ſi quæ n, o, non ſecuerit ***, K, eandem demonſtrabũtur.</s> <s xml:id="echoid-s449" xml:space="preserve"/> </p> <div xml:id="echoid-div34" type="float" level="2" n="1"> <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0029-01"/> </figure> </div> </div> <div xml:id="echoid-div36" type="section" level="1" n="26"> <head xml:id="echoid-head36" xml:space="preserve">OCTAVVS.</head> <p> <s xml:id="echoid-s450" xml:space="preserve">Recta portio rectanguli conoy dalis, quando axem habue <lb/>rit maiorem, quàm hemiolium eius, quæ uſque ad axem mi-<lb/>norem, autem ut ad eam, quæ ad axem habeat proportionẽ, <lb/>quam habet quindecim ad quatuor. </s> <s xml:id="echoid-s451" xml:space="preserve">Si grauis ad humidum <lb/>habeat proportionem minorem proportione, quam habet <lb/>tetragonum, quod ab ex ceſſu, quo axis eſt maior, quàm he-<lb/>miolius eius, quæ uſque ad axem ad tetragonum, quod ab <lb/>axe dimiſſa in humidum, it a ut baſis ipſius non tangat humi <lb/>dum, nec in rectum reſtituetur, nec manebit inclinata, niſi <lb/>quando axis ipſius ad ſuperficiem humidi fecerit angulum <lb/>æqualem ei qui dicendus eſt.</s> <s xml:id="echoid-s452" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s453" xml:space="preserve">_S_It portio qualis dicta est : </s> <s xml:id="echoid-s454" xml:space="preserve">& </s> <s xml:id="echoid-s455" xml:space="preserve">ſit quæ b, d,@ æquales axi, & </s> <s xml:id="echoid-s456" xml:space="preserve">quæ qui-<lb/>dem b, K, ſit duplaipſius k, d. </s> <s xml:id="echoid-s457" xml:space="preserve">Quæ autem r, k, æqualis ei, quæ uſ-<lb/>que ad axem. </s> <s xml:id="echoid-s458" xml:space="preserve">Sit autem, & </s> <s xml:id="echoid-s459" xml:space="preserve">quæ quidem e, b, hemiolia ipſius b, r. <lb/></s> <s xml:id="echoid-s460" xml:space="preserve">Quam autem proportionem habet portio in grauitate ad bumidum hãc <lb/>quod a, b, f, q, tetragonum ad id, quod a, d, b. </s> <s xml:id="echoid-s461" xml:space="preserve">Sit autem, & </s> <s xml:id="echoid-s462" xml:space="preserve">quæ f, dupla <lb/>ipſius q, palam, igitur quòd quæ f, g, ad ipſam d, b, proportionem habet <lb/>minorem proportione, quàm habet, quæ t, b, ad ipſam b, d, exceſſus enim <lb/>quòdg, d, eſt quo axis eſt. </s> <s xml:id="echoid-s463" xml:space="preserve">maior, quàm bemiolius eius, quæuſque ad <pb o="8" file="0031" n="31" rhead="LIBER II."/> axem. </s> <s xml:id="echoid-s464" xml:space="preserve">Quæ ergo f, q, erit minor ipſa b, c. </s> <s xml:id="echoid-s465" xml:space="preserve">Quare & </s> <s xml:id="echoid-s466" xml:space="preserve">quàm f, minor ipſæ <lb/>b, r. </s> <s xml:id="echoid-s467" xml:space="preserve">Sit autem ipſi f, æqualis, quæ r, x, & </s> <s xml:id="echoid-s468" xml:space="preserve">ſuper ipſa b, d, recta ducatur, <lb/>quæ x, e, quæ poſſit dimidium eius, quod ſub K, r, x, & </s> <s xml:id="echoid-s469" xml:space="preserve">copuletur quæ b, e, <lb/>demonſtrandum quòd portio dimißa in bumidum, vt dictum est, conſi-<lb/>ſtet inclinata ita, ut axis ad ſuperficiem bumidi faciat angulum æqualé <lb/>angulo e, b, x, demonstratur enim aliqua portio in bumidum, & </s> <s xml:id="echoid-s470" xml:space="preserve">baſis ip <lb/>ſius non tang at ſuperficiem bumidi. </s> <s xml:id="echoid-s471" xml:space="preserve">Et ſi poſſibile eſt axis ipſius ad ſu-<lb/>perficiem bumidi non faciat angulum æqualem angulo b, ſed primo ma-<lb/>iorem: </s> <s xml:id="echoid-s472" xml:space="preserve">ſecta autem portione per axem plano recto ad ſuperficiem bu-<lb/>midi. </s> <s xml:id="echoid-s473" xml:space="preserve">Sectio erit quàm apol. </s> <s xml:id="echoid-s474" xml:space="preserve">rectanguli coni ſectio. </s> <s xml:id="echoid-s475" xml:space="preserve">Superficies autem <lb/>bumidi, quæ x, s. </s> <s xml:id="echoid-s476" xml:space="preserve">Axis autem, & </s> <s xml:id="echoid-s477" xml:space="preserve">dyameter portionis, quæ n, o, duca-<lb/>tur autem, & </s> <s xml:id="echoid-s478" xml:space="preserve">quæ quidem p, y, æquediſtanter ipſi x, s, contingens ſectio <lb/>nem apol. </s> <s xml:id="echoid-s479" xml:space="preserve">ſecundum p. </s> <s xml:id="echoid-s480" xml:space="preserve">Quæ autem p, m, æquediſtanter ipſi n, o. </s> <s xml:id="echoid-s481" xml:space="preserve">Quæ au <lb/>tem p, i, perpendicularis, ſuper n, o, & </s> <s xml:id="echoid-s482" xml:space="preserve">quæ quidem b, r, ſit æqualis ipſi i, <lb/>***. </s> <s xml:id="echoid-s483" xml:space="preserve">Quæ autem r, K, ipſin, o, & </s> <s xml:id="echoid-s484" xml:space="preserve">quæ ***, b, rectam ſuper axem. <lb/></s> <s xml:id="echoid-s485" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0031-01a" xlink:href="fig-0031-01"/> Quoniam igitur ſupponitur axis portionis ad ſuperficiem bumidi facere <lb/>angulum maiorem angulo b, palam quòd angulo p, i, n, angulus, qui ad <lb/>p, i, m, _est_ maior angulo b, maiorem igitur proportionem habet tetrago <lb/>num, quod a, p, i, ad tetragonum quod ab i, quàm tetra-<lb/>gonum, quod ab e, x, ad tetragonum quòd a, x, o Sed quam quidem pro-<lb/>portionem habet tetragonum, quod a, p, i, ad id, quod ab i. <lb/></s> <s xml:id="echoid-s486" xml:space="preserve">hanc habet quæ K, r, ad i. </s> <s xml:id="echoid-s487" xml:space="preserve">Quam autem proportionem habet te <lb/>tragonum, quod ab e, x, ad tetragonum a, x, b, hanc habet medietas ip-<lb/>ſius K, r, ad x, b, maiorem ergo proportionem habet, quàm K, r, ad <lb/>i, quàm medietas ipſius k, r, ad x, b. </s> <s xml:id="echoid-s488" xml:space="preserve">Minor ergo eſt, quàm dupla, quæi, <pb file="0032" n="32" rhead="DE INSIDENTIBVS AQVAE"/> ipſius c, d. </s> <s xml:id="echoid-s489" xml:space="preserve">Ipſius autem o, i, dupla eſt, quæ ***, propter ſeptimum tbeore-<lb/>ma primi libri elementorum conoycorum A pollonij. </s> <s xml:id="echoid-s490" xml:space="preserve">Eſt ergo quæ o, i, <lb/>minor, quàm x, b. </s> <s xml:id="echoid-s491" xml:space="preserve">Quare quæ i, ***, eſt maior, quàm x, r, quæ autem x, r, <lb/>eſt æqualis ipſi f, maior ergo _est_, quæ i, ***, quàm f. </s> <s xml:id="echoid-s492" xml:space="preserve">Et quoniam ſupponi <lb/>tur portio ad humidum iu grauitate habere per portionem, quàm tetra-<lb/>gonum, quod ab f, q, ad tetragonum, quod a, b, d. </s> <s xml:id="echoid-s493" xml:space="preserve">Quam autem propor-<lb/>tionem, habet proportio ad humidum in grauitate, hanc habet propor-<lb/>tionem pars ipſius demerſa ad totam portionem, quam autem pars de-<lb/>merſa ad totam hanc habet tetragonum, quod a, p, m, a tetragonum, <lb/>quod ab o, n. </s> <s xml:id="echoid-s494" xml:space="preserve">Quam ergo proportionem babet tetragonum, quod a, b, f, q. <lb/></s> <s xml:id="echoid-s495" xml:space="preserve">ad tetragonum, quod a, b, d, hanc proportionem habet tetragonum, quod <lb/>a, b, m, b, ad tetragonum quod a, b, o, n, æqualis ergo eſt, quæ f, q, ipſi p, <lb/>m. </s> <s xml:id="echoid-s496" xml:space="preserve">Quæ autem p, b, demonstrata eſt eſſe maior, quàm f, palam ergo, quòd <lb/>quæ p, m, eſt minor, quàm dupla ipſius b, m. </s> <s xml:id="echoid-s497" xml:space="preserve">Sit igitur quæ p, Z, dupla ip <lb/>ſius Z. </s> <s xml:id="echoid-s498" xml:space="preserve">m, erit autem t, quidem centrum grauitatis ſolidi, eius auté, quod <lb/>intra bumidum Z. </s> <s xml:id="echoid-s499" xml:space="preserve">Reliquam autem magnitudinis centrum grauitatis <lb/>erit in linea Z, t. </s> <s xml:id="echoid-s500" xml:space="preserve">Copulata, & </s> <s xml:id="echoid-s501" xml:space="preserve">educta, & </s> <s xml:id="echoid-s502" xml:space="preserve">educatur ad g, demonſtrabitur <lb/>autem ſimiliter quæ t, b, perpendicularis exiſtens ad ſuperficiem humi-<lb/>di, & </s> <s xml:id="echoid-s503" xml:space="preserve">portio quidem quæ intra humidum fertur ad extra humidi, ſecun <lb/>dum perpendicularem ducta per Z, ſuperficiem humidi. </s> <s xml:id="echoid-s504" xml:space="preserve">Quæ autem ex-<lb/>tra humidum ferretur intra humidum, ſecundum ea, quæ per g, non ma-<lb/>net autem portio, ſecundum ſuppoſitam inclinationem, nec etiam in re-<lb/>ctum restituetur. </s> <s xml:id="echoid-s505" xml:space="preserve">palam enim propter hoc quoniam, quæ producuntur <lb/>per Z, g. </s> <s xml:id="echoid-s506" xml:space="preserve">perpendiculares. </s> <s xml:id="echoid-s507" xml:space="preserve">quæ quidem per Z, perducit ipſi g, <lb/>l, ad eaſdem partes cadit ad quas eſt, & </s> <s xml:id="echoid-s508" xml:space="preserve">ſecundum g. </s> <s xml:id="echoid-s509" xml:space="preserve">Quæ autem per <lb/>g, ad eaſdem ipſi Z, g. </s> <s xml:id="echoid-s510" xml:space="preserve">palam quòd propter prædicta Z, quidem centrũ <lb/>ſurſum ferretur:</s> <s xml:id="echoid-s511" xml:space="preserve">g, autem deorſum. </s> <s xml:id="echoid-s512" xml:space="preserve">Quare totius magnitudinis, quæ ex <lb/>parte a, deorſum ferretur, hoc antem erat inutile ad demonstrandum.</s> <s xml:id="echoid-s513" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <figure xlink:label="fig-0031-01" xlink:href="fig-0031-01a"> <image file="0031-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0031-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s514" xml:space="preserve">Supponatur rurſum alia quidem eadem axis autem portionis ad ſu-<lb/>perficiem humidi faciat angulum minorem eo, qui apud b, minorem au-<lb/>tem proportionem habet tetragonum, quod a, p, i, ad tetragonum, quod <lb/>ab i, ***, quàm ad a, b, x, ad id, quod a, x, b, & </s> <s xml:id="echoid-s515" xml:space="preserve">quæ K, r, ergo ad ***, i, mi <lb/>norem proportionem habet, quàm medietas ipſius K, r, ad x, b. </s> <s xml:id="echoid-s516" xml:space="preserve">Eſt ergo <lb/>quæ i, ***, maiorem quàm dupla ipſius x, b, ergo quæ ***, i, minor ipſius <lb/>autem o, i, dupla ergo ***, eſt, quæ o, i, ipſus x, b, eſt autem, & </s> <s xml:id="echoid-s517" xml:space="preserve">to <lb/>ta, quæ ***, t, æqualis ipſi r, b, & </s> <s xml:id="echoid-s518" xml:space="preserve">reliqua minor eſt, quàm ***, r, erit ergo, <lb/>& </s> <s xml:id="echoid-s519" xml:space="preserve">quæ p, h, minor, quàm f. </s> <s xml:id="echoid-s520" xml:space="preserve">Quæ autem m, p, ipſi f, q, eſt æqualis: </s> <s xml:id="echoid-s521" xml:space="preserve">palam <lb/>quòd p, m, eſt maior, quàm emiolia ipſius p, b, quæ autẽ p, h, minor, quàm <lb/>dupla ipſius h, m. </s> <s xml:id="echoid-s522" xml:space="preserve">Sit igitur, quæ p, z, ipſius z, m, dupla igitur rurſum. </s> <s xml:id="echoid-s523" xml:space="preserve">to <lb/>tius quidem cétrum grauitatis erit t, eius autem quod intra humidũ Z.</s> <s xml:id="echoid-s524" xml:space="preserve"> <pb o="9" file="0033" n="33" rhead="LBERI II."/> copulata autem z, t, inuenietur centrum eius, quòd extra humidum in <lb/>educta, & </s> <s xml:id="echoid-s525" xml:space="preserve">ſit g. </s> <s xml:id="echoid-s526" xml:space="preserve">& </s> <s xml:id="echoid-s527" xml:space="preserve">ducatur perpendicularis ad ſuperficiem bumidi per <lb/>z, g, æquediſtanter ipſi n, o, palam igitur, quòd non manet tota portio, <lb/>ſed reuoluetur ita, ut axis ad ſuperficiem humidi faciat angulum mino-<lb/>rem, quàm illo, quem nunc facit: </s> <s xml:id="echoid-s528" xml:space="preserve">quoniam nec axe faciente ad humidum <lb/>angulum maiorem, quàm b, conſiſtit portio, neque minorem. </s> <s xml:id="echoid-s529" xml:space="preserve">Manifeſtũ <lb/>quòd tantum angulum faciente conſistet. </s> <s xml:id="echoid-s530" xml:space="preserve">Sic enim erit quæ i, o, æqualis <lb/>ipſi x, b, & </s> <s xml:id="echoid-s531" xml:space="preserve">quæ ***, ipſi x, r, & </s> <s xml:id="echoid-s532" xml:space="preserve">quæ p h, ipſi f, erit igitur m, h, æmyolia <lb/>ipſius p, h, quæ autem p, b, ipſi b, ***, dupla quod autem ergo <lb/>eius, quod in humido centrum grauitatis eſt. </s> <s xml:id="echoid-s533" xml:space="preserve">Quare ſecundum eandem <lb/>perpendicularem ſurſum ferretur, et quod extra deorſum ferretur mane <lb/>bit ergo contra pellentur enim adinuicem.</s> <s xml:id="echoid-s534" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div38" type="section" level="1" n="27"> <head xml:id="echoid-head37" xml:space="preserve">NONVS.</head> <p> <s xml:id="echoid-s535" xml:space="preserve">Recta portio rectanguli conoydalis, quando axem habue <lb/>rit maiorem quidem, quàm hemiolium eius quæ uſque ad a-<lb/>xem, minorem autem ut hanc habeat proportione, quam <lb/>habent quindecim ad quattuor: </s> <s xml:id="echoid-s536" xml:space="preserve">& </s> <s xml:id="echoid-s537" xml:space="preserve">in grauitate ad humidũ <lb/>habeat proportionem maiorem proportione, quam habet <lb/>exceſſus, quo tetragonum quod ab axe eſt maius tetragono, <lb/>quod ab exceſſu, quo axis eſt maior, quàm hemiolius eius, <lb/>quæ uſq; </s> <s xml:id="echoid-s538" xml:space="preserve">ad axem ad tetragonum, quod ab axe demiſſa in hu <lb/>midũ, ita ut baſis ipſius tota, ſit in humido poſita inclinata, <lb/>nec ut axis ipſius ſecundum perpendicularem ſit, necmane <lb/>bit inclina ta, niſi quã do axis ipſius ad ſuperficiem humidife <lb/>cerit angulum<unsure/> æqu alem accepto ſimiliter, ut prius.</s> <s xml:id="echoid-s539" xml:space="preserve"/> </p> <figure> <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0033-01"/> </figure> <pb file="0034" n="34" rhead="DE INSIDENTIBVS AQVAE"/> <figure> <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0034-01"/> </figure> <p style="it"> <s xml:id="echoid-s540" xml:space="preserve">ES to portio, qualis dicta _est_, & </s> <s xml:id="echoid-s541" xml:space="preserve">ponatur, quæ d, b, æqualis axi por-<lb/>tionis, & </s> <s xml:id="echoid-s542" xml:space="preserve">quæ quidem b, k, ſit dupla ipſius K, d. </s> <s xml:id="echoid-s543" xml:space="preserve">Quæ autem K, r, <lb/>æqualis ei, quæ uſque ad axem, quę autem c, b, hemiolia ipſius b, r. <lb/></s> <s xml:id="echoid-s544" xml:space="preserve">Quam autem proportionem habet portio ad bumidum in grauitate, <lb/>banc habeat exceſſus, quo excedit tetragonum, quod a, b, d, tetragonum <lb/>quod a, b, f, q, ad tetragonum, quod a, b, d, ſit autem quæ f, dupla ipſius q. </s> <s xml:id="echoid-s545" xml:space="preserve"><lb/>Palam igitur, quòd exceßus, quo excidit tetragonum, quod a, b, d, tetra-<lb/>gonum, quod a, b, c, ad tetragonum, quod a, b, d, quo axis portionis eſt ma <lb/>ior, quàm hemiolius eius, quæ uſque ad axem minor eſt in maiori ergo te <lb/>tragonum, quod a, b, d, excedit id, quod a, b, f, q, quàm tetragonum quod <lb/>a, b, d, excedat tetragonum, quod a, b, c. </s> <s xml:id="echoid-s546" xml:space="preserve">Quare quæ f, q, eſt minor, quàm <lb/>b, c. </s> <s xml:id="echoid-s547" xml:space="preserve">Ergo & </s> <s xml:id="echoid-s548" xml:space="preserve">quæ f, quàm b, r. </s> <s xml:id="echoid-s549" xml:space="preserve">Sit igitur ipſi f, æqualis, quæ r, x, & </s> <s xml:id="echoid-s550" xml:space="preserve">quæ x, <lb/>e, recta ducatur ſuper b, d, potens medietatem eius, quòd continetur ſub <lb/>K, r, x, b, dico quòd portio demißa in bumidũ ita, ut baſis ipſius tota ſit <lb/>in humido conſistat, ita ut axis ipſius ad ſuperficiem humidt<unsure/> faciat angu <lb/>lum æqualem angulo b. </s> <s xml:id="echoid-s551" xml:space="preserve">Demittatur quidem enim portio in humidum, <lb/>ut dictum eſt & </s> <s xml:id="echoid-s552" xml:space="preserve">non faciat axis ad ſuperficiem humidi angulum æqua-<lb/>lem b, ſed maiorem primo. </s> <s xml:id="echoid-s553" xml:space="preserve">Secta autem ipſa plano recto ad ſuperficiem <lb/>bumidi porti<unsure/>onis ſectio ſit, quę apol. </s> <s xml:id="echoid-s554" xml:space="preserve">rectanguli coniſectio ſuperficiei au-<lb/>tem humidi, quæ c, i, axis autem portionis, & </s> <s xml:id="echoid-s555" xml:space="preserve">dyameter ſit quę n, o, & </s> <s xml:id="echoid-s556" xml:space="preserve"><lb/>ſit ſecta ſecundum ***, t, ut & </s> <s xml:id="echoid-s557" xml:space="preserve">prius ducatur autem quæ quidem y, p, <lb/>æquedistanter ipſi c, i, contingens ſectionem ſecundum p. </s> <s xml:id="echoid-s558" xml:space="preserve">Quæ autem m, <lb/>p, æquediſtanter ipſi n, o. </s> <s xml:id="echoid-s559" xml:space="preserve">Quæ uero p, s, perpendicularis ſuper axé, quo-<lb/>niam egit axis portionis ad ſuperficiem bumidi facit angulum maiorem <lb/>angulo b. </s> <s xml:id="echoid-s560" xml:space="preserve">Erit utique & </s> <s xml:id="echoid-s561" xml:space="preserve">angulus, qui ſub s, y, p, maior angulo b, tetrago <lb/>num ergo quod a, p, s, ad tetragonum quod a, b, s, y, habet proportionem <lb/>matorem, quàm tetragonum, quod a, x, e, ad tetragonum, quod a, x, b.</s> <s xml:id="echoid-s562" xml:space="preserve"> <pb o="10" file="0035" n="35" rhead="LIBER II."/> <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> Ergo & </s> <s xml:id="echoid-s563" xml:space="preserve">quæ K, r, ad s, y, habet proportionem maiorem, quàm medietas <lb/>ipſius K, r, ad x, b, minor ergo, quæ s, y, quàm dupla ipſius x, b, & </s> <s xml:id="echoid-s564" xml:space="preserve">quæ s, <lb/>o, quàm x, b, minor quæ s, ***, ergo maior, quàm r, x, & </s> <s xml:id="echoid-s565" xml:space="preserve">quæ p, b, quam f, <lb/>& </s> <s xml:id="echoid-s566" xml:space="preserve">ſi portio in grauitate ad humidum habet proportionem, quam exceſ <lb/>ſus, quo tetragonum, quod a, b, d, eſt maius tetragono, quod a, b, f, q, ad te <lb/>tragonum, quod a, b, d. </s> <s xml:id="echoid-s567" xml:space="preserve">Quam autem proportionem habet proportio in <lb/>grauitate ad humidum, hanc proportionem habet demerſa ipſius portio <lb/>ad totam palam, eandem habebit proportionem demerſa ipſius portio ad <lb/>totam portionem, quam exceſſus, quo tetragonum, quod a, b, d, excedit <lb/>tetragonum, quod a, b, f, q, ad tetragonum, quod a, b, d, habebit igitur, & </s> <s xml:id="echoid-s568" xml:space="preserve"><lb/>tota portio a d, eam quæ extra humidum proportionem, quam tetrago-<lb/>num, quod a, b, d, ad id, quod a, b, f, q. </s> <s xml:id="echoid-s569" xml:space="preserve">Quam autem proportionem habet <lb/>tota proportio ad eam, quàm extra bumidum hanc habet, quod ab n, o, <lb/>ad id, quod a, p, m, æqualis ergo quæ m, p, ipſi f, q. </s> <s xml:id="echoid-s570" xml:space="preserve">Quæ autem p, b, demõ <lb/>ſtrata eſt maior, quàm quæ ergom, b, eſt minor, quàm q, ergo quæ o, m, <lb/>eſt maior, quàm dupla ipſius b, m. </s> <s xml:id="echoid-s571" xml:space="preserve">Sit igitur quæ p, z, dupla ipſius z, m, <lb/>& </s> <s xml:id="echoid-s572" xml:space="preserve">copulata, quę 2, t, educatur ad g, erit ergo totius quidem portionis cé <lb/>trum grauitatis t, eius autem quæ extra humidum z, eius uero quæ in-<lb/>tra in linea t, g. </s> <s xml:id="echoid-s573" xml:space="preserve">Sit autem g, demonſtrabitur autem ſimiliter prioribus, <lb/>quæ t, h, perpendicularis ad ſuperficiem humidi, & </s> <s xml:id="echoid-s574" xml:space="preserve">quæ per z, g, æquedi-<lb/>ſtanter ipſit, n, productæ perpendiculares, & </s> <s xml:id="echoid-s575" xml:space="preserve">ipſæ ſuper ſuperficiem bu-<lb/>midi ferretur, ergo quæ quidem extra humidum portio deor ſum ſecundũ <lb/>eam quæ per z. </s> <s xml:id="echoid-s576" xml:space="preserve">Quæ autem intra ſecundum eam, quæ per g, eleuabitur <lb/>non manet ergo tota portio ſine inclinatione, nec etiam conuertetur ita <lb/>ut axis ſit perpendicularis ſuperficiem humidi, quoniam quæ ex parte <lb/>l, ad ſuperiora ferrentur, propter proportionalia dictis in præcedenti, ſi <pb file="0036" n="36" rhead="_DE INSIDENTIBVS AQVAE_"/> autem axis ad humidum faciat angulum minorem angulo b, conſimili-<lb/>ter prioribus demonstrabitur, quod non manebit portio ſed inclinabitur <lb/>donec utique axis ad ſuperficiem humidi faciat angulum æqualem, an-<lb/>gulo b.</s> <s xml:id="echoid-s577" xml:space="preserve"/> </p> <div xml:id="echoid-div38" type="float" level="2" n="1"> <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0035-01"/> </figure> </div> </div> <div xml:id="echoid-div40" type="section" level="1" n="28"> <head xml:id="echoid-head38" xml:space="preserve">DECIMVS.</head> <p> <s xml:id="echoid-s578" xml:space="preserve">Recta portio rectanguli conoydalis, quando leuior exi-<lb/>ſtens humido habuerit axem maiorem, quàm ut habeat pro-<lb/>portionem ad eam, quàm uſque ad axem, quam habent quin <lb/>decim ad quattuor demiſſa in humidum ita, ut baſis ipſius nõ <lb/>tangat humidum, quandoque quidem recta conſiſtet, quin-<lb/>que autem inclinata: </s> <s xml:id="echoid-s579" xml:space="preserve">& </s> <s xml:id="echoid-s580" xml:space="preserve">quandoque quidem ita inclinata, ut <lb/>baſis ipſius, ſecundum vnum ſignum tangat ſuperficiem hu-<lb/>midi: </s> <s xml:id="echoid-s581" xml:space="preserve">& </s> <s xml:id="echoid-s582" xml:space="preserve">hoc in duabus diſpoſitionibus faciet: </s> <s xml:id="echoid-s583" xml:space="preserve">& </s> <s xml:id="echoid-s584" xml:space="preserve">quandoq; <lb/></s> <s xml:id="echoid-s585" xml:space="preserve">ita inclinata conſiſtet, ut baſis ipſius ſecundum ampliorem <lb/>locum humefiat: </s> <s xml:id="echoid-s586" xml:space="preserve">quandoque autem ita, ut baſis@ipſius, nec ſe <lb/>cundum vnum tangat ſuperficiem humidi. </s> <s xml:id="echoid-s587" xml:space="preserve">Quam autẽ pro-<lb/>portionem habeant ad humidum in grauitate fingula ho-<lb/>rum demonſtrabuntur.</s> <s xml:id="echoid-s588" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s589" xml:space="preserve">SIt portio qualis dicta est, & </s> <s xml:id="echoid-s590" xml:space="preserve">ſecta ipſa plano recto ad ſuperficiem hu-<lb/>midi ſectio in ſuperficie ſit quæ apol. </s> <s xml:id="echoid-s591" xml:space="preserve">rectanguli coni ſectio axis autẽ <lb/>& </s> <s xml:id="echoid-s592" xml:space="preserve">dyameter ſectionis ſit quæ b, d. </s> <s xml:id="echoid-s593" xml:space="preserve">Secetur autem quæ b, d, ſecundũ <lb/>K, ita ut dupla ſit quæ b, d, ipſi K, d, ſecundum c, autem, ut quæ b, d, ad K, <lb/>c, habeat proportionem, quam habent quindecim ad quattuor. </s> <s xml:id="echoid-s594" xml:space="preserve">Palam <lb/>igitur quòd quæ K, c, eſt maior ea, quæ uſque ad axem ipſius autem K, r, <lb/>ſit hemiolia, quæ eſt autem, & </s> <s xml:id="echoid-s595" xml:space="preserve">quæ s, b, hemiolia ipſius b. </s> <s xml:id="echoid-s596" xml:space="preserve">r. </s> <s xml:id="echoid-s597" xml:space="preserve">Co-<lb/>puletur autem ipſa a, b, & </s> <s xml:id="echoid-s598" xml:space="preserve">ipſa c, e, recta producta ducaturq́ue e, Z, æque <lb/>diſtanter, ipſi b, d, & </s> <s xml:id="echoid-s599" xml:space="preserve">rurſum ipſa a,<unsure/> b, ſecta in duo æqualia penes t, duca-<lb/>tur æquediſtanter ipſi b, d, quæ t, h, & </s> <s xml:id="echoid-s600" xml:space="preserve">accipiatur rectanguli coni ſectio, <lb/>quæ a, e, circa dyametrum e, Z, & </s> <s xml:id="echoid-s601" xml:space="preserve">quæ a, t, circa dyametrum t h, ita, ut ſi <lb/>milis ſit, quæ a, e, i, a, t, h, portioni a, b, l, deſcribetur autem quæ a, e, i, co-<lb/>ni ſectio per K. </s> <s xml:id="echoid-s602" xml:space="preserve">Quæ autem a, b, r, recta producta ipſi b, d, ſecat ipſam a, <lb/>e, i, ſecet, ſecũdum y g, cum per y, g, ducantur æquediſtanter ipſi b, d, quæ <lb/>p, y, q. </s> <s xml:id="echoid-s603" xml:space="preserve">Secet antem ipſe ſectionem a, o, d, penes x, f, ducantur autem, & </s> <s xml:id="echoid-s604" xml:space="preserve"><lb/>quæ p, x, o. </s> <s xml:id="echoid-s605" xml:space="preserve">contingentes ſectionem apol. </s> <s xml:id="echoid-s606" xml:space="preserve">ſecundum o, p. </s> <s xml:id="echoid-s607" xml:space="preserve">Sunt <lb/>tres quædam portiones quæ apol. </s> <s xml:id="echoid-s608" xml:space="preserve">a, e, i, a, t, d, contentę arectis, & </s> <s xml:id="echoid-s609" xml:space="preserve">a ſe-<lb/>ctionibus rectangulorum conorum rectæ, & </s> <s xml:id="echoid-s610" xml:space="preserve">ſimiles, & </s> <s xml:id="echoid-s611" xml:space="preserve">inæquales, & </s> <s xml:id="echoid-s612" xml:space="preserve"><lb/>@angentes ſuper pnamquanque baſem a, b, n, autem ſurſum <lb/>ducta eſt, quæ n, x, p, n, o. </s> <s xml:id="echoid-s613" xml:space="preserve">o, g, ergo ad g, x, habet pro- <pb o="11" file="0037" n="37" rhead="_LIBER II._"/> portionem compoſitam ex proportione quam habet quæ i, l, ad l, a, & </s> <s xml:id="echoid-s614" xml:space="preserve"><lb/>quam habet quæ a, d, ad d, i, habet autem, & </s> <s xml:id="echoid-s615" xml:space="preserve">quæ l, i, ad l, a, quàm duo <lb/>ad quinque. </s> <s xml:id="echoid-s616" xml:space="preserve">Quæ enim c, b, ad b, d, habet proportionem, quàm ſex ad <lb/>quindecim, hoc eſt, quam duo ad quinque, & </s> <s xml:id="echoid-s617" xml:space="preserve">est ut quæ c, b, ad b, d, ita <lb/>quæ e, b, ad b, a, & </s> <s xml:id="echoid-s618" xml:space="preserve">quæ d, Z ad d, a, habeat autem d, Z, d, a, duplæ.</s> <s xml:id="echoid-s619" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s620" xml:space="preserve">l, i, l, a. </s> <s xml:id="echoid-s621" xml:space="preserve">Quæ autem a, d, ad d, i, proportionem habet, quam quinque <lb/>ad vnum. </s> <s xml:id="echoid-s622" xml:space="preserve">Proportio autem compoſita ex proportione, quam habẽt duo <lb/>ad quinque, & </s> <s xml:id="echoid-s623" xml:space="preserve">ex proportione, quam habent quinque ad vnum. </s> <s xml:id="echoid-s624" xml:space="preserve">eſt ean <lb/>dem cum proportione, quam habent duo ad vnum. </s> <s xml:id="echoid-s625" xml:space="preserve">Dupla ergo eſt, quæ <lb/>g, o, ipſius g, x, propter eandem autem, & </s> <s xml:id="echoid-s626" xml:space="preserve">quæ p, y, ipſius y, f, quoniam <lb/>igitur. </s> <s xml:id="echoid-s627" xml:space="preserve">quæ d, s, eſt hemiolia ipſius K, r, palam quòd quæ b, s, eſt exceſ <lb/>ſus, quo axis est maior, quàm hemiolius eius, quæ uſque ad axem, ſiqui-<lb/>dem igitur portio ad humidum in grauitate banc habet proportionem, <lb/>quam tetragonum, quod a, b, s, ad id, quod a, b, d, aut maiorem hac pro-<lb/>portione portio demiſſa in humidum ita, ut baſis ipſius non tangat humi <lb/>dum recta conſistet, demonſtratum est ei prius, quòd ſi portio habẽs ax ẽ <lb/>maiorem, quàm hemiolium eius, quæ uſque ad axem minorem propor-<lb/>tione ſi ad humidum in grauitate n, o, minorem proportionẽ habeat pro-<lb/>portione, quam habet tetragonum, quod ab exceßu, quo axis eſt maior, <lb/>quàm hemiolium eius, quæ ad axem ad tetragonum, quod ab axe. </s> <s xml:id="echoid-s628" xml:space="preserve">demiſ <lb/>ſa in humidum, ita ut dictum eſt, recta conſistet. </s> <s xml:id="echoid-s629" xml:space="preserve">Si autem portio ad hu-<lb/>midum in grauitate maiorem quidem proportionem habeat proportione <lb/>quàm habet tetragonum, quod a, b, s, b, ad tetragonum, quod a, b, d, ma-<lb/>iorem autem proportionem, quam habet tetragonum quod a, b, x, t, ad <lb/>id, quod a, b, demiſſain humidum inclinata ita, ut baſis contingat humi-<lb/>dum conſistet inclinata ita, ut baſis ipſius nibil tangat ſuperficiei humi-<lb/>di, & </s> <s xml:id="echoid-s630" xml:space="preserve">axis ipſius faciat ad ſuperficiem humidi angulum maiorem angu <lb/>lo m. </s> <s xml:id="echoid-s631" xml:space="preserve">Si autem portio ad humidum in grauitate hanc habet proportionẽ. <lb/></s> <s xml:id="echoid-s632" xml:space="preserve"> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> <pb file="0038" n="38" rhead="_DE INSIDENTIBVS AQV AE_"/> quam habet tetragonum, quod ab x, o, ad id, quod a, b, d, demiſſa in hu-<lb/> <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> midum inclinata, h, Z, vult diuidi in quinque æqualia, media quinta pars ſit t, K, t, i. </s> <s xml:id="echoid-s633" xml:space="preserve">m, n, vult eſſe æqualis o, n, & </s> <s xml:id="echoid-s634" xml:space="preserve">n, x, ſit media proportionalis inter m, n, & </s> <s xml:id="echoid-s635" xml:space="preserve">n, o, & </s> <s xml:id="echoid-s636" xml:space="preserve">quarta proportionalis c, n, ita, ut baſis ipſius non tã gunt humidum: </s> <s xml:id="echoid-s637" xml:space="preserve">conſiſtet, & </s> <s xml:id="echoid-s638" xml:space="preserve">manebit ita, ut baſis ipſius, ſecnndum am- pliorem locum humectetur ab humido. </s> <s xml:id="echoid-s639" xml:space="preserve">Si uero portio ad humidam in grauitate hanc proportionem habet, quam habet tetragonum, quod a.</s> <s xml:id="echoid-s640" xml:space="preserve"><unsure/> p, f, ad tetragonum, quod a, b, d, demiſſa in humidum, et poſita inclinata ita ut baſis ipſius non tangat humidum, conſistet inclinata ita, ut baſis ip- ſius, ſecundum vnum ſignnm tangat ſuperficiem humidi, & </s> <s xml:id="echoid-s641" xml:space="preserve">axis ipſi <pb o="12" file="0039" n="39" rhead="_LIBER II._"/> faciat angulum, æqualem angulo x. </s> <s xml:id="echoid-s642" xml:space="preserve">Si autem portio ad humidum ingra- uitate habeat proportionem minorem proportione, quam habet tetra- gonum, quod ab f, p, ad tetragonum, quod a, b, d, dimiſſa in humidum, &</s> <s xml:id="echoid-s643" xml:space="preserve"> poſita inclinata ita, ut baſis ipſius non tangat humidum conſiſtet inclina ta ita, ut axis quidem ipſius ad ſuperficiem humidi, faciat angulum mi- norem angulo x, baſis autem ipſius, nec ſecundum vnum tangat ſuperfi- ciem humidi. </s> <s xml:id="echoid-s644" xml:space="preserve">Demonſtrabitur itaque hæc deinceps.</s> <s xml:id="echoid-s645" xml:space="preserve"/> </p> <div xml:id="echoid-div40" type="float" level="2" n="1"> <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0037-01"/> </figure> <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0038-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s646" xml:space="preserve">Habeat itaque primo portio ad humidum in grauitate proportionem <lb/>quidem maiorem ea, quam habet tetragonum, quod ab x, o, ad id, quod a, <lb/>b, d, minore autem ea, quàm habet tetragonũ, quod ab exceſſu, quo axis <lb/>eſt maior, quàm hemiolius eius, quæ uſque ad axẽ ad tetragonum, quod <lb/>a, b, d, & </s> <s xml:id="echoid-s647" xml:space="preserve">ſupponatur prius diſpoſita figura. </s> <s xml:id="echoid-s648" xml:space="preserve">Quam autem proportionem <lb/>habet portio ad humidum in grauitate, hanc tetragonum, quod a, x, ad <lb/>id, qnod a, b, d, eſt autẽ quæ x, maior qui <lb/> <anchor type="figure" xlink:label="fig-0039-01a" xlink:href="fig-0039-01"/> dem quàm x, p, minor autem exceſſu, <lb/>quo axis eſt maior, quàm hemiolius eius <lb/>quæ uſque ad axem. </s> <s xml:id="echoid-s649" xml:space="preserve">Inaptetur autem <lb/>quædam inter media conicarum ſectio-<lb/>num apol. </s> <s xml:id="echoid-s650" xml:space="preserve">a, z, d, quæ u, o, æqualis ipſi x, <lb/>& </s> <s xml:id="echoid-s651" xml:space="preserve">ſecet ipſa reliquam coni ſectionem pe <lb/>nes ipſa autem r, s, rectam <lb/>penes b, demonſtr abitur autẽ quæ <lb/>o, u, ipſius a, n, ſicut demonstratum est, <lb/>quæ p, s, ipſius s, x, dupla ab o, autem du <lb/>catur, quæ o, s, cõtingens ſectionem apol <lb/>quæ autem o, c, perpendicularis ſuper <lb/>b, d, & </s> <s xml:id="echoid-s652" xml:space="preserve">ab a, ad n, copuletur, erunt autẽ <lb/>quę a, n, q, u, æquales inuicem. </s> <s xml:id="echoid-s653" xml:space="preserve">Quoniam <lb/>enim in ſimilibus portionibus apol. </s> <s xml:id="echoid-s654" xml:space="preserve">a, x, <lb/>d, producto ſunt ab axibus ad portiones, quæ a, n, a, q, æquales angulos fa <lb/>cientes ad baſes eandem proportionem habebunt quæ q, a. </s> <s xml:id="echoid-s655" xml:space="preserve">a, n, cum ipſis <lb/>l, a. </s> <s xml:id="echoid-s656" xml:space="preserve">a, d, propter ſecundam figuram præſcriptarum æqualis, ergo quæ a, <lb/>n ipſi q, n, & </s> <s xml:id="echoid-s657" xml:space="preserve">æquediſtans ipſi o, s, demonstrandum, quòd demiſſa in hu-<lb/>midum ita, ut baſis ipſius, non ſecundum vnum tangit axis ad <lb/>ſuperficiem humidi angulum acutum faciat maiorem exceſſu Di <lb/>mittatur enim, & </s> <s xml:id="echoid-s658" xml:space="preserve">conſiſtat ita, ut baſis ipſius tangat, ſecundum vnum ſi <lb/>gnum ſuperficiem humidi. </s> <s xml:id="echoid-s659" xml:space="preserve">Secta autem portione per axem plano recta <lb/>ad ſuperficiem humidi, ſuperficiei quidem portionis ſectio ſitq́ apol. </s> <s xml:id="echoid-s660" xml:space="preserve">re-<lb/>ctanguli coni ſectio, ſuperficiei autem humidi, quæ o, a, axis autem ſectio <lb/>nis, & </s> <s xml:id="echoid-s661" xml:space="preserve">dyameter, quæ b, d, & </s> <s xml:id="echoid-s662" xml:space="preserve">ſecetq́ b, d, penes K, r, ut dictum eſt duca- <pb file="0040" n="40" rhead="_DE INSIDENTIBVS AQV AE_"/> tur autem, & </s> <s xml:id="echoid-s663" xml:space="preserve">quæ quidem p, g, ęquediſtanter ipſi a, o, recta contingent <lb/>ſectionem apol. </s> <s xml:id="echoid-s664" xml:space="preserve">ſecundum p. </s> <s xml:id="echoid-s665" xml:space="preserve">Quæ autem p, t, ęquediſtanter ipſi b, d. <lb/></s> <s xml:id="echoid-s666" xml:space="preserve">Quæautem p, s, perpendicularis ſuper b, d. </s> <s xml:id="echoid-s667" xml:space="preserve">Quoniam igitur portio ad <lb/> <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> humidum in grauitate proportionem habet, quam tetragonum, quod a, <lb/>x, ad id, quod a, b, d. </s> <s xml:id="echoid-s668" xml:space="preserve">Quam autem proportionem habet portio ad humi-<lb/>dum, hãc habet demerſa ipſius portio ad totam, quàm autem demerſa ad <lb/>totam tetragonum, quod a, t, p, ad id, quod a, d, b, erit quæ x, ipſi t, p, æ-<lb/>qualis, & </s> <s xml:id="echoid-s669" xml:space="preserve">quæ n, o, ergo ipſi t, p, æqualis eſt. </s> <s xml:id="echoid-s670" xml:space="preserve">Quare, & </s> <s xml:id="echoid-s671" xml:space="preserve">portiones a, p, q, <lb/>a, p, f, inuicem ſunt æquales. </s> <s xml:id="echoid-s672" xml:space="preserve">Quoniam autem in portionibus æqualibus, <lb/>& </s> <s xml:id="echoid-s673" xml:space="preserve">ſimilibus apol. </s> <s xml:id="echoid-s674" xml:space="preserve">a, b, l, K, ab extremitatibus baſium productę ſunt, quæ <lb/>r, a. </s> <s xml:id="echoid-s675" xml:space="preserve">a, q, & </s> <s xml:id="echoid-s676" xml:space="preserve">portiones ablatæ faciunt ad dyametros angulos æquales, <lb/>propter tertiam figuram præſcriptarum. </s> <s xml:id="echoid-s677" xml:space="preserve">quare anguli qui apud y, g, <lb/>ſunt æquales, & </s> <s xml:id="echoid-s678" xml:space="preserve">quæ y, b, g, b. </s> <s xml:id="echoid-s679" xml:space="preserve">ergo æquales ſunt quare & </s> <s xml:id="echoid-s680" xml:space="preserve">quæ s, r, c, r, <lb/>& </s> <s xml:id="echoid-s681" xml:space="preserve">quæ p, Z, o, u, & </s> <s xml:id="echoid-s682" xml:space="preserve">quæ Z, t, s, K, n, quoniam minoré, quàm dupla quæ o, <pb o="13" file="0041" n="41" rhead="LIBER II."/> ipſius s, a, u, palã ꝙ quæ p, Z, ipſius Z, t, est minor, ꝗ̃ dupla. </s> <s xml:id="echoid-s683" xml:space="preserve">Sit igitur <lb/>quæ p, ***, ipſius, ***, t, dupla, & </s> <s xml:id="echoid-s684" xml:space="preserve">copulata quæK, ***, educa <lb/>tur ad e, totius quidem igitur cetrum grauitatis erit K, eius autẽ por <lb/>tionis, quæ inter humidũ centrũ, ***, eius autẽ quæ extrain linea K, e, <lb/>& </s> <s xml:id="echoid-s685" xml:space="preserve">ſite. </s> <s xml:id="echoid-s686" xml:space="preserve">Quæ auté K, Z, perpẽdicularis erit ſuꝑ ſuꝑſiciẽ humidi, quare <lb/>& </s> <s xml:id="echoid-s687" xml:space="preserve">quæ ꝑ ſigna, e, ***, æquediſtãter ipſi K, Z, non ergo manet portio ſed <lb/>inreclinabitur ut baſis ipſius, nec ſecundum unum tangat ſuperſiciem <lb/>humidi, quoniam nunc ſecundum unum tacta ipſa reclinatur. </s> <s xml:id="echoid-s688" xml:space="preserve">Mani-<lb/>feſtum ergo quòd portio conſiſtet, ita ut axis ad ſuperficiem humidi fa <lb/>ciat angulum maiorem anguloy.</s> <s xml:id="echoid-s689" xml:space="preserve"/> </p> <div xml:id="echoid-div41" type="float" level="2" n="2"> <figure xlink:label="fig-0039-01" xlink:href="fig-0039-01a"> <image file="0039-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0039-01"/> </figure> <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a"> <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0040-01"/> </figure> </div> <p style="it"> <s xml:id="echoid-s690" xml:space="preserve">HAbeat autem portio ad humidum in grauitate hanc proportio-<lb/>nem, quam habet tetragonum, quod a, b, x, o, ad id, quod a, b, d, & </s> <s xml:id="echoid-s691" xml:space="preserve"><lb/>dimittatur in humidum ita inclinata. </s> <s xml:id="echoid-s692" xml:space="preserve">Secta autem ipſa per axcm <lb/>plano recto ad ſuperficiem humidi ſolidi quidem, ſectio ſit quæ apol re <lb/>ctanguli coni ſectio. </s> <s xml:id="echoid-s693" xml:space="preserve">ſuperficiei autem humidi, quæ o, i, axis autem por <lb/>tionis & </s> <s xml:id="echoid-s694" xml:space="preserve">dyametris ſectionis quæ b, d, & </s> <s xml:id="echoid-s695" xml:space="preserve">ſecetur quæ b, d, ut prius & </s> <s xml:id="echoid-s696" xml:space="preserve"><lb/>ducatur. </s> <s xml:id="echoid-s697" xml:space="preserve">quæ quidem p, n, æquediſtanter ipſi i, o, cõtingens ſectionẽ ſe-<lb/>cundum p. </s> <s xml:id="echoid-s698" xml:space="preserve">Quæ autem p, t, æquediſtanter ipſi b, d, quæ autem p, s, per <lb/>pendicularis ſuper b, d. </s> <s xml:id="echoid-s699" xml:space="preserve">Demonstrãdum quòd portio non manet incli <lb/>nata ſic, ſed inclinatur donec utique baſis ſecundum unum ſignum tã <lb/>gat ſuperficiem humidi præiaceant auté & </s> <s xml:id="echoid-s700" xml:space="preserve">quæ in ſuperiori figura <lb/>prius diſpoſita ſunt, & </s> <s xml:id="echoid-s701" xml:space="preserve">quæ c, o, perpendicularis ducatur ſuper b, d, <lb/>& </s> <s xml:id="echoid-s702" xml:space="preserve">quæ a, x, copulata educatur ad q, erit autem quæ a, x, ipſi x, q, <lb/>æqualis, & </s> <s xml:id="echoid-s703" xml:space="preserve">ducatur ipſi a, q, quæ o, y, æquediſtans, & </s> <s xml:id="echoid-s704" xml:space="preserve">quoniam ſuppo <lb/>nitur portio ad humidum in grauitate hanc habere proportioné, quã <lb/>habet tetragonum quod ab x, a, ad id, quod a, b, d, habet autem hanc <lb/>proportionem & </s> <s xml:id="echoid-s705" xml:space="preserve">demerſa portio ad totam hoc eſt quòd a, t, p, ad id. <lb/></s> <s xml:id="echoid-s706" xml:space="preserve">quod a, b, d, æqualis utique erit, quæ p, t, ipſi x, o, et quoniam portionũ <lb/>i, b, o, a, b, q, dyametri ſunt æquales, & </s> <s xml:id="echoid-s707" xml:space="preserve">portiones rurſum quoniam in <lb/>portionibus æqualibus & </s> <s xml:id="echoid-s708" xml:space="preserve">ſimilib. </s> <s xml:id="echoid-s709" xml:space="preserve">apol a, o, q, l, productæ ſunt a, q, i, o, <lb/>æquales portiones auferentes, hoc quidem ab extremitate baſis hoc <lb/>autem non ab extremitate, palàm quòd minorem facit acutum <lb/>angulum ad dyametrum totius portionis, quæ ab extremitate baſis <lb/>producta eſt. </s> <s xml:id="echoid-s710" xml:space="preserve">Et quoniam angulus, qui apud y, eſt minor, qui apud h, <lb/>maior eſt, quæ b, c, quàm b, s. </s> <s xml:id="echoid-s711" xml:space="preserve">Quæ autem e, r, minor, quàm r, s, quare <lb/>& </s> <s xml:id="echoid-s712" xml:space="preserve">quæ o, y, minor quàm p, n, maior eſt quam du-<lb/>pla, & </s> <s xml:id="echoid-s713" xml:space="preserve">quoniam quàm o, y, dupla, est ipſius s, 3, palàm quod quæ p, a, <lb/>maior eſt, quàm dupla ipſis a, t. </s> <s xml:id="echoid-s714" xml:space="preserve">Sit igitur quæ p, h, dupla ipſius h, t, <lb/>& </s> <s xml:id="echoid-s715" xml:space="preserve">copuletur quæ h, K, & </s> <s xml:id="echoid-s716" xml:space="preserve">educatur ad, ***, erit autem totius quidem <pb file="0042" n="42" rhead="DE INSIDENTIBVS AQVAE"/> portionis centrum grauitatis K. </s> <s xml:id="echoid-s717" xml:space="preserve">Eius autem, quæ intra humidum h, <lb/>eius autem, quæ extra in linea K. </s> <s xml:id="echoid-s718" xml:space="preserve">***, & </s> <s xml:id="echoid-s719" xml:space="preserve">ſit***, demonſtrabitur autẽ <lb/>ſimiliter quæ K. </s> <s xml:id="echoid-s720" xml:space="preserve">2. </s> <s xml:id="echoid-s721" xml:space="preserve">ꝑpendicularis ſuper ſuperſiciem humidi, & </s> <s xml:id="echoid-s722" xml:space="preserve">quæ <lb/>per ſigna, h, ***, æqued ſtãter. </s> <s xml:id="echoid-s723" xml:space="preserve">ipſi K. </s> <s xml:id="echoid-s724" xml:space="preserve">Z, manifeſtũ igitur. </s> <s xml:id="echoid-s725" xml:space="preserve">quòd non ma <lb/>nebit portio, ſed inclinabiturd, onec utique baſis ipſius ſecundum unũ <lb/>ſignum tangat ſuperſiciem humidi, ſicut demonſtrabitur in tertia fi-<lb/>gura, quomodo ſe habet in tertio theoremate, & </s> <s xml:id="echoid-s726" xml:space="preserve">manebit portio ita cõ <lb/>ſiſtens. </s> <s xml:id="echoid-s727" xml:space="preserve">In portionibus h, æqualibus apol a, o, q, l, productæ erit ab ex-<lb/>tremitatibus baſium, quæ a, q, a, o, æquales auferentes demonſtrabitur <lb/>h, a, p, q, æqualis ipſi a, p, o, ſimiliter prioribus, æquales igitur facient <lb/>acutos angulos, quæ a, o, a, q, ad dyametros portionum. </s> <s xml:id="echoid-s728" xml:space="preserve">quoniam æqua <lb/>les ſunt qui apud n, y, anguli & </s> <s xml:id="echoid-s729" xml:space="preserve">Z, t, copulata autem ipſi Z, K, & </s> <s xml:id="echoid-s730" xml:space="preserve">edu <lb/>cta ad, ***, erit totius quidem portionis centrum grauitatis K, eius <lb/>autem quidem intra humidum, h, eius autem quæ extra in line a K, <lb/>***, & </s> <s xml:id="echoid-s731" xml:space="preserve">ſit, ***, & </s> <s xml:id="echoid-s732" xml:space="preserve">quæ K, h, perpendicularis eſt ſuper ſuper ficiem hu-<lb/>midi, ſecundum eaſdem igitur rectas quod quidem in humido ſurſum <lb/>feretur, & </s> <s xml:id="echoid-s733" xml:space="preserve">quod extra humidum deorſum feretur. </s> <s xml:id="echoid-s734" xml:space="preserve">Manebit autem <lb/>portio, & </s> <s xml:id="echoid-s735" xml:space="preserve">baſis, & </s> <s xml:id="echoid-s736" xml:space="preserve">magnitudo, & </s> <s xml:id="echoid-s737" xml:space="preserve">ſecũdum unum ſignum tanget ſu-<lb/>perficiem humidi & </s> <s xml:id="echoid-s738" xml:space="preserve">axis portionis ad ſuperficiem humidi faciet an-<lb/>gulum æqualẽ præſcripto. </s> <s xml:id="echoid-s739" xml:space="preserve">Similiter autem demonstrabitur, & </s> <s xml:id="echoid-s740" xml:space="preserve">ſi por-<lb/>tio ad humidum in grauitate habeat proportionem eandem, quã te-<lb/>tragonum quod h, p. </s> <s xml:id="echoid-s741" xml:space="preserve">ad id. </s> <s xml:id="echoid-s742" xml:space="preserve">quod a, b, d, dimiſſa in humidum ita, ut ba-<lb/>ſis ipſius non tangat ſuperficiem humidi. </s> <s xml:id="echoid-s743" xml:space="preserve">conſiſtet inclinata ita, ut ba-<lb/>ſis ipſius ſecumdum unum ſignũ tangat ſuperficiem humidi, & </s> <s xml:id="echoid-s744" xml:space="preserve">axis <lb/>ipſius ad ſuperficiem humidi faciat angulum æqualem angulo, quæ <lb/>apudf.</s> <s xml:id="echoid-s745" xml:space="preserve"/> </p> <figure> <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-01"/> </figure> <pb o="14" file="0043" n="43" rhead="LIBER II."/> <p style="it"> <s xml:id="echoid-s746" xml:space="preserve">SI autem rurſum portio ad humidum in grauitate habens quidem <lb/>proportionem maiorem illa, quam habet tetragonum, quod a, Z, p, <lb/>ad id, quod a, b, d, maiorem autem proportionem, quàm habet tetrago <lb/>num quod ab x, o, ad id, quod a, b, d, Quam autem proportionem ha-<lb/>bet portio ad humidum in grauitate, hanc habet tetragonum, quod a, <lb/>x, ad id, quod a, b, d, palàm igitur, quæ x, o, eſt quidem maior quàm Z, <lb/>p, minor autem quàm x, t, Inaptetur autem inter medio portionum <lb/>apol a, d, æqualis ipſi x, æquedistans autem ipſi b, d, quæf, i, ſecans ſe-<lb/>ctionem inter mediam coni penes y. </s> <s xml:id="echoid-s747" xml:space="preserve">Rurſum autem quæ f, y, dupla <lb/>ipſius y, i, demonstrabitur, ſicut quæt, ipſix, y, ut & </s> <s xml:id="echoid-s748" xml:space="preserve">prius de-<lb/>monstratum est. </s> <s xml:id="echoid-s749" xml:space="preserve">Ducatur autem a, b, f, ſectionem apol contingens <lb/>quæ f, ***, Similiter autem prioribus demonſtrabitur quæ quidem a, i, <lb/>ipſi q, i, æqualis. </s> <s xml:id="echoid-s750" xml:space="preserve">Quæ autem a, q, ipſi f, ***, æquedistans, Demonſtran <lb/>dum autem quòd portio demiſſa in humidum, ita ut baſis ipſius non <lb/>tang at humidum, & </s> <s xml:id="echoid-s751" xml:space="preserve">poſita inclinata ita inclinabitur, ut baſis ipſius <lb/>ſecundum ampliorem locum humectetur ab humido. </s> <s xml:id="echoid-s752" xml:space="preserve">Demittatur h, <lb/>in humidum, ut dictum eſt. </s> <s xml:id="echoid-s753" xml:space="preserve">& </s> <s xml:id="echoid-s754" xml:space="preserve">iaceat primo ſic inclinata ut baſis ip-<lb/>ſius neque ſecundum unum tang at ſuper ſiciem humidi. </s> <s xml:id="echoid-s755" xml:space="preserve">Secta autem <lb/>ipſa per axem plano recto ad ſuperſiciem humidi, in ſuperſicie quidem <lb/>portionis ſit ſectio, quæ a, b, g, in ſuperficie autem humidi, quæ e, Z, axis <lb/>autem ſectionis. </s> <s xml:id="echoid-s756" xml:space="preserve">& </s> <s xml:id="echoid-s757" xml:space="preserve">dyametrum portionis ſit quæ b, d, & </s> <s xml:id="echoid-s758" xml:space="preserve">ſecetur quæ <lb/>b, d, penes ſignum K, r, Similiter prioribus. </s> <s xml:id="echoid-s759" xml:space="preserve">ducatur auté & </s> <s xml:id="echoid-s760" xml:space="preserve">quæ quidẽ <lb/>h, l, æquediſtanter ipſi e, Z, cõtingens ſectionem a, b, g, penes h, quæ au-<lb/>tem h, t, æquedistanter ipſi b, d. </s> <s xml:id="echoid-s761" xml:space="preserve">Quæ autem h, s, perpendicularis ſu-<lb/>per b, d. </s> <s xml:id="echoid-s762" xml:space="preserve">Quoniā portio ad humidũ in grauitate proportionem babet <lb/>quam tetragonum, quòd a, x, ad id, quod a, b, d, palàm quod quæ x, eſt <lb/>æqualis ipſi h, t, demonſtrabitur h. </s> <s xml:id="echoid-s763" xml:space="preserve">Similiter prioribus, quare & </s> <s xml:id="echoid-s764" xml:space="preserve">quæ <lb/>h, t, eſt æqualis ipſi f, i, & </s> <s xml:id="echoid-s765" xml:space="preserve">portiones ergo a, f, q, e, b, Z.</s> <s xml:id="echoid-s766" xml:space="preserve"><unsure/> ſunt æquales in-<lb/>uicem, quoniam inequalibus, & </s> <s xml:id="echoid-s767" xml:space="preserve">ſimilibus portionibus apol a, b, g, ſunt <lb/>productæ, quæ a, q, e, Z, æquales portiones auferentes & </s> <s xml:id="echoid-s768" xml:space="preserve">hoc quidem <lb/>ab extremitate baſis, hoc autem non ab extremitate minorem faciet <lb/>acutum angulum ad dyametrum portionis quæ ab extremitate baſis <lb/>producta eſt. </s> <s xml:id="echoid-s769" xml:space="preserve">Et quoniam trigoni h, l, e, angulus eſt maior angulo, ***. <lb/></s> <s xml:id="echoid-s770" xml:space="preserve">palàm quòd minor eſt quæ b, s, quàm b, c. </s> <s xml:id="echoid-s771" xml:space="preserve">Quæ autem, s, r, <lb/>maior quàm r, c, & </s> <s xml:id="echoid-s772" xml:space="preserve">quæ h, l, maior quàm f, h, quæ a, t, mi-<lb/>nor est quàm h, i, & </s> <s xml:id="echoid-s773" xml:space="preserve">quoniam dupla est quæ f, y, ipſius y, i, palam ꝙ <lb/>quæ h, a, eſt maior, quàm dupla ipſius a, t, ſit igitur quæ h, l, dupla ip-<lb/>ſius l, t, palam autem ex hijs, ꝙ non @manebit portio, ſed inclinabitur <lb/>donec utique baſis ipſius tangat ſecundum unum ſignum ſuperficiem <lb/>humidi. </s> <s xml:id="echoid-s774" xml:space="preserve">Tangat autem ſecundum unum ſignum, ut in tertia figura <pb file="0044" n="44" rhead="DE INSIDENTIBVS AQVAE"/> ſcriptum eſt, et alia eadẽ diſponãtur, demõſtrabitur autẽ rurſum ꝙ t, <lb/>m, æquales exiſiens ipſi, f, i, & </s> <s xml:id="echoid-s775" xml:space="preserve">portiones a, f, q, a, b, Z, æquales inuicem <lb/>& </s> <s xml:id="echoid-s776" xml:space="preserve">quoniam in portionibus æqualibus, & </s> <s xml:id="echoid-s777" xml:space="preserve">ſimilib apol a, b, g, ſunt pro-<lb/>ductæ, quæ a, q, a, Z, æquales portiones auferentes æquales faciunt an-<lb/>gulos ad dyametros. </s> <s xml:id="echoid-s778" xml:space="preserve">portionum igitur a, h, b, Z, a, f, q, qui apud ſignal, <lb/>***, anguli ſunt æquales. </s> <s xml:id="echoid-s779" xml:space="preserve">& </s> <s xml:id="echoid-s780" xml:space="preserve">quæ b, s, recta ipſi b, c, æqualis & </s> <s xml:id="echoid-s781" xml:space="preserve">quæ s, <lb/>r, ipſi r, c. </s> <s xml:id="echoid-s782" xml:space="preserve">Et quæ h, a, ipſi f, h, & </s> <s xml:id="echoid-s783" xml:space="preserve">quæ a, t, ipſi m, i. </s> <s xml:id="echoid-s784" xml:space="preserve">Et quoniam dupla <lb/>est, quæ f, x, ipſiy, i. </s> <s xml:id="echoid-s785" xml:space="preserve">Manifeſtum quòd quæ h, a, eſt maior, quàm dupla <lb/>ipſius a, t. </s> <s xml:id="echoid-s786" xml:space="preserve">Sit igitur quæ h, a, ipſi l, t, dupla. </s> <s xml:id="echoid-s787" xml:space="preserve">Rurſum autem ex hijs pa <lb/>lam quòd non manet portio ſed inclinabitur ex parte a, quoniam ſup <lb/>ponebatur portio, ſecundum unum ſignum tangere humidum palam <lb/>quòd ſecundum ampliorẽ locum baſis ab humido comprehendetur.</s> <s xml:id="echoid-s788" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s789" xml:space="preserve">HAbeat etiam rurſum portio ad humidum in grauitate propor-<lb/>tionem minorẽ ea, quam habet tetragonum, quod ab n, o, ad id q đ <lb/>a, b, d. </s> <s xml:id="echoid-s790" xml:space="preserve">Quam autem proportionem habet portio ad humidum in gra-<lb/>uitate, hanc habeat tetragonum, quod a, x, minorem autem eſt, quæ x, <lb/>quàm o, n. </s> <s xml:id="echoid-s791" xml:space="preserve">Rurſum igitur in aptetur quædam intermedia portionum <lb/>a, m, d, apol quæ p, i, æquedistanter ipſi b, d, producta æqualis ipſi x. <lb/></s> <s xml:id="echoid-s792" xml:space="preserve">Secet autem ipſa intermedia coniſectione penes y, ipſam autem x, r, <lb/>rectam penes h, demonſtrabitur, autem quæ p, y, dupla ipſius y, i, ſicut <lb/>demonſtrata eſt, quæ, g, o, ipſius g, h, ducatur autem & </s> <s xml:id="echoid-s793" xml:space="preserve">quæ quidem <lb/>p, ***, contingens ſectionem apol ſecundum p, quæ autem p, e, perpen-<lb/>dicularis ſuper b, d, & </s> <s xml:id="echoid-s794" xml:space="preserve">a, i, copulata ducatur ad q. </s> <s xml:id="echoid-s795" xml:space="preserve">Erit autem quæ <lb/> <anchor type="figure" xlink:label="fig-0044-01a" xlink:href="fig-0044-01"/> <pb o="15" file="0045" n="45" rhead="LIBER II."/> <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> a, i, ipſi i, q, æqualis & </s> <s xml:id="echoid-s796" xml:space="preserve">quæ a, q, ipſi p, ***, æquediſtans. </s> <s xml:id="echoid-s797" xml:space="preserve">Demonſtran-<lb/>dum eſt autem quod portio demiſſa in humidum poſita inclinata ita, <lb/>ut baſis ipſis non tang at humidnm inclinata conſiſtet ita ut axis ip-<lb/>ſius ad ſuperficiem humidi faciat angulum minorem angulo f, baſis <lb/>autem ipſius nec ſecundum unum tangat ſuperficiem humidi. </s> <s xml:id="echoid-s798" xml:space="preserve">Demit <lb/>tatur h, in humidum, & </s> <s xml:id="echoid-s799" xml:space="preserve">conſiſtat ita, ut baſis ipſius ſecundum unum <lb/>ſignum tangat ſuperſiciem humidi. </s> <s xml:id="echoid-s800" xml:space="preserve">Secta autem portione per axem <lb/>plano recto ad ſuperficiem humidi, ſectio ſit ſuperficiei quidem por-<lb/>tionis, quæ a, h, b, l, rectanguli coni ſectio, ſuperficiei autem humidi, <lb/>quæ a, Z, axis autem portioni, & </s> <s xml:id="echoid-s801" xml:space="preserve">dyameter ſectionis, quæ b, d, & </s> <s xml:id="echoid-s802" xml:space="preserve">ſece-<lb/>tur quæ b, d, penes ſigna, K, r, conſimiliter ſuperioribus, ducatur au-<lb/>tem & </s> <s xml:id="echoid-s803" xml:space="preserve">quæ h, i, æquedistanter ipſi a, Z, contingens ſectionem conipe-<lb/>nes h. </s> <s xml:id="echoid-s804" xml:space="preserve">Quæ autem habet æquediſtanter ipſi b, d, quàm autem h, s, <lb/>perpendicularis ſuper b, d, quoniam igitur portio ad humidum in gra-<lb/>uitate hanc habet proportionem, quam tetragonum a, x, ad id quod <lb/>a, b, d. </s> <s xml:id="echoid-s805" xml:space="preserve">Quam autem proportionem habet proportio ad humidum in <lb/>grauitate, hanc habet tetragonum, quod ab h, t, ad id quod a, b, d, pro-<lb/>pter eandem prioribus. </s> <s xml:id="echoid-s806" xml:space="preserve">palam ꝙ quæ habet, eſt æqualis ipſi, x, qua-<lb/>re & </s> <s xml:id="echoid-s807" xml:space="preserve">portiones, a, m, Z, a, p, q, ſunt æquales. </s> <s xml:id="echoid-s808" xml:space="preserve">et quoniā in portioni-<lb/>bus æqualibus, & </s> <s xml:id="echoid-s809" xml:space="preserve">ſimilibus. </s> <s xml:id="echoid-s810" xml:space="preserve">apol a, K, h, l, K, ab extremitatib. </s> <s xml:id="echoid-s811" xml:space="preserve">baſium <pb file="0046" n="46" rhead="DE INSIDENTIBVS AQV AE"/> <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> ſunt productœ, quœ <lb/>a, q, a, z, æquales por <lb/>tiones auferétes, pa-<lb/>lam ꝙ æquales fa-<lb/>ciunt ad dy ametros <lb/>portionũ, ad buc au-<lb/>tem & </s> <s xml:id="echoid-s812" xml:space="preserve">trigonorũ b, <lb/>l, s, p, w, e, æquales <lb/>ſunt anguli ꝗ apud<unsure/> <lb/>l, w, erunt, ets, b, e, <lb/>b, œquales. </s> <s xml:id="echoid-s813" xml:space="preserve">Quare <lb/>et quœ, s, r, e, r, œqua <lb/> <anchor type="figure" xlink:label="fig-0046-02a" xlink:href="fig-0046-02"/> les & </s> <s xml:id="echoid-s814" xml:space="preserve">quœ b, a, p, h, & </s> <s xml:id="echoid-s815" xml:space="preserve"><lb/>quœ a, t, b, i, et quoniā eſt <lb/>dupla, quœ y, p, ipſius y, i, <lb/>manifeſtum, quòd minor <lb/>eſt, quœ dupla quœ b, a, <lb/>ipſius a, t. </s> <s xml:id="echoid-s816" xml:space="preserve">Sit igitur n, <lb/>y, dupla ipſius y, t, & </s> <s xml:id="echoid-s817" xml:space="preserve">co <lb/>pulata protrabatur, quœ <lb/>y, b, t. </s> <s xml:id="echoid-s818" xml:space="preserve">Sunt auté centra <lb/>grauitatum totius qui-<lb/>dem, K, eius auté quod <lb/>intra bumidumy, eius au <lb/>tem quod extra in linea K, c, <lb/>et ſit c, erit autem propter prœ <lb/>cedens theorema hoc mani-<lb/>feſtum quòd non manet portio, <lb/>ſed inclinabitur ita, ut baſis ip-<lb/> <anchor type="figure" xlink:label="fig-0046-03a" xlink:href="fig-0046-03"/> ſius nec ſecundum unum tan-<lb/>gat ſuperficiem humidi. </s> <s xml:id="echoid-s819" xml:space="preserve">Q uod <lb/>autem cõſiſtet ita, ut axis ip-<lb/>ſius ad ſuperficiem humidi fa-<lb/>ciat angulum minorem angulo <lb/>f. </s> <s xml:id="echoid-s820" xml:space="preserve">demonſtrabitur, Conſiſtat h, <lb/>ſi poſſibile est ita, ut faciat an-<lb/>gulum non minorem angulo f, <lb/>& </s> <s xml:id="echoid-s821" xml:space="preserve">alia diſponantur eãdem bijs <lb/>quœ in tertia figura. </s> <s xml:id="echoid-s822" xml:space="preserve">Simili-<lb/>ter autem demonſtrabitur, quœ <pb o="16" file="0047" n="47" rhead="LIBER II."/> t, m, œqualis ipſi, x, quare & </s> <s xml:id="echoid-s823" xml:space="preserve">ipſi, i, b, & </s> <s xml:id="echoid-s824" xml:space="preserve">quoniam, b, l, minor eſt quàm <lb/>f, non ergo maior eſt neque quœ, s, r, quám, s, r, neque n, a, <lb/>quàm, o, g, et quoniam quœ, i, b, eſt hemiolia ipſius, p, y, minor autem <lb/>quœ, p, y, quàm, g, o, & </s> <s xml:id="echoid-s825" xml:space="preserve">quœ quidem habet œqualis ipſi, p, c, eſt quœ au-<lb/>té, h, a, nõ eſt minor, quàm, o, g, maior ergo quœ, a, b, quàm, p, y, quœ <lb/>ergo, h, a, est maior, quàm dupla ipſius, t, a. </s> <s xml:id="echoid-s826" xml:space="preserve">Sit autem, b, y, dupla ip-<lb/>ſius, y, t, & </s> <s xml:id="echoid-s827" xml:space="preserve">copulat a quœ, y, K, educatur. </s> <s xml:id="echoid-s828" xml:space="preserve">palam autem ſimiliter prio-<lb/>ribus, quòd non manet portio, ſed uoluetur ita, ut axis ipſiur ad ſuper-<lb/>fictem bumidi faciat angulum minorem angulo f.</s> <s xml:id="echoid-s829" xml:space="preserve"/> </p> <div xml:id="echoid-div42" type="float" level="2" n="3"> <figure xlink:label="fig-0044-01" xlink:href="fig-0044-01a"> <image file="0044-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0044-01"/> </figure> <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0045-01"/> </figure> <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a"> <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0046-01"/> </figure> <figure xlink:label="fig-0046-02" xlink:href="fig-0046-02a"> <image file="0046-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0046-02"/> </figure> <figure xlink:label="fig-0046-03" xlink:href="fig-0046-03a"> <image file="0046-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0046-03"/> </figure> </div> </div> <div xml:id="echoid-div44" type="section" level="1" n="29"> <head xml:id="echoid-head39" style="it" xml:space="preserve">Archimedis de inſidentibus in bumido li-<lb/>ber ſecundus explicit, ad laudem Dei.</head> <p style="it"> <s xml:id="echoid-s830" xml:space="preserve">CHe a Curtio Troiano mercante de libri, ſia conceſſo, che altri che <lb/>lui, ò chi bauerà cauſa da lui, non poſſa in queſta città, & </s> <s xml:id="echoid-s831" xml:space="preserve">Do-<lb/>minio noſtro stampar, ne in quello ſtampate vender per ſpatio de an-<lb/>ni dieci proſſi. </s> <s xml:id="echoid-s832" xml:space="preserve">futuri, li libri intitulati Giordano de Ponderibus, & </s> <s xml:id="echoid-s833" xml:space="preserve"><lb/>il ſecondo libro d' Archimede de Inſidentibus aquœ, tradotti in lin-<lb/>gua uolgare. </s> <s xml:id="echoid-s834" xml:space="preserve">Et medeſimamente i ſopradetti libri Latini, ſotto pena <lb/>di perdere tutte le opere stampate, & </s> <s xml:id="echoid-s835" xml:space="preserve">di ducati dieci per una, lequali <lb/>opere ſiano del ſupplicante, ouero di cbi farà la ſpeſa, & </s> <s xml:id="echoid-s836" xml:space="preserve">la pena ſia <lb/>diuiſa in terzo, vn terzo all' Arſenale, vn terzo al Magistrato, che <lb/>farà l'eſſecutione, & </s> <s xml:id="echoid-s837" xml:space="preserve">uno terzo al denuntiante, eſſendo pero tenuto el <lb/>ſupplicante oſſeruar quanto è diſposto in materia de ſtampe.</s> <s xml:id="echoid-s838" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s839" xml:space="preserve">Angelus Cornelius, <lb/>Ducalis not, ex.</s> <s xml:id="echoid-s840" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s841" xml:space="preserve">10 Gaſparo comandador ai Pioueghi, ò intimado tutte le librarie, & </s> <s xml:id="echoid-s842" xml:space="preserve"><lb/>ſtamparie de Venetia.</s> <s xml:id="echoid-s843" xml:space="preserve"/> </p> <pb file="0048" n="48"/> <pb file="0049" n="49"/> <pb file="0050" n="50"/> </div></text> </echo>